Water of uniform depth rests on a flat horizontal bed

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow · Waves, acoustics and compressible flow

Fluids and Continuum Physics, Ch Surface gravity and capillary waves

Problem

Water of uniform depth

h

rests on a flat horizontal bed. The water is treated as inviscid, incompressible and of uniform density

\rho

, the motion as irrotational so that

\mathbf{u} = \nabla\phi

, and the wave amplitude as small enough that all products of two perturbations are dropped. Surface tension is neglected. Take

z

upward from the undisturbed surface, so the bed is at

z = -h

, and

g = 9.8\ \mathrm{m\,s^{-2}}

. The motion is two-dimensional in the

x

z

plane. Try

\eta = a\cos(kx - \omega t), \qquad \phi = A\cosh k(z+h)\,\sin(kx - \omega t),

with

a

A

k

\omega

constants. The linearised surface conditions are the kinematic one,

\partial\eta/\partial t = \partial\phi/\partial z

z = 0

, and the dynamic one,

\partial\phi/\partial t + g\eta = 0

z = 0

. ((a)) Show that this

\phi

satisfies

\nabla^{2}\phi = 0

and the condition of no flow through the bed. ((b)) Use the kinematic condition to express

A

in terms of

a

\omega

k

and

h

. ((c)) Use the dynamic condition to obtain the dispersion relation

\omega^{2} = gk\tanh kh

. ((d)) Evaluate the period and phase speed for a wave of wavelength

40\ \mathrm{m}

in water

8.0\ \mathrm{m}

deep, and compare the speed with the deep-water value

\sqrt{g/k}

. ((e)) From the dispersion relation of (c), find the group velocity

c_g = \mathrm{d}\omega/\mathrm{d}k

at finite depth and show that

\frac{c_g}{c} = \frac12\left(1 + \frac{2kh}{\sinh 2kh}\right),

using

\sinh 2u = 2\sinh u\cosh u

. Check that it gives the deep-water and shallow-water results in the two limits, and evaluate

c_g

for the wave of (d).

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