Often a linearly sloped beach is not a good model. Waves coming from deep water up a gradually shelving beach are better approximated by assuming a parabolic shape like that shown in Figure P7.6 below. If h ∝ x2:

a. Derive the differential equation describing the height of the wave as a function of space and time.

b. Solve the equation for an arbitrary initial surface, ƒ(x). Assume the wave height is zero at x = 0,1, periodic with period 1, and that the wave height is constant for all time t

Seek solutions of the form:

Transcribed Image Text:

8 = x" [a, cos(nhgt)+b, sin(nhgt)] an