Fun Wave Eqn, Seperable Solution, Wave Guide

autobot.d

1. The problem statement, all variables and given/known data
Boundaries at x=0,a y=0,b
This is a waveguide. w = angular frequency and k is the wavenumber.
Have the seperable solution to the wave equation.

[itex]\psi[/itex] = X(x)Y(y)e[itex]^{i(kz-wt)}[/itex]

Where w=c[itex]\sqrt{k^{2}+\pi^{2}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{a^{2}}\right) }[/itex]

I just need help with figuring out how to get this to an ODE and I should be able to figure out the boundary conditions. Thanks.


2. Relevant equations



3. The attempt at a solution

[itex]\psi[/itex][itex]_{xx}[/itex]+[itex]\psi[/itex][itex]_{yy}[/itex]-[itex]\frac{1}{c^{2}}[/itex][itex]\psi[/itex][itex]_{tt}[/itex]=o

Plugging in

X''Ye[itex]^{i(kz-wt)}[/itex] + Y''Xe[itex]^{i(kz-wt)}[/itex] - k[itex]^{2}[/itex]XYe[itex]^{i(kz-wt)}[/itex] + [itex]\frac{w^{2}}{c^{2}}[/itex]XYe[itex]^{i(kz-wt)}[/itex] = 0

Now dividing by XY and factoring out e[itex]^{i(kz-wt)}[/itex]

e[itex]^{i(kz-wt)}[/itex][itex][ \frac{X''}{X}[/itex] + [itex]\frac{Y''}{Y}[/itex] - k[itex]^{2}[/itex] + [itex]\frac{w^{2}}{c^{2}}][/itex] = 0

Does k[itex]^{2}[/itex] = [itex]\frac{w^{2}}{c^{2}}[/itex] necessarily?

Do I set [itex]\frac{X''}{X} = \alpha[/itex] and [itex]\frac{Y''}{Y} = \beta[/itex]
where [itex]\alpha , \beta[/itex] are constants and solve these ODE's. Doesn't seem to get me where I want to go though.

Any advice would be very awesome. Thanks!
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution