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UNIVERSITY OF SOUTHAMPTON

Interactions of a submerged

membrane with water waves and

its use in harnessing nearshore

wave power.

by

Nicolas Choplain

A thesis submitted in partial fulfillment for the

degree of Doctor of Philosophy

in the

Faculty of Engineering and the Environment

Energy and Climate Change Research Group

May 2012

http://www.soton.ac.uk

mailto:[email protected]

http://www.engineering.soton.ac.uk

http://www.civil.soton.ac.uk

UNIVERSITY OF SOUTHAMPTON

ABSTRACT

FACULTY OF ENGINEERING AND THE ENVIRONMENT

ENERGY AND CLIMATE CHANGE RESEARCH GROUP

Doctor of Philosophy

by Nicolas Choplain

Developed and developing countries need electricity, and this usage is increasing ev-

eryday. This constant increase cannot be satisfied with the current ways of electricity

generation that have shown themselves to be out of phase with environmental concerns.

Oceans yield great amount of energy that could be converted into electricity and the

current research deals with one portion of ocean energy, wave energy.

The wave energy converter studied in this thesis is a bottom-mounted, liquid-filled

rectangular duct, covered with a rubber membrane and aligned head to the waves prop-

agation direction. Two types of membrane were tested. The behaviour of this device

beneath waves was investigated with two configurations: one with both its ends closed,

the other one with a power take-off connected at its stern. The pressure at both ends

was characterised by means of pressure transducers and the pressure inside the duct by

means of laser sensors measuring the membrane displacement.

Results from experiments carried out on the closed version of the duct pointed out

a resonant behaviour of the system for wave frequencies at which bulges, propagating in

the rubber membrane, could travel an integer number of times along the duct’s length.

This resonance was characterised by pressure magnitudes at the stern up to 2.8 times

that acting on the membrane from the incident wave. Moreover, the membrane dis-

placement was for the first time mapped and the profile obtained showed characteristic

nodes and antinodes.

The performance of this device in harnessing wave power was evaluated by con-

necting a linear dashpot at its stern. Capture widths of up to 2.2 times the device width

were obtained and the bandwidth of maximum power capture not limited to a single

frequency.

The pressure behaviour in both configurations was explained with a one dimensional

theory of bulges propagating in distensible tubes with good agreement for the thicker

tested membrane. On the contrary, this was not the case for the thinner membrane,

suggesting that this model could not be used for configurations where bulge wavelengths

are much shorter than that of the incoming wave.

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mailto:[email protected]

Contents

Abstract iii

List of Figures xi

List of Tables xxiii

Declaration Of Authorship xxv

Acknowledgements xxvii

Nomenclature xxix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Wave energy glossary and relevance of the study 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Wave characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.3 Kinetic and potential energies . . . . . . . . . . . . . . . . . 7

2.3 Wave power background . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Wave power resource distribution . . . . . . . . . . . . . . . 10

2.3.3 Harnessing concepts of wave power . . . . . . . . . . . . . . 11

2.3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 11

2.3.3.2 Oscillating water column . . . . . . . . . . . . . . . 11

2.3.3.3 Overtopping devices . . . . . . . . . . . . . . . . . 12

2.3.3.4 Oscillating bodies . . . . . . . . . . . . . . . . . . . 12

2.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Relevance of the study . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 The Anaconda WEC . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2.1 Principle . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2.2 The bulge wave theory . . . . . . . . . . . . . . . . 15

v

vi CONTENTS

2.4.2.3 Current developments . . . . . . . . . . . . . . . . 18

2.4.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . 19

2.4.3 Nearshore wave power . . . . . . . . . . . . . . . . . . . . . 21

2.4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 21

2.4.3.2 Exploitable wave power . . . . . . . . . . . . . . . 21

2.4.3.3 Nearshore bottom-mounted wave energy converters 24

2.4.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Starting point of the project . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Aim and objectives . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.3 Novelty and advantages of the studied device . . . . . . . . . 27

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Literature review 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Use of rubber for marine applications . . . . . . . . . . . . . . . . . 31

3.3 Breakwaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Submerged elastic membrane breakwaters . . . . . . . . . . 34

3.3.3 Submerged fluid-filled membranes . . . . . . . . . . . . . . . 43

3.3.4 Bottom-mounted fluid-filled membranes . . . . . . . . . . . . 44

3.3.5 Conclusions on breakwaters studies . . . . . . . . . . . . . . 54

4 Experimental equipment 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Narrow wave flume . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.2 Wave-maker . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.3 Absorbing beach . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Wave basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 Wave-maker . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.3 Absorbing beach . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.2 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.3 Structural damping . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Power take-off system . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.3 Power calculation . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6.1 Data acquisition and equipment driving system . . . . . . . 70

CONTENTS vii

4.6.2 Wave probes . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6.3 Water pressure transducers . . . . . . . . . . . . . . . . . . . 71

4.6.4 Air pressure transducers . . . . . . . . . . . . . . . . . . . . 72

4.6.5 Digital camera . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.6.6 Laser sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.7 Tested model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 One dimensional model for bulges propagation in the duct 77

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Load-deflection of rectangular membrane and free bulge speed . . . 77

5.3 Bulge equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Closed ends configuration . . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Duct with a power take-off system . . . . . . . . . . . . . . . . . . . 84

5.5.1 Bulge pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.5.2 Power calculation . . . . . . . . . . . . . . . . . . . . . . . . 87

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 Narrow flume experiments: static deflection, free bulge speed and

pressure amplification 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Experimental layout . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.3 Static tests: 1mm thick membrane . . . . . . . . . . . . . . . . . . 91

6.3.1 Membrane deflection . . . . . . . . . . . . . . . . . . . . . . 91

6.3.2 Free bulge speed . . . . . . . . . . . . . . . . . . . . . . . . 93

6.4 Membrane beneath waves: 1mm thick membrane . . . . . . . . . . 94

6.4.1 Experimental conditions . . . . . . . . . . . . . . . . . . . . 94

6.4.2 Pressure amplification . . . . . . . . . . . . . . . . . . . . . 96

6.4.3 Membrane displacement and pressure in the duct . . . . . . 99

6.4.4 Bulge pressure components . . . . . . . . . . . . . . . . . . . 102

6.4.4.1 Separation of components . . . . . . . . . . . . . . 102

6.4.4.2 Components at resonance . . . . . . . . . . . . . . 104

6.4.5 Wave analysis and loss in the rubber . . . . . . . . . . . . . 111

6.4.6 Transverse motion of the membrane . . . . . . . . . . . . . . 116

6.4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Thin membrane results . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5.2 Experimental conditions . . . . . . . . . . . . . . . . . . . . 121

6.5.3 Static deflection and free bulge speed . . . . . . . . . . . . . 121

6.5.4 Pressure variation at the stern . . . . . . . . . . . . . . . . . 125

6.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Extractable power 131

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2 Experimental layout and test conditions . . . . . . . . . . . . . . . 131

viii CONTENTS

7.2.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2.2 Power take-off in experimental conditions . . . . . . . . . . . 133

7.3 Matching impedances configuration . . . . . . . . . . . . . . . . . . 137

7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.3.2 Power capture . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.3.3 Pressure variation in the duct . . . . . . . . . . . . . . . . . 140

7.3.4 Bulge components . . . . . . . . . . . . . . . . . . . . . . . 143

7.4 Mismatch impedances configuration . . . . . . . . . . . . . . . . . . 147

7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4.2 Power capture . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.4.3 Pressure variation in the duct and bulge components . . . . 150

7.5 Water-air interface displacement and pressure in the PTO . . . . . 152

7.6 Wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8 Two dimensional analytical model of a duct covered with a ten-

sioned membrane beneath waves 161

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.2 Problem formulation for the closed ends configuration . . . . . . . . 162

8.2.1 Diffraction problem . . . . . . . . . . . . . . . . . . . . . . . 164

8.2.1.1 Definitions of the potentials . . . . . . . . . . . . . 164

8.2.2 Radiation problem . . . . . . . . . . . . . . . . . . . . . . . 165

8.2.2.1 Definition of the potentials . . . . . . . . . . . . . 165

8.2.2.2 Membrane deformation . . . . . . . . . . . . . . . 167

8.2.3 Reflection and transmission coefficients . . . . . . . . . . . . 168

8.2.4 Conservation of volume in the duct . . . . . . . . . . . . . . 169

8.2.4.1 Uniform potential below the membrane . . . . . . . 170

8.2.4.2 Weighted least-square method . . . . . . . . . . . . 170

8.2.4.3 Matrix approximation . . . . . . . . . . . . . . . . 171

8.2.5 Code Layout . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.2.6 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.2.6.1 Diffraction problem . . . . . . . . . . . . . . . . . . 175

8.2.6.2 Complete configuration and comparison of the pro-posedmethods . . . . . . . . . . . . . . . . . . . . . . . . 175

8.3 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . 186

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9 Conclusions and future work 193

9.1 Achievements and perspectives . . . . . . . . . . . . . . . . . . . . . 193

9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

A Power take-off test in the narrow flume A1

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1

A.2 Experimental set-up and test conditions . . . . . . . . . . . . . . . A2

CONTENTS ix

A.3 Linearity of the power take-off . . . . . . . . . . . . . . . . . . . . . A3

A.4 Power capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5

B Analytical model of duct covered with a tensioned membrane:

the maths B1

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B1

B.2 Closed ends configuration . . . . . . . . . . . . . . . . . . . . . . . B1

B.2.1 Diffraction problem . . . . . . . . . . . . . . . . . . . . . . . B1

B.2.2 Radiation problem . . . . . . . . . . . . . . . . . . . . . . . B5

B.2.3 Membrane deformation . . . . . . . . . . . . . . . . . . . . . B8

C Pictures of the experimental set-ups C1

D Determination of rubber properties D1

D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D1

D.2 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . D2

D.3 Incompressibility of rubber . . . . . . . . . . . . . . . . . . . . . . . D4

D.4 Dynamic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . D5

List of Figures

2.1 Sketch of a water wave . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Energy absorbed per unit time T . . . . . . . . . . . . . . . . . . . 8

2.3 Global distribution of annual mean wave power, after Cornett (2008) 10

2.4 Wave energy converters classification, from Rahm (2010) . . . . . . 11

2.5 Anaconda WEC, after Checkmate SeaEnergy Ltd. . . . . . . . . . . 14

2.6 Bulge wave propagating in the rubber tube. . . . . . . . . . . . . . 16

2.7 Capture widths as functions of relative wave periods for (a) Z=0.64,(b) Z=1.05, (c) Z=3.3, (d) Z=7.7. Measurements are shown aspoints, one-dimensional theory with δ = 13 as continuous lines,after Chaplin et al. (2012) . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 Challenges facing the development of a new wave energy converter . 20

2.9 Shoaling of a 10 second energy period wave propagating orthogo-nal to depth contours for different seabed slopes, after Folley andWhittaker (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.10 Variation in percentage power loss from offshore to nearshore sitefor an initial sea-state of T=10s and Pi=50 kW/m travelling at anangle to the depth contours on a seabed slope of 1:100, after Folleyand Whittaker (2009). . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.11 The offshore Oyster 1 device, after Cameron et al. (2010). . . . . . 25

2.12 3D view of the tested duct. . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Example cross-section of a Fabriconda with 10 cells, showing thestructure at its minimum cross-section (left) and at its mediumpoint (right), after Hann et al. (2011). . . . . . . . . . . . . . . . . 32

3.2 Vertical membrane configuration. . . . . . . . . . . . . . . . . . . . 34

3.3 Reflection coefficient of hinged-hinged membrane breakwater as func-tion of dimensionless tension Tz/ρgh

2 and wave number kh, afterKim and Kee (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Horizontal membrane configuration. . . . . . . . . . . . . . . . . . . 38

3.5 Reflection coefficients of a submerged impermeable membrane break-water as function of length of membrane s/h and wavenumber k1hfor d/h=0.2, Tx/ρgh

2=0.1, after Cho and Kim (1998) . . . . . . . . 39

3.6 Response of the membrane (ξ/A) as a function of wavenumberk1h and horizontal coordinate x/L for d/h = 0.2, s/h = 0.5 andTx/ρgh

2=0.1, after Cho and Kim (1998). . . . . . . . . . . . . . . . 39

xi

xii LIST OF FIGURES

3.7 Modal response amplitude as function of wavenumber k1h and hor-izontal coordinate x/L for d/h=0.2, s/h=0.5, Tx/ρgh

2=0.1; upperplot: first symmetric mode, middle plot: second symmetric mode;lower plot: third symmetric mode. . . . . . . . . . . . . . . . . . . . 41

3.8 Modal response amplitude as function of wavenumber k1h and hor-izontal coordinate x/L for d/h=0.2, s/h=0.5, Tx/ρgh

2=0.1; upperplot: first asymmetric mode, middle plot: second asymmetric mode;lower plot: third asymmetric mode. . . . . . . . . . . . . . . . . . . 42

3.9 Wave interception by a pair of sea balloons, after Ijima and Uwatoko(1985). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.10 Sketch of the flexible mound device. . . . . . . . . . . . . . . . . . . 45

3.11 Wave energy loss in the flexible mound, after Tanaka et al. (1992a). 46

3.12 Amplitude of membrane normal deflection for p0/ρgh = 0.07 andd/h=0.5: (a) fundamental mode; (b) second mode, (c) third mode(– –, ρi/ρ = 0.9; —, ρi/ρ = 1.0; · · · , ρi/ρ = 1.1), after Phadke andCheung (1999); ρi, density of the inner fluid. . . . . . . . . . . . . . 47

3.13 Transmission coefficient as function of excitation dimensionless fre-quency L/λ: (a) p0/ρgh = 0.065, (b) p0/ρgh = 0.208; • • • datafrom Ohyama et al. (1989); – – – two dimensional linear model fromPhadke and Cheung (1999); · · · two dimensional non linear modelfrom Phadke and Cheung (2003); — three dimensional model fromDas and Cheung (2009) . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.14 Water-filled hemicircular membrane breakwater. . . . . . . . . . . . 50

3.15 Free surface wave amplitudes for normal waves over (a) rigid and(c) flexible structure for ω = π/6 and over (b) rigid and (d) flexiblestructure for ω = π/4, from Dewi et al. (1999). . . . . . . . . . . . . 51

3.16 Variation of shadow transmission coefficient KT with kR , afterDewi et al. (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.17 Side view of the hemi-cylindrical flexible composite breakwater model,after Stamos (2000). . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.18 Rapidly installed breakwater system XM99 (from Briggs (2001)) . . 55

4.1 Narrow flume at the University of Southampton. . . . . . . . . . . . 58

4.2 Wavemaker for the narrow flume. . . . . . . . . . . . . . . . . . . . 58

4.3 Wave basin at the University of Southampton. . . . . . . . . . . . . 60

4.4 Wave paddle for the wave basin. . . . . . . . . . . . . . . . . . . . . 60

4.5 Absorbing beach for the wave basin. . . . . . . . . . . . . . . . . . . 61

4.6 Measured absorbed reflection coefficients for the absorbing foambeach in the wave basin. . . . . . . . . . . . . . . . . . . . . . . . . 63

4.7 Stress in the strip of membrane as a function of the elongationfor the quasi-static Young’s modulus measurement. Each symbolsrepresent different loading. . . . . . . . . . . . . . . . . . . . . . . . 64

4.8 Cross section power take-off. . . . . . . . . . . . . . . . . . . . . . . 66

4.9 Schematic of the power take-off system. . . . . . . . . . . . . . . . . 66

4.10 Power take-off system set-up . . . . . . . . . . . . . . . . . . . . . . 67

4.11 Model for the power take-off. . . . . . . . . . . . . . . . . . . . . . . 68

LIST OF FIGURES xiii

4.12 Pressure transducers. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.13 Wheastone bridge for the pressure transducers. . . . . . . . . . . . . 72

4.14 Laser sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.15 Laser sensors set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.16 Influence of the plastic boxes used for the laser sensors on the re-flection coefficient for the experimental set-up as in chapter 7 fortwo sets of laser sensors positions. . . . . . . . . . . . . . . . . . . . 75

4.17 Cross section sketch of the rubber duct. . . . . . . . . . . . . . . . . 75

5.1 Cross section sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Theoretical components of the wave pressure in the rubber duct fora static internal pressure head of 7cm corresponding to a free bulgespeed of 3.06m/s in the configuration of chapter 6. . . . . . . . . . . 81

5.3 Theoretical internal velocities for the first four harmonics at 30 in-stants over one wave period for the duct with an internal static pres-sure head of 7cm; (a) f=0.306Hz; (b) f=0.612Hz; (c) f=0.918Hz(d) f=1.224Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Theoretical internal pressure for the first four harmonics at 30 in-stants over one wave period for the duct with an internal static pres-sure head of 7cm; (a) f=0.306Hz; (b) f=0.612Hz; (c) f=0.918Hz(d) f=1.224Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.5 Definition sketch of the stern. . . . . . . . . . . . . . . . . . . . . . 85

6.1 Experimental layout for the closed ends configuration. Dashed linescorrespond to different internal static pressure heads. . . . . . . . . 90

6.2 Experimental values of membrane centreline deflection (+ + +),load-deflection relation 5.9 (––) . . . . . . . . . . . . . . . . . . . . 92

6.3 Equivalent membrane submergence depth . . . . . . . . . . . . . . . 92

6.4 Time series of pressure transducers from free bulge speed measurement 93

6.5 Free bulge speed for the 1mm thick membrane, each symbols rep-resenting one set of measurement. . . . . . . . . . . . . . . . . . . . 94

6.6 Time series of laser sensors when rubber membrane is beneath waves. 95

6.7 Time series of pressure transducers (at bow and stern) when rubbermembrane is beneath waves. . . . . . . . . . . . . . . . . . . . . . . 96

6.8 Magnitude of the pressure at the stern relative to the wave pressureacting on the membrane with an internal static pressure head of7cm. symbols: experimental values; —: theory. . . . . . . . . . . . 97

6.9 Magnitude of the pressure at the stern relative to the wave pressureacting on the membrane with an internal static pressure head of10cm. symbols: experimental values; —: theory. . . . . . . . . . . . 97

6.10 Pressure in the duct at thirty instants over one wave period with aninternal static pressure head of 7cm; wave frequency f=0.95Hz; (a)theory; (b) inferred from laser sensors measurement of membranedisplacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

xiv LIST OF FIGURES

6.11 Pressure in the duct at thirty instants over one wave period with aninternal static pressure head of 7cm; wave frequency f=1.2Hz; (a)theory; (b) inferred from laser sensors measurement of membranedisplacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.12 Pressure in the duct at thirty different instants over one wave pe-riod with an internal static pressure head of 7cm; wave frequencyf=1.1Hz; (a) theory; (b) inferred from laser sensors measurementof membrane displacement. . . . . . . . . . . . . . . . . . . . . . . . 101

6.13 Pressure in the duct at thirty instants over one wave period with aninternal static pressure head of 10cm; wave frequency f=1.075Hz;(a) theory; (b) inferred from laser sensors measurement of mem-brane displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.14 Separation of bulge wave components from laser measurements. . . 103

6.15 Measured total internal pressure, normalised by the pressure actingon the membrane (a) and phase (b) for the duct with an inter-nal static pressure head of 7cm, plotted as points. Lines representamplitudes and phases computed as the sum of the componentsobtained from the least squares method. Wave frequency f=0.95Hz. 105

6.16 Pressure in the duct at thirty different instants over one wave pe-riod with an internal static pressure head of 7cm; wave frequencyf=0.95Hz; (a) inferred from laser sensor measurement of membranedisplacement amplitude; (b) sum of the three bulge wave compo-nents (and the wave pressure) obtained from a least squares method.105

6.17 Magnitude of P+b relative to the wave pressure acting on the mem-

brane with an internal static pressure head of 7cm. symbols: exper-imental values inferred from a least squares method; —: calculatedfrom (5.21). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.18 Magnitude of P−b relative to the wave pressure acting on the mem-


6.19 Magnitude of PAb relative to the wave pressure acting on the mem-


6.20 Measured total internal pressure, normalised by the pressure actingon the membrane (a) and phase (b) for the duct with an internalstatic pressure head of 10cm, plotted as points. Lines representamplitudes and phases computed as the sum of the componentsobtained from the linear squares method. Wave frequency f=1.075Hz.108

6.21 Pressure in the duct at thirty instants over one wave period with aninternal static pressure head of 10cm; wave frequency f=1.075Hz;(a) inferred from laser sensor measurement of membrane displace-ment amplitude; (b) sum of the three bulge wave components (andthe wave pressure) obtained for a least squares method. . . . . . . . 109

LIST OF FIGURES xv


brane with an internal static pressure head of 10cm. symbols: ex-perimental values inferred from a least squares method; —: calcu-lated from (5.21). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109





6.25 Ratio of the transmitted wave power, Pt, to the incident wave powerPi for the duct with an internal static pressure head of 7cm. Blackline and symbols: simple harmonic component of wave gauge sig-nals; Grey line and symbols: first three harmonic components ofwave gauge signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.26 Ratio of the reflected wave power, Pr, to the incident wave power Pi

for the duct with an internal static pressure head of 7cm. Black lineand symbols: simple harmonic component of wave gauge signals;Grey line and symbols: first three harmonic components of wavegauge signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.27 Ratio of the wave power loss, Ploss, to incident wave power Pi forthe duct with an internal static pressure head of 7cm. Line andsymbols: simple harmonic analysis of wave gauge signals. . . . . . . 113

6.28 Instantaneous elongation in the rubber at 20 instants (plotted aspoints) as a function of time normalised by the wave period T .Wave frequency is f = 0.775Hz. . . . . . . . . . . . . . . . . . . . . 115

6.29 Ratio of the power lost in the rubber, Ploss, to the incident wavepower Pi for the rubber duct with an internal static pressure headof 7cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.30 Cross-section view of the duct and transverse motion of the membrane.117

6.31 Experimental values of membrane centreline deflection (+ + +),load-deflection relation 5.9 (––) for εy,0 = 0.14 and εx,0=0 . . . . . . 122

6.32 Experimental values of membrane centreline deflection (+ + +),load-deflection relation 5.9 (––) for εy,0 = 0.28 and εx,0=0. . . . . . . 122

6.33 Experimental values of membrane centreline deflection (+ + +),load-deflection relation 5.9 (––) for εy,0 = 0.14 and εx,0=0.14. . . . . 123

6.34 Averaged values of free bulge speed measurement, for εy,0 = 0.14and εx,0=0 for the 0.2mm membrane; (––) calculated values; (+++)averaged values from pressure and laser measurements. . . . . . . . 124

6.35 Averaged values of free bulge speed measurement, for εy,0 = 0.28and εx,0=0 for the 0.2mm membrane; (––) calculated values; (+++)averaged values from pressure and laser measurements. . . . . . . . 124

xvi LIST OF FIGURES

6.36 Averaged values of free bulge speed measurement, for εy,0 = 0.14and εx,0=0.14 for the 0.2mmmembrane; (––) calculated values; (+++) averaged values from pressure and laser measurements. . . . . . 125

6.37 Magnitude of the pressure at the stern relative to the wave pressureat the free surface elevation with conditions of test 1; left: measuredfor the three runs (each symbols for one run); right: one dimensionaltheory with c = 1.11m/s. . . . . . . . . . . . . . . . . . . . . . . . . 126

6.38 As for Figure 6.37, but with conditions of test 2; left: measured forthe three runs (each symbols for one run); right: one dimensionaltheory with c = 0.88m/s. . . . . . . . . . . . . . . . . . . . . . . . . 126

6.39 As for Figure 6.37, but with conditions of test 3; left: measuredfor the two runs (each symbols for one run); right: one dimensionaltheory with c = 1.11m/s. . . . . . . . . . . . . . . . . . . . . . . . . 126



6.42 As for Figure 6.37, but with conditions of test 6; left: measured;right: one dimensional theory with c = 0.88m/s. . . . . . . . . . . . 127

6.43 Surface elevation at the middle point along the length of the duct,relative to the incident wave amplitude. . . . . . . . . . . . . . . . . 128

6.44 Ratio of the transmitted wave power, Pt, to the incident wave powerPi for all the tests and measured with the first three harmonic com-ponents of wave gauge signals. . . . . . . . . . . . . . . . . . . . . . 128

6.45 Membrane displacement at thirty instants over one wave period forconfiguration of test 1; wave frequency f=0.775Hz; (a) theory; (b)inferred from laser sensors measurement of membrane displacement. 129

7.1 Experimental layout for the rubber duct equipped with the PTO.Dashed lines correspond to different internal static pressure heads. . 132

7.2 PTO air pressure versus PTO water level for the rubber duct withan internal static pressure head of 0.02m; f=0.625Hz; (a) test 7;(b) test 6; (c) test 4; (d) test 5. . . . . . . . . . . . . . . . . . . . . 134

7.3 Phase lag between PTO water level and PTO air pressure as a func-tion of wave frequency for the duct with an internal static pressurehead of 0.02m ; • • • measured, — from (4.25); (a) test 7; (b) test6; (c) test 4; (d) test 5. . . . . . . . . . . . . . . . . . . . . . . . . . 135

7.4 Ratio of the flow rate in the copper pipes to that of the air in thechamber, for the duct with an internal static head of 0.02m. . . . . 135

7.5 Measured values of the bulk modulus of air, with configuration asin test 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.6 Capture width (in duct widths) for a static pressure head of 3cm;lines: computed from theory; symbols: measured; colour legend:test 2 (Z = 1.10) in grey and test 1 (Z = 1.03) in black. . . . . . . . 138

LIST OF FIGURES xvii

7.7 Capture width (in duct widths) for a static pressure head of 2cm;lines: computed from theory; symbols: measured; colour legend:test 6 (Z = 1.08) in grey and test 7 (Z = 1.02) in black. . . . . . . . 139

7.8 Pressure in the duct at thirty instants over one wave period with aninternal static pressure head of 2cm; wave frequency f=0.625Hz; (a)theory for Z = 1.02; (b) inferred from laser sensors measurementsof membrane displacement. . . . . . . . . . . . . . . . . . . . . . . . 140




7.12 Measured total internal pressure, normalised by the pressure actingon the membrane (a) and phase (b) for the duct with an inter-nal static pressure head of 2cm, plotted as points. Lines representamplitudes and phases computed as the sum of the components ob-tained from the linear squares method. Wave frequency f=0.625Hz,with conditions as in test 6 . . . . . . . . . . . . . . . . . . . . . . . 144

7.13 Measured total internal pressure, normalised by the pressure actingon the membrane (a) and phase (b) for the duct with an inter-nal static pressure head of 2cm, plotted as points. Lines representamplitudes and phases computed as the sum of the components ob-tained from the linear squares method. Wave frequency f=1.05Hz,with conditions as in test 6 . . . . . . . . . . . . . . . . . . . . . . . 144

7.14 Pressure in the duct at thirty instants over one wave period with aninternal static pressure head of 2cm; wave frequency f=0.625Hz; (a)inferred from laser sensor measurement of membrane displacementamplitude; (b) sum of the three bulge wave components (and thewave pressure) obtained for a least squares method. . . . . . . . . . 145



brane with an internal static pressure head of 2cm and conditions asin test 6; symbols: experimental values inferred from a least squaresmethod; —: calculated from (5.34). . . . . . . . . . . . . . . . . . . 146

xviii LIST OF FIGURES





7.19 Capture width (in duct widths) for a static pressure head of 3cm;lines: computed from theory; symbols: measured for the three runsof test 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.20 Capture width (in duct widths) for a static pressure head of 2cm;lines: computed from theory; symbols: measured for the run of test4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.21 Capture width (in duct widths) for a static pressure head of 2cm;lines: computed from theory; symbols: measured for the run of test5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149


7.23 Measured total internal pressure, normalised by the pressure actingon the membrane (a) and phase (b) for the duct with an inter-nal static pressure head of 3cm, plotted as points. Lines representamplitudes and phases computed as the sum of the components ob-tained from the linear squares method. Wave frequency f=0.625Hz,with conditions as in test 3. . . . . . . . . . . . . . . . . . . . . . . 151


7.25 Pressure (left column) in the PTO and amplitude (right column) ofthe water-air interface displacement. Lines are the calculated val-ues from the theory in section 5.5.2 and points represent measuredvalues for (a) test 1; (b) test 2; (c) test 3. . . . . . . . . . . . . . . . 153

7.26 Pressure (left column) in the PTO and amplitude (right column) ofthe water-air interface displacement. Lines are the calculated val-ues from the theory in section 5.5.2 and points represent measuredvalues for (a) test 7; (b) test 6; (c) test 4 and (d) test 5. . . . . . . 154

7.27 Wave analysis for the duct in with an internal static pressure headof 3cm and configurations as in (a) test 1; (b) test 2; (c) test 3 and(d) test 5. Left column: power carried by the first harmonic; rightcolumn: power carried in the first three harmonics of the signalsfrom the wave gauges. . . . . . . . . . . . . . . . . . . . . . . . . . 156

LIST OF FIGURES xix

7.28 Wave analysis for the duct in with an internal static pressure headof 2cm and configurations as in (a) test 7; (b) test 6; (c) test 4 and(d) test 5. Left column: power carried by the first harmonic; rightcolumn: power carried in the first three harmonics of the signalsfrom the wave gauges. . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.1 Definition sketch of a closed duct with elastic membrane on its top. 162

8.2 Code developed for the analytical model. . . . . . . . . . . . . . . . 172

8.3 Reflection coefficient for h/d = 2 and s/d = 2. — from Mei andBlack (1969), • • • diffraction part of the present method. . . . . . 173

8.4 As for Figure 8.3, but s/d = 4. . . . . . . . . . . . . . . . . . . . . . 173


8.6 As for Figure 8.3, but h/d = 2.78 and s/d = 4.43. . . . . . . . . . . 174

8.7 Reflection coefficient for h/d = 2 and s/d = 2. — diffraction only,Tx → ∞ for the following methods of conservation of volume: uniform potential; + + + least-squares; N N N matrix approxima-tion; • • • no correction. . . . . . . . . . . . . . . . . . . . . . . . . 176



8.10 As for Figure 8.7, but h/d = 2.78 and s/d = 4.43. . . . . . . . . . . 177

8.11 Symmetric (left column) and asymmetric (right column) of thediffracted potentials (upper plots) and velocities (lower plots) atthe interface x = −s; s = 2m, h/d = 2 with h = 0.4m andTx/ρgh

2 = 0.15 and wave frequency f = 1Hz. . . . . . . . . . . . . . 178

8.12 Symmetric (left column) and asymmetric (right column) of the ra-diated potentials (upper plots) and velocities (lower plots) at the in-terface x = −s; s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh

2 = 0.15and wave frequency f = 1Hz. . . . . . . . . . . . . . . . . . . . . . 179

8.13 Combined free surface dynamic condition (8.6) for the symmetric(left plot) and asymmetric (right plot) diffracted potentials in region(1); s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh

2 = 0.15 and wavefrequency f = 1Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.14 Combined free surface dynamic condition (8.6) for the symmetric(left plot) and asymmetric (right plot) radiated potentials in region(1); s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh


8.15 Combined free surface dynamic condition (8.6) for the symmetric(left plot) and asymmetric (right plot) diffracted potentials in region(2); s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh


8.16 Combined free surface dynamic condition (8.6) for the symmetric(left plot) and asymmetric (right plot) radiated potentials in region(2); s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh


xx LIST OF FIGURES

8.17 Dynamic condition (8.18) for the symmetric (left plot) and asym-metric (right plot) for the first modal (l = 1) radiated potential inregion (2); s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh


8.18 Dynamic condition (8.18) for the symmetric (left plot) and asym-metric (right plot) for the first modal (l = 1) radiated potential inregion (3); s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh


8.19 Kinematic condition (8.27) for the symmetric (left plot) and asym-metric (right plot) with no volume correction; s = 2m, h/d = 2with h = 0.4m and Tx/ρgh

2 = 0.15 and wave frequency f = 1Hz. . . 183

8.20 Kinematic condition (8.27) for the symmetric (left plot) and asym-metric (right plot) with the WLS volume correction; s = 2m,h/d = 2 with h = 0.4m and Tx/ρgh

2 = 0.15 and wave frequencyf = 1Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.21 Kinematic condition (8.27) for the symmetric (left plot) and asym-metric (right plot) with the MatApp volume correction (∆ = 10−15);s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh

2 = 0.15 and wave fre-quency f = 1Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.22 Comparison of the change of volume induced by the symmetric andefficiency of the proposed methods; (a) no correction of volume, (b)WLS method; (c) Unip method; (d) MatApp (∆ = 10−15); s = 2m,h/d = 2 with h = 0.4m and Tx/ρgh

2 = 0.15. . . . . . . . . . . . . . 185

8.23 Comparison of the error on the energy conservation in the systemfor the proposed methods; (a) no correction of volume, (b) WLSmethod; (c) Unip method; (d) MatApp (∆ = 10−15); s = 2m,h/d = 2 with h = 0.4m and Tx/ρgh

2 = 0.15. . . . . . . . . . . . . . 186

8.24 Ratio of the reflected power, Pr, to incident wave power Pi for theduct with an internal static pressure head of 2cm. Line: presentmodel; symbols: first harmonic analysis of wave gauge signals. . . . 187

8.25 Response of the membrane (ξ/A) as a function of wave frequencyf and horizontal coordinate x, parameters such as in test 8. . . . . 187

8.26 Modal response amplitude as function of wave frequency f and hor-izontal coordinate x; upper plot: first symmetric mode, middle plot:second symmetric mode; lower plot: third symmetric mode. . . . . . 188

8.27 As for Figure 8.26 for asymmetric modes. . . . . . . . . . . . . . . . 189

8.28 Membrane displacement (normalised by the incident wave ampli-tude A) for configuration of test 8 and f =1.05Hz. . . . . . . . . . . 190

8.29 Response of the membrane (ξ/A) as a function of wave frequencyf and horizontal coordinate x, parameters such as in test 8, withthe longitudinal tension used 2.7 times greater than that due toPoisson’s ratio effect only. . . . . . . . . . . . . . . . . . . . . . . . 191

8.30 As for Figure 8.29 with a the tension multiplied by a factor of 11. . 192

A.1 Power take-off set-up in the narrow flume. . . . . . . . . . . . . . . A2

LIST OF FIGURES xxi

A.2 Air pressure in the chamber as a function of the water-air inter-face elevation for several dimensionless impedances Z. The wavefrequency is f = 0.95Hz and incident wave amplitude A =0.025m. . A3

A.3 Phase lag between PTO water level aPTO and PTO air pressurepPTO as a function of dimensionless impedance Z for several wavefrequencies; • • • measured, — from (4.25). . . . . . . . . . . . . . A4

A.4 Capture widths for the duct equipped with the PTO and placedin the narrow flume for various dimensionless impedances Z; Greyline: calculated from (5.43), line and symbols: measured. . . . . . . A5

C.1 The duct, covered with the rubber 1mm membrane, in the wavebasin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1

C.2 Bow of the duct, with the connection for the water hose, in the wavebasin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C2

C.3 Stern of the duct, with the connection to the bent tube, in the wavebasin. Heavy masses are visible and were used to hold the bottomplate still. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C2

C.4 Power take-off system. . . . . . . . . . . . . . . . . . . . . . . . . . C3

C.5 Ramp and bow of the duct, in the narrow flume. . . . . . . . . . . . C3

C.6 The duct on the artificial seabed, covered with the 1mm membrane,in the narrow flume. . . . . . . . . . . . . . . . . . . . . . . . . . . C4

D.1 Sketch of the apparatus for testing rubber properties. . . . . . . . . D1

D.2 Three different loads applied to the strip of rubber; (a) F = 0N,(b) F = 78.86N, (c) F = 98.48N. . . . . . . . . . . . . . . . . . . . D2

D.3 Stress-strain relationship for successive loadings with two minutesintervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D3

D.4 Stress-strain relationship for successive loadings with thirty secondsintervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D3

D.5 Volume of the rubber strip for the different stresses applied. . . . . D4

D.6 Damped free oscillations of a rubber strip. . . . . . . . . . . . . . . D5

List of Tables

2.1 Comparison of wave power calculations for West Orkney site, afterFolley and Whittaker (2009). . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Average wave coefficients of flexible models, after Stamos et al. (2003). 53

3.2 Average wave coefficients of rigid models, after Stamos et al. (2003). 54

4.1 Measured characteristics of test rubber . . . . . . . . . . . . . . . . 65

6.1 Closed ends experiment: test conditions . . . . . . . . . . . . . . . 94

6.2 Comparison of experimental resonant frequencies fres with the the-oretical ones calculated as fn = nc/2L where c is the theoretical freebulge speed for the corresponding internal static pressure heads. . . 98

6.3 Experimental conditions for the closed ends experiment with the0.2mm membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.1 PTO experiment: test conditions . . . . . . . . . . . . . . . . . . . 133

A.1 PTO experiment in narrow flume: test conditions . . . . . . . . . . A2

xxiii

Declaration of Authorship

I, Nicolas Choplain, declare that the thesis entitled Interactions of a submerged

membrane with water waves and its use in harnessing nearshore wave power and

the work presented in the thesis are both my own, and have been generated by

me as the result of my own original research. I confirm that:

• this work was done wholly or mainly while in candidature for a research

degree at this University;

• where any part of this thesis has previously been submitted for a degree or

any other qualification at this University or any other institution, this has

been clearly stated;

• where I have consulted the published work of others, this is always clearly

attributed;

• where I have quoted from the work of others, the source is always given.

