Answer Happy

(a) A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density p, so that the equation of the surface is z = n(x, t) where z is measured vertically upwards from the undisturbed surface (z = 0). The two-dimensional linearised Euler equations governing the flow can be written 1 др де де 1 др .8 ді ди pdx' рдz ди + дх dw дz = 0, where p is the total pressure. Suppose that p can be written as P p = Pa - Pgz + po, where pa is the constant atmospheric pressure. (i) Derive the governing partial differential equation satisfied by 0. (ii) By relating w ton, derive the kinematic boundary condition on 0 at z = 0. (iii) By considering the pressure on the free surface, derive the dynamic boundary condition on at z = 0. (iv) By combining the results of (ii) and (ii), obtain a boundary condition in terms of alone at z = 0.