"The potential in the x-z plane is independent of z and given by a repeating step-function of magnitude 2(phi_0) and period 2a. The plane at y = y_0 is held at ground potential. Find the potential in teh region 0 < y < y_0." - Marion and Heald Classical Electromagnetic Radiation 3rd ed.
d2(X)/dx2*X^-1 + d2(Y)/dy2*Y^-1 = 0
The Attempt at a Solution
X(x) must be oscillatory, and I can determine X(x) by standard Fourier analysis of square wave.
If X(x) is oscillatory, d2(X)/dx2 = -k1* X -> d2(Y)/dy2 = +k1*Y, to meet laplacian(phi) = 0.
Thus, Y(y) is a linear combination of growing and decaying exponential functions in y. But, this can never satisfy Y(y_0)=0. This is my problem, am I missing something important?