With the exception of such quotations, this thesis is entirely my own work;

• I have acknowledged all main sources of help;

• where the thesis is based on work done by myself jointly with others, I have

made clear exactly what was done by others and what I have contributed

myself;

Signed: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Date

xxv

Acknowledgements

The work presented in this thesis would not have been possible without the support

of many people that ought to be given credits here. First of all, I would like to

thank Professor John R. Chaplin for giving me the opportunity to work on this

project, guiding me trough it and bearing with my many questions. Most of

this work being experimental, I owe a huge debt of gratitude to the laboratory

staff who helped me, one way or the other, during this study: Karl Scammell,

Rhys Jenkins, Harvey Skinner, Mike Rose, Dave Lynnock and Earl Peters. I also

benefited from interesting discussions or practical help from the Anaconda team

and I wish to thank its members for that: Francis Farley, Grant Hearn, Valentin

Heller, Steve Rimmer and Martyn Hann. I wish to thank my housemates for

making the different houses I lived in more than just a house: Patrick Wiemann,

John Sandoval, Diana Caicedo, Martha Climenhaga and Ramesh Satkurunath.

The friendship that is born from this sharing is truly valuable. I feel lucky for

meeting many people during my stay in the UK and I especially want to thank

a couple of them for different reasons: Salome Giao, Polyvios Polyviou, Tassos

Papafragkou, Nicholaos Bakker, Davide Magagna, Jack Giles, Pascal Galloway,

Mark Leybourne, Ghayth Abed, Peter Smith, Konstantinos Nestoridis and the

football mates. Back home, I wish to thank my “pillar” friends: Benjamin Msika,

Pascal Jacq, Flavien Gouillon, Adrien Biggi, Jerome Boisard, Sylvain Crequet,

Aurelien Collombet and Lauriane Pautot-Lemaire. Thanks for giving sense to the

word friendship. A special and sincere consideration goes to Loic and Aurelie

Jullion. Lastly, I would like to express my gratitude to my family who always

supported me through those years of study, and not only PhD. Life makes that

we do not see as much as we would like to, but their thoughts and gestures always

warm my heart up. My mother and father, for leaving me the choice to do what

I like and support me all the way long, mentally, emotionnaly, physically and

financially. Having such wonderful parents is a gift that I am more than lucky to

have received. Finally, Konstantina, with her constant love, support and positive

attitude made me believe it could happen. Thank you all!

xxvii

Nomenclature

Variables of importance are detailed in the table below. Other notations are used

in this thesis for mathematical simplification and are detailed when in use.

a Half-width of the duct (m)

ai Equivalent wave amplitude (m)

A Wave amplitude (m)

Ai Incident wave amplitude (m)

Ar Reflected wave amplitude (m)

b Depth of the duct (m)

c Free bulge speed (m)

cg Group velocity (m/s)

cp Phase speed (m/s)

chys Free bulge speed with hysteresis in the rubber (m/s)

d Membrane submergence (m)

d1 Equivalent submergence depth (m)

D Distensibility (Pa−1)

e Displacement of the water-air interface in the power take-off (m)

E Young’s modulus (Pa)

E Wave energy (J/m2)

Ep Potential energy of the membrane (J)

Ek Kinetic energy of the membrane (J)

f Wave frequency (Hz)

fres Resonant frequency of the fluid-membrane system (Hz)

fslug Resonant frequency of the slug of water (Hz)

F Pressure amplification factor

g Acceleration due to gravity (m/s2)

h Water depth (m)

H Wave height (m)

i Imaginary unit; index of summation

j Index of summation

k Wave number (m−1); index of summation

k1 Water wave number (m−1)

k2 Bulge wave number (m−1)

Kl Lost wave energy coefficient

Kr Reflected wave energy coefficient

Kt Transmitted wave energy coefficient

l Length of the slug of water in the rigid tube (m)

L Length of the membrane (m)

Lp Length of a copper pipe (m)

m mass per unit length of the membrane (kg/m); index of summation

mt sum of the mass of the membrane per unit area and its added mass (kg/m2)

M Number of natural modes for the membrane displacement

n Index of summation

N Number of eigenvalues in the expansion series

p Pressure (Pa)

p0 Internal static pressure head (m)

pD Wave dynamic pressure (Pa)

pa Atmospheric pressure (Pa)

pb Bulge pressure (Pa)

pw Wave pressure (Pa)

Phys Power lost in the rubber due to hysteresis (W)

Pi Incident wave power (W)

Ploss Lost wave power (W)

Pf Mean power absorbed by the power take-off

from flow rate measurement only (W)

Pr Reflected wave power (W)

Pt Transmitted wave power (W)

q Flow rate in the copper pipes (m3s−1)

r Radius of a stainless steel pipe in the power take-off (m)

R Gas constant (J/kg K)

R1 Resistance level for characteristic resistor 1 (Ω)




xxx

Rm Radius of curvature of the membrane deflection (m)

s Half length of the membrane (m)

S Instantaneous cross-section of the rubber tube (m2)

S0 Undisturbed cross-section of the rubber tube (m2)

Sd Cross-section of the duct (m2)

St Cross-section of the rigid tube (m2)

t Time (s)

T Wave period (s)

Tu Circular tension in the rubber (N/m)

Tx Tension in the rubber in the x-direction (N/m)

Ty Tension in the rubber in the y-direction (N/m)

Tz Tension in the rubber in the z-direction (N/m)

u Fluid particle velocity (m/s)

ux Fluid particle velocity in the x-direction (m/s)

uy Fluid particle velocity in the y-direction(m/s)

uz Fluid particle velocity in the z-direction(m/s)

v Instantaneous rubber thickness (m)

v0 Initial rubber thickness (m)

V Instantaneous volume of air in the PTO (m3)

VA Voltage level for characteristic point A (V)

VB Voltage level for characteristic point B (V)

w Membrane centreline deflection (m)

x Position along the x-axis (m) (wave propagation direction)

y Position along the y-axis (m)

z Position along the z-axis (m) (perpendicular to still water surface)

Z Dimensionless impedance

ZPTO Impedance of the power take-off (m3s−1)

Zd Impedance of the duct (m3s−1)

Z1 Impedance pf one copper pipe (m3s−1)

L Lagrangian (J)

β Hysteresis coefficient (s)

δ Loss angle

ε Strain

εx Instantaneous strain rubber in the x-direction

εy Instantaneous strain in the rubber in the y-direction

εx,0 Initial strain in the rubber in the x-direction

xxxi

εy,0 Initial strain in the rubber in the y-direction

η Water surface displacement (m)

θ Angle of wave incidence ()

λ Wavelength (m)

ν Poisson’s ratio

µ Ratio of the duct cross-section to that of the bent tube

µair Dynamic viscosity of air (Pa.s)

µm Mass per unit area of the membrane (kg/m2)

ξ Normalised (by the wave amplitude) membrane displacement

ρ Density of water (kg/m3)

ρr Density of rubber (kg/m3)

σ Stress (Pa)

σx Instantaneous stress in the rubber in the x-direction (Pa)

σy Instantaneous stress in the rubber in the y-direction (Pa)

σx,0 Initial stress in the rubber in the x-direction (Pa)

σy,0 Initial stress in the rubber in the y-direction (Pa)

τ Ratio of the flow rate in the copper pipes to that of the air in the chamber

φ Flow potential (m2s−1)

φi Phase of incident wave

φr Phase of reflected wave

φeq Phase lag between pressure and displacement in the power take-off

ω Angular frequency (s−1)

xxxii

To my parents and family

A mes parents et ma famille

xxxiii

Chapter 1

Introduction

1.1 Motivation

“I have a great many reasons why I think wave power can be put

to practical and profitable use, but I don’t wish to occupy too much of

your valuable time, or, perhaps, there are not many of you particularly

interested in the subject”

This statement made within the discussion of an article by Stahl (1892) seems ob-

solete. Nowadays, almost no one would disagree that the power contained in the

oceans, in many forms and particular in waves, could contribute to the ongoing

change in electricity production from fossil fuels to greener ways.

This has led to the development of various theories and corresponding devices

to harness this power over the last decades. Almost all of these structures were

solid, yielding high costs of operation and maintenance, and hampering the suc-

cessful wave energy conversion process.

Because they are made of rubber (and other components) and not solid material,

distensible devices are thought to contribute to the development of more reliable

systems. Rubber is cheap, environmental friendly and can adapt to even rough

sea states. One of the promising devices is the Anaconda, a rubber tube placed

beneath the surface and aligned head to waves. It is filled with water and its res-

onant conditions are used to maximise the magnitude of the pressure propagating

in it to extract the corresponding power. It is aimed to be deployed in deep waters

where wave power is expected to be of higher order of magnitude. As for any other

1

2 Chapter 1 Introduction

floating or submerged (but not on the seabed) structure, moorings can not only be

hazardous but also add to the cost of operation and maintenance in case of failure.

The goal of the current work was to investigate the applicability of the theory

developed for the Anaconda on a bottom-mounted version and provide the neces-

sary information on a potential commercial development. The appropriate model

was a long rectangular duct, with solid walls, aligned head to waves, resting on

the seabed and covered with a rubber membrane. Two configurations were tested,

one with closed ends and one with a power take-off system connected at its stern.

The work presented in this thesis was driven by the desire to compare theoreti-

cal results from the one dimensional model of pressure propagation in distensible

tubes, such as the one in the Anaconda, to the results of a series of experiments

carried on both versions of the duct. Values of pressure propagation in the duct,

membrane displacement and power captured by the power take-off (for the config-

uration with it) are then presented. Discrepancies between the above theory and

experimental data for one of the tested membrane are explained and a discussion

follows on the limitation of using this one dimensional model for specific configu-

rations.

The main assumption in the latter model is the absence of interactions between

the membrane and the wave propagating over it. In order to further study this

hypothesis, a two dimensional analytical model was developed. It is based on an

eigenfunction expansion method, and deals with the interactions of the studied

geometry where the membrane is stretched in the wave propagation direction. Its

results are applicable to the present work, and also to a wide range of fields, es-

pecially for breakwater studies.

A novel technique for the measurement of the membrane displacement (hence

the pressure in the duct) beneath waves, based on a set of laser sensors, placed in

waterproof boxes just beneath the water surface is also presented. Results from

this technique fill the gap of the current available experimental characterisation of

such fluid-filled system behaviour in waves, especially at resonance.

Chapter 1 Introduction 3

1.2 Thesis layout

The work carried is presented in this thesis according to the following layout:

• Chapter 2 supplies the reader with the basic concepts behind wave energy

conversion. The two main reasons on why this work is relevant are given,

together with the precise definition of the aims and objectives for the present

study.

• Chapter 3 is devoted to the literature review of previous studies dealing

with similar device. A technical background is also provided, and areas

where the present work can address gaps or improvement in the knowledge

are emphasised.

• Chapter 4 details the experimental facilities, techniques and instrumentation

used to fulfil the experimental part of the present investigation.

• Chapter 5 describes the theory for the free bulge speed calculation, pressure

in the duct for the closed ends and PTO-equipped versions, based on a

one dimensional model for wave propagation in compliant and liquid-filled

structures.

• Chapter 6 presents the experiments carried on the closed ends version of the

duct in a narrow flume and compares the values of membrane deflection, free

bulge speed in the membrane and pressure in the duct with those from the

theory from chapter 5. Discrepancies between results from this model and

that of a series of experiments carried out with the thinner of the two tested

membranes are explained.

• Chapter 7 presents the experiment carried on the duct equipped with the

power take-off connected at its stern, and starts with an analysis of the power

take-off system and its reliability as a linear dashpot. This done, power data

are presented for various configurations and there follows a similar study of

pressure characterisation as in chapter 6.

• Chapter 8 presents the two dimensional model developed within the course

of this study. It concerns the interactions of water waves with a bottom-

mounted duct, aligned head to waves and covered with a stretched mem-

brane. The challenging part of the fluid volume conservation is detailed and

solutions proposed with the aim of obtaining a best compromise solution.

4 Chapter 1 Introduction

Outputs from this model are compared as far as possible with experimental

data from the wide flume experiments.

• Chapter 9 closes this thesis by summarising the work done and its potential

contribution to science, and outlines the future work that could be carried.

Chapter 2

Wave energy glossary and

relevance of the study

2.1 Introduction

The author has continuously noticed during this work that the idea of harnessing

wave power receives an enthusiastic welcome from individuals not from this field.

This interest was either based on their environmental convictions, their technolog-

ical interests or simply by curiosity. However, the author also noticed that the first

reaction was that people were associating wave energy with the energetic impact

of a breaking plunging wave seen during their beach holidays.

This is why the present chapter is here to give a brief background on wave char-

acteristics and wave energy conversion. From that point, the justification of the

study follows, based on the consideration of the Anaconda wave energy converter

(WEC) and the new trend in capturing nearshore wave power. The aims and

objectives of the present work closes up this chapter.

2.2 Wave characteristics

2.2.1 Introduction

Due to the differential heating of the atmosphere by the sun, wind motion is

triggered in the atmosphere. When wind blows over oceanic area, forces (pressure

5

6 Chapter 2 Wave energy glossary and relevance of the study

and friction) disturbs the water surface and gravity tries to restore this disturbance

resulting in oscillations about the equilibrium state. Waves are created and are

the result of a transfer of energy from wind to the surface. For a wavelength more

than a couple of centimetres, ocean waves are categorised as gravity waves.

2.2.2 Definitions

A simple representation of a monochromatic (having a single wavelength and fre-

quency) gravity wave is given in Figure 2.1. It is characterised by:

. its frequency f and angular frequency ω = 2πf .

. its amplitude A and height H = 2A.

. its wavelength λ that is the distance between two wave crests (or troughs)

. its wavenumber k = 2π/λ with

ω = kcp, (2.1)

where cp is called the phase speed and represents the speed of a crest (or

a trough).

Wave period is defined as

T =1

f(2.2)

=λ

cp. (2.3)

The linear wave theory is used in this study. Linear waves have amplitudes much

smaller than the water depth they propagate in. This criteria is characterised as

Ak << 1 (orA/h << 1).

In that case, the dispersion relationship links the wavenumber and the frequency

as

ω2 = gk tanh(kh). (2.4)

Chapter 2 Wave energy glossary and relevance of the study 7

l

x

A

Direction ofpropagation

z

crest

troughh

O

hj

i

dz

Figure 2.1: Sketch of a water wave

from which k can be obtained and yielding the phase speed

cp =

√

g tanh kh

k. (2.5)

Finally the surface displacement and potential of the progressive monochro-

matic wave shown in Figure 2.1 is defined as

η(x, t) = A cos(kx− ωt). (2.6)

2.2.3 Kinetic and potential energies

Waves encompass two forms of energy: kinetic energy associated with the water

particles’ motion and potential energy as a result of a displaced mass of water

from its equilibrium position in a gravitational field.

The total wave energy per unit area is usually defined as

E =ρg

8H2. (2.7)

However for wave power absorption purpose, it is convenient to know how much

energy a device can absorb per unit time as shown in Figure 2.2. With Lc taken

as a unit length, Dean and Dalrymple (1991) defines this available energy as the

rate at which work is done by the dynamic pressure pD on a vertical section in

direction of wave propagation.


Lc

Area of energy

absorbed during

the time T

Incidentwaves

c Tg

Figure 2.2: Energy absorbed per unit time T

This instantaneous power is

I =

∫ η

−h

pDuxdz (2.8)

with

pD = p+ ρgz (2.9)

= −ρ(∂φ

∂t− gz), (2.10)

where φ is the flow potential and ux = ∂φ/∂x the fluid particle velocity. Averaging

this instantaneous work over one wave period T and for the monochromatic wave

in Figure 2.1 gives the average energy flux

I =1

2ρgA2

︸︷︷︸

E

ω

k︸︷︷︸

cp

1

2(1 +

2kh

sinh 2kh)

︸︷︷︸

n︸︷︷︸

cg

(2.11)

where cg is called the group velocity and is defined as the speed at which energy

is transported.


A dimensional analysis of I gives

[I] = [E].[cg].1 (2.12)

= M.T−2.L.T−1 (2.13)

= M.L.T−3 (2.14)

= M.L2.T−3︸︷︷︸

power.L−1 (2.15)

showing that the energy flux has the dimension of a power per unit length. The

definition in (2.11) will then be used to characterise themean water wave power

per unit crest length and I will therefore be denoted P .

The main advantage of using a definition of power relative to a unit crest length

is that it gives information on the width of crest from which a device will harness

wave power. This width is called capture width (CW) and is defined as

CW =mean power absorbed by a WEC

mean power per unit crest length(2.16)

2.3 Wave power background

2.3.1 Introduction

Waves carry energy as it was theoretically explained in the previous section. How-

ever, real sea states are far from the ideal monochromatic case shown in Figure

2.1. A typical wave elevation is due to waves of different amplitudes, frequencies

and directions. Offshore industry and wave energy developers require mapping of

the wave field in terms of wave characteristics, namely wave height and frequency.

It is quite straightforward with a monochromatic wave but not with the sea state.

However, several techniques exist for that purpose, based on the type of repre-

sentation of the sea-state (stochastic or deterministic) and the corresponding data

analysis of in-situ measurement. Wave models are also continuously developed

and improved either for large scale representation, or specific target site study.


Figure 2.3: Global distribution of annual mean wave power, after Cornett

(2008)

2.3.2 Wave power resource distribution

The theoretical global resource for wave energy is estimated to be 8, 000-80, 000

TWh/year, corresponding to a power of 0.91− 9.13 TW (Boud (2003)). It is usu-

ally estimated that the exploitable part of this power is in about 10% for practical

and economical reasons. To give an order of magnitude for comparison, the an-

nual world electricity consumption is around 16, 000 TWh/year corresponding to

a mean power of 1.826TW: so if all this portion of wave power were converted into

electricity, it could contribute up to 50% to the world electricity consumption.

This figure will be substantially lower, depending on technical developments of

wave energy converters and this is why wave energy should be seen as a contrib-

utor together with any other forms of renewable energies, marine or not, rather

than the only solution.

Wave climate, by its origin, is closely related to wind patterns. This is why

the best wave climates (for wave energy purpose) are found between the latitudes

of ∼ 30o and ∼ 60o, where the prevailing westerlies blow over a long oceanic

distance gifting the west coasts of Europe, North America, Australia and South

Africa (Figure 2.3). With high wave energy resource the wave climate in Europe

is particularly energetic: it has been estimated than the wave power resource for

the north-eastern Atlantic (including the North Sea) is about 320GW (Mørk et al.

(2010)).


a)

Column

Water level

Turbine

Turbine

Airflow

Basin water level

Waterflow

c)

b)

High Lowpressure pressureOWSC

PA

PA

Attenuator

OWCOvertoppingdevice

Submerged pressure differential

Figure 2.4: Wave energy converters classification, from Rahm (2010)

2.3.3 Harnessing concepts of wave power

2.3.3.1 Introduction

Although over 1,000 patents exist today regarding ways of extracting wave power,

a generic classification can be used to categorise the different concepts used for

that purpose. Differentiation between devices can be either based on intended

locations of use or modes of operations. The latter is used here to give the main

concept on wave energy conversion techniques and the differentiation criteria used

are the ones developed in Falcao (2010) and shown in Figure 2.4 as a) oscillating

water column, b) overtopping devices and c) oscillating bodies.

2.3.3.2 Oscillating water column

An oscillating water column is made of a solid structure enclosing a slug of water

whose heave motion is driven by the incoming wave pressure. Above it, a tur-

bine rotates due to the reciprocating flow associated with this displacement and

generates electricity.


2.3.3.3 Overtopping devices

Another way to extract wave energy is to apply the principles of hydropower to

wave energy conversion. Water wheels have always used the energy of flowing

or falling water to power any kind of work, as for milling flour in gristmills for

instance. Overtopping devices use this concept: they collect water of incident

waves in a elevated reservoir to drive low head turbine(s).

2.3.3.4 Oscillating bodies

Oscillating bodies represent any other devices that do not fit into the two previous

categories. As suggested by its name and unlike the two other types, an oscillating

device is in motion and wave-driven. It is this motion, mostly relative to other

component of the structure, or an immobile reference, that is used to extract wave

power.

This category can be sub-categorised into different categories regarding device

geometry and/or mode of motion. Looking at the oscillating bodies on Figure 2.4

from left to right, this sub-categorisation is:

• Oscillating Wave Surge Converters (OWSC). In deep water, as waves

pass over the surface, the particles of water do not move forward with the

waves: they gyrate in circles. As the depth increases, the radius of the circu-

lar motion decreases. At a depth equal or greater than half the wavelength,

they are quite still. On the contrary, in shallow waters, the particle trajec-

tories are compressed into ellipses due to the interaction between waves and

bottom. This leads to a back and forth motion of the water particles (bot-

tom waves). OWSC are driven by the surging action of waves, resulting in a

pitching motion of their constitutive plate relatively to the bottom. This is

also why they are set in shallow waters. They will be more detailed in section

2.4.3.3 as they are aimed to harness nearshore power as in the present work.

• Next two devices are point-absorbers (PA). The characteristic length of

such device is small compared to the wavelength. First device designated PA

converts the motion of the waves into rotational motion, causing an internal

fluid to run unidirectionally through a specially designed pump to generate

electricity. The second PA, and more widely used, consists of a floating buoy

whose relative displacement due to wave motion to a static reference is used


to drive a hydraulic pump. The advantage of the latter is that they are

not oriented into a particular direction so can absorb wave energy from all

directions of propagation.

• The penultimate device of Figure 2.4 is an attenuator. Attenuators are long

structures placed parallel to wave propagation. They are usually composed

of several sections that each follow the wave profile. The whole structure

length being of same order of magnitude or greater than the wavelength

these sections move relatively to each other. Connecting these sections with

hydraulic rams allows the conversion of this mechanical movement into elec-

tricity using hydraulic generators.

• The last illustrated device is a submerged pressure differential, using

the pressure variation below travelling waves. Low pressure (wave trough)

lifts the movable part of the device and high pressure (wave crest) pushes it

back, compressing the enclosed air in the chamber.

2.3.4 Conclusions

Through the years, many WECs have been fashioned by scientists, engineers or

inventors, using the concepts detailed in section 2.3.3. Yet, among the different

forms of marine renewable energies, wave energy seems late compared to offshore

wind energy for instance. The reasons are because of the rough and unpredictable

environment this field is dealing with and not because techniques and knowledge

are not mastered. However, because of the recent sea tests for various prototypes,

together with the cost reduction for material used and installation processes, there

are reasons to be optimistic for wave energy farms in the long-term.

Because of the vast majority of devices already existing, it is the belief of the

author that the study, development and experimentation of a new device should

be explained. Therefore, this section is to detail the reasons why this study is

relevant. This is achieved by considering the Anaconda project and the recent

developments of devices capturing nearshore wave power.


2.4 Relevance of the study

2.4.1 Introduction

This section is to present the two main arguments justifying the present work: a

promising device, the Anaconda, that is aimed to harness offshore wave power and

the reconsideration of capturing nearshore wave power.

2.4.2 The Anaconda WEC

2.4.2.1 Principle

The Anaconda is a new way of extracting energy from ocean waves, based on

bulge waves travelling along a distensible rubber tube (Farley and Rainey (2006);

Chaplin et al. (2007a,b, 2012)) (see Figure 2.5), filled with water, oriented in the

direction of wave travel and anchored head to waves (attenuator). Oscillations in

pressure beneath the waves generate travelling bulges in the tube. The bulge waves

grow in the down-wave direction, converting wave energy into internal oscillatory

flow that can be used to drive a turbine. One end of the tube is closed (the

bow) and the other one is linked to a power take-off (PTO). Tuning is achieved by

matching the speed of free bulge waves (a function of the tube’s material properties

and geometry) to the phase speed of water waves.

Figure 2.5: Anaconda WEC, after Checkmate SeaEnergy Ltd.


The main interest of this section on the Anaconda is not to detail all the theoretical

and experimental work that has been carried out so far by the Anaconda project

team. A set of interesting papers on the subject (Farley and Rainey (2006); Chap-

lin et al. (2007a,b)) has been written for that purpose. However, the aim here is

to briefly describe the theory behind it and to present some results agreeing with

the promising potential of this device.

2.4.2.2 The bulge wave theory

Bulges like the one shown in Figure 2.6 are waves propagating into the rubber

tube in the same way blood flows in arteries. They are driven by the change of

pressure difference across the tube (or channel) walls. The general assumption

is that this pressure difference is negligible over the cross-section and hence, the

wave propagation in the tube (or channel) is driven by the gradient of pressure

in the longitudinal direction (x with notations from Figure 2.6). Hence, all the

kinetic energy of the fluid is due to this longitudinal motion and the fluid particle

velocity in the channel is reduced to

u = uxi, (2.17)

where ux will therefore be noted u for clarity and with i the unit vector in the

x-direction.

The pressure acting on the compliant wall is noted pw and the pressure inside

the duct p = pb + pw. For the situation where no other forces are involved in the

fluid motion and neglecting the non-linear terms of the momentum equation,

ρ∂u

∂t= −∂p

∂x. (2.18)

On the other hand, assuming the flow to be uniform over the cross-section, the

linearised form of the continuity equation

∂ρS

∂t+

∂

∂x(ρSu) = 0 (2.19)

leads to∂S

∂t= −S0

∂u

∂x. (2.20)


Bulge wave

x

p (x,t)w

p (x,t)b +p (x,t)w

S

Wave front Water wave

propagation direction

Figure 2.6: Bulge wave propagating in the rubber tube.

Differentiating (2.20) with respect to t and using the definition of the distensibility

(with units of Pa−1)

D =1

S0

dS

dpb(2.21)

leads to the bulge wave equation obtained by Farley and Rainey (2006)

∂2pb∂t2

=1

Dρ

∂2

∂x2pb + pw. (2.22)

The distensibility represents the ability of the tube to respond to changes of inter-

nal pressure with a proportional change of its cross-sectional area from its undis-

turbed value S0.

This wave equation assumes no losses in the fluid or the rubber. Lighthill (1978)

detailed the wave attenuation by friction along the walls of the channels due to

viscous tangential stresses. This energy dissipation occurs in a boundary layer

(Stokes boundary layer) which Lighthill (1978) gives an approximate thickness a

a function of the fluid kinematic viscosity and frequency of fluid motion ω. Given

the dimensions of the problem, and especially the width of the duct, this dissipa-

tion is assumed negligible, on the contrary to what can happen when dealing with

smaller geometries, such as blood propagation into arteries.


More important are the losses in the rubber wall by hysteresis, a notion described

later in section 4.4.3. Those losses are represented by the extra term −β ∂3pb∂t3

added

to the left hand side of (2.22) such as

∂2pb∂t2

− β∂3pb∂t3

=1

Dρ

∂2

∂x2pb + pw, (2.23)

where the loss factor β being defined by the stress-strain relationship

σ = E(ε+ βε), (2.24)

with E the Young’s modulus of the rubber and δ = tan−1(βω) the loss angle due

to hysteresis in the rubber.

The bulge pressure pb can be written, according to Farley and Rainey (2006),

pb = ρgAF, (2.25)

with A the wave amplitude and F the pressure amplification factor that gives the

oscillating pressure inside the tube compared to the oscillating pressure in the sea

wave outside. In the absence of hysteresis, the pressure amplification factor is

F =k21

k∆ksin(x

∆k

2) ≈ 1

2k1x, (2.26)

which is valid for small ∆k where

∆k = k1 − k2, (2.27)

k =k1 + k2

2, (2.28)

with k1 = ωc1 is the wave number of the external wave and k2 is the bulge natural

wave number associated with the natural bulge wave speed first introduced by

Lighthill (1978) defined as

c =

√1

Dρ. (2.29)

The free bulge speed is affected by the hysteresis and the free bulge speed chys

with the hysteresis taken into account was defined by Chaplin et al. (2012) as

chys =c

√12(1 + sec δ)

. (2.30)


The measured values of the loss angle for the membranes tested in this work

(δ ≤ 2.69) do not result in significant modification of the free bulge speed due to

hysteresis. Hence this effect on the free bulge speed will be neglected.

It can be seen from (2.26) that the bulge amplitude grows linearly along the tube.

F is a sin(X)/X for variable ∆k so that the maximum amplification factor occurs

for ∆k = 0 when the natural bulge wave speed matches the external wave speed.

In this condition, the magnitude of oscillation of the bulge pressure is several times

stronger than the magnitude of the incoming wave amplitude. Bulges then carry

power extracted from the water wave above and the objective is to develop and

use a PTO that absorbs a maximum proportion of this power.

2.4.2.3 Current developments

The pressure amplification along the tube occurs when the free bulge speed matches

the above water wave speed. The idea recently developed by Chaplin et al. (2012)

is to use a bent slug of water at the end of the tube, whose natural frequency is

the same as the frequency at which the resonance in the tube occurs, hereinafter

called the resonant water wave frequency. This way, the horizontal motion of the

bulges at the stern of the rubber tube is completely converted into vertical motion

of the slug of water without any reflection. The power take-off is then an air power

take-off similar to what is found in an OWC. The water-air interface at the top of

the slug pushes back and forth air through a turbine.

In their work, Chaplin et al. (2012) model the air PTO as a linear dashpot,

meaning that the air pressure (force acting on the PTO) and the air-water in-

terface velocity are in phase. This is commonly used when modelling air PTO

in wave energy converters, also because turbines such as the Wells turbine are

aimed and designed to be linear. A more complete description on the linear dash-

pot used is detailed in section 4.5 as the same concept is used for the present work.

