A swell of period and deep-water height approaches a harbour over a gently sloping…

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

Signal Speeds in Fluids, Ch Waves of finite height: Stokes waves, shoaling and breaking

Problem

A swell of period

T = 8.0\,\mathrm{s}

and deep-water height

H_0 = 1.2\,\mathrm{m}

approaches a harbour over a gently sloping seabed whose depth contours are straight and parallel. In deep water its rays make

\theta_0 = 30^\circ

with the normal to the contours. At

h = 8.0\,\mathrm{m}

the finite-depth dispersion relation gives

k = 0.0968\,\mathrm{m^{-1}}

, and the group speed is

v_{\rm g} = n v_{\rm p}

with

n = \tfrac12[1+2kh/\sinh(2kh)]

. (a) Two neighbouring rays cross each depth contour a fixed distance

b_0

apart, measured along the contour. Show that their separation measured along the crest is

b_0\cos\theta

, and hence, by conserving the energy flux between them, that

H/H_0 = K_{\rm s}K_{\rm r}

with

K_{\rm s} = \sqrt{v_{\rm g0}/v_{\rm g}}

and

K_{\rm r} = \sqrt{\cos\theta_0/\cos\theta}

. (b) At

h = 8.0\,\mathrm{m}

, find the ray angle

\theta

, the coefficients

K_{\rm s}

and

K_{\rm r}

, and the wave height

H

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