The same swell of period (from the previous problem) enters water of depth

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

The same swell of period T=10.0 sT = 10.0\,\mathrm{s} (from the previous problem) enters water of depth h=5.0 mh = 5.0\,\mathrm{m}. The finite-depth dispersion relation is ω2=gktanh⁡(kh)\omega^2 = gk\tanh(kh), and the group speed is vg=nvpv_{\rm g} = n v_{\rm p} with n=12[1+2khsinh⁡(2kh)]n = \tfrac12\left[1 + \dfrac{2kh}{\sinh(2kh)}\right]. Find, by numerical root finding for kk, ((a)) the wavenumber kk; ((b)) the wavelength λ\lambda; ((c)) the phase speed vpv_{\rm p}; ((d)) the group speed vgv_{\rm g}.

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