A swell of period and deep-water height approaches a coast whose depth contours are…

PHYSICS · Multi-part · 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=10\,\mathrm{s}

and deep-water height

H_0=1.5\,\mathrm{m}

approaches a coast whose depth contours are straight and parallel. In deep water its rays make

\theta_0=40^\circ

with the normal to the contours. Take the bottom as slowly varying, with no reflection. ((a)) At

h=5.0\,\mathrm{m}

, find the ray angle

\theta

, the coefficients

K_{\rm s}

and

K_{\rm r}

, and the wave height

H

. ((b)) Find, by bisection, the depth

h_{\rm b}

at which

K_{\rm s}K_{\rm r}H_0=0.78h

, and compare it with the breaking depth of the same swell arriving straight onshore. ((c)) At

h=5.0\,\mathrm{m}

, compare the finite-depth second harmonic,

\tfrac14ka^2\cosh kh\,(2+\cosh2kh)/\sinh^3kh

with

a=H/2

, with the first harmonic

a

. Can the Stokes expansion be trusted there, and what does that say about the depth found in part (b)?

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