Question

2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with we obtain the linearized shallow-water equations Ot on O a) For the special non-rotating case (f -0 ) with constant depth (Ho - const.) show that the speed of gravity waves is c-VgHo Hint: set v-0 and derive a wave equation for the sea level η b) Given a harmonic wave η(x,t)=Asin(k-or) with amplitude A (again for f-0 and Ho= const.), derive the equation for the horizontal velocity u(x,t) and the vertical velocity w(x,z,t). Hint: to obtain w(x,z,t) remember the basic form of the incompressible continuity equation and that there is no flow into the sea floor (ie, w(x,0,t) 0)

Answer 1

Solution for one - dimensional flow , (nu = 0) partial differentialu/ partial differentialt = - g paratial differentialeta/partial differentialx = u = - g integral partial differentialn/partial differentialx dt = partial differentialu/partial differentialx = - g partial differential/partial differentialt . (dt/dx)^2 = - g/c^2 partial differentialn/partial differentialt = partial differentialu/partial differentialx = - g/c^2 [-H_0 partial differentialu/partial differentialx] [from eqn . 3] = partial differentialu/partial differentialx = gh_0/c^2 partial differentialu/partial differentialx = c = squareroot gh_0