# Frequency of vibration for non-uniform membrane - Partial Differential Equations

1. Jul 8, 2010

### ThomBoh

Frequency of vibration for non-uniform membrane -- Partial Differential Equations

Hi guys, i'm having a lot of trouble with a conceptual problem in my PDE's homework. I don't think the answer involves alot of work, I think I'm just not understanding something...

1. The problem statement, all variables and given/known data

Consider the 2-D wave equation with c (speed) being non-uniform across the membrane (whose shape is arbitrary). That is it is a function of x and y => c = c(x,y).

what is the frequency of vibration associated with each eigenvalue??

2. Relevant equations

2-d wave equation

3. The attempt at a solution

I've shown in a previous part of the question that after you separate out the time dependence of the solution, you're left with a 2-d spatial sturm-liouville problem of this form:

$$\nabla$$2u(x,y) + $$\lambda$$$$\frac{1}{(c(x,y))^2}$$u(x,y) = 0

where u(x,y) is the vertical displacement of the membrane at (x,y).

This PDE isn't seperable so I really don't think it's asking me to solve it to find the frequencies. It wants the nth frequency in terms of the nth eigenvalue lambda. I know that if it were uniform, the frequency should be c*$$\sqrt{\lambda}$$, but i doubt that's the case for spatially varying c... right? I really feel like i can't connect the dots on this one...

thanks for any suggestions you can offer!!