A tsunami crosses a continental shelf of depth as a single right-running hump

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

Signal Speeds in Fluids, Ch Long waves of finite height: bores, jumps and solitary waves

Problem

A tsunami crosses a continental shelf of depth h0=50 mh_0 = 50\,\mathrm{m} as a single right-running hump. At t=0t = 0 its elevation is η=Aexp⁡(−x2/ℓ2)\eta = A\exp(-x^2/\ell^2), with A=2.0 mA = 2.0\,\mathrm{m} and ℓ=4.0 km\ell = 4.0\,\mathrm{km}, and its velocity is tied to its elevation by u=2(c−c0)u = 2(c - c_0), as for a wave running in from undisturbed water. ((a)) Find how far ahead of the crest the characteristics that cross first start out, and the time tbt_{\rm b} (in minutes) and distance xb=c0tbx_{\rm b} = c_0t_{\rm b} (in kilometres) before the front first becomes vertical. ((b)) Taking k∼1/ℓk \sim 1/\ell, estimate the Ursell number and say whether dispersion can be expected to stop this steepening.

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