A regular swell of period travels from deep water towards a shore.

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Earth, Climate & Energy Physics · Wave motion, the wave equation and wave energy

Energy from Physics, Ch Natural energy flows: rotors, turbines, waves, channels and hot rock

Problem

A regular swell of period

T = 12\,\mathrm{s}

travels from deep water towards a shore. ((a)) Before calculating anything, state whether the wave height at a depth of

20\,\mathrm{m}

should be larger, smaller, or the same as its deep-water value, and likewise at

5\,\mathrm{m}

; give the one-sentence reasoning behind each answer. ((b)) Using

\omega^2 = gk\tanh(kh)

to find the wavenumber

k

at each depth, and

c_g = \tfrac12(\omega/k)(1 + 2kh/\sinh 2kh)

for the group velocity, together with conservation of energy flux along the ray (

H \propto c_g^{-1/2}

), compute the height at

20\,\mathrm{m}

and at

5\,\mathrm{m}

as a fraction of the deep-water height.

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