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13.42 Lecture #2Linear Waves

February 5, 2004Alexandra H. Techet

MIT Department of Ocean Engineering©A. H. Techet 2004

1. Recap of Linear Wave Problem2. Phase Speed3. Particle motion beneath waves

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Ocean Waves

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Linear Wave ProblemLinear free-surface gravity waves can be characterized by their amplitude, a, wavelength, λ, and frequency, ω.

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Conditions for Linear Waves

• Linear wave theory assumes that the ratio of the wave height to wavelength is less than 1/7. Above this value waves begin to exhibit non-linear behavior, eventually breaking.

/ 1/ 7h λ <

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Conservation of Mass

2 2

2 22 0

x zφ φφ ∂ ∂

∇ = + =∂ ∂

V φ= ∇

The velocity potential, φ, must satisfy the Laplace Equationin order for mass to be conserved:

( )0V∇× =Assuming ideal flow: incompressible, inviscid & irrotational

Define a velocity potential, φ, such that

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Boundary Conditions

• Body boundary condition: there is no fluid flux through (normal to) a solid surface such as a body or the ocean floor.

• On the ocean floor, @ , assuming a horizontal orientation:

0Bnφ∂=

∂

0wzφ∂

= =∂

z H= −

ˆBn

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Free Surface Conditions• Dynamic Boundary Condition: Pressure is constant

across the interface:

21 ( )2 atmp gz c t p

tφρ φ∂ = − + ∇ + + = ∂

21 ( , ) 02

g x ttφρ φ η∂ + ∇ + = ∂

21 1( , )2

x tg t

φη φ∂ = − + ∇ ∂

( , )z x tη=

Let the arbitrary Bernoulli constant equal atmospheric pressure: c(t) = patm . Substitute η(x,t) for z:

on

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Free surface Conditions• Kinematic Boundary Condition: Once a particle on the

surface, it remains there always. ( )( ) ( ), ,p p p p p pz z x x t t x t x t

x tη ηδ η δ δ η δ δ∂ ∂

+ = + + = + +∂ ∂

( , )p pz x tη=

p pz x tx tη ηδ δ δ∂ ∂

= +∂ ∂

,p px zu w

t tδ δδ δ

= =

We can write the velocity of a particle under the wave as:

w ux tη η∂ ∂

= +∂ ∂

( , )z x tη=, on

Therefore we have the kinematic boundary condition (KBC)

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Free surface Conditions• Kinematic Boundary Condition:

w ux tη η∂ ∂

= +∂ ∂

( , )z x tη=, on

Substitute the velocity potential into the KBC:

,u wx dzφ φ∂ ∂

= =∂

z t x xφ η φ η∂ ∂ ∂ ∂= +

∂ ∂ ∂ ∂( , )z x tη=, on

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Non-dimensional variables*aη η=Wave Elevation:

*u a uω=x-velocity:

*w a wω=z-velocity:

*t tω =Time:

*x xλ=Length:

*aφ ωλ φ=Velocity Potential: *d a dφ ωλ φ=

*1dt tω=

*dx dxλ=

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Linear Waves

/ 1/ 7h λRecall for linear waves:

We need to linearize the boundary conditions. For starters lets look at the terms from the dynamic boundary condition:

2 22 * *

* *2 2

* *2

* *

x xa a axa

t t t

φ φφω

φ φ φω λ λ λ

∂ ∂∂ ∂ ∂∂ = = ≈

∂ ∂ ∂∂ ∂ ∂

Therefore:2

x tφ φ∂ ∂

∂ ∂

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Linearized DBC & FBC:

21 1( , )2

x tg t

φη φ∂= − + ∇

∂ z t x x

φ η φ η∂ ∂ ∂ ∂= +

∂ ∂ ∂ ∂

On :( , )z x tη=DBC KBC

Neglecting higher order terms, and expanding φ(x,z,t) about z=0, these conditions become:

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2 0gt zφ φ∂ ∂+ =

∂ ∂; on 0z =

1and ( , )x tg t

φη ∂= −

∂

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Where is the dispersion relationship

A Solution to the Laplace Equation2 2

2 22 0

x zφ φφ ∂ ∂

∇ = + =∂ ∂

( )1( , , ) sinax z t f kx tkωφ ω ψ= − − +

( )( )1

coshsinh

k z Hf

kH + =

( )( )2

sinh

sinhk z H

fkH

+ =

( )1( , , ) cosdu x z t a f kx tdxφ ω ω ψ= = − − +

( )2( , , ) sindw x z t a f kx tdzφ ω ω ψ= = − +

( )( , ) cosx t a kx tη ω ψ= − +

2 tanhgk kHω =

sin cos , cos sin

sinh cosh , cosh sinh

d dx x x xdx dxd dx x x xdx dx

= = −

= =

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Pressure Under a Wave

nd

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hydrostatic2 order

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p gztφρ ρ φ ρ∂

= − + ∇ +∂

Dynamic pressure due to linear surface gravity waves:

( ) ( )2

1 cosdap f z kx wt

t kφ ωρ ρ ψ∂

= − = − +∂

( ) ( )2

1 ,dp f z x tkωρ η=

2

dispersion ret la atinh onshipg kHkω

= ⇔

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Motion of a Fluid Particle

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Particle Orbits

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Particle Orbits in Deep Water

Circles with exponentiallydecreasing radius

Particle motion extinct at z ≅ -λ/2

( )1H kH→∞

1 2( ) ( ) kzf z f z e≅ ≅

( )22 2 kzp p a eξ η+ =

2 dispersion relationshipgkω ⇔=

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Particle acceleration and velocity

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at the free surface…The intersection between the circle on which ξpand ηp lie and the elevation profile η(x,t) define the location of the particle.

This applies at all depths, z: η1(x,z,t) = a ekz cos(kx-ωt+ψ) = η(x,t) ekz

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Phase Speed

Velocity at which a wave crest is moving

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Group Speedη(

t, x

= 0)

t

Waves packets (or envelopes) move at the Group Speed

η(x,t) = a cos(ωt-kx)