The coefficient of proportionality between the air pressure and the air flow in

a linear dashpot is called the impedance. Figure 2.7 shows capture widths for

the PTO described above, placed at the stern of the 7m long and 250mm di-

ameter rubber tube, and for several dimensionless impedances Z (impedance of

the PTO/impedance of the tube) ratio. It can be seen that the maximum capture

width, of order of 1.75 tube diameter, is obtained for a ratio of Z=1.05. This means

that the maximum power absorption is obtained when the tube’s impedance (the


ratio pressure/flow, e.g. ρc/S0) matches the PTO impedance and occurs when

the wave period T equal the period the resonant water wave period T0. Capture

widths results agree with the 1D theory that is explained later in chapter 5.

Figure 2.7: Capture widths as functions of relative wave periods for (a)

Z=0.64, (b) Z=1.05, (c) Z=3.3, (d) Z=7.7. Measurements are shown as points,

one-dimensional theory with δ = 13 as continuous lines, after Chaplin et al.

(2012)

2.4.2.4 Conclusions

For any WEC project to be seen as feasible, the different challenges shown in

Figure 2.8 must be satisfied. The Anaconda is a distensible tube made of rubber

whose theory has been introduced in section 2.4.2.2 with a 1D model. This theory


has been experimentally demonstrated at a laboratory scale of 1:25 as detailed in

Chaplin et al. (2012) (Predictability). The main advantage of the Anaconda lies

in the fact that it is made of rubber. Rubber can be considered:

1. Cheap compared to the materials used for other mechanical devices (Afford-

ability).

2. A natural compound that does not corrode in saline environment (Surviv-

ability, Reliability and Operability).

3. A light material, making towing and anchoring processes easier(Installability).

Manufacturability

Challenges

Affordability

Reliability

Survivability Operability

Installability

Predictability

Figure 2.8: Challenges facing the development of a new wave energy converter

The industrial concern at this point is the Manufacturability challenge. At full

scale, the Anaconda is aimed to be about 200 m long and 7 m in diameter, mak-

ing it the largest tube of rubber ever designed. Such considerations are still in

development.

Located just beneath water surface, the Anaconda can however be hazardous

for ship passage or marine activities. Moreover, it is aimed to be anchored in

about 40-50 m depth of water. Mooring cables can represent a hazard for marine

life, especially whales’ migration. Main whales migration routes, located on the

west coast of North America, at the southern tip of South Africa and in Norway

are seen to be interesting spots regarding wave power resource as seen in Figure 2.3.


This is why the author aims to find out whether the bulge wave concept is appli-

cable on a seabed version, in order to avoid these issues. The bulges being created

by the dynamic wave pressure, the idea is to use it at a location where the seabed

is sensitive to the incoming wave pressure, hence in the nearshore zone. The next

section is to detail why this zone is of interest for wave power industry and the

current developments in term of bottom-mounted WEC.

2.4.3 Nearshore wave power

2.4.3.1 Introduction

Offshore locations have always attracted more interest from wave energy developers

due their highest gross energy resource. Onshore sites, mainly for OWC develop-

ment, have also been privileged for their easy access (operation and maintenance)

and the fact that no underwater cable is needed to carry back the electricity to

the mainland.

However, nearshore locations (usually defined as water depths of 10-25 m) have

often been of less interest for WEC developers. The main reason is that nearshore

wave climate have been evaluated using the definition of gross energy resource.

This section is to detail the reasons why this definition is not adequate for these

locations where the bathymetry and wave directionality have to be taken into

account.

2.4.3.2 Exploitable wave power

Folley and Whittaker (2009) have recently shown that the nearshore wave cli-

mate might be more useful than previously thought. Using the SWAN model,

the nearshore wave climate was predicted using the offshore wave climate as an

input in the model. As an example, they used a wave of 10 second period prop-

agating from offshore location (50 metre depth) to nearshore location, and using

different seabed slopes they showed that the wave power loss from one site to an-

other is relatively small (reproduced in Figure 2.9) as long as the water depth is not

smaller than 8 metre. After that the depth induces wave breaking and energy loss.


Figure 2.9: Shoaling of a 10 second energy period wave propagating orthogonal

to depth contours for different seabed slopes, after Folley and Whittaker (2009).

Figure 2.10: Variation in percentage power loss from offshore to nearshore

site for an initial sea-state of T=10s and Pi=50 kW/m travelling at an angle

to the depth contours on a seabed slope of 1:100, after Folley and Whittaker

(2009).


The usual wave energy resource uses the wave gross power, defined as the wave

power that crosses through a circle of 1 metre (Folley and Whittaker (2009)). This

definition includes all possible wave directions of propagation. This is why this

measure of the wave resource is not useful for evaluating a site where devices cap-

turing wave power in one direction only are to be installed. On the contrary, the

net wave power is defined as the wave power for one direction of propagation.

Since wave refraction is not a dissipative phenomenon, there is almost no net

wave power density loss from offshore location to nearshore site. However, the

percentage of wave power density loss from offshore to nearshore site increases

significantly due to directional dispersion (Figure 2.10).

In order to compare more accurately offshore and nearshore resource, Folley and

Whittaker (2009) defined a new representation for the wave resource called ex-

ploitable wave energy resource. Defining a “threshold wave power” for the

device/wave farm as four times the average power received (above this thresh-

old there is no increase in power capture for more energetic sea-states), the ex-

ploitable wave energy resource is defined as the average wave power in a fixed

direction as limited by the aforementioned threshold. This makes it possible to

get a better comparison (regarding the wave resource, rather than the economics

of the devices) between offshore gross power captured by devices (regardless of

wave direction) and nearshore power.

As an example, Table 2.1 compares the values for the three different definitions of

wave power resource. The table shows that the reduction of wave power from an

offshore to a nearshore site using the gross wave power is reduced from 20% to 7%

using the exploitable wave energy resource definition.

Deep water Offshore Nearshore

(200m) site site

Average gross wave power kW/m 30.8 27.2 21.8

Average net wave power kW/m 26.8 25.0 21.4

Average exploitable wave power kW/m 22.6 21.2 19.7

Table 2.1: Comparison of wave power calculations for West Orkney site, after

Folley and Whittaker (2009).


Another advantage of using nearshore power is that devices placed in this region

will not undergo the same hazards as offshore devices, such as storms or rogue

waves. This has a considerable impact upon the cost of the structure and hence

of the electricity generated.

In summary, it may be the time to reconsider the potential of nearshore regions

for harnessing wave power. Some concepts, rapidly introduced in section 2.3.3.4,

are aimed for that purpose and are presented in greater detail next.

2.4.3.3 Nearshore bottom-mounted wave energy converters

The OYSTER (Whittaker et al. (2007)), developed by Aquamarine Power (Scot-

land) is a hinged flap attached to the seabed that is driven by the back and forth

movement of the waves. The oscillation of the flap (shown in Figure 2.11) com-

presses and extends the two hydraulic cylinders, which in turn, pump water ashore

at high-pressure supplying the hydroelectric turbine. After tank testing at scale

1/40th and later 1/20th at Queen’s Univerity of Belfast, the first full-scale proto-

type, Oyster 1, was pinned to the seabed at EMEC in summer 2009. It was later

connected to the national grid in October of the same year.

The 18m wide flap was constructed of five cylinders with a height of 11m. It

was set in a mean water level of 10m, so that the top flap emerged. The analysis

carried by Folley et al. (2007a) has shown that the power capture depends pri-

marily on the incident wave force. For that reason, the flap is surface piercing in

order to maximise the force acting on it. This 350kW rated prototype operated

for 6000 hours, giving valuable information for the next built generation of Oyster

2.

Oyster 2 consists of three Oyster 800 devices (rated at 800kW). The width of

the flap has been extended to 26m, as a wider flap was shown to be more efficient

(Folley et al. (2007b)). However, too wide a flap experiences greater structural

load (Cameron et al. (2010)) and this is why this width was not extended more.

The first Oyster 800 has been installed at EMEC and two others, one in 2012 and

one in 2013, will make the 2.4MW wave energy farm complete.


Figure 2.11: The offshore Oyster 1 device, after Cameron et al. (2010).

The WaveRoller designed by AW-Energy uses the same concept as the Oyster but

is not surface piercing, using only the (less powerful) bottom-waves. It is devel-

oped by AW-Energy (Finland) and has continuously been developed since 1999

with small-scale tests in real sea and laboratory for validating the concept. This

led to the first version of WaveRoller, a 3.5m high and 4.5m plate anchored in wa-

ter depths of 10-15m off the coast of Peniche, Portugal, in 2007 and rated at 10kW.

Successful results led to the development of the next generation of WaveRoller

prototype. It showed good capability in harnessing wave power for a wide range

of wave period.

The second step, helped by EU funding, is to develop a first full-scale demo consist-

ing of three plates rated at 300kW and linked together. This full scale prototype

has arrived (January 2012) at Peniche and is awaiting installation at the same

location as their smaller prototype.

2.4.3.4 Conclusions

Among all the new extraction techniques, more and more are aimed at nearshore

locations for the reasons mentioned. These are mainly easy access and less rough

(yet interesting) wave climate and short underwater electrical cable (conversion

can be done onshore), both arguments leading to a cost reduction of the electricity

produced.


2.5 Starting point of the project

2.5.1 Introduction

It has been shown that the move into using nearshore power has taken place in

the last years. This is mainly due to the fact that these locations are easier to

access and much less prone to severe storm occurring in offshore sites. In an effort

to use these locations, the available energy has been reevaluated and reconsid-

ered. Nearshore sites have always been seen as less energetic than their offshore

counterpart because the offshore power spectrum considers contributions from all

wave directions. However, most devices have one or two predominant directions

for absorption and disregard the energy coming from other directions. In con-

trast, wave energy is concentrated in the nearshore in a narrow band of directions,

strengthening the potential use of directionality dependent WEC.

In the Anaconda, bulge waves travels back and forth in a long rubber tube placed

beneath the water surface, extracting the energy form the wave above. The theory

and the main advantages of using and developing such a concept have been pre-

sented, and recent developments show the potential of the Anaconda in harnessing

wave power.

These two aspects justify the investigation of the potential of a seabed-mounted

version. In order for the membrane to be under the influence of dynamic pressure

the device must be set in shallow waters and hence addresses harnessing nearshore

wave power.

2.5.2 Aim and objectives

This project will be driven by different aims to be achieved through either exper-

imental work or theory. It is worth mentioning again here that the aim is not to

develop another WEC that would supposedly be more efficient than the others,

less noisy or have less impact on the surrounding environment. The main aim

is to find out whether the bulge wave concept developed through the

Anaconda project is applicable in a seabed version. In other words, be

able to explain why, or why not, it can harness wave power. This aim is achieved

with precise experimental and theoretical objectives in mind.


Regarding the theoretical study of the present geometry, the objectives consist

of

• determining the membrane deflection for different pressure heads in static

water.

• determining the theoretical free bulge speed.

• having a validated theoretical model for predictions of pressure amplifica-

tions (closed ends) and power capture (PTO) regarding parameters involved

(membrane tension, membrane submergence, water depth and PTO charac-

teristics).

The intended experimental work can therefore be summarised as:

• designing and building a long rectangular duct covered with an elastic rubber

membrane to be tested in the available facilities. Equip it appropriately for

measurements of static deflection, free bulge speed, membrane deflection,

and pressure oscillation beneath waves.

• test a closed end configuration beneath waves with the purpose of investi-

gating pressure oscillation at its stern.

• equip it with a power take-off at its stern for measurement of extractable

power.

2.5.3 Novelty and advantages of the studied device

The design studied is a rectangular duct enclosing water and closed on its top face

with an elastic rubber membrane as shown in Figure 2.12. It rests on the seabed

and is placed in the direction of wave propagation. This configuration results in

the following novel features:

1. use of a distensible device for harnessing nearshore wave power.

2. non-surface piercing bottom-mounted device, like the WaveRoller but with-

out any movable solid parts.


If possible power capture is to be measured within this study and the potential

of such a device demonstrated, the latter would have the main advantages of

being sited nearshore (cost reduction), made of relatively cheap material and not

interfering with marine and leisure activities.

stern

bow

x y

z

Direction ofwave propagation

rubbermembrane

Figure 2.12: 3D view of the tested duct.

2.6 Conclusions

In the effort to rethink the way electricity is produced in societies, wave power

is seen as a viable source. Resource is abundant, renewable and its predictability

ensured by the constant development of spectral models and data assimilation

techniques.

This resource is unevenly spread around the world and this is why some coun-

tries more than the others (UK, Ireland, Portugal, USA, Canada, Australia and

Japan) launched research and developments programmes in the 1970s, mainly trig-

gered by the oil crisis of 1973. Those efforts have paid off with the development

of full-scale testing and wave farms projects.

The cost of the produced electricity is still unattractive compared to traditional

ways (oil and coal), or even offshore wind energy, for reasons deliberately not men-

tion here. Techniques, on the other hands, are known and have been detailed in


section 2.3.3. Unconventional devices are created each year but leading technolo-

gies are usually quite simple. The main effort is in reducing cost of operation and

maintenance.

The two main motivations of the current study have been given and a descrip-

tion of how a combination of them could possibly lead to the development of a

device that could contribute to this cost reduction task. The studied device, a long

rectangular duct enclosing water with its top face made of rubber was fashioned

and aimed to be tested following precise tasks.

The advantages of using rubber have been detailed. However, such use is not

new for marine applications. The next chapter gives a review of studies dealing

with interactions of water waves with submerged elastic membranes or with closely

related configurations.

Chapter 3

Literature review

3.1 Introduction

The general consideration of a fluid-filled membrane covered duct can relate to a

great number of similar studies, in various areas of science. The aim of the litera-

ture review is to detail previous investigations carried out and theories developed

that can be of use for the present study. This is also a way to synthesise the

information provided and to spot areas where the present work can contribute in

extending the current knowledge. A first section gives some generalities on the

use of rubber in the marine environment. The studied device presented in Figure

2.12 has various similarities with previous studies dealing with use of submerged

membranes for marine coastal applications, and in particular, flexible breakwaters

of various shapes, and this is detailed in another section.

3.2 Use of rubber for marine applications

Rubber use is not limited to the Anaconda (and present work). One of the main

uses of rubber in wave power technologies is for hose pumps and hydraulic pipes in

which water (or oil) is compressed to drive a turbine. An example of this concept

is given by the AquaBuOY (Finavera Renewables). This is a buoy composed of a

tube, enclosing a hose pump. This hose pump contains a piston that is driven by

the vertical movement of the buoy. It stretches the hose-pump, resulting in pres-

surising the seawater contained in it. This is then used to drive a turbine (the first

prototype sank off the coast off Oregon and the whole wave energy project was

31

32 Chapter 3 Literature review

abandoned for economic reasons. Finavera Renewables now focus on wind energy).

Rubber is also used on the Pelamis wave power device where it is used in spring

system technology. It provides an ultra-flexible and hard-wearing seal between the

steel cylinders and the hydraulic units.

Another device based on the bulge wave concept is the Fabriconda (Hann et al.

(2011)). In this case, the tube is made of several cells made of inelastic fabric and

joined together to form a larger central tube, as seen in Figure 3.1. Each cells

is filled with elastic tube in which water can be pumped at desired pressure for

tuning the device. Bulges propagate in the cells and in the central tube and power

is captured as in the Anaconda. The main advantage of this configuration is that

the elastic material is constrained in shape by the inelastic fabric, avoiding any

undesirable aneurysm effects.

Figure 3.1: Example cross-section of a Fabriconda with 10 cells, showing the

structure at its minimum cross-section (left) and at its medium point (right),

after Hann et al. (2011).

Rubber has also been used for WECs enclosing air. Some devices with this com-

mon approach can be seen in the AWS-III (AWS Ocean) technology, which is an

updated version of the SEA Clam, that is a ring-shaped array of twenty intercon-

nected cells. Wave energy is transferred into pneumatic energy in the air pumped

in and out of the cells due to wave motion. This moving air spins a turbine which

produces electricity. A 1/9th scale prototype has been deployed in Loch Ness,

Scotland and a full-scale prototype is aimed to be deployed at sea within the year

2012.

Chapter 3 Literature review 33

An earlier similar structure is the Lancaster flexible bag, made of a long spine

aligned head to waves, equipped with air-filled bags on each of its sides, giving

the structure enough buoyancy to float. The up and down motion of the waves

compress the bags and the air inside, and the resulting flow is used to drive a

turbine.

Outside wave energy conversion, rubber is used for various marine applications.

For instance, inflatable flexible membrane dams (IFMD) have been deployed in

various regions. They consist of bottom-mounted (usually on a concrete block)

rubber bags. They can be inflated with either air or water or a combination of

both, although air is the most common type as it is faster to inflate/deflate the

membrane with air than water. They are commonly placed across rivers and in

coastal regions to prevent flood, control irrigation, or for water storage. General

theories for the static shape under load have been studied by various researchers

and are nicely summarised in Chu et al. (2011) for various bottom topography

and either air or water inflation. Practical uses have been studied by Chanson

(1996) for instance, who also detailed the design techniques used for controlling

the flow over and behind the membrane to avoid instabilities of the whole sys-

tem. Many of these IFMDs exist today (more than 2000) at every scale, from a

small design application (rice cultivation at Baghamari, Odisha state, India) to

large scale, such as the ones in Ramspol, Netherlands (Jongeling and Rovekamp

(1997)) or at Linyi (Shandong Province, China). Some of these IFMDs burst as

the one in Temple Lake (Arizona, USA) last year that is to be replaced by a steel

dam. Those membranes are usually not made of rubber only, but together with

other variety of geotextiles.

More closely related to the present work are the previous investigations dealing

with the use of flexible membrane for breakwater purposes. They are described in

the next section.

3.3 Breakwaters

3.3.1 Introduction

In order to protect harbours and coasts from the negative influence of strong

waves, especially during storms, use of breakwaters has been studied as a possi-

ble solution for wave attenuation. Conventional breakwaters are structures that


are made to firmly stay on the sea-floor. When incoming waves hit them, wave

energy is partly reflected and partly dissipated. A calmer sea state is obtained

behind the breakwater. These breakwaters are efficient but require high costs of

installation/maintenance and are designed to stay in one specific location. Al-

though submerged horizontal plates can be seen as a potential solution (McIver

(1985); Patarapanich and Cheong (1989); Neelamani et al. (1992)), studies have

also focused on the use of flexible structures to counter these drawbacks. They

could be used as single breakwaters or be placed seaward of the solid structures

to reduce the amount of energy reaching them by dissipating the energy carried

in long waves.

3.3.2 Submerged elastic membrane breakwaters

Relevant work has been done regarding the use of elastic membranes as possible

breakwaters. An important part of the literature deals with vertical membranes

as one of the configurations shown in Figure 3.2. The configurations correspond

to: (a) surface piercing and hinged-hinged or hinged-elastically supported (at the

top) membrane, (b) submerged buoy-membrane system and (c) floating buoy-

membrane system.

h

l

(a) (b) (c)

buoy

d

Figure 3.2: Vertical membrane configuration.


Figure 3.3: Reflection coefficient of hinged-hinged membrane breakwater asfunction of dimensionless tension Tz/ρgh

2 and wave number kh, after Kim andKee (1996).

Kim and Kee (1996) studied the first configuration theoretically. They developed

an analytical model, based on an eigenfunction expansion method for the fluids

in both regions and represented the membrane as a tensioned spring being driven

by the difference of potentials in each regions. They also developed a boundary

element model using a simple source distribution method coupled with the discrete

form of the membrane equation. These models were used to theoretically repre-

sent the behaviour of the membrane in waves, and a study of best efficiency of the

structure as a wave barrier was carried using the different parameters involved.

Kim and Kee (1996) used their model to study the influence of the membrane ten-

sion, boundary condition (hinged-hinged or hinged-elastically supported) at the

membrane ends and mass of the membrane. The general conclusion is that such

system can be efficient as a wave barrier. Complete reflection of incident waves

could be obtained. While the tension in the membrane plays an important role in

the efficiency of this system, the mass of the membrane was seen to be without

influence, because the membrane mass is negligible compared to its added mass.

An example of reflection coefficient has been reproduced in Figure 3.3 where the

overall efficiency of the system can be appreciated. An interesting observation is

that the reflection coefficient drops at a specific frequency and that this frequency

increases as the membrane tension increases. Those peaks were found to be re-

lated to the lowest natural frequency of the system (π/h)√

Tz/mt, where Tz is the

tension in the membrane and mt the sum of the membrane mass and its added

mass due to the surrounding fluid (the latter was approximated as the added mass

of a solid plate of unit width, ρπ/4). The mesh plot of the membrane response as


a function of wavenumber and position along the membrane was presented, and

at the resonant frequencies, the amplitude of this displacement was maximum,

yielding large transmission coefficient, hence minimum reflection.

Such a configuration is unlikely to be used in practice. This is why Kee and

Kim (1997) extended the previous study to more feasible situations ((b) and (c)),

where the top part of the membrane is attached to a buoy, either floating or sub-

merged. The buoyancy of this buoy provides the tension in the membrane. Their

study was theoretical and experimental. They compared the results from their

boundary element method (with different boundary due to the presence of the

buoy, the membrane equation being the same) with experiments carried out in a

2D tank. These experiments are more detailed in Kim et al. (1996). They studied

the influence of mooring type (two types are shown in Figure 3.2 and a third one

is with both at the same time), buoy radius and membrane parameters. The over-

all conclusion is that for both configuration (floating and submerged), optimum

conditions on buoy radius and membrane parameters exist for wave-blocking ef-

ficiency. This study was extended later by Cho et al. (1997) who used the same

model but using oblique incident incoming waves, yielding the same kind of study

for best efficiency.

It is interesting to note at this point that these previous configurations have been

studied modelling the wave barrier as a tensioned membrane. Other authors have

considered the same geometries, using a beam plate. Among them, Lee and Chen

(1990); Williams (1991); Abul-Azm (1996). As a matter of fact, the tensioned

membrane case is the same as that of a tensioned beam-plate with zero bending

stiffness. The thickness of the membrane in the present work is thin enough to

model it as a membrane. Therefore those studies are mentioned here, but not

developed.

As an extension of these studies, the use of dual vertical membranes was investi-

gated. Cho et al. (1998) investigated analytically the interactions of oblique inci-

dent waves with dual hinged-hinged membranes (configuration (a) with notations

from Figure 3.2) and extended it to a more realistic case (configuration (b) and

(c)) by developing a boundary element model using a simple source distribution

coupled with the discrete form of the membrane equation. This computer program

was used to evaluate the performance of the two above-mentioned configurations

and study the influence of several parameters like the separation between each


membrane, the buoy physical properties, mooring stiffness and membranes ten-

sion. Results were compared with a single membrane system. The hinged-hinged

dual membrane system showed itself more effective, especially in long waves, and

this efficiency was increased by reducing the gap between the two membranes. In-

terestingly, asymmetric systems (membranes of different tensions, whatever their

sequence was) were seen to be more efficient for a wide range of wave frequencies

and headings. The dual buoy-membrane system was also evaluated. As for the

hinged-hinged dual membrane, the system performed better when the gap between

the two buoy-membranes is small (compared to the wavelength). When the gap is

large, using two membranes instead of one does not seem advantageous. The use

of asymmetric systems (different stiffness between front and rear moorings) was

also observed to widen the bandwidth of best efficiency.

The hinged-hinged dual membrane was also investigated by Lo (1998) using eigen-

function expansion series for the fluid potential in the three regions. Influence of

the gap between the two membranes, their tensions and their mass was studied.

As for the single membrane case, varying the masses of the membranes seemed

without influence on the transmission of the system as those masses were anyway

negligible compared to the added masses of the membranes. Lo (1998) plotted

the modal amplitudes of the two membranes in order to study the influence of the

membranes displacement on the transmission coefficient. The membrane response

was seen to be dominated by the two lowest modes and the largest modal ampli-

tudes corresponded to maximum transmission coefficient (hence lower reflection

coefficient). This is similar to what happened for the single membrane of Kim

and Kee (1996) as mentioned earlier. Interest was given also to the viscoelastic

losses in the rubber, which were found to be negligible even for large membrane

displacement.

Lee and Lo (2002) studied the case of successive vertical surface piercing mem-

branes with variable draft. An eigenfunction expansion method was used for ex-

pressing the the velocity potential in each regions separated by a membrane. Influ-

ence of the protrusion and draft of the membranes were evaluated, together with

different membrane tensions. This was compared with experimental values. Good

agreement between model and experiment were obtained when Lee and Lo (2002)

included in their model the energy loss due to their experimental configuration

(spacing between tank walls and membranes).


The main problem with any vertical configurations is the large wave loading on

the membrane and a possible blockage of currents, with everything that it can

include on the surrounding ecosystem.

That led Cho and Kim (1998) to study the possibility of using the membrane

horizontally rather than vertically, as shown in Figure 3.4, where the tension Tx is

applied in the direction of wave propagation. Cho and Kim (1998) developed an

analytical model (expansion series with matching of the potentials at the interface)

and numerical one (boundary element method). Both methods yielded the same

results and the ones from the analytical model were compared with the results ob-

tained from experiments. As will be detailed in chapter 8, their analytical model

was coded and checked during the course of this study with the aim of using it

for the duct configuration. This is why all figures from Cho and Kim (1998) are

indeed not a reproduction, but another computation as described in chapter 8.

h

l d

s

Tx

Figure 3.4: Horizontal membrane configuration.

While the model with infinite tension was successfully compared with McIver

(1985) results for the rigid plate, the influence of the different parameters were

studied in order to predict conditions for an efficient wave-blocking system. For

instance, the membrane tension was varied for fixed membrane length and sub-

mergence (s/h = 0.5 and d/h = 0.2 respectively). It was seen that these flexible

membranes perform better than rigid plates as the bandwidth of best efficiency for

the case of a membrane with infinite tension (corresponding to a rigid plate) was

extended by varying the membrane tension and allow its motion. The length of

the membrane was also varied and the reflection coefficients for various configura-

tions are given in Figure 3.5. Regions of best efficiency are noticed for all of them


and interestingly, the largest bandwidth is obtained for s = 0.4h meaning that the

efficiency was not necessarily improved by the longer size of the membrane.

Figure 3.5: Reflection coefficients of a submerged impermeable membrane

breakwater as function of length of membrane s/h and wavenumber k1h for

d/h=0.2, Tx/ρgh2=0.1, after Cho and Kim (1998)

−0.50

0.50 1

2 34 5

0

1

2

3

4

x/L kh

ξ/ A

Figure 3.6: Response of the membrane (ξ/A) as a function of wavenumber

k1h and horizontal coordinate x/L for d/h = 0.2, s/h = 0.5 and Tx/ρgh2=0.1,

after Cho and Kim (1998).


Figure 3.6 shows the response of the membrane ξ (normalised by the incident wave

amplitude A) for the case s/h = 0.5 with the other parameters fixed as in Figure

3.5. It is seen that the efficiency of this membrane as a wave barrier for that

configuration is linked to its displacement not being too small.

Because of the geometry, this efficiency does not lie in an actual wave block-

ing but, as will be seen later for the bottom-mounted cases, into the cancellation

of the transmitted waves by the radiated waves from the membrane displacement.

The first peak for the reflection coefficient for that configuration corresponds to a

shape characteristic of the membrane first mode: this is the contrary to what hap-

pened for the vertical membrane mentioned earlier, where the reflection coefficient

dropped down at the membrane natural frequency(ies). Figures 3.7 and 3.8 shows

that the membrane displacement at this first peak is mainly composed of the first

symmetric mode, while the contribution of the higher mode is more pronounced

for the higher k1h for the total membrane displacement.

Later Cho and Kim (1999) studied the same configuration but with a circular,

flexible, horizontal and stretched (uniform tension Tu) membrane. Using a similar

expansion as for the case of the rectangular membrane but in polar form, they ex-

pressed the velocity potentials ahead, above and beneath the membrane. Results

from this analytical model were used to evaluate the difference between circular

rigid and flexible membranes. As before, the case with infinite tension in the mem-

brane was successfully compared with the results from Yu and Chwang (1993) for

the case of a rigid submerged circular disk. Cho and Kim (1999) computed the free

surface elevation for various tensions and submergence depths and noticed that

for small membrane tension (0.02 < Tu/ρgh2 < 0.05), pronounced peaks of wave

focusing (free surface amplitude three to four times the incident waves amplitude)

in front of the membrane occurred even for deeply submerged membranes. For

those frequencies, the membrane response exhibited interesting patterns (two or

four peaks, sombrero shape) that are characteristic of the different modal shapes.

For higher tension in the membrane (Tu/ρgh2 = 0.1) this wave focusing occurred

for low values of membrane submergence. The influence of this submerged disk

was made more noticeable by plotting (contour map) the free surface elevation

around the disk for various disk size, submergence and tension. For different yet

close conditions, free surface patterns were found to be totally different and again,

those patterns were linked to the membrane response and which mode was pre-

dominantly excited.


−0.50

0.50 1

2 34 5

0

1

2

3

4

x/L kh

−0.50

0.50 1

2 34 5

0

0.02

0.04

x/L kh

−0.50

0.50 1

2 34 5

0

2

4

x 10−3

x/L kh

Figure 3.7: Modal response amplitude as function of wavenumber k1h and hor-izontal coordinate x/L for d/h=0.2, s/h=0.5, Tx/ρgh

2=0.1; upper plot: firstsymmetric mode, middle plot: second symmetric mode; lower plot: third sym-

metric mode.


−0.50

0.50 1

2 34 5

0

0.1

0.2

x/L kh

−0.50

0.50 1

2 34 5

0

5

10

x 10−3

x/L kh

−0.50

0.50 1

2 34 5

0

1

2

3

x 10−3

x/L kh

Figure 3.8: Modal response amplitude as function of wavenumber k1h andhorizontal coordinate x/L for d/h=0.2, s/h=0.5, Tx/ρgh

2=0.1; upper plot: firstasymmetric mode, middle plot: second asymmetric mode; lower plot: third

asymmetric mode.


3.3.3 Submerged fluid-filled membranes

An important amount of studies dealing with rubber use is linked to the concern

of storage or transport of oil or freshwater at sea, early introduced by Hawthorne

(1961). Such devices are usually floating fluid-filled membranes. Importance has

to be given on the dynamic tension in the flexible container (resistance) and on

how to identify the resonant response of such system. These concerns have been

studied for instance by Zhao (1995); Zhao and Aarsness (1998) (first concern) and

Phadke and Cheung (2001) (second concern) and many more researchers. In prac-

tice, those bags are made not only of rubber, but combined with other types of

textiles to their design.

Closer to the present submerged configuration are the submerged fluid-filled mem-

branes used for breakwater purposes. Frederiksen (1971) proposed a flexible fluid-

filled bag placed head to waves just beneath the water surface. The bag was made

of rubber, but slightly reinforced with nylon fabric. Bags of different thicknesses

and lengths were tested, filled with fluid with different densities. Values of wave

attenuation (ratio of the difference of the incident wave height and the transmit-

ted wave height to the incident wave height) between 0.8 and 1 were obtained for

bags with length two or more times greater than the wavelength. Others values

lay in the range of [0.3:0.8], showing the effectiveness of this device in attenuating

incident wave amplitude.

Broderick and Jenkins (1993) carried out experiments on a submerged fluid-filled

cylinder, placed horizontally across a wave tank (with its axis perpendicular to the

direction of wave propagation). This system was tested for two submergence depth

and was observed to be efficient in reducing the transmitted wave energy. Broder-

ick and Jenkins (1993) used string pots to measure the membrane displacement at

three points on the cylinder and noticed that the maximum displacement occurred

for wave frequency at which the transmitted wave energy is a minimum. They also

equipped the cylinder with a pressure transducer to measure the internal pressure,

but the obtained data did not show any extreme for the low transmission frequency.

Later, Broderick and Leonard (1995) developed a boundary element method, cou-

pled with the finite element model for the membrane equation discretisation, to

study the behaviour of the previous cylinder in waves. The free water surface on

top and aft the cylinder was computed and compared with the case of a rigid cylin-

der, showing the positive effect of using a flexible cylinder as a wave attenuator.


Results from the numerical model were also compared to the experimental ones:

the free surface elevation was under predicted by the model while the displacement

of the membrane was over predicted, the general trend being still satisfactory. This

discrepancy might come from the fact that Broderick and Leonard (1995) assumed

the internal pressure as constant in their model.

3.3.4 Bottom-mounted fluid-filled membranes

Bottom-mounted membrane devices can either be fully or partially filled with

water. For the latter case, Frederiksen (1971) described the experimental work

carried on what he called a rubber mattress. It was basically made of two mat-

tresses made of rubber-nylon, enclosing approximately 50% water and 50% air,

and connected by a duct. When incoming waves were propagating above it, air

pockets were formed and propagated forth and back, altering the wave above.

The study mentioned an attenuation of 40% of the incoming wave amplitude for

wavelength equal to twice the device length.

Incident waveTransmitted wave

Incident wave Transmitted wave

Reflected wave

Reflected wave

Radiation wave Radiation wave

Figure 3.9: Wave interception by a pair of sea balloons, after Ijima and Uwa-

toko (1985).


