# Exercises on Wave theory, MEK 4600 J. Rabault 5th February 2016

Exercises on Wave theory, MEK 4600 J. Rabault 5th February 2016 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 command 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. 1 Waves exercises 2. Compute the phase velocity cp (k) = ω/k and the group velocity cg (k) = ∂ω/∂k for deep water gravity waves. 3. Using Matlab or Python, plot the dispersion relation for gravity waves (horizontal axis: wave period in seconds, vertical axis: wavelength in meters). 4. Using Matlab or Python, plot the curves for c and cg (horizontal axis: wave period in seconds, vertical axis: m/s). 5. You are doing an experiment in the wave tank. The wave tank depth is h = 0.7 meter. You use the paddle to generate waves of period T = 2 seconds. Can you consider those waves as deep water gravity waves? 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 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 Ex. 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: 1 Z Tm Vm (z) = 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. 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 a −∞ us (z)dz = Z a −∞ Vm (z)dz J. Rabault 2 (7) Waves exercises 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. 2. The maximum wave steepness before waves breaking is H/λ ≈ 0.1412. Compute the corresponding maximum max . 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 3

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