Fun Wave Eqn, Seperable Solution, Wave Guide

[autobot.d]

Homework Statement

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.

Homework Equations

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!

 

[mighty2000]

Hi there! It looks like you're on the right track. Let's break down the steps to get to the ODE and see if we can find a solution.

Step 1: Start with the wave equation

The wave equation is given by \psi_{xx}+\psi_{yy}-\frac{1}{c^{2}}\psi_{tt}=0. In this case, we are dealing with a wave in a waveguide, so we can assume that the wave is traveling in the z-direction and can be represented as \psi = X(x)Y(y)e^{i(kz-wt)}. Plugging this into the wave equation, we get:

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, we can divide by XY and factor out e^{i(kz-wt)} to get:

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

Step 2: Simplify the equation

In order to simplify this equation, we can assume that k^{2} = \frac{w^{2}}{c^{2}}. This assumption is valid because in a waveguide, the wave travels at a constant speed c, so the relationship between the angular frequency w and the wavenumber k is given by w = ck. With this assumption, our equation becomes:

e^{i(kz-wt)}[ \frac{X''}{X} + \frac{Y''}{Y}] = 0

Step 3: Separating the variables

Now, we can set \frac{X''}{X} = \alpha and \frac{Y''}{Y} = \beta, where \alpha and \beta are constants. This will give us two separate ODEs to solve:

\frac{X''}{X} = \alpha

\frac{Y''}{Y} = \beta

Step 4: Solving the ODEs

Solving these ODEs will give us the solutions for X(x) and Y(y