Another partially fluid device is the so-called sea balloon, as proposed by Ijima and

Uwatoko (1985). This consisted of a vertical tube, made of rubber, with its axis

perpendicular to the direction of wave propagation. About 60% of its volume was

filled with air, giving the cylinder a bulb shape. Sea balloons could be used in pairs

by being connected to each other by an air pipe . Beneath the waves the volume

of air in one balloon changed, affecting the shape and response of the following

one. Numerical analysis (boundary integral method) and experimental work were

carried out to determine the possibility of attenuating incoming waves with this

configuration and the effects of size, number of balloons, gap between them and

the coefficient of resistance of the air-flow in the pipe on this damping. Those pairs

of balloon could be used on a artificial submerged plate or on the seabed. The

overall configuration was seen to be effective in long waves, where the transmitted

wave is cancelled by the radiated wave from the balloons’ displacements by the

process shown in Figure 3.9.

h

d

p= gh+pr 0

l

L

Figure 3.10: Sketch of the flexible mound device.

Most of the flexible breakwaters were thought to be completely water-filled mem-

brane. For instance, Ohyama et al. (1989) proposed a device called flexible mound

that consisted of a thin elastic membrane bag filled with water and mounted on

the seabed. A sketch of it has been reproduced in Figure 3.10 with notations

used in the present study. The motions of the internal and external fluids were

computed using two boundary elements models, which were coupled through a

force-equilibrium relationship for the membrane, the latter being modelled as a

series of spring. Experiments were carried as well in order to validate the math-

ematical model and study the influence of internal pressure, submergence of the

membrane or length of the device on the reflection and transmission coefficient.


Peaks of maximum reflection (hence minimum transmission) were obtained the-

oretically and experimentally at some specific frequencies. At those frequencies,

the membrane displacement was computed, and exhibited nodes and antinodes

specific of the natural mode shapes of the membrane. However, the analysis car-

ried by Ohyama et al. (1989) could not predict those resonant frequencies. The

general conclusion regarding this geometry was that this concept was efficient un-

der certain circumstances to attenuate the incoming waves and create a calmer

sea-state behind it. That is why later two of these were installed to protect a har-

bour (Tanaka et al. (1992a)). They were 9m wide, 40m long and 3m high (when

inflated). They were normally deflated so liners could pass over them, and inflated

in case of emergency, as during a storm, to protect the harbour.

Figure 3.11: Wave energy loss in the flexible mound, after Tanaka et al.

(1992a).

For the same device, Tanaka et al. (1992b) extended the information they obtained

from the experimental tests mentioned above and the power lost in the system.

This has been reproduced in Figure 3.11 where it is seen that up to 50% of the

incident wave power was lost. However, no explanation is given on the mechanisms

of this energy loss.

Phadke and Cheung (1999) used a second-order differential equation governing the

membrane deformation, derived from the membrane theory of cylindrical shells to

deal with the same geometry as the flexible mound. Because they extended the

study of Ohyama et al. (1989) to cases where the density ratio between the in-

ner and outer fluid is different from one, they studied first the static shape of

the membrane which was a function of the hoop stress, which is the circumferen-

tial stress resulting from the difference between internal and external pressures.


The inner and outer flows were computed using a boundary element method and

coupled to the finite element model obtained from the membrane equation. This

study was theoretical and the computed values of reflection and transmission co-

efficients compared with the experimental values of Ohyama et al. (1989). Peaks

of zero transmission are theoretically observed for specific frequencies and over a

wide range of conditions. For the same conditions, the dimensionless membrane

response was given for the peak frequencies (Figure 3.12). Node and antinodes

are observed yielding the conclusions that those those frequencies were in fact the

natural frequencies of the fluid-membrane system and completing the study of

Ohyama et al. (1989). Finally, they performed analyses on the influence of the

density ratio between the inner and outer fluid and concluded that the response

amplitude of the membrane deformation decreased with increasing density ratio

due to an increase in both the fluid mass and the membrane hoop stress.

Later, Phadke and Cheung (2003) carried out the same study in the time do-

main to incorporate the geometric and material non-linearity of the membrane.

Frequencies of zero transmission were shifted towards higher frequencies due to

hysteresis. Normalised responses of the membrane exhibited the same patterns as

in Figure 3.12 while their magnitudes were smaller than the magnitudes of the

normalised response obtained with the linear model. Moreover, Phadke and Che-

ung (2003) showed that the magnitude of the normalised response decreased with

increasing wave amplitude due to a phenomena called strain stiffening, coming

from the nonlinear term in the strain-stress relationship of the membrane, making

the membrane stiffer when it was more and more deflected.

Figure 3.12: Amplitude of membrane normal deflection for p0/ρgh = 0.07

and d/h=0.5: (a) fundamental mode; (b) second mode, (c) third mode (– –,

ρi/ρ = 0.9; —, ρi/ρ = 1.0; · · · , ρi/ρ = 1.1), after Phadke and Cheung (1999);

ρi, density of the inner fluid.


Despite the inclusion of the membrane non-linearity in their model, the theory from

Phadke and Cheung (2003) could not fairly agree with the experimental results

from Ohyama et al. (1989). For that purpose, Das and Cheung (2009) extended

the initial work of Phadke and Cheung (1999) by extending it to three dimensions.

The problem was formulated as before for the membrane and outer and inner flow,

taking into account the supplementary dimension. An interesting comparison is

to plot, for the same configuration, the transmission coefficient obtained from the

two dimensional model (Phadke and Cheung (1999)), the two dimensional model

with the non-linearity of the membrane, the linear model from Das and Cheung

(2009) and to compare it with the experimental values from Ohyama et al. (1989).

This comparison has been reproduced in Figure 3.13. The main comment is that

peaks of complete zero transmission were non longer observed and that was more

consistent with the laboratory data. The other interesting fact is that the low

transmissions frequencies are obtained for higher frequencies than the two other

models. This is a result of Poisson’s effect on the membrane, yielding a lateral

stress affecting the natural frequencies of the system. The obtained model seems to

fairly agree with the data from Ohyama et al. (1989), yielding the conclusion that

non linear effect are less prone to explain the discrepancies between Phadke and

Cheung (1999) model and Ohyama et al. (1989) than the three dimensional effects.

By mapping the free surface wave ahead, above and behind the membrane, Das

and Cheung (2009) could characterise the modification of the incoming wave by

the membrane displacement and the reason of low transmission coefficients for the

resonant frequencies. The motion of the membrane at those resonant frequencies

is associated with the generation of radiated waves that travel opposite to the

incoming waves in front of the structure, generating standing waves, but travelling

in the the same direction behind a structure, but with a 180 lag, cancelling out

the transmitted waves.


Figure 3.13: Transmission coefficient as function of excitation dimensionless

frequency L/λ: (a) p0/ρgh = 0.065, (b) p0/ρgh = 0.208; • • • data from

Ohyama et al. (1989); – – – two dimensional linear model from Phadke and

Cheung (1999); · · · two dimensional non linear model from Phadke and Cheung

(2003); — three dimensional model from Das and Cheung (2009)

The same type of device was investigated theoretically by Liapis et al. (1996), who

considered waves over a submerged, bottom-mounted water-filled membrane that

was semicircular in shape, as shown in Figure 3.14. The membrane deformation

was computed by finding the hydrodynamic pressure acting on the membrane at

each time step and using this pressure to compute its deformation at the next

time step. The efficiency of the structure was, as before, tested by considering the

reflection and transmission coefficients obtained from varying the parameters of

the problem (length of the structure, submergence). These values were compared

with that obtained from the rigid structure of the same shape. The general con-

clusion was that the flexible one was more effective as a wave barrier than its rigid

configuration. The membrane displacement was not included in this analysis of

best efficiency.


h

d

l

Rm

x

z

Figure 3.14: Water-filled hemicircular membrane breakwater.

Dewi et al. (1999) extended the configuration of Liapis et al. (1996) to three

dimensions, considering the membrane as a shell. The tested model was a 150m

long shell of 4m radius placed in a water depth of 6m. A boundary integral

method was used for the inner and outer fluid and coupled with a finite element

method for the shell. Before dealing with the structure in waves, the different

modes of vibrations of the fluid-membrane system were computed for an internal

hydrostatic pressure 5% higher than the external hydrostatic pressure. “Wet” and

“dry” modes were computed by solving the membrane equation of motion without

external force. Wet modes corresponded to the natural modes of vibrations when

the system was in water. On the other hand, dry modes corresponded to the

natural modes of vibrations if the system was placed in air and the enclosed

volume was subject to the same pressure difference. When the system was subject

to incoming waves, the free surface elevation was computed as in Figure 3.15

where the incident water waves have an amplitude of unity. In this figure, the

free surface elevation obtained for the rigid structure of the same geometry ((a)

and (b)) was compared to the one obtained from the flexible version ((c) and (d)).

The shell expands from x=-4m to x=4m and from y=0m to y=150m. Yellowish

regions were reduced from upper plots to lower ones, especially for the higher

frequency, yielding the conclusion that the flexible structure was more effective

in reducing the transmitted wave. Red regions are visible ahead of the structure

and correspond to the combination of incident and diffracted (and reflected for

the flexible structure) to form partial standing waves in front of the structure.


There is however no clear explanation why the two frequencies used for this plot

(ω = π/4 and ω = π/6) were chosen rather than any of the wet modes for the

fluid system. A shadow-zone transmission coefficient was defined as the average

wave amplitude along the line y=75m (middle point of the membrane in the y

direction) from x =4m to x =50m. Results have been reproduced in Figure 3.16

for the flexible structure and three different incident angles θ (θ = 0 representing

waves parallel to the x-axis). It can be seen that the optimum efficiency is obtained

when this structure is placed perpendicular to wave propagation direction.

1.4

1.51.3

1.41.3

1.41.2

1.31.2

1.31.1

1.21.1

1.2

1.11

1.1 1

10.9

10.9

0.9

0.8

0.80.7

0.90.8

0.8

0.7

0.70.6

0.70.6

0.6

0.60.5

-50 0 50 -50 0 50(m) (m)x x

150

150

170

170

0

0

-20

-20

20

20

75

75

(a) (b)

(c) (d)

y(m

)y(

m)

Figure 3.15: Free surface wave amplitudes for normal waves over (a) rigid and

(c) flexible structure for ω = π/6 and over (b) rigid and (d) flexible structure

for ω = π/4, from Dewi et al. (1999).


Figure 3.16: Variation of shadow transmission coefficient KT with kR , after

Dewi et al. (1999).

Stamos (2000) compared experimentally the effectiveness of rectangular and hemi-

cylindrical, flexible and rigid submerged and bottom-mounted breakwaters by

measuring reflection and transmission coefficients ahead and behind the model.

An example of his hemi-cylindrical model is given in Figure 3.17 where the axis of

this bottom-mounted structure was perpendicular to the direction of wave prop-

agation. Internal added pressure (for the flexible models) and submergence ratio

(for the rigid one) were the variables of the problem and a parameterisation on

the influence of those parameters was conducted. An energy loss coefficient was

calculated and explained as wave breaking over the structure or turbulent flow in-

side the flexible breakwaters. Interesting values are shown in Table 3.1 and Table

3.2 where the flexible and rigid transmission and reflection coefficients are given,

respectively. It is worth noticing that an important amount of energy was lost in

the system and that the role of the reflection was not the principal reason of the

efficiency of the system in term of wave attenuation. Rectangular models (rigid or

flexible) were seen to be the most effective structure to dissipate energy (and not

reflect it).


Figure 3.17: Side view of the hemi-cylindrical flexible composite breakwater

model, after Stamos (2000).

h (cm) d/h p0/ρgh Kr Kt Kl

Hemi-cylindrical 22.5 0.24 0.007 38% 54% 69%

Rectangular 22.5 0.24 0.007 32% 35% 87%


Rectangular 22.5 0.24 0.141 33% 53% 74%


Rectangular 22.5 0.24 0.282 36% 52% 74%


Rectangular 27.5 0.38 0.007 29% 51% 79%


Rectangular 27.5 0.38 0.141 29% 60% 72%


Rectangular 27.5 0.38 0.282 33% 61% 70%

Table 3.1: Average wave coefficients of flexible models, after Stamos et al.

(2003).


h (cm) d/h Kr Kt Kl

Hemi-cylindrical 22.5 0.24 40% 72% 55%

Rectangular 22.5 0.24 38% 64% 63%

Hemi-cylindrical 27.5 0.38 24% 78% 55%

Rectangular 27.5 0.38 40% 69% 57%

Table 3.2: Average wave coefficients of rigid models, after Stamos et al. (2003).

3.3.5 Conclusions on breakwaters studies

Further developments for flexible breakwaters studies would be the use of porous

membranes, as studied by Cho and Kim (2000) for the horizontal case and Kee

et al. (2003) for two vertical membranes attached at the top of a floating pontoon.

This is not of interest here.

The advantages of using a flexible (light, easy to carry) breakwater rather than

a rigid one led to the ocean-scale deployment of flexible rapidly installed break-

water systems (RIBS). A practical example is given with the RIBS XM99 (Briggs

et al. (2002)) which is a combination of vertical membranes and steel frames. This

breakwater is a V-shaped structure, with each side like the panel shown in Figure

3.18.

It was tested offshore Cape Canaveral, Florida in May 1999 (Briggs (2001)), at

full-scale. Wave transmission in the lee side (inside the V) was evaluated, to-

gether with the forces on the structures (pressure transducers were mounted on

each side). A threshold of 0.5 for the transmission coefficient in the lee side was

used, above which the system was considered inefficient. Transmission coefficients

below or around this value were measured during several days and different wave

environment, giving full satisfaction in the use of the XM99 as a temporary break-

water. This efficiency was more pronounced in long waves.

The interactions of water waves with flexible membranes have been widely studied

for the purpose of breakwaters. The studied geometries were considering the mem-

branes as either a tensioned vertical or horizontal sheet, or enclosing water. The

reviewed studies pointed out that such a structure can attenuate the transmitted

waves when properly tuned.


Figure 3.18: Rapidly installed breakwater system XM99 (from Briggs (2001))

A common point from the reviewed studies is that peaks of theoretical complete re-

flection (or zero transmission) coefficient were observed at the natural frequencies

of the system (for the vertical membrane, the inverse is true). This was confirmed

by computing the membrane displacement at these frequencies. However, no ex-

perimental data was given for the membrane displacement. Broderick and Jenkins

(1993) used string pots but only at three locations along a 3.66m long cylinder.

Kee and Kim (1997) mentioned measuring the membrane displacement using a

video set-up but did not present their results.

The present study does not have a breakwater objective. However, among the

objectives stated in section 2.5.2, the membrane behaviour beneath waves need

to be measured adequately and for that, an efficient technique using laser sensors

was developed. Using such a technique will provide more consistency in the link

between maximum reflection and resonance of the structure. This would also be

useful for measuring the losses in the rubber due to hysteresis, the membrane dis-

placement being known, and what part of the power losses presented in Figure

3.11 or in Table 3.1 can be attributed to these viscoelastic losses.

In a same way, no experimental (and theoretical) consideration has been given

on the pressure variation inside the fluid-membrane system, in particular at reso-

nance. Again, Broderick and Jenkins (1993) measured the pressure in their flexible


cylinder but the trend of the presented results did not show any extrema. Ohyama

et al. (1989) and Tanaka et al. (1992b) noticed an increase of the static component

of the internal pressure when waves passed over their flexible mound but gave no

further details. In the present work, the information regarding pressure oscillation

in the duct is important and will be analysed thoroughly by developing adequate

techniques that could be useful for future similar studies on membranes in waves.

The next chapter gives a description of the instrumentation used for that purpose

and the facilities in which experiments took place.

Chapter 4

Experimental equipment

4.1 Introduction

This chapter describes the experimental apparatus used during this work, together

with the instrumentation. The power take-off system, briefly described in section

2.4.2.3, is detailed, and power calculations are explained.

4.2 Narrow wave flume

4.2.1 Dimensions

The narrow flume at the University of Southampton, School of Civil Engineering

and the Environment, is 17m long, 0.44m wide and about 1m deep. The wave-

maker detailed in section 4.2.2 operates with a water depth of 0.7m. The flume’s

side walls are made of glass, as can be seen in Figure 4.1. The end of the flume is

equipped with high density foam to minimise wave reflections.

57

58 Chapter 4 Experimental equipment

Figure 4.1: Narrow flume at the University of Southampton.

Figure 4.2: Wavemaker for the narrow flume.

Chapter 4 Experimental equipment 59

4.2.2 Wave-maker

The wave-maker is of a flap-type and is manufactured by Edinburgh Designs. The

flap is dry back and driven by an electric servo-motor via a stainless steel belt

running over the curved top of the paddle (Figure 4.2). Hydrostatic forces acting

on the paddle are compensated by springs located behind it. While running, the

system measures the forces acting on the paddle from any reflected waves and

corrects the paddle motion accordingly.

4.2.3 Absorbing beach

In order to prevent any undesirable reflections, the end of the flume is equipped

with an absorbing beach made of foam blocks (in the same way as for the wave

basin shown in Figure 4.5). The efficiency of this beach is given by Blenkinsopp

(2007) with a reflection coefficient varying from 2% to 13% for the wave conditions

covered in chapter 6.

4.3 Wave basin

4.3.1 Dimensions

The wave basin (Figure 4.3) is 1.5m wide, 0.8m deep and 11m long. It is operated

with a water depth of 0.6m. Its side walls and bottom are made of glass, except

over a distance of 2 meters ahead and behind the wave paddle where the bottom

is made of Grey Plastic. One side is equipped with the wave-maker detailed in

section 4.3.2 and the other with an absorbing beach made of foam blocks.

4.3.2 Wave-maker

The wave-maker shown in Figure 4.4 is of piston type. The paddle is driven by

hydraulic piston rod and moves backwards and forwards horizontally. Four wave

gauges are placed on the front part of the paddle to give a feedback signal to the

wave monitor that includes a dynamic absorption servo loop.


Figure 4.3: Wave basin at the University of Southampton.

Figure 4.4: Wave paddle for the wave basin.


4.3.3 Absorbing beach

The absorbing beach was built during the course of this study with wedge shaped

foam blocks as seen in Figure 4.5. Its efficiency was tested by running some

waves in the empty basin and using three pairs of waves gauges for measuring the

reflection coefficient. The two wave gauges constituting a pair were separated by

0.5m (in the width direction of the basin). The first pair was located 2.7m from

the equilibrium position of the paddle, the second 0.4m from the first pair, and

the third pair 0.5m from the second.

Figure 4.5: Absorbing beach for the wave basin.

A Fast Fourier Transform (FFT) was used to obtain the amplitude A and phase φ

of the signals at the three positions (each pair being averaged) and the reflection

coefficient

Kr =Ar

Ai

(4.1)


was calculated. The reflected and incident wave amplitudes, Ar and Ai respec-

tively, were calculated following den Boer (1981) as

φr = arctan (A2 cos φ2 − A1 cos k∆l

A2 sin φ2 − A1 sin k∆l), (4.2)

Ar =A1 cos k∆l − A2 cosφ2

2 sin k∆l sin φr

, (4.3)

and

φi = arctan (A1 cos k∆l −A2 cos φ2

A1 sin k∆l + A2 sin φ2), (4.4)

Ai =A1 cos k∆l − A2 cosφ2

2 sin k∆l sin φi

, (4.5)

where the subscripts stand for the pair of wave gauges used and ∆l is the distance

between the two pairs of gauges used. It can be seen that the calculation of Ar and

Ai using this method requires that ∆l 6= nλ/2 (n=1,2,3...) for the denominator of

(4.3) and (4.5) not to be zero.

The values of Kr were calculated using the first two pairs for wave height used in

this basin (in the same layout used as in chapter 7) and are shown in Figure 4.6.

The second and third pairs were also used to calculate Kr and values were in good

agreement with the ones presented in Figure 4.6.

The primary comment from Figure 4.6 is that the beach was seen to be effec-

tive with a reflection coefficient varying from 4% to 15% for wave frequencies

higher than 0.7Hz and equal to 20% for longer waves. This will be discussed in

chapter 7.

4.4 Rubber

4.4.1 Introduction

Rubber being the key material of this work, its different properties need to be ac-

curately defined and determined. The complete procedure for determining those

characteristics is given in Appendix D, leading here to a summary of the method-

ology used and the obtained values.


Figure 4.6: Measured absorbed reflection coefficients for the absorbing foam

beach in the wave basin.

4.4.2 Young’s modulus

To get the Young’s modulus, sample strips were clamped at both ends. One end

was fixed, the other one designed with a hook to place successive loads. First

load was applied and the change of length was recorded, as well as the change of

width and thickness (averaged over three positions along the length of the strip).

The process was repeated with ten successive increasing loads. A linear fit, pass-

ing trough the origin was then used to determine the ratio stress/elongation, and

hence the Young’s modulus.

It is worth mentioning that the successive masses were placed with two minute

intervals. A quicker test, with the same masses placed at less than thirty sec-

onds intervals was also carried out and results detailed in Appendix D show few

differences in the obtained Young’s moduli.


4.4.3 Structural damping

While successively unloading the strips from their successive masses, the unloading

path was observed not to be the same as the loading path due to the hysteresis

effect. However, the hysteresis loop was not of paramount importance for the

quasi-static Young’s modulus determination as this test required only one cycle of

deformation. This is visible in Figure 4.7 where the stress in the strip is plotted as a

function of its elongation. This is also why the hysteresis was not of relevant effect

for the free bulge speed as the local deformation due to a bulge was happening

fast enough.

Figure 4.7: Stress in the strip of membrane as a function of the elongation

for the quasi-static Young’s modulus measurement. Each symbols represent

different loading.

However, when rubber is repeatedly deformed and restored to its original state

over a certain number of cycles, the role of the hysteresis is to dissipate the energy

involved in this deformation. Natural oscillations of a strip of rubber were damped

in the same way as oscillations are damped in a mass-spring-dashpot system.

This hysteresis damping was obtained by measuring the decaying oscillation of

a mass attached to a strip of rubber released from an initial position correspond-

ing to the strip being stretched. The hysteresis parameter β was obtained from the


formula detailed in Appendix D. In order to characterise the effect of the hystere-

sis damping regardless of the frequency of oscillation, it is common to introduce

the loss angle defined as

loss angle = arctanβω, (4.6)

with ω being the angular frequency of oscillation.

4.4.4 Summary

The complete characteristics of the tested rubber, obtained from the above-mentioned

procedures are given in Table 4.1.

Thickness (mm) Density ρr (kg/m3) Young’s modulus (MPa) Loss angle ()

0.2 1052 0.92 1.81 1143 1.87 2.5

Table 4.1: Measured characteristics of test rubber

4.5 Power take-off system

4.5.1 Introduction

The primary concern regarding the PTO design is how to use the horizontal in-

ternal oscillatory flow due to the propagation of bulges. One option was to have a

set of high and low pressures hydraulic accumulators at the stern. One-way valves

would allow water to flow into the high pressure accumulator and flow out from

the low pressure accumulator back to the tube, hence maintaining the enclosed

volume of water in the tube constant. A turbine could be placed between the two

accumulators and would be driven by the flow from the high pressure accumulator

to the low pressure accumulator.

The idea developed by Chaplin et al. (2012) and used hereinafter was to transfer

the horizontal internal oscillatory flow into the motion of a slug of water. The

heave motion of this slug of water was used to push air trough a turbine. Experi-

mentally, this turbine was modelled as a linear dashpot, in the form of 17 copper

pipes, each of them containing 140 stainless steel pipes of 1.6mm internal diameter


140 pipes, ID=1.6mm

copper pipe, ID=26.2mm

Figure 4.8: Cross section power take-off.

and 0.8m long as seen in Figure 4.8. The air flow through those pipes lost energy

by friction at a rate equal to power. This way, the power captured by a turbine

was modelled as the rate of energy lost by this flow in the pipes.

rubber

rigidbent tube

slug ofwater

Air

horizontalmotion

Verticalmotion

Capillarypipes

Figure 4.9: Schematic of the power take-off system.


Rubber

Rectangularbent tube

Connector

Copperpipes

Pressuretransducers

Figure 4.10: Power take-off system set-up

4.5.2 Components

The slug of water was enclosed in a rigid and bent aluminum tube, one side being

connected to the duct’s stern, the other to the copper pipes detailed later. This

bent tube was rectangular, having a width of 0.276m and a height of 0.153m, cov-

ering a cross section of St =0.042m2. A two-dimensional representation is given

in Figure 4.9.

The top part of the aluminium tube is connected to the dashpot, via a distortable

aluminium tube as can be seen in Figure 4.10. The vertical displacement of the

slug of water induces an excess of pressure in the chamber above the water-air

interface. This difference in pressure under and above the pipes induce a flow

in the pipes. This flow is assumed to be linear and the flow rate q and pressure

difference ∆p = pa + p − pa = p (notations from Figure 4.11) are related by the


Flow rate =q p/ZPTO

p +pa

V

paSt

SpF

e

Figure 4.11: Model for the power take-off.

Hagen-Poiseuille relationship

p =8µairLp

πr4q, (4.7)

with r the radius of one stainless steel pipe, Lp the length of a stainless steel pipe

and µair the dynamic viscosity of air. Using the value of µair = 1.85 × 10−5Pa.s

at room temperature, the ratio p/q for one copper pipe (140 stainless steel pipes)

is Z1 = 657kPa/m3s−1. Those copper pipes are in parallel, so the equivalent

impedance for a number N of them open is

1

ZPTO

=N

Z1, (4.8)

or in other words,

ZPTO =Z1

N. (4.9)

4.5.3 Power calculation

The power take-off can be modelled as in Figure 4.11, where the water-air interface

displacement e acts as a piston, increases the pressure in the chamber and pushes

the air through an exhaust pipe of impedance ZPTO.

Assuming air as an ideal gas leads to

p = ρRT (4.10)

where T is the temperature of the air inside, assumed to be equal to room tem-


perature and R the gas constant. The rate of change of the mass density is

ρ =m

V− m

V 2V (4.11)

=m− ρV

V. (4.12)

The rate of change of mass due to the outflow is

m = −ρq (4.13)

= − ρp

ZPTO

, (4.14)

while the rate of change due to piston motion is

V = −Ste (4.15)

So the rate of change of pressure in the chamber is

p =F

St

(4.16)

=m− ρV

VRT (4.17)

= (− p

ZPTO

+ Ste)ρRT

V(4.18)

= (− F

StZPTO

+ Ste)paV, (4.19)

so from (4.16), multiplying by St

F = (− F

ZPTO

+ S2t e)

paV, (4.20)

which leads to the equation for the flow rate

q + αq = Stαe, (4.21)

with

α =pa

ZPTOV. (4.22)

This is the equation that relates the water-air interface displacement to the flow

rate in the pipes. Substituting

e = e0e−iωt (4.23)


and

q = q0e−i(ωt+φeq) (4.24)

into (4.21) yields the phase shift between water-air interface displacement and flow

rate

φeq = arctanpa

ωZPTOV(4.25)

and the amplitude of the flow rate as a function of the water-air interface displace-

ment amplitude

q0 =Stαω

ω cosφeq + α sin φeq

e0 (4.26)

Energy is dissipated in the pipes as air flows through it. The rate at which this

dissipation takes place (hence power) is

Pf =1

2p0q0 (4.27)

=1

2ZPTOq

20, (4.28)

where p0 is the amplitude of p. In the case φeq = π/2, that corresponds to low

values of impedances, (4.26) yields

q0 = Stωe0, (4.29)

meaning that the the flow rate that goes through the pipes is exactly the same to

that of the air above the water-air interface. However, as the impedance increases,

air is more and more compressed and the phase lag due to this compression is

taken into account by using (4.28).

4.6 Instrumentation

4.6.1 Data acquisition and equipment driving system

A PCI data acquisition card was installed on a Personal Computer and could

accommodate up to 32 input channels and 2 outputs. Purpose made TestPoint

programs were used to either acquire signals from the wave monitors, amplifiers

or laser sensors, or to drive the two wave-makers.


4.6.2 Wave probes

The wave gauges used in the experimental work were resistance-type gauges. They

consisted of two wires of 3.2mm in diameter, 620mm in length and separated by

20mm. Wave monitors energised the two wires and output a signal proportional to

the submergence depth. This signal was simultaneously sampled by the indicated

data acquisition card.

Wave probes were used for recording free surface elevation (incident, reflected

and transmitted wave calculation) or the water-air interface displacement of the

slug of water in the aluminium tube shown in Figure 4.10. They were calibrated

daily when in use.

4.6.3 Water pressure transducers

Pressure transducers used in this study are manufactured by Omegadyne (model

PX42G7 as shown in Figure 4.12) and were strain-gauge-based transducers. They

had four strain gauges bonded into the diaphragm (Figure 4.12) forming a Wheat-

stone bridge circuit as seen in Figure 4.13.

The bridge was excited by an amplifier (Excitation⊕ and Excitation). When

no pressure was applied on the diaphragm, the bridge was balanced and VA = VB.

Pressure acting on the diaphragm resulted in physical deformation of the gauges

and a change in their electrical resistance (R1 to R4). The bridge was then un-

balanced (VA 6= VB) and the analog electrical signal produced was amplified and

recorded.

The water pressure transducers were found not to need recalibration often. Start-

ing from a arbitrary zero condition with water level above the transducers, the

water level was increased in the flume (or basin) and the signal delivered by the

above-mentioned bridge was recorded for three more water heights.


Figure 4.12: Pressure transducers.

A B

Excitation

Excitation

Signal Signal

+

+

-

-

R1

R3R4

R2

Figure 4.13: Wheastone bridge for the pressure transducers.

4.6.4 Air pressure transducers

For the configuration with the PTO system detailed in 4.5, two air pressure trans-

ducers were used to record the pressure variations in the chamber above the water-

air interface. They were mounted on a wooden box and linked to the aluminium

tube as shown in Figure 4.10.


They were calibrated using a purpose made manometer. Pressure was measured

via the head difference between the two air-water interfaces in the manometer

pipes, with the head difference being increased by blowing more air into the sys-

tem, and vice versa. This process was repeated every two days when in use.

4.6.5 Digital camera

Pictures presented within this thesis were taken with a 12 Megapixel Pentax K-x

digital camera.

4.6.6 Laser sensors

Five laser sensors from Leuze Electronic (model ODSL 9/V6-450-S12) were used

for measuring membrane displacement when subject to incoming waves, or centre-

line deflection when inflated and free bulge speed in static water. One of them is

shown in Figure 4.14. These laser sensors had a operating range of measurement

of 50-450mm, a measurement time of 2ms and a resolution of 0.01mm.

For each of them, a 3-Pin DIN plug was used: two wires for the power (25 Volts

DC) and the third one for the signal. This signal was then connected to a BNC

in order to plug it on an oscilloscope or to connect to the data acquisition board.

Figure 4.14: Laser sensor.


These laser sensors were not meant to be used under water. Hence, they were

placed in transparent boxes submerged by a depth slightly higher than the wave

amplitude, to be sure the free water surface did not cause undesirable diffraction

of the laser beam. The influence of these boxes on the surrounding flow was seen

to be negligible as the reflection coefficient from the duct system was not changed

by the presence of these boxes, as seen in Figure 4.16.

Figure 4.15: Laser sensors set-up.

These sensors were calibrated for every set of the five positions they were used at.

The first recorded voltage was for the flat membrane. Then successively, wooden

sticks of known thickness (20 mm each) were used to lift up the whole frame (or

just each frame for the set-up in the narrow flume). Output voltage was recorded

and the process repeated. If they were moved, they were calibrated again.

4.7 Tested model

A 5m long rectangular duct was initially built with the corresponding dimensions

shown in Figure 4.17. From those dimensions, the cross-section of the rubber duct

was Sd=0.055m2. Its length was reduced to 2m for the tests described in chapter

7. The bottom plate was made of Grey Plastic while its side walls were made of

transparent Plexiglas. The side walls were incorporated in the bottom plate in


Figure 4.16: Influence of the plastic boxes used for the laser sensors on thereflection coefficient for the experimental set-up as in chapter 7 for two sets of

laser sensors positions.

15 mm

Bottom plate(Grey plastics)

Side walls(Plexiglas)

Rubber layer

Angle brackets(aluminum)

200mm

300mm

8mm

v0

L-shaped metalbracket

Figure 4.17: Cross section sketch of the rubber duct.


a depth of approximately 5mm and glued inside. All the joints were sealed with

silicone to prevent any leaks. Some L-shaped brackets were also screwed in the

bottom plate to avoid lateral motions of the side walls. The rubber membrane of

thickness v0 (at rest) was clipped using office-style binder clips that proved to be

strong enough to secure the membrane. Photos of the duct with the membrane

are provided in Appendix C.

4.8 Conclusions

Experimental set-ups using the equipment described in this chapter will be detailed

in chapters 6 and 7. Along with this experimental work, a theory for free bulge

speed and rubber duct behaviour beneath waves needs to be detailed: this is the

purpose of the next chapter.

Chapter 5

One dimensional model for bulges

propagation in the duct

5.1 Introduction

The basic theory for bulge propagation in elastic tubes has been introduced when

introducing the Anaconda in section 2.4.2. This chapter details the use of this

theory for the present geometry, in terms of free bulge speed and bulge propagation

in the closed ends when the membrane is beneath waves. The inclusion of a linear

power take-off at the duct stern is then considered, and the method for calculating

the power absorbed is presented.

