Exercises on Wave theory, MEK 4600
J. Rabault
26th January 2017
Notations for the exercises
In the following exercises, we will write a the wave amplitude, ω = 2π/T the wave angular
frequency (rad/s) where T is the wave period, g the acceleration of gravity, k = 2π/λ the
wavenumber where λ is the wavelength, and h the water depth. The horizontal direction is x,
the vertical direction is z (pointing upwards), and the free undisturbed surface lies at z = 0.
The wave phase is θ = kx − ωt .
1 Plot of the wave velocity field
The velocity potential and surface elevation for water gravity waves are:
φ(x, z, t) =
aω cosh[k(z + h)]
sin(kx − ωt)
k
sinh(kh)
η(x, t) = acos(kx − ωt)
(1)
(2)
1. Show that in the deep water approximation, i.e. kh >> 1, the velocity potential becomes:
φ=
aω kz
e sin(kx − ωt)
k
(3)
~ for deep water gravity waves.
2. Compute the velocity field (vx , vz ) = ∇φ
3. Plot a few examples of velocity fields in Matlab or Python, using the pcolor and quiver
commands to get an intuition of what the velocity field under a wave looks like.
2 Dispersion relation for gravity waves
The dispersion relation for gravity waves is:
ω 2 = gk.tanh(kh)
(4)
1. Show that in the deep water approximation, i.e. kh >> 1, the dispersion relation becomes
ω 2 = gk . Explain why this relation is actually true already for h > λ/2.
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Waves exercises
2. Compute the phase velocity cp (k) = ω/k and the group velocity cg (k) = ∂ω/∂k for deep
water gravity waves and intermediate depth waves.
3. Using Matlab or Python, plot the dispersion relation for gravity waves in the laboratory
(water depth in the wavetank 0.7 m, horizontal axis: wave period in seconds, vertical axis:
wavelength in meters).
4. Using Matlab or Python, plot the curves for cp and cg in the laboratory (horizontal axis:
wave period in seconds, vertical axis: m/s).
5. You are doing an experiment in the wave tank. You use the paddle to generate waves of
frequency f = 2 Hz. Can you consider those waves as deep water gravity waves? What
is the wavelength you expect from theory? You generate such waves during 20 seconds
before stopping, and the wave tank is long enough so that the first waves do not reach the
opposite end of the wave tank when the last waves are generated. If you take a snapshot
of the wave tank, how many waves can you count? If you initially stand at a fixed point
in space, ahead of the first wave, and count the number of wave crests that pass in front
of you, how many wave crests will you count?
3 Stokes drift
Note for all this exercise: all the results you derive should be exact, i.e. formal calculations
and not numerical results. However, we are interested here in physics and not computing integrals, so feel free to use Maple, Python SymPy or any other formal calculation tool you would like.
The Stokes drift (from the Lagrangian point of view, i.e. following the particles) for deep
water gravity waves is:
us = ωka2 e2kz
(5)
The Stokes drift can also be computed in the Eulerian point of view, i.e. looking at a
fixed point in space. According to the results you found in section 1, the mean velocity
(over one period) at a fixed point under z = −a is 0. But for z positions higher than
z = −a and lower than z = a, any point in space is part of the time occupied by water, part of the time occupied by air. Therefore, the mean velocity of the water at a fixed
point should be computed only on the part of the wave period when there is water at that point.
In the following, we compute the mean value of the water velocity at the position x = 0
for a fixed point P (z) of elevation −a < z < a over the period [−T /2; T /2]. We will use the
results of Ex. 1.
1. Compute the value Tm so that their is water at the point P during the time interval
[−Tm ; Tm ]
2. Compute the mean water velocity at the fixed point P, as a function of z:
Vm (z) =
1 Z Tm
vx (z, t)dt
T −Tm
(6)
3. Plot in the same figure the Stokes drift us and the mean velocity profile at a fixed point
Vm that you obtained, both as a function of z.
J. Rabault
2
Waves exercises
4. Changing between Eulerian and Lagrangian, the Stokes drift looks different and this is
perfectly normal. But changing between Eulerian and Lagrangian point of view should
not change the underlying physics. In particular, mass conservation should be the same in
both cases. Show that this is the case and that the mean mass flow rate Mm is the same
in both descriptions, i.e. show that:
Mm =
Z 0
−∞
us (z)dz =
Z a
−∞
Vm (z)dz
(7)
Note: up to which order in the steepness does it make sense to check this equality?
Limit yourself to this order of in your (otherwise, analytically exact) calculation.
4 Third order Stokes waves
The third order deep water gravity Stokes waves are given by:
(
)
1
3
η(x, t) = a cos(θ) + (ka)cos(2θ) + (ka)2 cos(3θ)
2
8
φ=
aω kz
e sin(kx − ωt)
k
ω 2 = gk 1 + k 2 a2
(8)
(9)
(10)
The parameter = ka describes how stong non linearities are.
1. Compute the wave heigth H for third order Stokes waves, which is the difference in heigth
between the crest (highest point) and trough (lowest point) of a wave. Note: this should
be an exact result.
2. The maximum wave steepness before waves breaking is H/λ ≈ 0.1412. Compute the
corresponding maximum max . Note: you should write an analytical equation to solve,
and use a numerical method to find its roots.
3. Plot the wave elevation for third order Stokes waves at = max /2 and = max . What can
you observe compared to the usual sinusoidal wave elevation? (you can plot a sinusoidal
wave in addition).
J. Rabault
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