(a) A small-amplitude wave is progressing in the positive x-direction on the surface of water of constant density p, so

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A A Small Amplitude Wave Is Progressing In The Positive X Direction On The Surface Of Water Of Constant Density P So 1 (100.62 KiB) Viewed 38 times

(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 ди 1 др ді р дх dw =- ot 1 др р дz. 8 ди dw + = 0, дх дz where p is the total pressure. Suppose that p can be written as 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 to n, 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 (iii), obtain a boundary condition in terms of alone at z = 0. (b) An earthquake-generated ocean wave may be modelled by considering water of depth h, with the lower boundary condition on the vertical velocity w(-h, t) = b cos(kx - ot) at z = -h, where b is constant. The surface boundary conditions are unchanged. Consider a pressure disturbance of the form 0(x, z, t) = Z(z) sin(kx - ot), where is as given in part (a). (i) What does this lower boundary condition become in terms of Z(z)?