5.2 Load-deflection of rectangular membrane and

free bulge speed

Consider a long submerged and liquid-filled rectangular duct, covered by an elastic

membrane. Figure 5.1 defines key parameters and shows the membrane bulging to

a height w in response to a positive internal pressure pb, measured relative to the

external pressure pw. When pb is zero, the deflection w is zero, the stress in the

membrane is σy,0, and the membrane thickness is v0. The profile of the membrane

is the circular arc

z =√

R2m − y2 + w − Rm, (5.1)

77

78 Chapter 5 One dimensional model for bulges propagation in the duct

with

Rm =a2 + w2

2w. (5.2)

To calculate the speed at which bulges in the membrane would travel along the

length of the duct it is first necessary to determine the relationship between pres-

sure and cross-sectional area, or in other words the distensibility

D =1

Sd

dS′

dpb=

1

Sd

dS′

dw

dw

dpb, (5.3)

where

S′

=

∫ a

−a

(√

R2m − y2 + w − Rm)dy. (5.4)

fm

b

2a a

Sd Sd

S’

Rm

z

y

pb=0p >b 0

w

Figure 5.1: Cross section sketch

Resolving forces in the vertical direction, the circumferential stress in the mem-

brane is found to be

σy =pbRm

v, (5.5)

where v, the thickness of the stretched membrane, differs from the initial thickness

v0 because of the effect of the Poissons ratio ν.

Assuming that the longitudinal strain is zero, and that the material of the mem-

brane (like rubber) is for practical purposes incompressible, then

v =av0

Rmφm

, (5.6)

where sinφm=a/Rm (see Figure 5.1). The circumferential strain in the membrane

is

ε =Rmφm

a− 1. (5.7)

Chapter 5 One dimensional model for bulges propagation in the duct 79

Using the plane strain condition σ = εE/(1− ν2), it follows that

pb =v

Rm

[E

1− ν2(Rmφm

a− 1) + σy,0], (5.8)

including the effect of the initial stress in the membrane σ0. Using a Taylor series

expansion for φm about w/a = 0 in (5.8) (with σy,0 << E) yields

pb =2v0a

σy,0(w

a) +

2v03a

(2E

1− ν2)(w

a)3 + ..., (5.9)

as obtained for long rectangular membranes by Xiang et al. (2005); Vlassak (1994);

Huang et al. (2007); Larson et al. (2007). The linear form in w/a of (5.9) will be

used in subsequent analysis.

The distensibility follows from (5.1)-(5.4) and (5.9) and the free bulge wave speed

following Lighthill (1978)

c =1√ρD

(5.10)

by neglecting the effects of longitudinal curvature.

5.3 Bulge equation

The internal (or bulge) pressure pb, the external pressure pw, and the displacement

of the membrane are now considered to be functions of distance along the duct x

and time t. The wave equation for a long duct with a tensioned membrane cover

is essentially the same as that for the Anaconda’s rubber tube (Chaplin et al.

(2012)):∂2pb∂t2

− β∂3pb∂t3

=1

ρD

∂2

∂x2(pb + pw). (5.11)

The motion of the membrane is assumed harmonic at frequency ω, hence represent-

ing the bulge pressure pb(x, t) as the real part of Pb(x)e−iωt and the external pres-

sure pw(x, t), generated by the incident waves as the real part of Pw(x)e−iωt. The

internal velocity is also written with the harmonic notation as u(x, t) = U(x)e−iωt

following (2.18)

U(x) =i

ρω

d

dx(Pb(x) + Pw(x)) (5.12)

With this notation, (5.11) can be written as

d2Pb

dx2+K2Pb = −d2Pw

dx2, (5.13)


with K = k2√1− iβω and k2 = ω

√ρD.

Assuming the wave pressure to be of the form

Pw(x) = aie−ik1x, (5.14)

with

ai =A

cosh(k1d), (5.15)

where d is the submergence of the membrane, the general solution of (5.13) is

Pb(x) = C1 cos(Kx) + C2 sin(Kx) +ρgaik

21

(K2 − k21)e−ik1x, (5.16)

where C1 and C2 depend on the boundary conditions.

5.4 Closed ends configuration

For the case of a duct of finite length L closed at both ends, the velocity at the

bow and stern are zero. Following from (5.12), this yields

d

dx(Pb + Pw) = 0 x = 0, (5.17)

d

dx(Pb + Pw) = 0 x = L. (5.18)

The solution of (5.13) with boundary conditions (5.17) and (5.18) can be written

as

Pb(x) = P+b e−iKx + P−

b eiKx + PAb e−ik1x, (5.19)

with the three components obtained using Maple software as

PAb =

ρgaik21

(K2 − k21), (5.20)

P+b =

ρgKaik1(e−ik1L − eiKL)

2 sinh(iKL)(K2 − k21)

, (5.21)

P−b =

ρgKaik1(e−ik1L − e−iKL)

2 sinh(iKL)(K2 − k21)

, (5.22)


The bulge wave pressure is seen to be made of three distinct wave components:

one travelling from stern to bow with wavenumber K (P+b ), one from bow to stern

with wavenumber K (P−b ) and one from bow to stern with wavenumber k1 (P

Ab ).

Figure 5.2: Theoretical components of the wave pressure in the rubber duct

for a static internal pressure head of 7cm corresponding to a free bulge speed of

3.06m/s in the configuration of chapter 6.

The bulge pressure at the stern can also be expressed using Maple as

Pb(x)x=L =ρgaik1(iK − iCLKe−ik1L + k1SLe

−ik1L)

SL(K2 − k21)

, (5.23)

with CL = cos(KL) and SL = sin(KL).

Providing the loss angle is small, resonant peaks are expected in the following

cases:

. k1 = k2 or in other words, the water wave speed matches the free bulge

speed, or

. k2 = nπ/L (n = 1, 2, ...) corresponding to the different harmonics of the

membrane.


The magnitudes of the three components (5.20)-(5.22) are plotted in Figure 5.2

for the same range of experimental frequencies used in chapter 6. Resonant peaks

are observed for frequencies corresponding to the above second case. A better

understanding of this will be given in 6.4.2.

Figure 5.3: Theoretical internal velocities for the first four harmonics at 30

instants over one wave period for the duct with an internal static pressure head

of 7cm; (a) f=0.306Hz; (b) f=0.612Hz; (c) f=0.918Hz (d) f=1.224Hz.


Figure 5.4: Theoretical internal pressure for the first four harmonics at 30

instants over one wave period for the duct with an internal static pressure head

of 7cm; (a) f=0.306Hz; (b) f=0.612Hz; (c) f=0.918Hz (d) f=1.224Hz.

For these resonant peaks, the magnitudes of P−b and P+

b , two waves travelling in

opposite directions at the same speed, are the same, leading to the presence of

a standing wave. For those frequencies, the total pressure is then seen to be the

sum of a standing wave and a travelling wave PAb , making it a partial standing

wave. The characteristic patterns of these different harmonics can be seen in


Figures 5.3 and 5.4, where the theoretical internal velocity and pressure in the

duct, respectively, are plotted at 30 different instants over one wave period, for

the configuration of the rubber duct with an internal static pressure head of 7cm

corresponding to a free bulge speed of 3.06m/s.

5.5 Duct with a power take-off system

5.5.1 Bulge pressure

When including a dashpot Power Take-Off (PTO) at the stern, the boundary

condition (5.17) remains the same whereas (5.18) changes to

Pb + Pw = iCd

dx(Pb + Pw), (5.24)

where the dashpot rate is CSdρω.

The idea used in this study is that of a bent tube fixed at the duct’s stern (Figure

5.5), containing a slug of water as Chaplin et al. (2012). The internal horizontal

flow at the stern is transferred into the vertical motion of the slug of water. Similar

to what happens in an oscillating column, the vertical motion of this water surface

is used to push the air above it through a PTO. This PTO has been detailed in

section 4.5.

Theoretically, consider the slug of water of length l connected at the duct’s stern,

where the turbine is modelled as the PTO described above and shown in Figure

5.5. From the momentum equation,

p1 = p2 + ρgh2 + ρldu2

dt. (5.25)

The conservation of the flow rate yields

u2 = µu1, (5.26)

where µ = Sd/St, the ratio of the duct cross-section to that of the bent tube.


stern x=L

1)l

h2

2)

Figure 5.5: Definition sketch of the stern.

Differentiating (5.25) with respect to time and using the harmonic motion no-

tation yields

P1 = P2 +ρg

iω

dh2

dt︸︷︷︸

u2

+iµρlωU1 (5.27)

yielding

P1 = P2 + µ(g

ω2− l)

dP1

dx |x=L. (5.28)

Recalling that the impedance of the PTO ZPTO is the ratio (pressure/flow rate)

and that the duct impedance is

Zd =ρc

Sd

, (5.29)

the pressure at 2) can be written as

p2 = Stu2ZPTO = Stµu1ZPTO

= Sdu1ZPTO

Zd︸︷︷︸

Z

Zd

= Sdu1Zρc

Sd

= Zρcu1. (5.30)


Using (5.12), this yields

P2 = ZρcU(x = L) = Zρci

ρω

dP1

dx |x=L

=i

k2ZdP1

dx |x=L. (5.31)

Substituting (5.31) into (5.28) and substituting g/ω2 = 1/k1 tanh k1h leads the

boundary condition at the stern x = L

Pb + Pw =

[iZ

k2+ µ

( 1

k1 tanh k1h− l

)] d

dx(Pb + Pw). (5.32)

In a similar way to the the closed end situations, it is possible to solve (5.11) with

the boundary conditions of zero velocity at the bow (5.17) and with the dashpot

condition at the stern (5.32). The resulting bulge pressure Pb(x) can be written

as the sum of three wave components as in (5.19), with the three components

obtained with Maple as

PAb =

ρgaik21

(K2 − k21), (5.33)

P+b =

ρgaiK(Kk1C(e−ik1L − eiKL)−Ke−ik1L − k1e

iKL)

2(K2 − k21)(iCSLK + CL)

, (5.34)

P−b =

ρgaiK(Kk1C(e−ik1L − e−iKL)−Ke−ik1L + k1e

−iKL)

2(K2 − k21)(iCSLK + CL)

, (5.35)

with C from (5.24) being

C =

[Z

k2− iµ

( 1

k1 tanh k1h− l

)]

. (5.36)

For practical reasons, the impedance of the PTO cannot be completely equal to

the duct’s impedance. The same remark applies for the duct’s cross-section being

different to that of the rigid tube containing the slug of water. This formula takes

those mismatches into account, in the Z and µ coefficients for the impedance and

cross-section, respectively.

It is interesting to notice that having C → ∞ in (5.24) corresponds to the

closed ends configuration. Making C → ∞ in (5.33)-(5.35) and recalling that

sinh(iKL) = i sin(KL) yields the obtained respective components for the closed

ends as in (5.20)-(5.22).


5.5.2 Power calculation

As for the closed ends case, the bulge pressure at stern can also be expressed with

Maple as

Pb(x)x=L =ρgai

((K2(k1C − 1) + k2

1

)CLe

−ik1L −Kk1C(K − ik1SLe

−ik1L))

(K2 − k21)(iCSLK + CL)

,

(5.37)

while the internal velocity at the same position is

U(x)x=L =gaiK

2(

e−ik1L(iKSL + k1CL

)− k1

)

ω(K2 − k21)(iCSLK + CL)

. (5.38)

The flow rate (4.26) of the air flowing in the pipes is then expressed as

q0 = τ(Stω)e0 (5.39)

where

τ =α

ω cosφeq + α sin φeq

, (5.40)

and e0 the magnitude of the complex surface displacement e obtained from

e =i

ωu2. (5.41)

The velocity u2 is calculated from (5.26) and (5.38). This leads the expression of

the (complex) flow rate as

q =τStiµgaiK

2(

e−ik1L(iKSL + k1CL

)− k1

)

ω(K2 − k21)(iCSLK + CL)

. (5.42)

The power absorbed by the linear dashpot is then calculated as

Pf =1

2ZPTO|q|2. (5.43)

5.6 Conclusions

A model for the free bulge speed has been developed, based on a load-deflection

analysis of the rectangular membrane. The free bulge speed is seen to be a func-

tion of the membrane properties (Young’s modulus and dimensions). The change


of membrane thickness on the free bulge speed due to its deflection was introduced

and will be evaluated with measurements in chapter 6.

Bulge propagation was explained with a one dimensional model. The closed ends

rubber duct is seen to be subject to resonant peaks, for which the bulge pressure

inside the duct take the form of a partial standing wave. This will be compared

and analysed with experimental values from pressure transducers measurement

and laser sensor analysis of the membrane displacement in chapter 6.

The power absorbed by the PTO, modelled as a linear dashpot in which air flows

loses its energy, can be calculated by taking into account the physical constraints

of the problem, namely mismatches of impedances between PTO and duct, and

different cross-sectional areas between the rigid tube and the duct. Moreover and

unlike previous work using this PTO, the air compressibility is now part of the

problem. Theoretical capture widths presented in chapter 7 are calculated with

(5.43) and compared with experimental data inferred from wave gauges and air

pressure transducers measurement detailed in section 4.5.

Chapter 6

Narrow flume experiments: static

deflection, free bulge speed and

pressure amplification

6.1 Introduction

This chapter describes the tests carried on the models with closed ends in the

narrow flume. For these tests, the total length of the duct was 5m. The static

membrane deflection, free bulge speed and pressure amplification beneath waves

were monitored with this configuration. The membrane displacement was also

measured with the laser sensors, and the pressure in the duct was deduced from

these measurements. Measured values are compared with the calculated ones from

the theory described in 5.2 and 5.4.

6.2 Experimental layout

In order for sufficient pressure variations to act on the membrane, the duct was

raised from the flume floor. For this reason a false seabed and a ramp were in-

stalled, as shown in Figure 6.1. The ramp and the artificial seabed were held in

place using suction lifters attached to the glass wall/floor of the flume. Pictures

of the complete set-up are given in Appendix C. The bottom part of the duct was

also drilled so it could be attached to the aluminum profiles at several location

along its length. This way, no undesirable motion of the ramp, aluminum profiles

89

90Chapter 6 Narrow flume experiments: static deflection, free bulge speed and


d[1

] d[2

] 0

.16m

5

.0m

0.9

m 6

.01m

2.0

m 1

.0m

Abso

rbin

gbea

ch

Wav

e-m

aker

Pt1

Pt2

Wg

1W

g2

Wg

3

0.5

m

Figure 6.1: Experimental layout for the closed ends configuration. Dashedlines correspond to different internal static pressure heads.

Chapter 6 Narrow flume experiments: static deflection, free bulge speed and


or duct was observed when the system was beneath waves.

Two water pressure transducers, Pt1 and Pt2, were mounted inside the duct at its

bow and stern respectively. Two wave gauges Wg1 and Wg2 were installed ahead

of the model for the purpose of identifying incident and reflected waves by means

of the theory described for example by den Boer (1981). A third was installed

behind the model to measure transmitted waves. The distance d[2] between Wg1

and Wg2 was changed with frequency in order to avoid it being an integer multiple

of half the water wave length (den Boer (1981)). A hydraulic pump pressurised

the duct with water, maintaining the desired internal static pressure head and

making up for any leaks.

When the membrane displacement was to be measured, the laser sensors were

placed in rigid transparent boxes that were submerged enough to avoid any un-

desired diffraction of the laser beam by the free water surface. When they were

not used, a fourth wave gauge (not drawn in Figure 6.1) was placed in the middle

length of the duct for the purpose of evaluating the evolution of the wave acting

on the membrane.

6.3 Static tests: 1mm thick membrane

6.3.1 Membrane deflection

The static centreline deflection was measured for several internal pressure heads.

Measurements were carried out either manually, with a ruler measuring the mem-

brane submergence (hence its deflection) at ten points along the length of the

membrane, or electronically with the laser sensors at six points. These points

were chosen not to be to close to the duct edges to be sure to measure the deflec-

tion on a flat portion of the membrane.

Averaged values for the deflection shown in Figure 6.2 reflect fair agreement with

the linearised form of (5.9).

As stated earlier, the point was to determine the membrane submergence in or-

der to determine the dynamic wave pressure acting on it. The submergence of the

membrane was not the same away from its centreline so an equivalent submergence



Figure 6.2: Experimental values of membrane centreline deflection (+ + +),load-deflection relation 5.9 (––)

d

b

wS’S’

a

w1

d1

Figure 6.3: Equivalent membrane submergence depth

depth d1 was defined. It was chosen so that the cross-section of the duct stayed

the same, meaning

2ab+ S ′ = 2ab+ 2aw1 (6.1)

⇔ w1 =S ′

2a, (6.2)

with the analogy shown in Figure 6.3 and S′

obtained from (5.4).



6.3.2 Free bulge speed

Bulges were manually triggered by suddenly displacing the membrane at the bow.

They propagated in the duct and were reflected at the stern. Time series of pres-

sure at the bow and at the stern, together with the displacement time series of the

laser sensors were used to measure the free bulge speed. An example of time series

from the two pressure transducers is given in Figure 6.4, where the bulge is created

at the bow, reflected at the stern and back to the bow. The free bulge speed was

then calculated knowing the time the bulge took to travel a known distance.

Data obtained are compared in Figure 6.5 with predicted values from the the-

ory developed in section 5.2 for several internal static pressure heads in the duct,

corresponding to different circumferential stresses. Calculated and measured val-

ues follow the same trend and are in good agreement, although the latter ones

seem to be slightly higher (by up to 12%). This can be the result of a slight error

in the Young’s modulus measurement.

Figure 6.4: Time series of pressure transducers from free bulge speed mea-

surement



Figure 6.5: Free bulge speed for the 1mm thick membrane, each symbolsrepresenting one set of measurement.

6.4 Membrane beneath waves: 1mm thick mem-

brane

6.4.1 Experimental conditions

Two configurations of the duct with closed ends were studied with the test con-

ditions being summarised in Table 6.1. Waves were generated in the range of

[0.7Hz-1.5Hz] with increment of 0.025Hz and wave amplitudes varied from [0.1m-

0.04m] with increment of 0.0025m. Each configuration was tested a total of three

times, once without the laser sensors and twice with the laser sensors in place (at

different locations between the two runs).

Internal static Wave Number of Number

Test no pressure head frequencies wave of

(m) (Hz) amplitudes runs

1 0.07m 0.7-1.5 13 3

2 0.1m 0.7-1.5 13 3

Table 6.1: Closed ends experiment: test conditions



Figure 6.6: Time series of laser sensors when rubber membrane is beneath

waves.



Figure 6.7: Time series of pressure transducers (at bow and stern) when

rubber membrane is beneath waves.

Example of time series from the laser sensors and the pressure transducers are

given in Figures 6.6 and 6.7, respectively. The signals are predominantly simple

harmonic, hence the data analysis of other frequencies were initially neglected.

This was justified also by measuring reflected and transmitted wave power with

up to three frequency components (see section 6.4.5) where it was seen that the

power carried by higher harmonics was negligible. The noise on the signals from

the laser sensors in Figure 6.6 are characteristic of ripples propagating on the

membrane. The effect of these ripples will be discussed in section 6.5.

6.4.2 Pressure amplification

Pressure recordings at the stern made it possible to express the total internal pres-

sure amplitude at the stern as a linear function of the pressure due to the undis-

turbed incident wave at the mean elevation of the membrane, |P (x)|x=L/ρgai,

where the submergence d for ai was taken as d1.

Figures 6.8 and 6.9 show this ratio for internal static pressure heads of 0.07m and

0.1m, corresponding to a natural free bulge speeds in the membrane of 3.06m/s

and 3.28m/s, respectively. Experimental values are compared with the theoreti-

cal pressures at the stern calculated from (5.23) (on which the wave pressure was

added).



Figure 6.8: Magnitude of the pressure at the stern relative to the wave pressure

acting on the membrane with an internal static pressure head of 7cm. symbols:

experimental values; —: theory.

Figure 6.9: Magnitude of the pressure at the stern relative to the wave pressure

acting on the membrane with an internal static pressure head of 10cm. symbols:

experimental values; —: theory.



Resonance peaks are observed for the frequencies set out in Table 6.2. These

resonant frequencies are compared with the theoretical resonant frequencies from

the bulge pressure maxima (5.23) for the case k2 = nπ/L (n = 1, 2, ...).

Internal static

pressure head fn fres ‖(fn − fres)/fn‖of 7cm (c=3.06m/s)

n=3 0.918Hz 0.95Hz 3.4%

n=4 1.224Hz 1.2Hz 1.9%

n=5 1.53Hz 1.425Hz 6.8%

Internal static

pressure head

of 10cm (c=3.28m/s)

n=3 0.984Hz 1.075Hz 9.2%

n=4 1.312Hz 1.325Hz 0.9%

Table 6.2: Comparison of experimental resonant frequencies fres with the

theoretical ones calculated as fn = nc/2L where c is the theoretical free bulge

speed for the corresponding internal static pressure heads.

Experimental and theoretical resonant frequencies seem to be in fair agreement,

and discrepancies of less than 10% are noticed. Those are possibly due to the

practical difficulty of maintaining a constant pressure (hence a constant free bulge

speed c) in the duct, yielding resonant frequencies that were not entirely equally

spaced. This is also consistent with the results from the free bulge speed mea-

surements in section 6.3.2 where the measured free bulge speeds were higher than

the predictions. Despite this, the repetitive occurrence of resonance in the duct is

fairly well predicted, with respect to the frequency.

The theoretical amplitudes of the total internal pressure were calculated using

(5.23) and the wave pressure acting on the membrane. Discrepancies are noticed

between those calculated values and the measured ones at the stern. This can be

the result of the omission in the model of the interaction of radiated waves by the

membrane with the incoming waves.

Despite those two main discrepancies, the general trend of the pressure varia-

tion at the stern is seen to be fairly well predicted with the model developed in



chapter 5. The resonant frequencies can be calculated in the same way as the

case of a stretched rope, on which an impulse will travel along the rope at a speed

depending on the tension in it. For those frequencies, the rope exhibits node and

antinodes characteristic of standing waves (or partial standing waves). Studying

the membrane displacement, related to the internal pressure variation, helps to

explain these observations. This was not done here.

6.4.3 Membrane displacement and pressure in the duct

Laser sensors recorded membrane displacements beneath waves at 11 locations

along the membrane, w being now a function of time. Using again the linearised

form of (5.9), the total internal pressure was inferred at those points from this

measurement as

p(x, t) ≈ 2v0a

σy,0(w(x, t)

a), (6.3)

the membrane deflection being due in that case to pb(x, t) + pw(x, t) (and not just

pb(x, t) as in the static case). The stress σy,0 was due to the static deflection from

the internal static pressure. A spline interpolation passing through the measure-

ment points was used to smooth out the line.

The total pressure in the rubber duct (bulge pressure and wave pressure) is plot-

ted in Figures 6.10 for an internal static pressure head of 7cm corresponding to a

theoretical free bulge speed of c =3.06m/s. The wave frequency was f = 0.95Hz,

corresponding to the first peak observed in Figure 6.8 for this configuration.

Theoretical and experimental values are in fair agreement. Repetitive patterns

are observed, and measurements are seen to agree in shape and amplitude with

the predictions. Those patterns are characteristics of a standing wave - or as will

be justified by separating the three bulge components later, a partial standing

wave. The number of repetitive patterns increases with frequency as seen in Fig-

ure 6.11 where the total internal pressure is plotted for the second peak observed

in Figure 6.8 (f = 1.2Hz) for the same internal static pressure head and free bulge

speed.



Figure 6.10: Pressure in the duct at thirty instants over one wave period with

an internal static pressure head of 7cm; wave frequency f=0.95Hz; (a) theory;

(b) inferred from laser sensors measurement of membrane displacement.






For those frequencies close to the resonant frequencies fn, the wavelength of the

bulge wave is seen to be an integer multiple of the length of the duct, agreeing

with all the above regarding standing/partial standing waves. When the wave

frequency is between two resonant frequencies, this is not the case as can be seen

in Figure 6.12 where the total internal pressure is plotted for f = 1.1Hz for the

same conditions as above and in this case, measurements and predictions do not

agree as well as before.

Figure 6.12: Pressure in the duct at thirty different instants over one wave pe-

riod with an internal static pressure head of 7cm; wave frequency f=1.1Hz; (a)

theory; (b) inferred from laser sensors measurement of membrane displacement.

The same total pressure mapping is given in Figure 6.13 for the rubber duct with

an internal static pressure head of 10cm corresponding to a theoretical free bulge

speed of c =3.28m/s and a wave frequency of f =1.075Hz. This corresponds to

the first peak observed for this configuration as seen in Figure 6.9. This peak was

earlier seen to be shifted up towards higher frequencies due to experimental free

bulge speeds being higher than the predicted ones. This is why the theoretical

total internal pressure was calculated with a free bulge speed of c =3.48m/s. In

the same way as for the other studied configuration, the total internal pressure is

seen to exhibit similar repetitive patterns at those frequencies and the measured

values are in good agreement, in shape and amplitude, with the calculated ones

using experimental values of free bulge speed.






The total internal pressure inferred from the laser sensor measurements of the

membrane displacement is seen to be in fair agreement with the calculated values

from the theory. To go further in the verification of this theory, it is necessary

to verify that this computed total internal pressure is actually made up of the

three components of the bulge wave introduced in section 5.4 (on which the wave

pressure has to be added to). The following task was to separate those components

and reconstruct the total internal pressure in the duct as in Figures 6.10-6.12 by

summing up these three waves.

6.4.4 Bulge pressure components

6.4.4.1 Separation of components

Laser sensors recorded membrane displacements at 11 points. The displacement

(hence the pressure in the duct) was found to be predominantly harmonic, at the

wave frequency. Using an FFT on the obtained signal for the three points shown



Rubbermembrane

LS[1] LS[2] LS[3]

x1

x2

Figure 6.14: Separation of bulge wave components from laser measurements.

in Figure 6.14, the total internal pressure p of those points could be written as

pLS[j] = ALS[j] cosωt+BLS[j] sinωt j = 1, 2, 3...11. (6.4)

As introduced in section 5.4, the total pressure in the duct is expected to be

made of one wave of amplitude P+b =

√

a21 + b21 travelling from stern to bow with

complex wavenumber K, one wave of amplitude P−b =

√

a22 + b22 travelling from

bow to stern with complex wavenumber K, one wave of amplitude PAb =

√

a23 + b23

travelling from bow to stern with k1. Using these notations

pj = a1 cos(Kxj−1 − ωt) + b1 sin(Kxj−1 − ωt) + a2 cos(Kxj−1 + ωt) +

b2 sin(Kxj−1 + ωt) + a3 cos(k1xj−1 − ωt) + b3 sin(k1xj−1 − ωt). (6.5)

Setting equal those pressures at each point and separating cosine and sine terms

leads to a linear system of 6 equations and 6 unknowns (a1, b1, a2, b2, a3, b3). The

signal from the laser sensors was pre-processed in order to subtract the wave pres-

sure from the total internal pressure, hence dealing with only the bulge pressure.

The first measurement point was taken as the origin for the phase calculation.

Several combinations of (LS[1],LS[2],LS[3]) triplets could be used to separate the

three absolute amplitudes of the bulge pressure component.

This technique can be successfully applied and amplitudes of each components

can be obtained for specific triplets of points. However, those solutions are sought

with constant amplitude all over the length of the membrane as seen in (5.20)-

(5.22). This means that the obtained values should not depend on the chosen

triplet.



The idea is then to write the system obtained for all the points j. That yields an

overdetermined system (6 unknowns for 22 equations if 11 points are taken). Such

systems have usually no exact solution and a best fit solution can be obtained

using a least squares method. In practice, this means that the obtained solution

is that which minimises the sum Sq of the squares of the errors at each point of

measurement j as

Sq =

11∑

j=1

(pLS[j] − p(j))2. (6.6)

This was solved with an iterative method using Maple software.

6.4.4.2 Components at resonance

Using this technique, the three components were obtained for the rubber mem-

brane with an internal static pressure head of 7cm for the wave frequency f =

0.95Hz corresponding to the first experimental resonant peak observed for this

configuration.

Figure 6.15 shows the measured amplitude and phase of the total pressure in

the duct and compares them with the values of amplitude and phases from the

total internal pressure along the duct calculated with (5.19), the different pressure

components being those computed from the least squares method. The magni-

tudes of pressure in the duct seem to be in fair agreement. The repetitive patterns

observed in Figure 6.10 are confirmed here. Measured phases do not seem in

perfect agreement with the computed ones. However, the kind of nodes visible in

Figure 6.10 can also be characterised here as each of them corresponds to the phase

crossing φ = π. This change of phase is similar between measured and computed

values, the difference in magnitude of the phase affecting only the amplitude of

the instantaneous pressure.



Figure 6.15: Measured total internal pressure, normalised by the pressure

acting on the membrane (a) and phase (b) for the duct with an internal static

pressure head of 7cm, plotted as points. Lines represent amplitudes and phases

computed as the sum of the components obtained from the least squares method.

Wave frequency f=0.95Hz.

Figure 6.16: Pressure in the duct at thirty different instants over one wave pe-

riod with an internal static pressure head of 7cm; wave frequency f=0.95Hz; (a)

inferred from laser sensor measurement of membrane displacement amplitude;

(b) sum of the three bulge wave components (and the wave pressure) obtained

from a least squares method.



Figure 6.16 shows those two ways of mapping the total internal pressure over

several instants of one wave period: the first one obtained from the laser sensor

measurement as before, and the second one by reconstructing this total inter-

nal pressure by summing up the three bulge wave components (amplitudes and

phases) obtained at any point on the membrane. The two obtained mappings

seem to agree with this approach of decomposing the bulge pressure as the sum

of three components and the slight mismatch of measured and computed phases

seen in Figure 6.15 is consistent with Figure 6.16.

Finally, the pressure amplitudes are compared with those predicted from (5.20)-

(5.22) for the same configuration and results are shown in Figures 6.17, 6.18 and

6.19. Measured and calculated results similarities around the first resonant peak

f=0.95Hz are in quite fair agreement with respect to their amplitudes. Keeping

in mind that those amplitudes were obtained from a least squares method, it is

quite satisfactory.

Figure 6.17: Magnitude of P+b relative to the wave pressure acting on the

membrane with an internal static pressure head of 7cm. symbols: experimental

values inferred from a least squares method; —: calculated from (5.21).



Figure 6.18: Magnitude of P−b relative to the wave pressure acting on the



Figure 6.19: Magnitude of PAb relative to the wave pressure acting on the





The same idea has also been verified with the duct with an internal static pressure

head of 10cm and the first observable resonant frequency f =1.075Hz from Figure

6.9. Figure 6.20 shows the measured amplitude and phase of the total pressure in

the duct and compares it with the values of amplitude and phases computed as

before from the least squares method while Figure 6.21 shows the total internal

pressure inferred from the laser measurements and the total internal pressure re-

constructed by summing the different components obtained from the least squares

method. Finally Figures 6.22, 6.23 and 6.24 compare the computed amplitudes of

the three components of the bulge wave with those computed from (5.20)-(5.22)

for this condition.

Figure 6.20: Measured total internal pressure, normalised by the pressure

acting on the membrane (a) and phase (b) for the duct with an internal static

pressure head of 10cm, plotted as points. Lines represent amplitudes and

phases computed as the sum of the components obtained from the linear squares

method. Wave frequency f=1.075Hz.



Figure 6.21: Pressure in the duct at thirty instants over one wave period

with an internal static pressure head of 10cm; wave frequency f=1.075Hz; (a)

inferred from laser sensor measurement of membrane displacement amplitude;

(b) sum of the three bulge wave components (and the wave pressure) obtained

for a least squares method.






6.4.5 Wave analysis and loss in the rubber

Signals of the three wave gauges were used to measure the incident (Pi), reflected

(Pr) and transmitted (Pt) wave powers. Figures 6.25 and 6.26 show the ratio of

the transmitted and reflected wave power to the incident wave power.

The transmitted wave coefficient increased with frequency. The higher the wave

frequency, the less the membrane is affected by the incident wave, hence most of

the wave power is transmitted behind the rubber duct. Peaks of minimum trans-

mission occurs for f = 0.95Hz and f = 1.2Hz that were seen earlier to be resonant

frequencies of the system.

Figure 6.25: Ratio of the transmitted wave power, Pt, to the incident wave

power Pi for the duct with an internal static pressure head of 7cm. Black line

and symbols: simple harmonic component of wave gauge signals; Grey line and

symbols: first three harmonic components of wave gauge signals.



Figure 6.26: Ratio of the reflected wave power, Pr, to the incident wave

power Pi for the duct with an internal static pressure head of 7cm. Black line

and symbols: simple harmonic component of wave gauge signals; Grey line and

symbols: first three harmonic components of wave gauge signals.

In contrast, the reflected ratio seems to exhibit the peaks at the resonant frequen-

cies. This is due to maximum membrane deflection at those frequencies, triggering

radiated waves. However, this reflected ratio is seen to be very low. Those peaks

of maximum reflection and minimum transmission at the resonant frequencies are

characteristic of such system, as developed in section 3.3.4.

The energy in the system should be conserved. In theory, this means

Pt + Pr = Pi. (6.7)

However, losses can occur for various reasons as described later and a power loss

Ploss is defined as

Ploss = Pi − Pr − Pt (6.8)

and is plotted in Figure 6.27.



Figure 6.27: Ratio of the wave power loss, Ploss, to incident wave power Pi

for the duct with an internal static pressure head of 7cm. Line and symbols:

simple harmonic analysis of wave gauge signals.

It can be seen that about 50% of the incident wave power is dissipated in the

system over one wave period. Here is a explanation of the possible causes of this

loss:

Power in higher wave harmonics- The power loss shown in Figure 6.27 was cal-

culated using only the first harmonic of the signals from the wave gauges. Some

power is often expected to be found in higher harmonics, especially in the trans-

mitted power with such a step configuration. A description of this phenomenon is

given by Massel (1983), for instance. This was not the case here, as can be seen

in Figure 6.25 where the transmitted wave power was calculated using two more

harmonics. Some power was found in the higher harmonics of the reflected wave

but not enough to explain the magnitude of lost power.

