## Deep Water Gravity Waves - Finite Depth

The equations satisfied by waves are

for water of depth (1)

at (2)

at (3)

whereis the velocity potential.

(1) can be solved by separation of variables technique. Assume(there will also be an arbitrary factor ofwhich we deal with later). (1) becomessince the left hand side is a function ofonly and the right hand side is a function ofonly, so both sides are equal to the same constant

Ifthe solution will be exponential, tending toasand the same problem occurs ifsoto givewhere

This equation has solutions

The corresponding equation foriswhich has solutionshence

atso

hence

In the same way we can solve (2) by the separation of variables method to findAssume (ignoring the factor)to giveas before, and as beforeelseasso putto give

Now writeandand sincewe haveso by picking a suitable point on the wave surface and a suitable time we have