Relationship between Probability density and Spectrum

An interconnection of the statistical properties of random

wave field

Motivations

• We have seen the proposed spectral forms mostly consisted of products of a power function and an exponential terms.

• How can we be sure that the functional form indeed is of such type?

• From probability properties of the wave field, we have various distribution of frequency, amplitude…. Are there any connections between the probability density and spectral function?

Joint Amplitude and Period Density Functions

• If we know the joint amplitude-period distribution, we could derive a quantity that would have the physical property of a spectrum:

2

0

2 2 2 2

0 0

E ( n )dn

E p ,n d E H H p H ,n dH

Can we relate (n) with p( ,n) or a transform of it?

Joint density : Longuet-Higgins (1983)

22o

L 1 / 2 2o 1o 1

21 / 221

o 2

mL( ) H 1p ( H , ) exp 1 1 ,

2m m2 m m

in which

m 1 1 and 1 1

m m L( ) 2

Verification

Joint density : Yuan (1982)

222 3 22 o

Y 1 / 23 / 2 2o2 y

1 / 2 1 / 2

22 4y o 4 2

o 2

o2o

o

m m2H H 1p ( H , ) exp 1 ,

2m2 m N

in which

m m1N ; = m m m ;

4 m m

t ; t is the time at wave peaks

t

Verification

Verification

Verification

Marginal Probability: Longuet-Higgins

2

0

5 / 222L o o

L 21 1

2 1 / 2L 3

HDefine S( ) = p H , dH

2

3 m L( ) m 1S ( ) 1 1

4 m m

2( 1 R )Conservation of energy requires: ; R=(1+ )

3R R 2

Marginal Probability: Yuan

23L o

Y 5 / 22 4 222

2

2 2o 4 2 2 o y

o

1 / 22

2

1 13 mS ( )

1 1

2m m / m ; m / m ;

1

7 m

m 2 25 21 1

Verification

Verification

Verificationi

i0

i i

i i

1. Define M S( ) d

relationships between M and m can be estblished.

With 1, M m . So the two approaches are equivalent

for narrow band case.

2. For a wide band (signal as white noise

ii

i

1 / 2

o 2 4

)

Mthe ratios, r = , for Yuan's formula are these:

m

7 14r 1; r ; r = .

3 3

3. Longuet-Higgins formula is for narrow band only.

Conclusion

• So, there is a relationship between probability density function and spectral function based on Fourier analysis.

• This relationships exists only from the energy containing point of view. In other words, it works only for energy containing range of the spectra.

• Fourier spectra works for wider bands than the probability density functions, which are all derived under narrow or near narrow band assumption.

A turning point in my research

My effort to study the nonlinear Schrödinger equation

Types of Waves

• Water wave motion was amongst the first fluid mechanics treated successfully by mathematics.

• John Scott Russell observed a solitary wave in a barge canal in 1834: Shallow water Waves.

• George Gabriel Stokes derived deep water periodic wave of permanent shape in 1847: Deep Water Waves.

Shallow Water Waves

Solitary Waves

Solitary waves• His experimental observations were viewed with

skepticism by George Airy and George Stokes because their linear water wave theories were unable to explain them.

• Joseph Boussinesq (1871) and Lord Rayleigh (1876) published mathematical theories justifying Scott Russell’s observations.

• In 1895, Diederik Korteweg and Gustav de Vries formulated the KdV equation to describe shallow water waves.

• The essential properties of this equation were not understood until the work of Kruskal and his collaborators in the 1960's.

Solitary wave and soliton• The name soliton was coined by Zabusky and

Martin Kruskal. However, the name solitary wave, used in the propagation of non-dispersive energy bundles through discrete and continuous media, irrespective of whether the KdV, sine-Gordon, non-linear Schrödinger, Toda or some other equation is used, is more general.

• Kruskal received the National Medal of Science in 1993 “for his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution.”

Other soliton equations• Sine-Gordon equation:

• Nonlinear Schrödinger equation:

• Kadomstev Petviashvili (KP) equation:

0sin uuu xxtt

0|| 2 uuuiu xxt

03)6( yyxxxxxt uuuuu

KdV: solitary wave solution• Korteweg and de Vries (1895) discovered the eq

uation possesses the solitary wave solution:• KdV equation:

• Traveling wave solution (Kruskal):

06 xxxxt uuuu

)(),( ctxftxu

)](2

1[sec

2),( 0

2 ctxxchc

txu

Solitary Wave: 2 solitons

Two solitons travel to the right with different speeds and shapes

Solitary Waves

Solitary Waves

Solitary Waves

Shallow Water Waves

The governing equations

What are the assumptions?

Are they reasonable?

Shallow Water Waves: Governing Equations I

Starting from the KdV equation,

We can derive the Nonlinear Schrödinger Equation,

Provided that the wave number and frequency are constant to the third order.

oit x xxx 2 o

au uu u 0 ; let u = e + u ( , x, t) + cc

2

22 *

og 0kk 020

a a 1 a 1i c a a k a 0 ,

t x 2 x 24k

Governing Equations II

We will have a different Nonlinear Schrödinger Equation,

Provided that the wave number and frequency are constant to the second order.