Energy dissipated in the flow - A portion of the incident wave energy can be lost

through the following processes: wave breaking over the structure, turbulence in

the inner flow or around the corners which were present in this configuration.

Typical values of 50% of incident energy dissipated are quite common for those

type of configuration as seen in Table 3.1.



Loss in the rubber - The membrane displacement beneath waves stretches the rub-

ber. This oscillatory stretching is characterised by some loss of the energy required

for the deformation into heat (hysteresis) in the rubber.

Consider the strip of membrane (length L, thickness v0 and width 2a) under-

going oscillatory loading in its length direction, so that the longitudinal strain is

ε = ε0 sin(ωt). Following the stress-strain relationship (2.24), the stress in the

rubber is

σ = Eε0(sin(ωt) + βω cos(ωt)), (6.9)

yielding the force in the rubber

Fε = 2aσv0 (6.10)

= 2av0Eε0(sin(ωt) + βω cos(ωt)). (6.11)

The mean power loss due to hysteresis is then calculated as the mean product of

the force and the velocity of the point of action of the force as

Phys =1

T

∫ T

0

(2av0Eε0(sin(ωt) + βω cos(ωt)))(dε

dtL)dt (6.12)

=1

2VrEε20βω

2, (6.13)

with Vr = v02aL being the total volume of rubber.

An intuitive approach would consist in mapping the membrane deflection at several

instants over one wave period, as previously done for the pressure, and measure

the arc-length of the envelope of the obtained plot. However, this would be jus-

tified if the pressure wave in the duct was a pure standing wave. It was seen in

section 6.4.4.2 that this is not the case, and that at best, the pressure wave is a

partial standing wave at resonance only.

Instead, the instantaneous elongation was plotted at twenty instants over one wave

period, as seen in Figure 6.28. It allowed the measurement of ε0, the amplitude

of the elongation with respect to the incident wave amplitude. It is interesting to

note that the frequency of membrane stretching over one wave period is twice the

wave frequency, and this should be considered accordingly in (6.13), where ω is

the frequency of oscillation of the strip of rubber and not the wave.



Figure 6.28: Instantaneous elongation in the rubber at 20 instants (plotted as

points) as a function of time normalised by the wave period T . Wave frequency

is f = 0.775Hz.

Figure 6.29: Ratio of the power lost in the rubber, Ploss, to the incident wave

power Pi for the rubber duct with an internal static pressure head of 7cm.

Figure 6.29 shows the ratio of the mean power loss due to hysteresis in the rubber to

the incident wave power. Two distinctive peaks are observed and correspond once

again to the resonant frequencies. At those resonant frequencies, the amplitude



of membrane deformation is maximum, yielding a maximum elongation of the

rubber. However, the magnitude of this power loss is seen to be negligible in the

concern of power conservation in this system.

6.4.6 Transverse motion of the membrane

The membrane motion in its length is characterised by characteristics patterns, in

a similar way as for the velocity profile in Figure 5.3 for instance. At resonance,

an integer multiple number of complete patterns are visible, while this number of

patterns goes from one integer to the successive one between resonance.

A concern can arise with regard to the motion of the membrane in its width

direction, namely the general form of deformation beneath waves. Consider the

cross-sectional view of the duct in the narrow flume as seen in Figure 6.30. A

close configuration was studied by Bauer (1981) who investigated the hydroelastic

vibrations of a liquid in a rectangular container, with an elastic bottom and a

free liquid surface. He also studied the hydroelastic vibrations of a liquid in a

rectangular container covered by a flexible membrane. The idea here is to use a

similar method by adapting the three dimensional potentials he used for the two

dimensional configuration shown in Figure 6.30. The potential in the upper region

is sought to satisfy the Laplace equation, the zero velocity on the flume’s wall and

the free surface boundary condition. It is then expressed as

φ2 =∑

n

(cosh[δn(z − d)] +

ω2

gδnsinh[δn(z − d)]

)An cos

nπy

2ae−iωt, (6.14)

(6.15)

where δn = nπ/2a. The potential in the lower region is sought to satisfy the

Laplace equation, together with the zero velocity on the duct’s wall and bottom,

yielding

φ1 =∑

n

cosh[δn(z + b)]

cosh(δnb)Bn cos

nπy

2ae−iωt. (6.16)

Consider the displacement of the membrane as

w(y, t) =∑

n

Wn(t) sin(nπy

2a). (6.17)



z

y

flumewalls

2a

duct

w

b

h

(0,0)

(2)

(1)

Figure 6.30: Cross-section view of the duct and transverse motion of the

membrane.

The kinetic energy of the membrane associated with this displacement is

Ek =µm

2

∫ 2a

0

(∂w

∂t

)2

dy (6.18)

=µma

2

∑

n

W 2n , (6.19)

with µm the mass per unit area of the membrane. The potential energy associated

with this displacement is

Ep =1

2

∫ 2a

0

Ty

(∂w

∂y

)2

dy (6.20)

=a

8

∑

n

n2Tyπ2

a2W 2

n , (6.21)

from which we can deduce the Lagrangian of the system L = Ek−Ep. The loading

on the membrane due to the pressure p(y, t) makes it possible to determine the



generalised force Qn with the help of the virtual work concept,

δW =∑

n

∫ 2a

0

p(y, t) sin(nπy

2a)dy

︸︷︷︸

Qn

δWn. (6.22)

The equation of motion of the membrane is then

d

dt

( ∂ L∂Wn

)

− ∂ L∂Wn

= Qn. (6.23)

The pressure acting on the membrane is

p(y, t) = ρ(∂φ1

∂t− ∂φ2

∂t)∣∣∣z=0

. (6.24)

Using the continuity of the velocities ∂φ/∂y at the interface z = 0 to express An

as a function of Bn and using

cos(kπy

2a) =

∑

j

αkj sin(

jπy

2a) (6.25)

from Bauer (1981) with

αkj =

2

π

j[1− (−1)j+k]

j2 − k2, (6.26)

(6.23) yields

Wn + ω2nWn =

−2ρi

µπ

∑

k

ωBkΘk

n[1− (−1)n+k]

n2 − k2. (6.27)

Substituting

Wn = Cneiωt (6.28)

in (6.27) and the kinematic free surface condition

∂w

∂t=

∂φ1

∂z

∣∣∣z=0

(6.29)

and setting equal the two obtained Cn leads to

∑

k

n[1− (−1)n+k]

n2 − k2

(ω2(S)k

g(ω2

n − ω2) +ρ

µΘk

)

Bk = 0, (6.30)

with ω2n corresponding to the natural frequencies of the stretched membrane in

air, that is

ω2n =

π2Tyn2

4µma2. (6.31)



Furthermore

ω2(S)k = gδk tanh(δkh1), (6.32)

Θk = −δkg sinh

(

δk(b+ d))

− ω2 cosh(

δk(b+ d))

cosh(δkb)(

ω2 cosh(δkd)− sinh(δkd)) . (6.33)

Solving the determinant of (6.30) yields the natural frequencies of the fluid-membrane

system, with the effect of the surrounding water. The formulation of the membrane

displacement does not verify the conservation of volume, in term of

∫ 2a

0

wdy 6= 0. (6.34)

However, this is not a requested condition as here the concern is about a cross-

section of the duct. If at this cross-section the membrane deflects in a way, further

down (or up) the duct, it deflects in a way so that the volume is conserved.

The effect of the tension in the length direction (x−direction) is also neglected con-

sidering the dimensions of the problem, the length of the duct being way greater

that its width. Considering the length of the membrane L (hence modes in the

length direction), this all comes down to saying that

π2

µm

(

Ty

n2

4a2+ Tx

m2

L2

)

≈ π2

µm

Ty

n2

4a2, (6.35)

meaning that the obtained frequencies would be slightly modified if considering

the 3D problem, but not in a significant way.

With the parameters from the experiment (a=0.15m, Ty = 170.9N/m, b = 0.185m,

and taking here d = d1=0.171m, µm = 1.14kg/m2) and the physical values

(g=9.81m/s2, ρ=1000kg/m3), the first three resonant frequencies of the system

are found to be

f1 = 1.615Hz, (6.36)

f2 = 2.279Hz, (6.37)

f3 = 2.793Hz. (6.38)

Given the range of wave frequencies used ([0.7Hz:1.5Hz]), it seems reasonable to

say that higher modes than the fundamental ones are unlikely to be excited. The

membrane shape in the y-direction is predominantly made of the form sin(πy2a).



This is not a circular shape as introduced in chapter 5, although it was checked

that using a sine shape for the membrane deflection does not make much difference

in the distensibility and free bulge speed calculation.

6.4.7 Conclusions

The duct was placed beneath waves and pressures at its bow and its stern were

recorded. It exhibited resonant peaks for frequencies at which k2 = nπ/L as pre-

dicted by the theoretical model. Pressure amplifications were found to be up to

2.7 times the wave pressure acting on the membrane (or to be precise, the wave

pressure acting at the equivalent submergence depth defined from the static tests)

for those frequencies.

The total internal pressure in the duct could be inferred from the laser sensors

measurement of the membrane displacement. It was plotted over the length of the

membrane for 30 instants over one wave period. For the frequencies mentioned

above, these plots exhibit nodes and anti-nodes. By expressing the bulge wave

pressure in the duct as the sum of three components, and separating them using

a least squares method, the wave inside the duct at those frequencies was seen

to be made of two components of almost similar amplitude travelling in opposite

directions and one component travelling with the wavenumber of the water wave.

The characteristic patterns observed from the total internal pressure mapping were

then proved to be characteristics of a partial standing wave.

Loss in rubber is one main concern when dealing with this material. The mean

power loss due to hysteresis was found to be noticeable only at these resonant

frequencies at which the displacement (hence elongation) of the membrane is max-

imum.

6.5 Thin membrane results

6.5.1 Introduction

The work described above concerned the experimental work undertaken with the

1mm thick membrane (hereinafter referred to as thick membrane). The 0.2mm

thick membrane (hereinafter referred to as thin membrane) was also tested using



the same experimental set-up. Unlike the thick membrane, there was no agreement

between the one dimensional theory for bulge propagation described in section 5.4.

This section present the results obtained for the static deflections, free bulge speed

and pressure amplification and tries to give an explanation of the discrepancies

between the calculation and measurements, and then discuss the applicability of

the one dimensional model used so far.

6.5.2 Experimental conditions

The tests conditions are the same as before. Due to its thickness, this membrane

was initially stretched in both directions. This is summarised in Table 6.3.

Strain in Strain in Internal static Number

Test no y-direction x-direction pressure head of

εy,0 εx,0 (m) runs

1 0.14 0 0.02 3

2 0.14 0 0.05 3

3 0.28 0 0.02 2

4 0.28 0 0.05 2

5 0.14 0.14 0.02 2

6 0.14 0.14 0.05 1

Table 6.3: Experimental conditions for the closed ends experiment with the

0.2mm membrane.

6.5.3 Static deflection and free bulge speed

Before running waves, static tests were carried out as before to measure the cen-

treline deflection w and free bulge speeds, and to compare the obtained results

with the calculated ones. The centreline deflection is shown in Figures 6.31, 6.32

and 6.33 for the three configurations tested.

Deflection values are seen to be of a higher magnitude for the 0.2mm membrane, as

it is less resistant to stretching compared to the 1mm rubber sheet. Experimental

and theoretical values follow the same trend with fair agreement, and the influence

of εx,0 (hence σx,0) is negligible, as can be seen in (5.9), where the influential stress

is in the y-direction.



Figure 6.31: Experimental values of membrane centreline deflection (+++),

load-deflection relation 5.9 (––) for εy,0 = 0.14 and εx,0=0


load-deflection relation 5.9 (––) for εy,0 = 0.28 and εx,0=0.




load-deflection relation 5.9 (––) for εy,0 = 0.14 and εx,0=0.14.

The influence of the initial longitudinal stress σx,0 can be further studied with

the free bulge speed measurements. Figures 6.34, 6.35 and 6.36 show the average

measured free bulge speeds for tests 1, 2 and 3 respectively, for various internal

static pressure heads p0. No major differences between the three figures are ob-

served, except near p0 = 0.0m where the influence of the lateral initial strain εy,0

is more pronounced. This can be explained by the fact that when p0 increases, the

stress due to the deflection σy, such as the one in (5.5) takes over the initial stress

σy,0. This yields close values of free bulge speeds on each figures for the different

internal static pressure heads tested.

The free bulge speed is expected to be independent of the longitudinal stress σx,

as detailed in section 5.2. This is verified when comparing the measured free bulge

speeds from tests 1 and 2, that do not seem to be influenced by this longitudinal

stress.



Figure 6.34: Averaged values of free bulge speed measurement, for εy,0 = 0.14

and εx,0=0 for the 0.2mm membrane; (––) calculated values; (+ + +) averaged

values from pressure and laser measurements.


and εx,0=0 for the 0.2mm membrane; (––) calculated values; (+ + +) averaged





and εx,0=0.14 for the 0.2mm membrane; (––) calculated values; (+++) averaged


6.5.4 Pressure variation at the stern

As before, pressure recordings at the stern made it possible to express the total

internal pressure amplitude at the stern as a linear function of the pressure due

to the undisturbed incident wave and results are shown in Figure 6.37 to Figure

6.42 for tests 1 to 6.

The measured pressures at the stern do not agree with the calculated ones. The

different peaks in the calculated values (right column) correspond, as before, to

the different harmonics of the membrane fn = nc/2L. Measured values are of

much smaller magnitude and an explanation of cause is given next.

6.5.5 Discussion

The main assumption made for the one dimensional model for bulge propagation

in the duct was that the external wave is undisturbed during its travel over the

membrane. This argument can be challenged with the data obtained from the

wave gauge located above the middle point of the duct.



Figure 6.37: Magnitude of the pressure at the stern relative to the wave

pressure at the free surface elevation with conditions of test 1; left: measured

for the three runs (each symbols for one run); right: one dimensional theory

with c = 1.11m/s.

Figure 6.38: As for Figure 6.37, but with conditions of test 2; left: measured

for the three runs (each symbols for one run); right: one dimensional theory

with c = 0.88m/s.


for the two runs (each symbols for one run); right: one dimensional theory with

c = 1.11m/s.





c = 0.88m/s.



c = 1.11m/s.

Figure 6.42: As for Figure 6.37, but with conditions of test 6; left: measured;

right: one dimensional theory with c = 0.88m/s.



Figure 6.43: Surface elevation at the middle point along the length of the

duct, relative to the incident wave amplitude.

Values of the water surface elevation (normalised by the incident wave amplitude)

are shown in Figure 6.43 for tests 1 and 3 and compared to those obtained with

the thick membrane and an internal static pressure head of 7cm. Values of surface

elevation for the thin membrane are much lower than that of the thick one. The

average ratio η/A for the thick membrane is 0.84 whereas for the thin membrane,

it is 0.44 and 0.42 for tests 1 and 3, respectively. There is a significant difference

between the two membranes regarding the behaviour of the wave propagating

above them.

Figure 6.44: Ratio of the transmitted wave power, Pt, to the incident wave

power Pi for all the tests and measured with the first three harmonic components

of wave gauge signals.



This can be further observed in Figure 6.44 where the transmitted power (nor-

malised by the incident wave power) is plotted for all the tests carried out. Values

are found to be of a lower magnitude than that for the thick membrane (Figure

6.25), where in the latter case, Pt/Pi was less than 0.5 for the peaks of minimum

transmission.

Figure 6.45: Membrane displacement at thirty instants over one wave period

for configuration of test 1; wave frequency f=0.775Hz; (a) theory; (b) inferred

from laser sensors measurement of membrane displacement.

The membrane displacement measured from the laser sensors is compared in Figure

6.45 with that obtained from the one dimensional model, for the wave frequency

of f = 0.775Hz, for the configuration of test 1. For that one, the free bulge speed

is 1.11m/s and so f = 0.75Hz is the closest available frequency to the resonant

one, that is the 7th harmonic f7 = 7c/2L = 0.777Hz. It is worth noticing the

agreement in amplitude and space from the calculated and measured values of

displacement from x = 0m to the first node x = L/7 = 0.71m. At points further

along the membrane, the second node to node bump is visible, although its mag-

nitude is already seen to be much lower than the predicted one. Further down the

membrane, the magnitude of the displacement vanishes completely and there is no

agreement between theory and measurement. Ripples propagating forth and back

on the membrane were observed after one or two wavelengths of the membrane

displacement. The energy carried in the bulge waves can be transferred to these

ripples in the same way to what happen on ocean surface waves for step gravity



waves, and is thought here to be the cause of the mismatch between calculated

and measured values of membrane displacement.

With regards to the information provided by Figures 6.43, 6.44 and 6.45, an expla-

nation is provided regarding the reasons why the one dimensional model cannot

be applied to the thin membrane.

For free bulge speeds as low as those propagating in the thin membrane, the

bulge wavelength is experimentally observed to be small, the membrane profile

exhibiting, for almost the lowest frequency tested, already six antinodes. In the

case of the thick membrane, where only the third and fourth harmonic (and the

corresponding membrane profile) were observed, it seems appropriate to consider

that the incoming waves propagate over the membrane without being significantly

perturbed, the bulge wavelength remaining larger than the water wavelength. This

way, the incoming waves “sees” the rubber membrane as a flat bed. Conversely,

the displacement profile exhibited by the thin membrane is made of many more

wavelengths, and this simulates a sort of sinusoidal seabed. Waves propagating

over such a bottom are more prone to lose their energy by various physical phe-

nomena that are beyond the scope of the present work.

Finally, it can be argued that a thin membrane cannot handle great values of

stress. Observations included values of membrane deflection more than 4 times

the incoming wave amplitude, as can be seen in Figure 6.45 over the first metre

where the membrane is still subject to an almost undisturbed wave. Given the

submergence of the membrane for the tests conditions (d < 0.15m), the effect of

such displacements is more than likely to affect the incoming wave significantly.

The effect of increasing the initial stress in the membrane increases the free bulge

speed, so it goes back to the argument on the negative effects of having low free

bulge speeds in the one dimensional model.

Chapter 7

Extractable power

7.1 Introduction

This chapter details the tests carried on the version of the duct equipped with

the PTO system detailed in section 4.5 and placed in the wave basin presented

in section 4.3. For this test, the total length of the duct was reduced to 2m and

the membrane thickness was 1mm. Membrane deflection and pressure in the duct

were measured as in chapter 6. The extractable power was also measured using

(4.28). Results are compared with the predicted values from 5.43 and discussions

follow on the general efficiency of this system as a wave energy converter.

7.2 Experimental layout and test conditions

7.2.1 Description

The experimental sketch is given in Figure 7.1. The duct was placed on the bottom

of the wave basin (in the middle in the width direction) and kept in position using

heavy masses. The same hydraulic pump system used previously maintened the

desired internal static pressure head. Laser sensors (not drawn) were mounted

as in Figure 4.15 to measure the membrane displacement. Resistive wave gauges

(two per position in the width direction) were used to measure incident, reflected

and transmitted wave power. Two air pressure transducers and three wave gauges

were mounted in the bent tube.

131

132 Chapter 7 Extractable power

Wav

e pad

dle

Hig

hden

sity

foam

2.7

m

0

.5m

0

.44

m

2

.0m

0

.58

0m

2.0

m

Air

pre

ssure

tran

sduce

rs

Wav

egau

ges

Pt1

Copper

pip

es

Abso

rbin

gbea

ch

Wg1

Wg2

Wg3

Figure 7.1: Experimental layout for the rubber duct equipped with the PTO.

Dashed lines correspond to different internal static pressure heads.

Chapter 7 Extractable power 133

The test conditions are summarised in Table 7.1 where the range of impedances,

wave frequencies and number of waves amplitudes used are specified. Waves were

generated in the range of [0.45Hz-1.2Hz] with increments of 0.025Hz. For this

range (and this water depth), it was not necessary to change the distance between

Wg1 and Wg2. Pictures of this set-up are provided in Appendix C.


Test no pressure head Z = ZPTO/Zd frequencies wave of

(m) (Hz) amplitudes runs

1 0.03m 1.03 0.45-1.2 5 3

2 0.03m 1.10 0.5-0.875 5 1

3 0.03m 1.93 0.45-1.075 3 3

4 0.02m 1.73 0.45-1.2 5 1

5 0.02m 3.47 0.45-1.2 5 1

6 0.02m 1.08 0.45-1.2 5 3

7 0.02m 1.02 0.45-1.2 5 3

8 0.02m ∞ 0.5-1.1 3 1

Table 7.1: PTO experiment: test conditions

As in the experiment carried in chapter 6, the signals obtained from the wave

gauges (in the flume and in the PTO), the pressure transducers (water and air) and

the laser sensors were found to be predominantly monochromatic. Hence the data

analysis carried out hereinafter neglects harmonics other than the fundamental

wave frequency.

7.2.2 Power take-off in experimental conditions

The PTO used was aimed to be viscous, meaning that pressure and flow rate are

in phase, hence the water-air interface displacement and the pressure lagged by

90. A way to cross check this assumption is to plot one signal against the other.

Such plots are shown in Figure 7.2 for the four tests with Z finite (Table 7.1) and

the duct set with an internal static pressure head of 0.02m.


Figure 7.2: PTO air pressure versus PTO water level for the rubber duct with

an internal static pressure head of 0.02m; f=0.625Hz; (a) test 7; (b) test 6; (c)

test 4; (d) test 5.

PTO water level and air pressure are seen to be in quadrature for tests 6 and 7

when Z=1.02 and Z=1.08 respectively. As the impedance increases, the phase lag

between the water level and the air pressure decreases, meaning that the phase

lag between flow rate and pressure, on the other hand, increases.

The measured phase lag is compared in Figure 7.3 with the theoretically phase lag

calculated using (4.25). The values seem to be in fair agreement and the simplified

model from section 4.5.3 is found to give an accurate explanation of the phase lag

happening in the chamber.

Equation (4.26) can be rewritten as

q0 = τ(Stωε0), (7.1)

where the term Stωε0 represents the flow rate of the air due to the water-air in-

terface displacement. If the air was not compressed, this flow rate would be equal

to q0. The ratio τ is plotted in Figure 7.4 for the four previous impedances in the

same frequency range as before. It is seen to be close to 1 for impedances Z =1.02


Figure 7.3: Phase lag between PTO water level and PTO air pressure as afunction of wave frequency for the duct with an internal static pressure head of0.02m ; • • • measured, — from (4.25); (a) test 7; (b) test 6; (c) test 4; (d)

test 5.

Figure 7.4: Ratio of the flow rate in the copper pipes to that of the air in thechamber, for the duct with an internal static head of 0.02m.


and Z =1.08, while for the two other impedances Z =1.73 and Z =3.47, it is seen

to decrease by up to 20%. For those higher impedances, the flow rate in the pipes

is then slightly different from that due to the water displacement in the chamber.

One of the main explanations for that to occur is that air is being compressed at

those impedances. Power required for this compression is stored in the air (as in the

case of pumping air in a tire) and mostly restored when air expands, the remaining

power being used into heating up the air during compression. This latter power

is assumed to be negligible hereinafter, and even though this power is present at

higher impedances, it is not seen as useful power potentially absorbed by the PTO.

Data from test 8, where the ∞ sign stands for the fact that all pipes were closed,

allows measurement of the bulk modulus of the air enclosed in the chamber, that

is

KT = −VdP

dV. (7.2)

KT characterises the resistance of the air to a uniform compression. This bulk

modulus is shown in Figure 7.5 and yields an average value of KT = 0.53×105Pa,

while the bulk modulus of air in the literature is found to be KT = 1 × 105Pa.

The main explanation for this difference is the existence of air leaks in the system.

This differential is considered to be of minor influence on the PTO impedance

calculation and therefore will be neglected.

This explanation has been carried out with data from the situation where the

internal static pressure head was 0.02m. The same demonstration can be un-

dertaken with data from the situation where the internal static pressure head is

0.03m. Moreover, a wider range of impedances has been studied in a previous

experiment and the corresponding data are presented in Appendix A.


Figure 7.5: Measured values of the bulk modulus of air, with configuration as

in test 8.

7.3 Matching impedances configuration

7.3.1 Introduction

Tests were carried for the two internal static pressure heads provided in Table 7.1,

with two sets of PTO configurations. In the first case, the impedance of the PTO

was as close as possible to that of the duct and in the second case, a deliberate

mismatch between the PTO and duct impedances was set up to study the influ-

ence on the power absorbed by the PTO.

The first configuration (matching) is described in this section. Capture widths

are presented, and in a similar way to what has been done in sections 6.4.3 and

6.4.4, characterisation of the inner flow is made is term of bulge pressure compo-

nents.


7.3.2 Power capture

Wave gauges recorded the water level e0 in the rigid tube so that the associated

flow rate in the copper pipes could be evaluated, leading to the measured power

absorbed by the PTO following (4.28). Resistive gauges in the basin, ahead of the

duct, provided the measurements of the incident wave power from which capture

widths were calculated.

Capture widths are shown in Figures 7.6 and 7.7 for tests 1-2 and 6-7, respec-

tively. They are compared with the calculated values of capture widths obtained

from (5.43) for those corresponding impedances.

For both internal static pressure heads, the difference in impedances between the

two corresponding tests does not seem to be of a strong influence and can be

thought to represent the case Z = 1. Calculated and measured values seem to

be in fair agreement. Discrepancies are noticeable in term of absolute values of

capture width, but the general trend is similar.

Figure 7.6: Capture width (in duct widths) for a static pressure head of

3cm; lines: computed from theory; symbols: measured; colour legend: test 2

(Z = 1.10) in grey and test 1 (Z = 1.03) in black.


Figure 7.7: Capture width (in duct widths) for a static pressure head of

2cm; lines: computed from theory; symbols: measured; colour legend: test 6

(Z = 1.08) in grey and test 7 (Z = 1.02) in black.

Mathematically the peaks of maximum power capture correspond to frequencies

minimising the denominator of |q|2 where q is obtained with (5.42). A physical

explanation can be given to them, as they correspond to one of several interest-

ing phenomena happening in the duct or the chamber. The theoretical peak of

maximum capture widths in Figure 7.6 was for f=0.725Hz. For this configuration

the length of the slug of water was l =0.683m, yielding a resonant frequency of

fslug = (1/2π)√

g/l =0.6Hz.

Two peaks are visible in Figure 7.7. The first theoretical peak was for f =0.68Hz.

The length of the slug of water was, for this configuration, l =0.672m, yielding a

resonant frequency of fslug=0.61Hz. Moreover, at f =0.585Hz, k1 = k2 meaning

that the free bulge speed in the duct and the water wave speed match. This cor-

responds to a condition of maximum internal velocity at the stern, and hence flow

rate in the pipes. Finally, at f =0.625Hz, k1 = k2√1 + βω which is a solution of

<(K2 − k21)

2 = 0, maximising |q|2.

The second visible peak in Figure 7.7 occurred for f =1.05Hz. This corresponds

to the second harmonic of the duct as studied in section 6.4.2. The free bulge

speed was c =2.1m/s so f2=2.1/2=1.05Hz. This is again a condition of maximum


internal velocity at the stern.

Experimental capture widths of up to 2.2 times (tests 7-8) and 2 times (tests

1-2) the duct’s width are obtained with this experimental configuration. Due to

the simultaneous occurrence of physical phenomenon maximising the flow rate in

the pipes around one frequency, maximum power capture is not concentrated at

this frequency. The bandwidth of these non-dimensional capture widths for which

it is greater than one are found to be [0.575Hz;0.75Hz] and [0.65Hz;0.8Hz] for

tests 7-8 and tests 1-2, respectively. Attention on how these resonant peaks are

characterised in term of pressure variation in the duct is given in the next section.

7.3.3 Pressure variation in the duct

The internal pressure was mapped in the manner described in section 6.4.3, using

this time the adjusted pressure from the air pressure transducers for the pressure

at the stern. This is shown in Figures 7.8 and 7.9 for tests 7-6 respectively and

the frequency of maximum power output f =0.625Hz.


an internal static pressure head of 2cm; wave frequency f=0.625Hz; (a) the-

ory for Z = 1.02; (b) inferred from laser sensors measurements of membrane

displacement.



an internal static pressure head of 2cm; wave frequency f=0.625Hz; (a) the-

ory for Z = 1.08; (b) inferred from laser sensors measurements of membrane

displacement.

Calculated and measured values of this internal pressure follow the same trend,

although the measured values seem slightly smaller in amplitude than the theo-

retical values. This can be improved in the model by increasing the loss angle

by a couple of degrees. Doing so is justified by saying that a higher hysteresis

coefficient can represent the energy lost in the system by wave radiation from the

membrane, a phenomenon not otherwise taken into account. Unlike in the closed

ends configuration, the pressure is not reflected at the stern and its amplitude is

seen to be increasing along the length of the rubber duct. Figure 7.10 shows a

similar plot for the duct with a internal static pressure head of 3cm and, again,

for the first frequency of maximum power output.

Moreover, Figure 7.11 shows the total internal pressure for the second peak ob-

served in Figure 7.7 (for test 6). Nodes and anti-nodes as observed, in a similar

way to the closed ends configuration. This peak occurred for the second harmonic

of the membrane, as stated earlier.




theory for Z = 1.03; (b) inferred from laser sensors measurements of membrane

displacement.




displacement.


The total internal pressure is seen to be well predicted by the model from section

5.5. As in the case of closed ends, this total pressure was decomposed by a linear

least squares method to verify whether the bulge pressure could be written as the

sum of the three components (5.33)-(5.35).

7.3.4 Bulge components

A least-squares method was used to express the total pressure in the duct as a

sum of three distinct components, as for the case of the duct with closed ends.

An example of values obtained for the amplitude and phase of this inner pres-

sure wave is given in Figures 7.12 and 7.13 for the two peaks of maximum power

capture mentioned above and for the experimental conditions of test 6.

As before, points represent the measured values of amplitude and phase of the

internal total pressure at 10 points along the membrane (and the pressure at the

stern and bow), while the line is the sum of the components obtained with the

least-squares method. In Figure 7.12, points follow the reconstructed internal pres-

sure amplitude and phase and the amplitude is seen to increase along the length

of the membrane as previously noticed in Figures 7.8 and 7.9. The sum of these

three components (plus the wave pressure) were used to map the internal pressure

in a similar way as before, and this is shown in Figure 7.14.

In Figure 7.13, the general shape of the amplitude profile along the length of

the membrane follows that of a natural mode. However, the node and antinodes

of the amplitude obtained from the least-squares seem to be slightly shifted to-

wards points further along the membrane. This can be well appreciated in Figure

7.15 where the location of nodes and antinodes are for slightly different positions.

Also the second minimum is not in fact a node as the phase tends to π around

x =1.6m, but does not completely cross the line φ = π.


Figure 7.12: Measured total internal pressure, normalised by the pressure act-

ing on the membrane (a) and phase (b) for the duct with an internal static pres-

sure head of 2cm, plotted as points. Lines represent amplitudes and phases com-

puted as the sum of the components obtained from the linear squares method.

Wave frequency f=0.625Hz, with conditions as in test 6





Wave frequency f=1.05Hz, with conditions as in test 6

The magnitudes of the three bulge components obtained at each experimental

frequency were compared with the predicted ones from (5.33)-(5.35) and results are

shown in Figures 7.16-7.18. The choice of the logarithmic axes is to distinguish the

peak occurring in the region f=1.05Hz for P+b and P−

b , from that at f=0.625Hz.



an internal static pressure head of 2cm; wave frequency f=0.625Hz; (a) inferred

from laser sensor measurement of membrane displacement amplitude; (b) sum

of the three bulge wave components (and the wave pressure) obtained for a least

squares method.





squares method.



membrane with an internal static pressure head of 2cm and conditions as in

test 6; symbols: experimental values inferred from a least squares method; —:

calculated from (5.34).










The agreement between the three amplitudes obtained from the least-squares

method and the predicted ones can be judged as satisfactory. Unlike the closed

ends configuration from chapter 6, the amplitude of P−b seems much lower than

the amplitudes of P+b and PA

b , especially around f=0.625Hz. These two last com-

ponents being those travelling from bow to stern; this is consistent with Figure

7.11 where the pressure in the duct is seen to increase along its length. On the

contrary, for f =1.05Hz, the amplitudes of P+b and P−

b are the same. Moreover,

those two components are travelling with the same speed, in opposite direction,

giving another justification of the partial standing wave pattern in the duct for

that frequency.

7.4 Mismatch impedances configuration

7.4.1 Introduction

The analysis carried out above stands for the configuration with the impedance

of the PTO as close as possible to that of the duct. More tests were carried with


a mismatch in impedances in order to investigate the applicability of the model

detailed in section 5.5, and a similar analysis as in the previous section is carried

out here.

7.4.2 Power capture

Values of capture widths are shown in Figures 7.19-7.21 for tests 3, 4 and 5, respec-

tively. Measured and calculated power capture widths seem to agree quite well and

any discrepancies in frequencies of maximum power capture can be attributed to

the experimental difficulty of maintaining the internal static pressure head (hence

the free bulge speed) constant through the whole experimental run. Mismatches

between the impedance of the PTO and that of duct do not seem to be of strong

influence since for both internal static pressure heads, the general trend of the

capture width is quite similar to that of the case of matching impedance for the

same configuration.

