(a) The Laplace's Equation ay ay + 0, Әr? 822 (1) governs the surface waves in water. Show that the dispersion relation is given by w? = gk tanh kd -
if the propagating wave-like solution of Equation (1) takes the form (4$,t) = F(2) cos(kx - wt). (b) The one-dimensional shallow-water inertia gravity wave dispersion rela- tion is given by wa+cawhere -H, where g is the gravity, k is the wave number, / is the inertial frequency, and H is the depth of the water below the surface. (1) Compute the phase velocity = w/k and the group velocity C = dw/dk and show that they satisfy (ii) Do packets composed of longer waves travel faster or slower than those composed of shorter waves? (iii) For a packet composed of waves of a given wavelength, how does in- creasing change the travel speed ?
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