22 *

og 0kk 20 0kk okk2

*

*

0k 20

a a 1 a i k 1i c 3 a a a aC 0 ;

t x 2 x 2 x 6

C 1 aa 0 ; and

t 2 x

k k aa,

t x x

Governing Equations IIIWe will have still another form of NSE,

Provided that the wave number and frequency are constant to the first order.

22 *

og 0kk 20 0kk2

2

okk2

*

*

0k 20

a a 1 a i ki c 3 a a a

t x 2 x 2 x

k a k 13 a aC 0 ;

x x x 6

C 1 aa 0 ; and

t 2 x

k k aa,

t x x

Deep Water Waves

Periodic waves

Stokes

• George Gabriel Stokes 1819-1903

• Lucasian Professor • President of Royal

Society• Navier-Stokes

equations• Stokes theorem

  

The Classical Stokes Waves

• Stokes waves are periodic waves of permanent shape.

• The solution was derived in 1847 by Stokes using perturbation method.

2 3 21 3 = a cos + a k cos 2 + a k cos 3 + ...

2 8where = k x - t .

The Classical Stokes Waves• Stokes waves was treat as the standard solution

for more than hundred years.• Phillips’s (1960) theory on the dynamics of

unsteady gravity waves of finite amplitude opened a new paradigm: through 3rd order resonant interactions, the wave profiles could change over a long time compared to the wave period.

• At that time, the Ship Division of the National Physical Laboratory built a wave tank, but the wave maker could not generate periodic wave of permanent shape.

Unstable Wave Train : National Physical Laboratory

The Classical Stokes Waves

• There was almost a law suit for the incompetence of the wave maker contractor.

• Then, Brook-Benjamin and Feir (1967) found that the Stokes wave was inherently unstable.

• Later, it was found that this instability was a special case of Phillips’s 3rd order resonant interactions.

• A new era was down for deep water periodic wave studies.

NASA Wind-Wave Experimental Facility

Data

What should be the governing equations

• Based on Phillips’s resonant theory, Hasselmann (1962, 1963, 1966) formulate the resonant interaction in spectral form.

• Meanwhile, the TRW group (Lake and Yuan etc. 1975, 1878) formulated the wave evolution in nonlinear Schrödinger equation, and claimed Fermi-Ulam-Pasta recurrences … They are the most successful wave research group in the US.

Unstable Wave Train : Su

What should be the governing equations

• Unknown to the west, Zakharov (1966, 1968) had already derived the nonlinear Schrödinger equation from Hamiltonian approach.

• The nonlinear Schrödinger equation is in terms of envelope.

• The carrier should be water waves; the envelope, a soliton.

What should be the governing equations

• This form is not just a group formed by beating of two independent free wave trains.

• An example of the Sech envelope soliton is given below:

Deep Water Waves:

Dysthe, K. B., 1979: Note on a modification to the nonlinear Schrodinger equation for application to deep water waves. Proc. R. Soc. Lond., 369,

105-114.

Equation by perturbation up to 4th order.

But ω = constant.

Governing Equations I:

2

22 2

2

, 0,

1, 0,

2

, .

z

z gt z t

zt z

Governing Equations II:

2

2

2

2

2 22 22

2

0, 0,

10, 0,

2

, , , 0 ,

1 1.

2 2

h h h h

h

z L L Qz t

zt z z

where L gt z x y

Q Lz t z

Governing Equations III:

2 22

22

..... . .

.... . .

.

kz i kz i

i i

Ae e A e e c c

Be B e c c

wherer k x t

Governing Equations IV: The 4th order Nonlinear Schrödinger

Equation

2 22

2 2

3 3

2 3

2

1 1 12

2 2 4

1 36

8 2

1.

2

A A A Ai A A

t x y x

A A A Ai iA A A

x y x x x

Ai A A i

x x z

Observations

• All the published governing equations are in terms of envelope, which is governed by a cubic nonlinear Schrödinger equation.

• The carriers are assumed to be of constant frequency.

• The constant carrier frequency assumption is untenable.

Conclusions

• Most people studying waves are actually studying mathematics rather than physics.

• But it is physics that we should understand.

• We need new paradigm for wave studies.

My experiences

• I started to explore the envelopes.

• A natural way was to turn to Hilbert Transform.

• Once I used Hilbert transform, I found solutions as well as problems; it open a new view point not only of water waves but also the whole world.

Reminiscence• By the time (1990) I finished these studies, I

thought I had found the key to ocean wave study: the significant slope, S.

• Then, I studied the governing equations of water wave motion, the nonlinear Schrödinger equation, and conducted laboratory experiments to compare with the theoretical results.

• In that effort, I used the Hilbert Transform to analyze the laboratory data; the results shocked me. Fortunately, I made a mistake in the processes. HHT, born through my efforts to correct that mistake. The rest is history. and the subject of this course.