Figure 7.19: Capture width (in duct widths) for a static pressure head of 3cm;

lines: computed from theory; symbols: measured for the three runs of test 3.



lines: computed from theory; symbols: measured for the run of test 4.


lines: computed from theory; symbols: measured for the run of test 5.


7.4.3 Pressure variation in the duct and bulge components

The total internal pressure can be inferred as before from the displacement of the

membrane. This is shown for a wave frequency f=0.625Hz in Figure 7.22 with

the experimental conditions of test 3. As for the matching case configuration,

the pressure is seen to increase along the length of the second half of the duct.

General trends are similar between calculated and measured values, and the lower

amplitude in the measure internal pressure can be attributed to the influence of

the radiated waves from the membrane displacement affecting the incoming wave

pressure, the latter being assumed not to be disturbed in the theoretical model

(chapter 5).




displacement.

As before, a linear squares method was used to separate the three bulge com-

ponents and a similar mapping of the internal pressure shown in Figure 7.22 was

performed with the values obtained. This is shown in Figure 7.23 for the amplitude

and phase of the internal pressure and in Figure 7.24 for the reconstruction.






Wave frequency f=0.625Hz, with conditions as in test 3.





squares method.


7.5 Water-air interface displacement and pres-

sure in the PTO

Values of measured normalised pressure in the PTO chamber and of the vertical

displacement of the slug of water are seen in Figure 7.25 and 7.26 for internal static

pressure heads of 3cm and 2cm, respectively and for all the tests carried. They

are compared with those obtained from the one dimensional theory of section 5.5.2.

Measured and calculated values seems to agree, although the measured values

for test 2 are of smaller magnitudes than the calculated ones. Both trends (pres-

sure and amplitude) are similar to those of the capture width plots for the corre-

sponding configuration, in the sense that they have their maximum values in the

frequency range where the power absorbed by the PTO is maximum.

The pressure in the chamber is seen to be increasing as the impedances increases,

while the opposite is true for the amplitude of the water-air interface motion.

Interestingly, there seems to be a slight downwards shift of the theoretical peak

frequency of the maximum pressure as the impedance increases, in the same way

as for the capture width in Figure 7.21, for instance. On the other hand, the

experimental peak of pressure in the chamber (around f = 0.625Hz) seems to be

affected in amplitude, but not in frequency, by the change of impedance.

Measured (and calculated) values of the motion amplitude inside the rigid bent

tube indicates amplitudes of motions up to 3 times (Figure 7.26) and 2.5 times

(Figure 7.25) the incoming wave amplitude. The bandwidth of the frequency range

for which the normalised amplitude of motion is greater than one is seen to become

narrower as the impedance increases. For the duct with an internal static pressure

head of 2cm, this (experimental) range is 0.45Hz < f < 0.8Hz for Z=1.02 and

Z =1.08, 0.45Hz < f < 0.75Hz for Z=1.73 and 0.45Hz < f < 0.65Hz for Z=3.47.

For the duct with an internal pressure of 3cm, this range is 0.45Hz < f < 0.8Hz

for Z=1.03, 0.5Hz < f < 0.8Hz for Z=1.10 and 0.45Hz < f < 0.7Hz for Z=1.93.

On the other hand, as the pressure in the chamber increases for increasing impedances,

the frequency bandwidth for which the normalised pressure is greater than one gets

larger. For the duct with an internal static pressure head of 2cm, this (experimen-

tal) range is 0.45Hz < f < 0.775Hz for Z=1.02, 0.45Hz < f < 0.8Hz for Z =1.08,

0.45Hz < f < 0.825Hz for Z=1.73 and 0.45Hz < f < 1.075Hz for Z=3.47. For


the duct with an internal pressure of 3cm, this range is 0.45Hz < f < 0.8Hz for

Z=1.03, 0.5Hz < f < 0.8Hz for Z=1.10 (except the points for f = 0.525Hz and

f = 0.55Hz) and 0.45Hz < f < 0.85Hz for Z=1.93. However, as for the am-

plitude of motion, the change of bandwidth is significant only for high values of

dimensionless impedance and in this case, for Z =3.47.

Figure 7.25: Pressure (left column) in the PTO and amplitude (right column)

of the water-air interface displacement. Lines are the calculated values from the

theory in section 5.5.2 and points represent measured values for (a) test 1; (b)

test 2; (c) test 3.


Figure 7.26: Pressure (left column) in the PTO and amplitude (right column)

of the water-air interface displacement. Lines are the calculated values from the

theory in section 5.5.2 and points represent measured values for (a) test 7; (b)

test 6; (c) test 4 and (d) test 5.


7.6 Wave analysis

The two pairs of wave gauges placed in front of the duct made it possible to

measure the incident and reflected wave powers Pi and Pr, respectively. The two

pairs of gauges behind the duct enabled the measurement of the transmitted wave

power Pt. Unlike the experiment for the closed ends, the energy conservation

includes a term related to the power captured Pf by the PTO, yielding

Pt + Pr + Pf + Ploss = Pi (7.3)

Figures 7.27 and 7.28 show the measured values of the transmitted, reflected and

captured power, normalised by the incident wave power, for all the tests taking

only the first harmonic of the wave gauge signals, or the first three harmonics.

The first comment to be made is that the higher wave harmonics do not seem

to be carrying much power, as graphs from left and right column are similar for

both figures. This supports the present approach of neglecting higher harmonics

in the data analysis.

The sum of the normalised transmitted, reflected and absorbed power is seen

to be greater than one (by up to 25%) for both set of duct pressurisation and for

the range of wave frequency where the power output is maximum. For the same

range of wave frequencies, values of normalised transmitted power are seen to be

also close to unity even though neither the normalised reflected nor the capture

widths are zero. This is the result of the poor efficiency of the absorbing beach

for that range of frequency, usually f ≤ 0.85Hz as seen in Figure 4.6, where the

reflection coefficient from the foam is seen to be up to 20%. Taking this error into

account, it can be deducted that the magnitude of the power lost (Ploss = 1− Σ)

in the system is small.


Figure 7.27: Wave analysis for the duct in with an internal static pressure

head of 3cm and configurations as in (a) test 1; (b) test 2; (c) test 3 and (d)

test 5. Left column: power carried by the first harmonic; right column: power

carried in the first three harmonics of the signals from the wave gauges.


Figure 7.28: Wave analysis for the duct in with an internal static pressure

head of 2cm and configurations as in (a) test 7; (b) test 6; (c) test 4 and (d)

test 5. Left column: power carried by the first harmonic; right column: power

carried in the first three harmonics of the signals from the wave gauges.


7.7 Conclusions

The information presented in this chapter relates to the the experiment carried

on the duct equipped with a power take-off system. This PTO was investigated

and its characteristics were checked for the experimental conditions tested. It was

observed that increasing the impedance of this PTO results in a phase lag between

pressure in the chamber and the flow rate due to the water-air interface motion.

This yields a difference between this flow rate and the one in the copper pipes that

is of interest for measuring the power absorbed by the PTO. This phase difference

was explained with fair agreement with the simple mechanical model described in

section 4.5.3.

Power data were presented for various configurations, in terms of internal static

pressure heads and PTO impedances. Capture widths up to 2.2 times the width

of the duct were obtained for impedances of the PTO close to that of the duct.

Maximum power output occurred for any physical phenomenon maximising the

velocity at the stern (and hence the flow rate) such as k1 = k2 or k2 = nc/2L. For

the first case, the pressure was seen to increase along the length of the duct, while

in the second case corresponded as in chapter 6 to a natural mode of the mem-

brane. Again, this pressure mapping was enabled by the use of the laser sensors

to measure membrane displacement beneath waves.

The bulge pressure was characterised as the sum of three components using a least

squares method as introduced in chapter 6. For the maximum power absorbed by

the PTO corresponding to the case k1 = k2, the two components travelling in

the same direction as the wave were of higher amplitudes than the reflected wave

amplitude. However, similar results to that of the closed ends duct were obtained

for the frequency at which k2 = c/L, namely a partial standing wave in the duct

for that frequency.

The amplitudes of these components, together with the capture widths, pressure

in the duct, and amplitude of motion of the slug of water were compared with

the calculated corresponding values from the one dimensional theory such as in

section 5.5. The agreement was satisfactory and discrepancies could be attributed

to the incoming wave being affected by the radiated waves from the membrane, a

phenomenon otherwise not included in the model.


In comparison to the case of the raised duct in the narrow flume, the power lost

in the system was checked to be small. It is more likely that in the latter case,

the power lost in the system was due to the experimental set-up, e.g. the gaps

and sharp corners of the artificial seabed. The pressure inside the duct was of the

same order of magnitude as with closed ends, and the length of the duct (hence the

rubber sheet) was 60% less, making as before the losses in the rubber negligible.

Chapter 8

Two dimensional analytical model

of a duct covered with a tensioned

membrane beneath waves

8.1 Introduction

The results presented for the duct with closed ends and with the two types of

membrane emphasised the limits of the one dimensional theory from chapter 5.

The main hypothesis that waves were undisturbed during their travel over the

membrane was experimentally shown to be invalid for the thin membrane. The

purpose of this chapter is to present a two dimensional analytical model developed

for the interactions of water waves with a submerged rectangular duct, enclosing

water beneath its top rubber face, and placed beneath waves propagation. Results

are compared with those of the experiments in the wide flume.

The idea is to expand the work initiated by Cho and Kim (1998) on a stretched hor-

izontal submerged membrane, mentioned in section 3.3.2. The diffracted and radi-

ated potentials in each region of the fluid are expressed using the same eigenfunc-

tion expansion method and the series coefficients obtained with the solid boundary

conditions on the duct wall or by matching the different potentials at the interface.

The membrane equation is solved analytically to obtain the modal amplitude of

membrane displacement.

161

162Chapter 8 Two dimensional analytical model of a duct covered with a tensioned


Before any comparison with experimental values is made, the model is checked

with previous studies of wave interactions by rectangular submerged obstacles, as

this corresponds to the limiting case of infinite tension in the membrane. The main

problem encountered with the development of such a model for the configuration

of the duct with closed ends is the conservation of the volume of water in it. Three

methods are presented and discussed for that purpose.

This chapter is intended to present the model and the code developed while the

different equations and other mathematics can be found in Appendix B.

8.2 Problem formulation for the closed ends con-

figuration

The interactions of a submerged flexible membrane with monochromatic incident

waves are considered. A definition sketch is given in Figure 8.1. The coordinate

system is (x,y) with y pointing upwards and y = 0 being at the mean free surface

and x = 0 the middle point of the membrane. The length of the membrane is

L = 2s and it is fixed at both ends, x = ±s. The study is carried out in two

dimensions. The submergence depth of the membrane is d and the total water

depth is h. Tension is applied uniformly in the membrane in the x-direction.

x

y

O

s

(1) (2)

(3)

h

d

Figure 8.1: Definition sketch of a closed duct with elastic membrane on its

top.

Chapter 8 Two dimensional analytical model of a duct covered with a tensioned

membrane beneath waves 163

The geometry studied by Cho and Kim (1998) does not enclose any fluid, meaning

that there is no wall between regions (1) and (3). In this study, the domain was

closed on this boundary x = ±s for −h ≤ y ≤ −d, leading to the shape of a

submerged step with an elastic top. The aim is to use the method developed by

Cho and Kim (1998) but for this new geometry.

The domain of study is separated into three regions as detailed in Figure 8.1.

Wave amplitude and membrane motions are assumed to be small in order to use

linear theory. The problem can be decomposed into a diffraction problem (in this

case region (3) does not exist and the problem is that of diffraction by a rectan-

gular submerged obstacle) and a radiation problem. The total velocity potential

φ at point (x,y) can then be written as

φ(x, y) = φD(x, y) + φR(x, y), (8.1)

with

φD(x, y) = φi(x, y) + φS(x, y), (8.2)

where φD, φR, φi and φS represents the diffraction, radiation, incident and scat-

tering potentials respectively.

The incident wave used can be expressed as

φi(x, y) = −igA

ω

cosh k1(y + h)

cosh k1he−ik1x, (8.3)

where g is the gravitational acceleration, A the wave amplitude (taken as unity)

and i the imaginary unit. Assuming harmonic motion of frequency ω, the velocity

potential φ(x, y, t) can be expressed as

φ(x, y, t) = <(φ(x, y)e−iωt), (8.4)

where φ is one of the potentials mentioned above.



8.2.1 Diffraction problem

8.2.1.1 Definitions of the potentials

For the diffraction problem, the region (3) does not exist and the duct can be seen

as a concrete block. The boundary conditions for the diffraction velocity potential

are the Laplace equation, the combined linear free surface boundary condition,

the bottom boundary condition of zero velocity on y = −d and y = −h.

∂2φD

∂x2+

∂2φD

∂y2= 0, (8.5)

∂φD

∂y− ω2

gφD = 0 on y = 0, (8.6)

∂φD

∂y= 0 on y = −h, x ≤ s, (8.7)

∂φD

∂x= 0 on x = −s, − h ≤ y ≤ −d, (8.8)

limx→±∞

(∂φD

∂x± ik1φD) = 0, (8.9)

∂φD

∂y= 0 on y = −d, − s ≤ x ≤ s. (8.10)

Given the geometry of the problem, the potentials in each region could be expressed

using symmetric functions. However, symmetric potentials alone can not represent

progressive waves, hence the necessity to also include asymmetric potentials. The

diffraction potential is then split into symmetric and asymmetric parts as

φD(x, y) = φSD(x, y) + φA

D(x, y). (8.11)

The symmetric diffraction potentials are given by Cho and Kim (1998) for their

geometry (region (3) not closed) as

φS(1)D = −ig

ω12e−α10xf10(y) +

∞∑

n=0

aSneα1n(x+s)f1n(y), (8.12)

φS(2)D = −ig

ω

∞∑

n=0

bSn cosh(α2nx)f2n(y), (8.13)

with the eigenfunctions f1n and f2n detailed in Appendix B together with the α1n

series. The first part of φS(1)D represents half of the incident wave (the other hlaf

being carried by the assymetric potential for the same region) and the second part,

evanescent waves that vanish at x = −∞. The symmetric potential in region (2)



is expressed so that is satisfy the boudary condition at y = −d.

Matching φS(1)D and φ

S(1)D on x = −s over [0,−d] yields expressions for the bSm

as functions of the aSm. Matching the velocities at the same boundaries and set-

ting zero velocity on [−d,-h] leads to a system of equations from which the aSm can

be obtained, hence the bSm. Those governing equations and their solutions are set

out in Appendix B.

Using the same procedure for the asymmetric potentials

φA(1)D = −ig

ω12e−α10xf10(y) +

∞∑

n=0

aAn eα1n(x+s)f1n(y), (8.14)

φA(2)D = −ig

ω

∞∑

n=0

bAn sinh(α2nx)f2n(y), (8.15)

the total scattered potentials in each regions can be obtained.

8.2.2 Radiation problem

8.2.2.1 Definition of the potentials

The radiation problem is approached using a series expansion for the membrane

displacement in the form of its natural modes:

ξ(x) =

∞∑

j=1

ςjfj(x), (8.16)

where ςj is the unknown complex modal amplitude for the jth mode and fj(x) is

the associated modal function given by

fj(x) =

fSj (x) = cos

λSj x

s, λS

j =2j − 1

2π j = 1, 2, 3...

fAj (x) = sin

λAj x

s, λA

j = jπ j = 1, 2, 3...,

(8.17)

where S stands for symmetric mode and A for asymmetric mode. The symmetry is

about x = 0 as can be seen in Figure 8.1. The radiation potential φj,R is governed

by the same boundaries conditions as the diffracted potential and by the dynamic



condition

∂φj,R

∂y= −iωfj(x) on y = −d, (8.18)

stating that the membrane modal normal velocity is the same as the fluid ve-

locity around it. As for the diffraction problem, φlR is split into symmetric and

asymmetric parts as

φj,R(x, y) = φSj,R(x, y) + φA

j,R(x, y), (8.19)

and the radiation problem is treated using separately these two potentials as in

the diffraction case.

The symmetric modal radiation potentials are expressed as

φS(1)j,R = −ig

ω

∞∑

n=0

aSjneα1n(x+s)f1n(y), (8.20)

φS(2)j,R = −ig

ω

∞∑

n=0

bSjn cosh(α2nx)f2n(y) +iω

gφS(2)j,R (x, y), (8.21)

φS(3)j,R = −ig

ω

∞∑

n=0

cSjn cosh(α3nx)f3n(y) +iω

gφS(3)j,R (x, y), (8.22)

with the ˜φj,R for each region is as defined in Appendix B and used to satisfy the

dynamic condition (8.18).

In a similar way as for the scattered potentials (matching of velocities and po-

tential at the x = −s interface) the aSjm and bSjm series can be obtained. The cSlmseries are obtained from the zero velocity condition

∂φS(3)j,R

∂x(−s, y) = 0 on x = −s, − h ≤ y ≤ −d. (8.23)

In a similar way with the asymmetric modal radiated potentials

φA(1)j,R = −ig

ω

∞∑

n=0

aAjneα1n(x+s)f1n(y), (8.24)

φA(2)j,R = −ig

ω

∞∑

n=0

bAjn sinh(α2nx)f2n(y) +iω

gφA(2)j,R (x, y), (8.25)

φA(3)j,R = −ig

ω

∞∑

n=0

cAjn sinh(α3nx)f3n(y) +iω

gφA(3)j,R (x, y), (8.26)



the aAjm and bAjm series can be obtained. The cAlm series are obtained from the zero

velocity condition as in the symmetric case.

8.2.2.2 Membrane deformation

The equation for the elastic membrane, in absence of physical damping can be

written following Cho and Kim (1998)

Tx

d2ξ

dx2(x) +mω2ξ(x) = −iρω[φ(3)(x,−d)− φ(2)(x,−d)], (8.27)

where Tx is the membrane tension, ρ the fluid density and m the mass of the

membrane per unit length.

The kinematic condition (8.27) can be written in a discrete form as detailed in

Appendix B as

∞∑

j=1

[KSij − ω2(MS

ij + aSij)− iωbSij ]︸︷︷︸

Λij

ςSj = F Si i = 1, 2, 3.... (8.28)

Truncating (8.28) to the desired numbers of modes M , the symmetric modal am-

plitudes ςSj can be obtained. The same type of system is used for the asymmetric

modes.

The membrane motion can be expressed as

ξ(x) =∞∑

j=1

(ςSj fSj (x) + ςAj f

Aj (x)), (8.29)

and the potential in each region (k) using

φ(k)(x, y) = φA(k)D + φ

S(k)D +

M∑

j=1

(ςSj φS(k)j,R (x) + ςAj φ

A(k)j,R (x)) : k = 1, 2, 3. (8.30)



8.2.3 Reflection and transmission coefficients

Combining the symmetric and asymmetric parts, the total potential in region (1)

can be expressed as:

φ(1)(x, y) = −ig

ωe−α10xf10(y)

︸︷︷︸

incident wave

−ig

ω

N∑

n=0

(aAn + aSn)eα1n(x+s)f1n(y)

︸︷︷︸

scattered wave

−ig

ω

M∑

j=1

N∑

n=0

(ςSj aSjn + ςAj a

Ajn)e

α1n(x+s)f1n(y)

︸︷︷︸

radiated wave

, (8.31)

where the sum over the eigenvalues has been truncated to N . The reflection

coefficient is therefore

Kr =∣∣∣

ηradiated+scatteredηincident

∣∣∣ as x → −∞ (8.32)

where η is the surface elevation of the wave referred to.

Using the dynamic free-surface condition and taking x = −2s,

ηincident =iω

gφi(x, 0)

= e2α10s. (8.33)

Noticing that for n ≥ 1, the α1n are real, the terms for n ≥ 1 in the scattered and

radiated waves vanish for x → −∞, yielding

ηradiated+scattered = ((aS0 + aA0 ) +L∑

j=1

(ςSj aSj0 + ςAj a

Aj0)) (8.34)

and

Kr = |[(aS0 + aA0 ) +

M∑

j=1

(ςSj aSj0 + ςAj a

Aj0)]e

2α10s|. (8.35)

The transmission coefficient is given by Cho and Kim (1998) as

Kt = |[(aS0 − aA0 ) +M∑

j=1

(ςSj aSj0 − ςAj a

Aj0)]e

2α10s|. (8.36)



8.2.4 Conservation of volume in the duct

The volume of water enclosed in the duct should be conserved. Using the definition

of the membrane displacement, this is expressed as

∫ s

−s

ξ(x)dx = 0 (8.37)

⇔∫ s

−s

ςAj fAj (x)dx+

∫ s

−s

ςSj fSj (x)dx = 0 ∀j. (8.38)

The first term of the l.h.s of (8.38) is always true due to the definition of the fAj

modal functions in (8.17). This is not the case for the symmetric modes and for

that purpose, a method should be developed so that the symmetric modes satisfy

both the kinematic condition (8.28) and the conservation of volume

W =

M∑

j=1

∫ s

−s

ςSj cosλSj x

adx (8.39)

=

M∑

j=1

−4s cos(jπ)

(2j − 1)πςSj (8.40)

=4s

π(ςS1 − ςS2

3+

ςS35

+ ...) (8.41)

= 0. (8.42)

This change of volume was the challenging part of this model and slowed down its

development. However, three solutions have been attempted and coded and there

follows a description of the methods used to solve this problem of conserving the

inner fluid volume constant.

An ideal solution would be to have a set of trapped modes whose amplitudes

could be adjusted to cancel out the change of volume from the symmetric modes.

Such modes would radiate energy only into the fluid near the membrane, hence

would not affect the energy conservation of the system. However for such mode(s)

to exist at a specific frequency, the determinant of the system (8.28) should be

null. This was checked and found not to be the case, necessitating a search for

other ways to deal with this problem.



8.2.4.1 Uniform potential below the membrane

The first idea developed was to solve the system for the symmetric radiated modes

and then look for a set of modes that would cancel this out. In other words, find

a potential that satisfies all the physical boundary conditions of the problem but

whose effect on the membrane would be a displacement triggering a change of

volume opposite to that the one triggered by the symmetric modes.

A change of volume in the duct can be characterised by a uniform potential acting

on the membrane from below. The idea was to solve the kinematic equation (8.27)

with a uniform pressure of unit amplitude beneath the membrane. The change of

volume due to these modes is calculated and the process repeated with the uniform

pressure scaled by a factor so that the new potential obtained implies a change of

volume equal and opposite to that in the original calculation.

This was the first method developed and coded. It will be denoted hereinafter

as the UniP method The two others deal directly with finding symmetric modes

that simultaneously solve (8.28) and that do not imply a change of volume.

8.2.4.2 Weighted least-square method

The idea here is to try solving simultaneously (8.28) and (8.41). This brings an

over-determined system of M+1 equations for M unknowns. Writing this system

explicitly this gives for M = 3 (this will be used for explanation)

Λ11 Λ12 Λ13

Λ21 Λ22 Λ23

Λ31 Λ23 Λ33

4sπ

−4s3π

4s5π

︸︷︷︸

Z

ς1

ς2

ς3

︸︷︷︸

ς

=

F1

F2

F3

︸︷︷︸

F

. (8.43)

As previously stated in 6.4.4, such a system does not accept an exact solution. A

best-fit solution is obtained by using a least-square method which looks for the

solution that minimises the sum of the errors. This solution is found to be of the

form

ς = (ZT .Z)−1.ZT .F (8.44)



The above stands for the case that the errors have the same variance, meaning

that the same importance is given to any equations to be validated. The weighted

least-squares is the extension of the least-squares and a diagonal matrix of weight

W is included in the calculation of ς

ς = (ZT .W.Z)−1.ZT .W.F (8.45)

By tuning the coefficients wii, it is possible to force one condition relative to the

others.

8.2.4.3 Matrix approximation

The last idea developed and coded is purely a mathematical approximation. Con-

sider again the matrix of the system Z and the system obtained (8.43). The reasons

this system is not directly solvable with a traditional Gauss approach is that di-

mensions Z is not square and its line dimensions do not match the dimension of

ς. One way to tackle this is to solve the following system

Λ11 Λ12 Λ13 ∆

Λ21 Λ22 Λ23 ∆

Λ31 Λ23 Λ33 ∆4sπ

−4s3π

4s5π

0

ς1

ς2

ς3

ς4

=

F1

F2

F3

0

. (8.46)

with ∆ as small as the computational resource allows it in order to avoid overflow.

This way approaches the case of (8.43) but with a system that is solvable. This

will be denoted the MatApp method.

8.2.5 Code Layout

The previous model was coded on Iridis, the supercomputer at the University of

Southampton. The code, made of several routines detailed in Figure (8.2), was

written in Fortran 90. Inputs were the physical parameters of the problem: water

depth, submergence of the membrane, length of the duct and tension in the mem-

brane. The number of eigenvalues N for the expansion series and the number of

modes M for the membrane deformation were also defined at this stage.

The matching conditions at x = −s used to determine the aS,Am , bS,Am , aS,Alm , cS,Alm , cS,Alm ,

were numerically checked, together with the zero velocity on the bottom and on the



Defintion of the parameters , , , , ,s h d M N Tx

Determination of the wavenumbersseries , and and eigenfunctions

series

k k k

f , f and f .1n 2n 3n

1n 2n 3n

Calculation of the and coefficientsa bm m

Evaluation of the diffracted potentials

for regions (1) and (2)

fD

Evaluation of the radiated potentials

for regions (1), (2) and (3) and the total

potential

f

f

R

Calculation of the and coefficientsa , blm lm clm

Evaluation of the modal radiated potentials

for regions (1) and (2)flR

Calculation of the asymmetric complexmodal amplitudes from (8.22)

Calculation of thesymmetric complexmodal amplitudesfrom (8.38)


Corrective potentialto cancel out thechange of volume


Figure 8.2: Code developed for the analytical model.



outer and inner side walls of the duct. The other boundary conditions (kinematic

and dynamic free surface boundary conditions) were also numerically checked. A

example is given with the kinematic condition (8.27) later.

Figure 8.3: Reflection coefficient for h/d = 2 and s/d = 2. — from Mei and

Black (1969), • • • diffraction part of the present method.

Figure 8.4: As for Figure 8.3, but s/d = 4.




Figure 8.6: As for Figure 8.3, but h/d = 2.78 and s/d = 4.43.



8.2.6 Case studies

8.2.6.1 Diffraction problem

The diffraction part of the model can be tested with previous studies made on the

diffraction of water waves by submerged obstacles. One key study is that of Mei

and Black (1969) in which scattering of normal incident waves by either a surface

or submerged rectangular obstacle is investigated. Potentials in different regions

are expressed by means of eigenfunction expansions in which the coefficients are

found using the matching and boundary conditions. Their numerical results for

the reflection coefficients are compared with the present model (N = 10) in Fig-

ures 8.3-8.6 and are in fair agreement with the present model.

The main comment from these results is the oscillatory nature of the reflection

coefficients and the effects of the dimensions of the problem on the period and

amplitude of these oscillations. Mallayachari and Sundar (1996) used a boundary

element method for the same kind of problem but with submerged obstacles of

various shapes (rectangular, trapezoidal and half-cylindrical). Their results agree

with those from Mei and Black (1969) for the rectangular obstacle.

8.2.6.2 Complete configuration and comparison of the proposed

methods

One intuitive test to carry out was the one with infinite tension in the membrane.

Figures 8.7-8.10 shows the computed reflection coefficient for that situation with

the complete code (three with correction and no correction) and compares it with

the computed values obtained for the diffraction case alone. The four configura-

tions tested are the same as that of the diffraction situation.

Figures 8.3-8.7, 8.4-8.8, 8.5-8.9, and 8.6-8.10 are the same, the membrane with

infinite tension in the model behaving like a rigid plate. So far, there seems not to

be much difference between the three correction methods and the results without

any of them used.



Figure 8.7: Reflection coefficient for h/d = 2 and s/d = 2. — diffraction only,

Tx → ∞ for the following methods of conservation of volume: uniform po-

tential; + + + least-squares; N N N matrix approximation; • • • no correction.





Figure 8.10: As for Figure 8.7, but h/d = 2.78 and s/d = 4.43.



The boundary conditions at the free surface and on the duct walls were checked nu-

merically, together with the continuity of potentials and velocities at the interface

between regions (2) and (3). For all the four methods (three for the conservation

of volume concern and the one without), this was seen to be valid.

An example of such numerical check is given in Figures 8.11 to 8.18 for the config-

uration s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and a wave frequency

f = 1Hz. The number of eigenvalues was chosen as N = 10 and the number of

modes M = 10. The code used for presentation of these figures is the one with no

correction of internal volume of fluid.

Figure 8.11: Symmetric (left column) and asymmetric (right column) of the

diffracted potentials (upper plots) and velocities (lower plots) at the interface

x = −s; s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave

frequency f = 1Hz.



Figure 8.12: Symmetric (left column) and asymmetric (right column) of the

radiated potentials (upper plots) and velocities (lower plots) at the interface

x = −s; s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave

frequency f = 1Hz.

Numerical verification of the matching of the potentials and velocities at the in-

terface x = −s and 0 ≥ y ≥ −d are shown in Figures 8.11 and 8.12 with

φS,AD,x =

∂φS,AD

∂x, (8.47)

φS,AR,x =

∂φS,AR

∂x, (8.48)

for the diffracted and radiated potentials, respectively. The potentials (or veloci-

ties) are calculated at x = −s using the expression for region (2) (points) and for

region (1) (line).



Diffracted and radiated potentials are seen to be continuous at the interface be-

tween the two regions. The diffracted and radiated velocities are null on the

outside wall of the duct (x = −s and −h ≤ y ≤ −d) and continuous at the inter-

face, except over a small distance above it. This is typical of a singularity due to

the presence of the sharp corner, where the velocity at this point needs to use an

infinite number of eigenvalues to minimise the error.

Figure 8.13: Combined free surface dynamic condition (8.6) for the symmetric

(left plot) and asymmetric (right plot) diffracted potentials in region (1); s =

2m, h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency f = 1Hz.


(left plot) and asymmetric (right plot) radiated potentials in region (1); s = 2m,

h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency f = 1Hz.



Figures 8.13 and 8.14 show the the dynamic condition (8.6) at the free surface in

region (1) for the diffracted and radiated potentials, respectively, where

φS,AD,y =

∂φS,AD

∂y, (8.49)

φS,AR,y =

∂φS,AR

∂y, (8.50)

and Figures 8.15 and 8.16 show the same verification for region (2).


(left plot) and asymmetric (right plot) diffracted potentials in region (2); s =



(left plot) and asymmetric (right plot) radiated potentials in region (2); s = 2m,

h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency f = 1Hz.



The boundary condition at the free surface (8.6) is validated for both potentials

and both regions of interest. The symmetric and asymmetric pattern of the po-

tentials are visible for the plots concerning the region above the membrane, e.g.

region (2).

Figures 8.17 and 8.18 show the numerical verification on the dynamic condition

(8.18) for regions (2) and (3), respectively.

Figure 8.17: Dynamic condition (8.18) for the symmetric (left plot) and asym-

metric (right plot) for the first modal (l = 1) radiated potential in region (2);

s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency

f = 1Hz.

Figure 8.18: Dynamic condition (8.18) for the symmetric (left plot) and asym-

metric (right plot) for the first modal (l = 1) radiated potential in region (3);

s = 2m, h/d = 2 with h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency

f = 1Hz.



All those previous numerical verifications (and figures) were carried for all the

methods described above. At this stage, the UniP method was seen not to satisfy

the continuity of potentials and velocities at the interface x = −s. This is likely

to be due to the inclusion of the uniform potential in the three regions. The

comparison of methods continues by taking into account the two main remaining

conditions, e.g. the energy conservation and the kinematic boundary condition on

the membrane (8.27), together with the volume conservation with the aim to try

a best fit (or compromise) to the problem.

Figure 8.19: Kinematic condition (8.27) for the symmetric (left plot) and

asymmetric (right plot) with no volume correction; s = 2m, h/d = 2 with

h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency f = 1Hz.


asymmetric (right plot) with the WLS volume correction; s = 2m, h/d = 2

with h = 0.4m and Tx/ρgh2 = 0.15 and wave frequency f = 1Hz.



Figures 8.19, 8.20 and 8.21 show the numerical verification of the kinematic equa-

tion (8.27) for the code with no correction of volume, with the WLS method and

with the MatApp method, respectively.


asymmetric (right plot) with the MatApp volume correction (∆ = 10−15); s =


The asymmetric verification is unchanged as the different methods for the volume

conservation problem deals only with the symmetric modes. For both symmetric

and asymmetric parts, discrepancies occur around x = ±s and this is consistent

with the numerical verification of the continuity of potential and velocities where

it was seen that the potentials at these locations needed an infinite number of

eigenvalues to counter the singularity due to the sharp corner. For all three cases,

the membrane equation is verified along the length of the membrane, although

discrepancies centred around x = 0 and over a length of x = ±s/20 are visible for

the MatApp method.



Figure 8.22: Comparison of the change of volume induced by the symmetric

and efficiency of the proposed methods; (a) no correction of volume, (b) WLS

method; (c) Unip method; (d) MatApp (∆ = 10−15); s = 2m, h/d = 2 with

h = 0.4m and Tx/ρgh2 = 0.15.

