A surface at z = 0 is held at potential V (x, y) = V0 cos(qx) sin(qy). Find the
potential in the region z > 0.
Laplace's equation in Cartesian coordinates
The Attempt at a Solution
I wrote at least a page of my past 2 attempts at a solution. For some reason I was logged out during writing it and when I logged back in, everything was gone....
Anyway to summarize: (note I need only boundaries, I can do the math)
Attempt 1: Make a geometry and assume boundary conditions. That is, find the potential inside this geometry that extends only to points above the origin and with it carries the solution to z > 0
Attempt 2: Solve it directly from laplace's equation.
I think I knw V(x) and V(y) at the surface potential from the problem set up, so all I have to do is find V(z) from laplace's equation.
X(x) = Cos(qx)
Y(y) = sin(qy)
Z(z) = ???
Plugging these in I get:
Z(z) = e^(-qz) or e^(qz)
But these must have some coefficients and I suppose I can work backwards and find the boundary conditions that satisfy general solutions to each 2nd order Diff Eq with these potentials. Then find the coefficients from Fourier's trick.
I have been working on this for days....Any help would be greatly greatly and I mean greatly appreciated!