Here is a brief outline of how you might approach solving this problem. The symbols do not survive HTML and some of them are not labeled very well. If you would like more information, you should look at:

Timman, Hermans and Hsiao. Water Waves and Ship Hydrodynamics. Matinus Nijhoff Publishers: Boston, 1985.

Solving the Boundary Value Problem for a Moving Pressure Point -Timman, Hermans and Hsiao

Since we want to look at water waves on a calm surface, we are going to examine the problem for small amplitudes. The linearized problem is defined by Laplaces equation:

Fxx + Fyy + Fzz = 0

where: Fy = 0   at   y = -h

and U2 Fxx + 2 UFxt +Ftt + gFy =0 at y=0

In the moving object or moving pressure point we need to use the Fourier or Laplace transform method to help us find a solution. The boundary value problem for the moving pressure point can be formulated as follows:

Fxx + Fyy + Fzz = 0

where: Fxx +Fy = 0   at   y = 0, x and z cannot = 0

and F is finite as y approaches infinity.

We look for a simple solution in a steady flow, for which everywhere at y=0 except at x = z =0 the pressure vanishes. Eventually, we would like to find a solution of this boundary value problem in the form of a Fourier integral in x:

F(x,y,z)=  (2{pi})-1 int[-oo to oo]  e-iax F(a, y, z) da

where F(a,y,z)=  integral e -iax F(x,y,z) dx

Thus, the boundary value problem for G(x,y,z) is:

Gyy + Gzz - a2G= 0

where: -a2G + Gy = 0   at   y = 0

and G is finite as y approaches infinity

This has the solution:

G= ea2y f(z)

Where F(z) safisfies the equation:

(a4 - a2)f+fzz= 0

Thus, there is a solution which looks like:

F(x,y,z)=  A (2{pi})-1 int[-oo to oo]  exp {-iax +a2y +ia(a2-1)1/2z}da

The free surface elevation is:

h(x,z)=  Ai(2{pi}U)-1 lim(y goes to 0) int[-oo to oo] {aexp (a2y) exp (-i(ax +a(a2-1)1/2z)}da

The phase of the wave is very important. The phase of the wave is given by:

Y= -ax + (cos s)-1(a2-1)1/2z