Figure 8.22 shows the change of volume induced by the symmetric radiated modes

of the membrane vibration, as a percentage of the undisturbed volume (in two

dimensions). The MatApp method seems to be the one treating the volume con-

servation issue with best efficiency, for which almost no change of volume due to

the symmetric modes can be obtained. As can be seen in Figure 8.23, the energy

relation

K2r +K2

t = 1 (8.51)

is numerically verified for the range of studied wave frequencies with error of less

than 1% for methods (a), (b) and (c) with the notations of the same figure. The

Unip method, on the contrary, does not satisfy it, as the inclusion of a (uniform)

potential corresponds to the inclusion of more energy in the system. The MatApp

correction method was then the one chosen to compare the results of the analytical

model with experimental data.



Figure 8.23: Comparison of the error on the energy conservation in the system

for the proposed methods; (a) no correction of volume, (b) WLS method; (c)

Unip method; (d) MatApp (∆ = 10−15); s = 2m, h/d = 2 with h = 0.4m and

Tx/ρgh2 = 0.15.

8.3 Experimental verification

Outputs from this model were compared with results of the experiment carried

out in the wide basin for the configuration (test 8) for which all pipes of the PTO

were closed, tending to the limit case of a submerged closed duct as for the present

modelled geometry. The internal static pressure head was p0 = 0.02m, determining

the coefficients cSj0 in (8.22). The membrane submergence d was approximated as

the equivalent membrane submergence depth for that internal static pressure head,

so for that case, d = 0.363m. The tension Tx used to run the code was the one

needed to counter the Poisson’s effect from the elongation in the y-direction due

to the internal pressure applied. Values of N=10 and M=10 were chosen. More

runs with higher values for both of these parameters were seen not to change the

results presented below.



Figure 8.24 shows the predicted reflection coefficient and the measured one with

conditions such as test 8. The orders of magnitude are similar, but there is no

clear evidence of agreement between calculated and measured values.

Figure 8.24: Ratio of the reflected power, Pr, to incident wave power Pi for

the duct with an internal static pressure head of 2cm. Line: present model;

symbols: first harmonic analysis of wave gauge signals.

−1−0.5

00.5

10

0.51

0

20

40

x (m)f (Hz)

ξ/ A

Figure 8.25: Response of the membrane (ξ/A) as a function of wave frequency

f and horizontal coordinate x, parameters such as in test 8.



−1−0.5

00.5

10

0.51

0

2

4

6

x (m)f (Hz)

−1−0.5

00.5

10

0.51

0

5

10

15

20

x (m)f (Hz)

−1−0.5

00.5

10

0.51

0

2

4

x (m)f (Hz)

Figure 8.26: Modal response amplitude as function of wave frequency f and

horizontal coordinate x; upper plot: first symmetric mode, middle plot: second

symmetric mode; lower plot: third symmetric mode.



−1−0.5

00.5

10

0.51

0

10

20

30

40

x (m)f (Hz)

−1−0.5

00.5

10

0.51

0

5

10

15

x (m)f (Hz)

−1−0.5

00.5

10

0.51

0

1

2

3

x (m)f (Hz)

Figure 8.27: As for Figure 8.26 for asymmetric modes.



The other available data was the membrane displacement. It is shown in Figure

8.25, together with the modal responses of the first three symmetric (Figure 8.26)

and asymmetric (Figure 8.27) modes, for all frequencies. The free bulge speed

for that configuration was c = 2.10m/s, leading to a natural frequency of f2 =

1.05Hz as detailed earlier. This is verified by mapping the experimental membrane

displacement for that frequency, as shown in Figure 8.28. Three distinct bulge

patterns are visible as expected, and the difference in their shapes comes from the

fact that this configuration is not exactly the same as one with purely closed ends.

Again there is no agreement with the calculated values of membrane displacement.

Figure 8.28: Membrane displacement (normalised by the incident wave am-

plitude A) for configuration of test 8 and f =1.05Hz.

8.4 Conclusions

An analytical model based on an expansion series for the different potential con-

cerned for the geometry shown in Figure 8.1 has been developed, following the

work initiated by Cho and Kim (1998). One of the important issue to be ad-

dressed was that of the conservation of volume in this closed geometry. For that

purpose, three methods have been proposed to tackle the problem of change of

volume induced by the symmetric modes of the membrane vibration. With re-

gards to the analyses carried out, the MatApp method seemed to be the best fit

for this problem. All the boundary conditions were numerically checked, together

with the energy conservation and the change of volume. This is why this method

was the one chosen for comparison with experimental results. Other methods were

seen to be invalid for one or more of these numerical tests.



One disadvantage of this method is that increasing the number of modes M of the

membrane vibration increases the dimension of the matrix of the system (8.46).

This yields an increase of error when approximating the membrane equation (8.27)

by the linear system(8.46). This can be avoided by taking a smaller ∆ coefficient,

if computational resources allow it.

Results of reflection coefficient and membrane displacement obtained from this

model did not agree with the experimental results obtained with the configuration

of chapter 7 where all copper pipes were closed. This configuration was the closest

one to that of Figure 8.1.

The mapped displacement in Figure 8.25 suggests that the bulge wavelengths

obtained with this model are much shorter than those measured. One possible

explanation is that the membrane stress used to run this code was too low, even

though it has a physical meaning in a two dimensional analysis. Three dimensional

effects may have to be considered, and in particular the fact that the restoring force

due to the bending of the membrane in the width direction yields a much higher

tension in the rubber than that used so far.

−1−0.5

00.5

10

0.51

0

5

10

15

20

x (m)f (Hz)

ξ/ A

Figure 8.29: Response of the membrane (ξ/A) as a function of wave frequency

f and horizontal coordinate x, parameters such as in test 8, with the longitudinal

tension used 2.7 times greater than that due to Poisson’s ratio effect only.



−1−0.5

00.5

10

0.51

0

2

4

6

x (m)f (Hz)

ξ/ A

Figure 8.30: As for Figure 8.29 with a the tension multiplied by a factor of

11.

Figures 8.29 and 8.30 show the displacement obtained by multiplying the tension

used before by factors of 2.7 and 11 respectively. The membrane displacement

profile exhibits longer wavelengths, approaching the experimental case. However

there is so far no clear understanding of the longitudinal tension needed to charac-

terise the experimental case so far. One possibility could be to work out an width

averaged membrane model for the determination of a more appropriate tension to

run the code with, as part of future work.

Chapter 9

Conclusions and future work

9.1 Achievements and perspectives

This thesis reported experiments carried out on a rectangular duct covered with

a rubber membrane, set on the seabed and aligned with the wave propagation

direction. Two different versions were investigated: the first one with closed ends

and the second one with a power take-off attached at the duct stern. The main

goal of this work was to verify the one dimensional model used for the Anaconda

wave energy converter for a novel configuration, and explain any mismatch due to

its limitations.

As mentioned in section 3.3.5, the available experimental data regarding behaviour

of such a structure beneath waves was poor. The methodology developed within

this study has filled this gap by means of pressure transducers placed at the bow

and stern of the duct, together with laser sensors carefully set along the length of

the membrane. For the first time to the author’s knowledge, a complete experi-

mental characterisation of the membrane displacement and pressure variation was

obtained for such a structure.

The experiments carried out on the closed end version of 5m in a narrow flume

with the thick (1mm) membrane exhibited resonant behaviours for frequencies

fn = nc/2L, at which the bulge wavelength is an integer times 2L. The membrane

displacement was made of nodes and antinodes characteristic of such resonance.

This measured displacement yielded the measurement of the amplitude of the pres-

sure in the duct, following the load-deflection relationship. The magnitude of this

193

194 Chapter 9 Conclusions and future work

pressure at the stern for these resonant frequencies was seen to reach up to 2.8

times that of the incident wave at a depth equal to the membrane submergence.

By means of a least squares method, the internal pressure was decomposed into

three components and the pressure wave explained as a partial standing wave near

resonance. This was explained with a one dimensional model where the incoming

(water) wave was assumed to be undisturbed along its travel above the membrane.

On the contrary, this model failed to predict the membrane displacement and

pressure behaviour for the thinner (0.2mm) of the tested membranes. For that

membrane, values of free bulge speeds were much lower than those of the thicker

membrane, yielding a profile of membrane displacement like a ripple bed. This

measured displacement agreed with the predicted one over a length of the duct

equal to the bulge wavelength. After that, the incoming wave lost its energy by its

interactions with this non uniform bed and this was confirmed by measuring the

surface displacement at mid-length of the duct. The main conclusion from this

disagreement was that the one dimensional model used was not effective for devices

whose length is of a much higher order of magnitude than the bulge wavelength,

that is, the hypothesis of non disturbance of the incoming wave is no longer valid.

Moreover, the experimental observation of ripples, of much higher amplitude than

those propagating on the thick membrane, were found to play an important role

in the way the energy carried by the bulges is dissipated during their travel.

The same duct, reduced to a length of 2m, was equipped with a PTO and tested

in a wide basin with the thick membrane. Capture widths of up to 2.2 times the

duct width were obtained and explained with the one dimensional theory. Maxi-

mum power outputs were obtained for frequencies maximising the flow rate that

went through the pipes of the PTO. These frequencies corresponded to either the

case that the water wave speed and bulge speed matched, or the wave frequency

matched a natural frequency fn. For the first situation, the amplitude of the

internal pressure was seen to increase along the length of the duct, while as be-

fore, the second situation exhibited characteristic patterns of a resonant behaviour.

This showed that the theory previously tested on the Anaconda rubber tube works

for another configuration. Using a seabed configuration (hence in shallow waters)

broadens the possible usage of such a concept. Bottom-mounted, this device would

not interfere with commercial and leisure activities present nearshore. Because of

wave refraction, there is no concern in capturing power contained in waves coming

from many different directions. Practically, this rectangular structure could be

Chapter 9 Conclusions and future work 195

towed on site and then sunk by adding ballast. The rubber membrane could then

be installed.

To go one step forward in the general study of the interactions of water waves

with a submerged duct covered with a stretched membrane, a two dimensional

analytical model was developed. It is based on expansion series for the diffracted

and radiated potentials in each regions, the series coefficients being obtained by

matching the potentials and velocities at the boundaries. The problem of conserv-

ing the inner water volume was addressed and several solutions were proposed.

One of them was chosen as a best compromise and its outputs compared with the

experimental data obtained from the tests carried on the configuration presented

in chapter 7 with all pipes of the PTO closed. There was very poor agreement with

regards to membrane displacement and reflection coefficient. Calculated profile of

the membrane displacement was seen to be many more wavelengths greater than

the experimental case. The tension in the membrane used to run the code was

seen to be of importance in this number of wavelengths, suggesting that results

obtained when using a higher tension in the membrane could potentially match

the experimental ones.

9.2 Future work

Testing a new wave energy converter at a laboratory scale is only the first step

within the overall development, and this is what has been done in this study. The

tests carried on the duct equipped with the PTO showed that the bulge wave

concept can be used to harness wave power when used on a seabed configuration.

However, such potential needs to be further examined if a commercial full-scale

prototype is intended for design. Particular issues to be addressed are:

• Scale effects if longer devices are intended to be used, in particular the ratio

of the water wavelength to that of the bulge wave to avoid situations observed

in the thin membrane test. An useful experiment would be one with the thin

membrane clipped on top of a closed ends duct shorter than the one tested

in this study. The ratio of the free bulge speed in the thick membrane to its

length was around

c

2L≈ 0.3 (9.1)

196 Chapter 9 Conclusions and future work

so an experiment with the same ratio for the thin membrane would induce

a length of L = 0.6m. This would be a way to check the validity of the

argumentation developed in section 6.5 regarding the influence of the ripple-

like profile of the membrane displacement on the incident water wave.

• Design of a more appropriate (bottom-mounted and fully submerged) power

take-off. The PTO used in this study was to quantify the power that could

be absorbed by a linear dashpot but not to use it. A new design, aimed at

harnessing this power, needs to be bottom-mounted and fully submerged for

taking full advantages of a non-visible wave energy converter.

• Rubber life in seawater compared to freshwater and the role of this salty

environment on rubber properties. Will the rubber properties and behaviour

be the same in seawater? What is the influence of the micro-organisms

growing on the membrane? Will this affect the cost of the rubber, initially

thought to be cheap? These questions are among those that need to be

addressed by further research.

Another aspect that needs to be addressed by further studies is that of ripples

propagating on the tensioned membrane. For the thinner of the tested membranes,

faster travelling ripples were experimentally observed ahead of bulge waves. Such

ripples have been widely studied for the case of surface waves as for instance by

Munk (1955); Longuet-Higgins (1963, 1995); Jiang et al. (1999) and many more.

When gravity waves reach their maximum steepness, the surface tension is un-

evenly distributed and localised near the wave crests and this travelling stress is

the source of capillary waves triggered ahead of the crests. The quoted previ-

ous works were carried out with respect to the surface tension at the free surface

(Ts=0.0744N/m) yielding a characterisation of maximum steepness, ripple ampli-

tude and wavelength based on that value. A similar study for different surface

tensions, closer to the thinner membrane ought to be carried out to understand

specifically the influence of ripples on bulge propagation. The displacement of

the thinner membrane was of a higher order of magnitude than the thicker mem-

brane, leading to the generation of steeper bulges and capillary waves. Because

the displacement of the thicker membrane was lower, no capillary waves were ex-

perimentally observed.

Finally, attention has to be given on to theoretical model with more experiments

carried out with the exact same configuration. The main task to be undertaken is

that of finding the correct longitudinal tension to run the code with. The effects

Chapter 9 Conclusions and future work 197

of including the third dimension in such a model was mentioned in section 3.3.4.

The advantages of this inclusion might be similar for the present model. Working

out a width averaged membrane model to determine an equivalent tension to use

in the membrane can be the step before that.

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Appendix A

Power take-off test in the narrow

flume

A.1 Introduction

The first attempt at using the power take-off system described in section 4.5 was

for the 5m duct in the narrow flume. For this test, the slug of water was enclosed

in a circular rigid tube of cross-sectional area Sd = 0.021m2, yielding a ratio of

cross-sectional area µ = 2.75. Figure A.1 shows this and the power take-off instal-

lation. The complete set-up is similar to that of Figure 6.1, the only difference is

that the PTO was connected at the duct stern.

Power capture values were very low for this test. This is explained fairly well

with the 1D analytical model. This test with a wider range of impedances than

those used in the wider basin helped to provide a better understanding of the

PTO. This appendix shows plots of phase lag in the PTO and capture widths

obtained for the test conditions detailed in Table A.1. The general explanation

has been given in section 7.2.2.

A1

A2 Appendix A Power take-off test in the narrow flume

copperpipescircular

rigid tube

membrane

Figure A.1: Power take-off set-up in the narrow flume.

A.2 Experimental set-up and test conditions


Test no pressure head Z frequencies wave of

(m) (Hz) amplitudes tests

1 0.07m 12.33 0.7-1.7 5 1

2 0.07m 6.16 0.7-1.7 5 1

3 0.07m 4.11 0.6-1.7 5 1

4 0.07m 3.08 0.6-1.7 5 1

5 0.07m 1.54 0.6-1.7 5 1

6 0.07m 1.02 0.6-1.7 5 1

7 0.07m 0.77 0.7-1.7 5 1

8 0.07m ∞ 0.7-1.7 5 1

Table A.1: PTO experiment in narrow flume: test conditions

Appendix A Power take-off test in the narrow flume A3

As before, the signals obtained from the wave gauges (in the flume and in the

PTO), the pressure transducers (water and air) and the laser sensors were found to

be predominantly monochromatic. Hence the data analysis carried out hereinafter

neglects higher harmonics.

A.3 Linearity of the power take-off

Figure A.2: Air pressure in the chamber as a function of the water-air interface

elevation for several dimensionless impedances Z. The wave frequency is f =

0.95Hz and incident wave amplitude A =0.025m.

A4 Appendix A Power take-off test in the narrow flume

Figure A.3: Phase lag between PTO water level aPTO and PTO air pressure

pPTO as a function of dimensionless impedance Z for several wave frequencies;

• • • measured, — from (4.25).

Figure A.2 shows the air pressure in the chamber as a function of the water-air

interface displacement for various dimensionless impedances. As the impedance

increases, the air pressure and water displacement are no longer in quadrature.

For the case where all pipes are closed, the magnitude of both values seems higher

than for lower impedances.

The phase lag between water-air displacement and pressure is plotted for various

impedances and wave frequencies in Figure A.3 and compared with the theoretical

values calculated from 4.25) with again fair agreement.

Appendix A Power take-off test in the narrow flume A5

A.4 Power capture

Capture widths for the tested impedances are presented in Figure A.4 and com-

pared with the values calculated with (5.43). The magnitudes of the captured

power are very low and can be explained with the 1D model used in this study,

although the agreement between calculated and measured values is poorer in this

case than for the experiment in the wide flume.

Figure A.4: Capture widths for the duct equipped with the PTO and placed in

the narrow flume for various dimensionless impedances Z; Grey line: calculated

from (5.43), line and symbols: measured.

Appendix B

Analytical model of duct covered

with a tensioned membrane: the

maths

B.1 Introduction

This section is to provide the details of the analytical model from chapter 8.

B.2 Closed ends configuration

B.2.1 Diffraction problem

The symmetric diffraction potentials are given by Cho and Kim (1998) for their

geometry (region (3) not closed) as

φS(1)D = −ig

ω12e−α10xf10(y) +

∞∑

n=0

aSneα1n(x+s)f1n(y), (B.1)

φS(2)D = −ig

ω

∞∑

n=0

bSn cosh(α2nx)f2n(y), (B.2)

with

α1n =

−ik1 n = 0

k1n n ≥ 1(B.3)

B1

B2Appendix B Analytical model of duct covered with a tensioned membrane: the

maths

α2n =

−ik2 n = 0

k2n n ≥ 1(B.4)

α3n = k3n n ≥ 0 (B.5)

where the eigenfunctions f1n and f2n are defined as

f1n =cos k1n(y + h)

cos k1nhn ≥ 0, (B.6)

f2n =cos k2n(y + d)

cos k2ndn ≥ 0, (B.7)

and the eigenvalues k1n and k2n are solutions of

k1 tanh(k1h) =ω2

g,

k1n tanh(k1nh) =− ω2

gn ≥ 1,

(B.8)

k2 tanh(k2d) =ω2

g,

k2n tanh(k2nd) =− ω2

gn ≥ 1,

(B.9)

and k10 and k20 are set as

k10 = −ik1

k20 = −ik2

. (B.10)

The coefficients aSn and bSn can be calculated by matching the two solutions at

x = −s. The continuity of potential at x = −s leads to

1

2eα10sf10(y) +

∞∑

n=0

aSnf1n(y) =∞∑

n=0

bSn cosh(α2ns)f2n(y) − d ≤ y ≤ 0. (B.11)

Multiplying (B.11) by f2m(y) and integrating over [−d,0] allows determination of

bSn as:

bSm =

12eα10sCm0 +

∞∑

k=0

aSmCmk

cosh(α2ms)N(2)m

, (B.12)

where∫ 0

−d

f2n(y)f2m(y)dy =

N (2)m m = n

0 m 6= n(B.13)

Appendix B Analytical model of duct covered with a tensioned membrane: the

maths B3

N (2)m =

d

2 cos2(k2md)1 + sin(2k2md)

2k2md, (B.14)

Cmn =

∫ 0

−d

f1n(y)f2m(y)dy

= (2 cos(k1nh) cos(k2md))− 1

22k1n sin[k1n(h− d)]

k22m − k2

1n

+

sin(dk2m + hk1n)

k2m + k1n+

sin(dk2m − hk1n)

k2m − k1n. (B.15)

The continuity of velocity ∂φSD/∂x and the boundary conditions (8.8) at x = −s

leads to the equality:

−1

2α10e

α10sf10(y) +∞∑

n=0

α1naSnf1n(y) =

−∞∑

n=0

α2nbSn sinh(α2ns)f2n(y) − d ≤ y ≤ 0

(B.16)

0 − h ≤ y ≤ −d (B.17)

Defining

µnm =

∫ 0

−d

f1n(y)f1m(y)dy (B.18)

and∫ 0

−h

f1n(y)f1m(y)dy =

N (1)m m = n

0 m 6= n(B.19)

N (1)m =

h

2 cos2(k1mh)1 + sin(2k1mh)

2k1mh, (B.20)

the aSm coefficients can be written after multiplying (B.17) by f1m(y) and integrat-

ing it over [−h,−d] as:

aSm −∞∑

k=0

α1kµk0

α10N(1)0

aSk =− 1

2eα10(

µS00

N10

− 1) m = 0

aSm −∞∑

k=0

α1kµkm

α1mN(1)m

aSk =− 1

2

α10eα10

α1mN(1)m

µ0m m ≥ 1

(B.21)


maths

Next, multiplying (B.16) by f1m(y) and integrating it over [−d,0] using (B.12)

leads to

aSm +

∞∑

k=0

F S0k − α1kνk0

α10N(1)0

aSk =− 1

2eα10(

F S00

N10

− 1)− 1

2eα10a

ν00

N(1)0

m = 0

aSm +

∞∑

k=0

F Smk − α1kνkm

α1mN(1)m

aSk =− 1

2

eα10

α1mN(1)m

(F Sm0 + α10ν0m) m ≥ 1

(B.22)

where

νnm =

∫ −d

−h

f1n(y)f1m(y)dy (B.23)

and

F Smk =

∞∑

n=0

α2n tanh(α2ns)CnmCnk

N(2)n

. (B.24)

The coefficients aSm make the two systems consistent. Adding (B.21) and (B.22)

leads to:

(S1)

2aSm +

∞∑

k=0

F S0k − α1kδk0N

(1)k

α10N(1)0

aSk =− 1

2eα10(

F S00

α10N(1)0

− 1) m = 0

2aSm +

∞∑

k=0

F Smk − α1kδkmN

(1)k

α1mN(1)m

aSk =− 1

2eα10

F Sm0

α1mN(1)m

m ≥ 1

(B.25)

where δkm is the usual Kronecker symbol:

δkm =

1 k = m

0 k 6= m. (B.26)

By solving (S1) the aSm (m = 0,1,2...) can be evaluated and hence the bSm(m = 0,1,2...) determined using (B.12).

The asymmetric parts of the diffracted potentials used are the ones from Cho

and Kim (1998)

φA(1)D = −ig

ω12e−α10xf10(y) +

∞∑

n=0

aAn eα1n(x+s)f1n(y), (B.27)

φA(2)D = −ig

ω

∞∑

n=0

bAn sinh(α2nx)f2n(y). (B.28)


maths B5

In a similar way to what was done for the symmetric part, the aAm (m = 0,1,2...)

can be evaluated by solving the system

(S2)

2aAm +∞∑

k=0

FA0k − α1kδk0N

(1)k

α10N(1)0

aAk =− 1

2eα10(

FA00

α10N(1)0

− 1) m = 0

2aAm +

∞∑

k=0

FAmk − α1kδkmN

(1)k

α1mN(1)m

aAk =− 1

2eα10

FAm0

α1mN(1)m

m ≥ 1

(B.29)

where

FAmk =

∞∑

n=0

α2n coth(α2ns)CnmCnk

N(2)n

(B.30)

and the remaining bAm using

bAm = −12eα10aCm0 +

∞∑

k=0

aAmCmk

sinh(α2ma)N(2)m

. (B.31)

B.2.2 Radiation problem

The symmetric radiation modal potentials are expressed as

φS(1)lR = −ig

ω

∞∑

n=0

aSlneα1n(x+s)f1n(y), (B.32)

φS(2)lR = −ig

ω

∞∑

n=0

bSln cosh(α2nx)f2n(y) +iω

gφS(2)lR (x, y), (B.33)

φS(3)lR = −ig

ω

∞∑

n=0

cSln cosh(α3nx)f3n(y) +iω

gφS(3)lR (x, y), (B.34)

with

φS(2)lR (x, y) =

−iω cos(λSl x/s)[m1 cosh(m1y) + ν sinh(m1y)]

m1[−m1 sinh(m1d) + ν cosh(m1d)], (B.35)

φS(3)lR (x, y) =

−iω cos(λSl x/s) cosh[m1(y + h)]

m1 sinh[m1(h− d)], (B.36)

and

m1 =λSl

s, (B.37)

ν =ω2

g. (B.38)


maths

In a similar way to the diffraction potential, aSlm can be determined by solving the

system

(S3)

2aSlm +∞∑

k=0

F Smk − α1kδkmN

(1)k

α1mN(1)m

aSlk =XS

lm

α1mN(1)m

m ≥ 0, (B.39)

where

XSlm =

iω

g

∫ 0

−d

∂φS(2)lR

∂x(−s, y)f1m(y)dy. (B.40)

The bSlm are obtained using the continuity of potential on x = −s as done earlier

with

bSlm =

∞∑

k=0

aSlkCmk

cosh(α2ms)N(2)m

. (B.41)

Finally using the condition of vanishing normal velocity (8.8) for region (3)

∂φS(3)lR

∂x(−s, y) = 0 on x = −s, − h ≤ y ≤ −d (B.42)

leads to

∞∑

n=1

cSlnα3n sinh(α3ns)f3n(y) =iω

g

∂φS(3)lR

∂x(−s, y). (B.43)

Multiplying (B.43) by f3m(y) and integrating over [-h,-d] allows the calculations

of cSlm as

cSlm =iω

gα3mN(3)m sinh(α3ms)

∫ −d

−h

∂φS(3)lR

∂x(−s, y)f3m(y)dy, (B.44)

with

f3n =cos k3n(y + h)

cos k3n(h− d)n ≥ 0 (B.45)

and∫ −h

−d

f3n(y)f3m(y)dy =

N (3)m m = n

0 m 6= n, (B.46)

N (3)m =

(h− d) m = 0

1

2(h− d) m 6= 0

. (B.47)


maths B7

The asymmetric radiated modal potentials are given as

φA(1)lR = −ig

ω

∞∑

n=0

aAlneα1n(x+s)f1n(y), (B.48)

φA(2)lR = −ig

ω

∞∑

n=0

bAln sinh(α2nx)f2n(y) +iω

gφA(2)lR (x, y), (B.49)

φA(3)lR = −ig

ω

∞∑

n=0

cSln sinh(α3nx)f3n(y) +iω

gφA(3)lR (x, y), (B.50)

with

φA(2)lR (x, y) =

−iω sin(λAl x/s)[m2 cosh(m2y) + ν sinh(m2y)]

m2[−m2 sinh(m2d) + ν cosh(m2d)], (B.51)

φA(3)lR (x, y) =

−iω sin(λAl x/s) cosh[m2(y + h)]

m2 sinh[m2(h− d)](B.52)

and

m2 =λAl

s. (B.53)

In a similar way to solving the symmetric problem, the aAlm coefficients can be

calculated by solving

(S4)

2aAlm +∞∑

k=0

FAmk − α1kδkmN

(1)k

α1mN(1)m

aAlk =XA

lm

α1mN(1)m

m ≥ 0 (B.54)

where

XAlm =

iω

g

∫ 0

−d

∂φA(2)lR

∂x(−s, y)f1m(y)dy (B.55)

and hence the bAlm, using the continuity of potential at x = −s,

bAlm = −

∞∑

k=0

aAlmCmk

sinh(α2ms)N(2)m

. (B.56)

The zero velocity condition is again used to determine the cAlm leading to

cAlm =−iω

gα3mN(3)m cosh(α3ms)

∫ −d

−h

∂φA(3)lR

∂x(−s, y)f3m(y)dy. (B.57)


maths

B.2.3 Membrane deformation

The equation for the elastic membrane, in absence of physical damping can be

written as

Tx

d2ξ

dx2(x) +mω2ξ(x) = −iρω[φ(3)(x,−d)− φ(2)(x,−d)], (B.58)

where Tx is the membrane tension, ρ the fluid density and m the mass of the

membrane per unit length. φ(2)(x, y) and φ(3)(x, y) are the total potential in

regions (2) and (3) respectively and can be written as

φ(2)(x, y) = φ(2)D (x, y) +

∞∑

j=1

ςjφ(2)lR (x, y) (B.59)

φ(3)(x, y) =

∞∑

j=1

ςjφ(3)lR (x, y) (B.60)

and the membrane displacement ξ(x) is expressed as

ξ(x) =∞∑

j=1

ςjfj(x). (B.61)

Splitting potentials and displacement into symmetric and asymmetric modes,

(B.58) can be solved successively for both modes. Substituting

φS(2)(x, y) = φS(2)D (x, y) +

∞∑

j=1

ςSj φS(2)lR (x, y), (B.62)

φS(3)(x, y) =∞∑

j=1

ςSj φS(3)lR (x, y), (B.63)

and

ξS(x) =∞∑

j=1

ςSj fSj (x) (B.64)

into (B.58) leads to

∞∑

j=1

ςSj [−Td2fS

j

dx2−mω2fS

j (x)− pSjR(x)] = pSD(x). (B.65)


maths B9

where

pSjR(x) = iωρ[φS(3)jR (x,−d)− φ

S(2)jR (x,−d)], (B.66)

pSD(x) = iωρ[φS(3)D (x,−d)− φ

S(2)D (x,−d)]. (B.67)

Multiplying (B.65) by fSi and integrating over [-s,s] yields

∞∑

j=1

[KSij − ω2(MS

il + aSil)− iωbSij]ςSj = F S

i i = 1, 2, 3... (B.68)

with

KSij = −

∫ s

−s

Td2fjdx2

(x)fi(x)dx, MSij =

∫ s

−s

mfj(x)fi(x)dx (B.69)

aSij = <( 1

ω2

∫ s

−s

pSjR(x)fSi (x)dx), bSij = =( 1

ω

∫ s

−s

pSjR(x)fi(x)dx) (B.70)

F Si =

∫ s

−s

pSD(x)fi(x)dx. (B.71)

Appendix C

Pictures of the experimental

set-ups

Figure C.1: The duct, covered with the rubber 1mm membrane, in the wave

basin.

C1

C2 Appendix C Pictures of the experimental set-ups

Figure C.2: Bow of the duct, with the connection for the water hose, in the

wave basin

Figure C.3: Stern of the duct, with the connection to the bent tube, in the

wave basin. Heavy masses are visible and were used to hold the bottom plate

still.

Appendix C Pictures of the experimental set-ups C3

Figure C.4: Power take-off system.

Figure C.5: Ramp and bow of the duct, in the narrow flume.

C4 Appendix C Pictures of the experimental set-ups

Figure C.6: The duct on the artificial seabed, covered with the 1mm mem-

brane, in the narrow flume.

Appendix D

Determination of rubber

properties

D.1 Introduction

A stripe of rubber of length L0 = 0.4m, width W0 = 0.2m and thickness at rest v0

(1mm or 0.2mm) was clamped on one its top side to a fixed item while the other

side was fashioned to receive successive loads, as shown in Figure D.1.

F

L W h0 0 0

Figure D.1: Sketch of the apparatus for testing rubber properties.

D1

D2 Appendix D Determination of rubber properties

(a) (b) (c)

Figure D.2: Three different loads applied to the strip of rubber; (a) F = 0N,

(b) F = 78.86N, (c) F = 98.48N.

This apparatus was aimed for determining Young’s modulus of the rubber and its

damping properties. Notations used in this appendix are specific to it.

D.2 Young’s modulus

Measurement of the Young’s modulus was carried by applying successive loads on

the stripe of rubber. Time interval between each new load was two minutes so

the load had to time to be completely effective. At each step, the new length L

was measured manually and electronically by using a laser sensor recording the

displacement of the bottom frame. The new thickness v (averaged over three posi-

tions lengthwise) and the new width W (averaged over three positions lengthwise)

were also measured manually. A view of the apparatus with the rubber strip sub-

ject to three different loads is shown in Figure D.2.

The stress in the rubber F/A where A is the cross-section of the strip (A = Wv)

was then measured together with the elongation and the stress-strain relation

plotted. The Young’s modulus was calculated as the slope of the best fit passing

trough the origin as shown in Figure D.3.

Appendix D Determination of rubber properties D3

Figure D.3: Stress-strain relationship for successive loadings with two minutes

intervals.

Figure D.4: Stress-strain relationship for successive loadings with thirty sec-

onds intervals.

D4 Appendix D Determination of rubber properties

The same process was repeated but with time intervals between each loads being

thirty seconds. Result is shown in Figure D.4 where a slightly higher value of the

Young’s modulus was obtained as for the same stress, the rubber did not have

time to stretch completely, yielding lower values of elongation. However, the two

obtained values for the Young’s modulus from this test were close enough not to

make any difference in the chosen value.

D.3 Incompressibility of rubber

One assumption used while developing the load-deflection model in (5.2) was that

rubber is incompressible. This was checked in the same time than measuring

the Young’s modulus. Figure D.5 shows the measured volume V of the strip

(V = WvL) for the different stresses applied. This total volume is seen to be

constant, hence justifying the incompressibility of the rubber used in this work.

Figure D.5: Volume of the rubber strip for the different stresses applied.

Appendix D Determination of rubber properties D5

D.4 Dynamic damping

The coefficient β used is related to the energy dissipated in the mechanism of

deformation of the rubber. It can be calculated, with the notations from Figure

D.1 as

β =cL0

W0v0E(D.1)

where c is a decay coefficient.

Consider a oscillating mass m attached to the rubber strip. The amplitude of

motion d is recorded as shown in Figure D.6 from which several following peaks

can be determined as d1 > d2 > ... > d9. This decay in amplitude can be charac-

terized by a coefficient δ such as

d1+n

d1= exp(−nδ) n = 1, 2...8 (D.2)

This coefficient is then calculated as the slope of the plot (n,ln(d1+n/d1)). The

coefficient c can then be calculated as

c = 2mfδ (D.3)

yielding the calculation of β from (D.1)

Figure D.6: Damped free oscillations of a rubber strip.

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