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

  1. Jul 8, 2010 #1
    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:

    [tex]\nabla[/tex]2u(x,y) + [tex]\lambda[/tex][tex]\frac{1}{(c(x,y))^2}[/tex]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*[tex]\sqrt{\lambda}[/tex], 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!!
     
  2. jcsd
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