A tsunami crosses an ocean deep as a wave long with amplitude

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

Modes, Packets and Media, Ch Water waves and de Broglie waves

Problem

A tsunami crosses an ocean 4000 m4000\,\mathrm{m} deep as a wave 200 km200\,\mathrm{km} long with amplitude A=0.50 mA = 0.50\,\mathrm{m}. It then runs up a gently sloping continental shelf toward the coast, neither reflecting nor losing energy. Take g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. ((a)) Find the wave's period. Find its wavelength and khkh where the water is 50 m50\,\mathrm{m} deep, and say whether it is still a shallow-water wave there. ((b)) Use Green's law to find its amplitude at that 50 m50\,\mathrm{m} depth. ((c)) Green's law is linear theory, but the crest of a hump of height aa outruns its base by the fraction 3a/2h3a/2h. Taking the amplitude from Green's law, find the depth at which this fraction reaches 0.100.10, and the wavelength there. ((d)) At that depth, compare the fractional dispersive correction to the speed, (kh)2/6(kh)^2/6, with 3a/2h3a/2h. Compare the wavelength with the width hh/ah\sqrt{h/a} that a solitary wave of that height would have. What happens to the front of the wave next, and why can Green's law not be trusted nearer the shore?

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