A ship-model towing tank holds clean fresh water of density and surface tension to a…

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

Fluids and Continuum Physics, Ch Surface gravity and capillary waves

Problem

A ship-model towing tank holds clean fresh water of density

\rho = 1000\ \mathrm{kg\,m^{-3}}

and surface tension

\gamma = 0.073\ \mathrm{N\,m^{-1}}

to a uniform depth

h = 0.60\ \mathrm{m}

over a flat floor. Take

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

, and treat the water as inviscid and incompressible with small-amplitude irrotational waves. ((a)) Find the speed of the very long waves of the tank (those with

kh \ll 1

), and hence the depth-based Froude number

\mathrm{Fr} = U/\sqrt{gh}

of a model towed at

U = 1.45\ \mathrm{m\,s^{-1}}

. ((b)) Find the smallest possible phase speed of a wave on this surface and the wavelength at which it occurs, quoting the capillary length

\ell_c = \sqrt{\gamma/(\rho g)}

of the water. ((c)) The wavemaker at one end produces a wave of wavelength

1.00\ \mathrm{m}

whose velocity potential is

\phi = C\,\mathrm{e}^{kz}\sin(kx - \omega t)

with

C = 0.0250\ \mathrm{m^{2}\,s^{-1}}

, where

z

is measured upward from the mean free surface. Verify that this wavelength is in the deep-water range and that its surface-tension term is negligible, then use the linearised kinematic condition to find the amplitude

a

of the surface elevation and hence the energy flux per metre of crest.

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