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Airy Wave Theory 2: Wave Orbitals and Energy

Compilation of Airy Equations

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Orbital Motion of Water Particles

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

Airy Wave Theory also predicts water particle orbital path trajectories. Orbital path divided by wave period provides the wave orbital velocity.

0 10 20 30 40 50 60−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time (sec)

velo

city

(m/s

)

HorizontalVerticalTangential

Orbital Motion of Water Particles

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

H=2m, T=10s, h=4000m

Where is the wave crest? The trough?

A B C D

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Orbital Motion of Water Particles

Deep water (h>L/2): s=d=Hekz, circular orbits whose diameters decrease through water column to zero at h = L/2. At water surface, diameter of particle motion = wave height, H Intermediate water (h<L/2): elliptical orbits, whose size 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.

Orbital Motion at the Bed in Shallow Water

The horizontal diameter at the bed simplifies to … And the maximum horizontal velocity at the bed, which relates conveniently to the shear stress, is

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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”

[units]

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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

The expression Cn (sometimes written Cg) is known as the group celerity. In deep water, the first wave in a group decreases in height until it disappears and the second wave now becomes the leading wave (Figure). A new wave develops behind the last wave, thus maintaining the number of waves. The effect of this process is that the group of waves travels at a speed equal to ½ the speed of the individual waves in the group. This is important in forecasting wave propagation and in particular the travel time of waves generated by a distant storm (hint for a problem on Assignment 3).

Stokes’s 2nd Order Wave Theory

Airy (linear) wave theory which makes use of a symmetric wave form, cannot predict the mass transport phenomena which arise from asymmetry that exists in the wave form in intermediate-to-shallow water. The wave form becomes distorted in shallower water. The crest narrows and the trough widens. Shoreward-directed horizontal velocity becomes higher under the wave crest than the offshore-directed velocity under the trough. Waves steepen and relative depth decreases, so that

these waves are no longer considered “small-amplitude”. Instead they are called “finite-amplitude”.

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Orbital Motion in Finite-Amplitude Wave Theory

Due to the asymmetry of the wave form, orbital paths are not closed. There is a net motion of the water particle in the direction of wave advance, called Stokes drift. Stokes drift is important because it provides a mechanism of sediment transport on beaches, independent of current-driven transport. Can divide drift distance by wave period to obtain drift velocity.

Shallow Water - Cnoidal and Solitary Wave Theories

Wave speed in shallow-water is influenced more by wave amplitude than water depth. The water particle motion is dominated by horizontal flows - vertical accelerations are small, and Stokes's theory becomes invalid. Mathematically complex formulations have emerged that predict shallow water wave forms well – Cnoidal and Solitary theory, which originates from the shallow water Boussinesq equation.

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Limits of Application