# Fun Wave Eqn, Seperable Solution, Wave Guide

by autobot.d
Tags: guide, seperable, solution, wave
 P: 68 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. $\psi$ = X(x)Y(y)e$^{i(kz-wt)}$ Where w=c$\sqrt{k^{2}+\pi^{2}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{a^{2}}\right) }$ 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 $\psi$$_{xx}$+$\psi$$_{yy}$-$\frac{1}{c^{2}}$$\psi$$_{tt}$=o Plugging in X''Ye$^{i(kz-wt)}$ + Y''Xe$^{i(kz-wt)}$ - k$^{2}$XYe$^{i(kz-wt)}$ + $\frac{w^{2}}{c^{2}}$XYe$^{i(kz-wt)}$ = 0 Now dividing by XY and factoring out e$^{i(kz-wt)}$ e$^{i(kz-wt)}$$[ \frac{X''}{X}$ + $\frac{Y''}{Y}$ - k$^{2}$ + $\frac{w^{2}}{c^{2}}]$ = 0 Does k$^{2}$ = $\frac{w^{2}}{c^{2}}$ necessarily? Do I set $\frac{X''}{X} = \alpha$ and $\frac{Y''}{Y} = \beta$ where $\alpha , \beta$ 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