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

[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 a lot of work, I think I'm just not understanding something...

Homework Statement

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??

Homework Equations

2-d wave equation

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!

 

[kurt.physics]

The answer is that the frequency of vibration associated with each eigenvalue is given by f=\sqrt{\lambda/\rho}, where \rho is the effective mass density, which is defined as \rho = \int c^{-2}(x,y) \mathrm{dA}, where dA is the area element. This formula follows from the definition of the frequency of vibration in terms of the effective mass density and the eigenvalue.