AN INTRODUCTIONWAVE DYNAMICS

Scope

• Wave generation• Regular Linear waves• Wave Charecteristics

Introduction

• Ocean surface waves cause periodic loads on all man-made structures in the sea

• Responses: accelerations, displacements, internal loads

• Effects of waves & resulting motions on ships:– Added resistance– Impaired safety– Affect operations of weapons & equipment– Affect aircraft/ helo operations– Affect humans

Wave generation• Waves generated by a ship or any other floating structure which is

moving, either at a constant forward speed or by carrying out an oscillatory motion.

• Waves generated by the interaction between wind and the sea surface.• Waves generated by astronomical forces: Tides.• Waves generated by earthquakes or submarine landslides: Tsunamis.• Interaction of ocean currents can create very large wave system• Free surface waves generated in fluids in partially filled tanks; such as

fuel or cargo tanks on a ship.

• No single mathematical solution• Approximations required: be aware of simplifications

Tsunami

Wind generated wave systems• The size of the wave system is dependent on the following

factors• Wind Strength :

– The faster the wind speed, the larger the energy transfer to the sea.– Larger waves are generated by strong winds.

• Wind Duration :– The longer wind blows, the greater the time the sea has to become

fully developed at that wind speed.• Water Depth :

– Wave heights are affected by water depth.– Waves traveling to beach will turn into breaking wave by a depth

effect.• Fetch

– Fetch is the area of water that is being influenced by the wind.– The larger the fetch, the more efficient the energy transfer between

wind and sea.

Wind Energy

Energy Dissipationdue to viscous friction

Fully Developed Wave(Wind energy =Dissipation Energy)

Swell (low frequency long wave)

Small Wave or dying out(Wind energy <Dissipation Energy)Ripple

(high freq.)

Wind energy >Dissipation Energy

Wave creation sequence

Wind-generated waves

• Sea– Train of waves driven by the prevailing local

wind field– Short-crested with the lengths of the crests only

a few (2-3) times the apparent wavelength– Very irregular– Multi-directional– Crests are fairly sharp– Apparent wave period & apparent wave length

vary continuously

Wind-generated waves

• Swell– Waves which have propagated out of the area

and local wind in which they were generated– No longer dependent upon the wind– Individual waves are more regular and the

crests are more rounded– Lengths of the crests are longer: several (6-7)

times the virtual wave length– Wave height is more predictable

Superposition principle• Wind waves are very irregular • Can be seen as a superposition of

many simple, regular harmonic wave components, each with its own amplitude, length, period or frequency and direction of propagation

• To analyze complicated wave systems, it is necessary to know the properties of the simple harmonic components– time and location-dependent pressure in

the fluid– relation between wave length and wave

period– energy transport, etc.

Regular Waves: Definitions

• Origin & conventions• Crest, Trough, Amplitude (a ), Height (H= 2 a )• Wave length (), Wave Period (T)• Wave steepness = H/ • Zero crossings• Wave number (k=2/ ); Circular frequency (= 2/ T)• Phase velocity (c = /T = /k)

Basic Categories• Deep water waves (short waves)

– The water is considered to be deep if the water depth, h, is more than half the wavelength,

– Thus, h/ > 1/2 or /h < 2– These (relatively) short waves do not ’feel’ the sea

floor.• Shallow water waves (long waves)

– The water is considered to be shallow if the water depth, h, is less than 1/20 of the wave length,

– Thus, h/ < 1/20 or /h > 20. – The sea floor has a very large influence on the

characteristics of these (relatively) long waves.

Linear Wave theory• Progressive harmonic wave: = a cos(kx- t)• Linear wave theory: water

surface slope is very small• Wave steepness is small• Harmonic displacements,

velocities, accelerations & pressures have linear relation with wave surface elevation

• Profile of such a wave looks like sine/ cosine

• Motion of water particle in wave depends on depth below SWL

Relations for Linear Waves• Continuity (Laplace equation)• Boundary Conditions

– Sea bed– Free surface dynamic– Free surface kinematic

• Dispersion relation: 2 = g.k. tanh(kh)- Deep water: 2 = g.k or ≈ 1.56 T2

- Shallow water: =k.√gh or = T.√gh

• Phase velocity:- Deep water: c = √(g/k) or c ≈1.25√ ≈ 1.56 T - Shallow water: c= √gh (‘critical velocity’)

Velocity field of water particles

In shallow water wave In deep water wave

Trajectories of water particles

Wave group

Group Velocity

• In deep water, cg = c/2

• In shallow water, cg = c