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Linear (Airy) Wave Theory

Mathematical relationships to describe wave movement in deep, intermediate, and shallow (?) water

We’ll obtain expressions for the movement of water particles under passing waves - important to considerations of sediment transport --> coastal geomorphology.

Works v. well, but only applicable when L >> H

Originates from Navier Stokes --> Euler Equations

Solution is eta relationship - write eqn. and draw on blackboard - show dependence on x,t

Wave Number: k = 2π/L

Radian Frequency: σ = 2π/T

Water Surface Displacement Equation

What is the wave height? What is the wave period?

2

Dispersion Equation

Fundamental relationship in Airy Theory - put eqns. 5-8, 5-9 on blackboard

These are tough to solve, as L is on both sides of equality and contained within hyperbolic trigonometric function.

Compilation of Airy Equations - Table 5-2, p. 163 in Komar

Door Number 1 = Relationship for wavelength

Door Number 2 = Relationship for celerity

Effect of the Hyperbolic Trig Functions on Wave Celerity

What’s the relationship for celerity in deep water?

What’s the relationship for celerity in shallow water?

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So the celerity illustrated is…

DWS, T=16 s

DWS, T=14 s

DWS, T=12 s

DWS, T=10 s

DWS, T=8 s

SWS, only depth dependent

Gen’l Soln., T=16 s

Gen’l Soln., T=14 s

Gen’l Soln., T=12 s

Gen’l Soln., T=10 s

Gen’l Soln., T=8 s

General Expression:

Deep-water expression:

Shallow-water expression:

Wave Speed - Tsunami

In Shallow Water

wave speed C = (gh)1/2

Deep Ocean Tsunami

C = (10m/s2*4000 m)1/2 ~200 m/s

~450 mph!

(Alaska to Hawaii in 4.7 hours)

How fast does a tsunami travel across the ocean?

What classification is this wave?

Deep water? Intermediate? Shallow water?

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Wave Speed - Nearshore

wave speed C = (gh)1/2

tow-in waves: H = ~8 m

C = (10 m/s2 * 10 m)1/2 ~ 10 m/s

~25 mph!

waves “surfable” by mortals:

C = (10 m/s2 * 2 m)1/2 ~ 4.4 m/s

~9 mph!

How fast does a Laird Hamilton surf?

Derivation of Deep & Shallow water Equations

Deep water - L, C depend only on period

Shallow water - L, C depend only on the water depth

Summarize regions of applications of approximations

Behavior of normalized variables.

5

Airy Wave Theory Continued

Orbital Motion in Waves

Show code for this: /Users/pna/Work/mFiles/pna_library/wave_pna_codes/waveOrbVelDeep.m

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Orbital Motion in Waves

Show code for this: /Users/pna/Work/mFiles/pna_library/wave_pna_codes/waveOrbVelDeep.m

Orbital Motion in Waves Deep water: s=d=Hekz, circular orbits whose diameters decrease EXPONENTIALLY (truly) through the water surface – at water surface the diameter of particle motion is obviously the wave height, H.

Intermediate water: ellipse sizes decrease downward through water column

Shallow water: s=0, d=H/kh; ellipses flatten to horizontal motions; orbital diameter is constant from surface to bottom.

Airy assumptions not valid in shallow water.

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z

Total Energy =

Potential Energy + Kinetic Energy

€

E = Ep + Ek

€

=1L

ρgzdzdx−hη∫ +0

L∫1L

12ρ u2 + w2( )dzdx−h

η∫0L∫

=116

ρgH 2 +116

ρgH 2

=18ρgH 2

[units] = M L L2 = joules/m2 or ergs/m2 L3 T2

Derivation of Wave Energy Density

€

P =1T

Δp(x,z,t)[ ]udzdt−hη∫0

T∫

€

=18ρgH 2c 1

21+

2khsinh(2kh)

⎡

⎣ ⎢ ⎤

⎦ ⎥

= Ecn[dimensions] = M L L2 L

L3 T2 T

= joules/sec/m = Watts/m

Deep Water n=1/2

Shallow Water n=1

Wave Energy Flux Energy density carried along by the moving waves.

a.k.a. “Power per unit wave crest length”

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Consider two waves with the same height beating together

average wave

η = η1 + η2 η = H/2 cos(k1x - σ1t) + H/2 cos(k2x - σ2t) = H cos[(k1+k2)/2x - (σ1+σ2)/2t] * cos[(k1-k2)/2x - (σ1-σ2)/2t] = H cos(kx-σ t)*cos[1/2Δk(x-Δσ/Δk*t)]

wave 1 wave 2

group envelope

Groupiness looks like a wave:

ηg = cos(Δk/2 x - Δσ/2 t) With group velocity:

cg = Δσ/ Δk

Groupiness / Group velocity

Group velocity approx. cg = Δσ/ Δk ~ ∂σ/∂k

Deep Water σ2 = gk

cg = ∂σ/∂k = g/2σ = 1/2 c

Shallow Water σ2 = ghk2

cg = ∂σ/∂k = (gh)1/2 = c

Group Velocity and n

⎥⎦

⎤⎢⎣

⎡+=

)2sinh(21

21

khkhn

9

Radiation Stress - introduced

“the excess flow of momentum due to the presence of the waves”

Komar, 1998

Nonlinear Waves - Stokes Theory

10

Cnoidal and Solitary Wave Theory

Limits of Application