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Circular membrane, PDE, separation of variables, coefficients.

  1. Nov 20, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    I must find the oscillations of a circular membrane (drum-like).
    1)With the boundary condition that the membrane is fixed at r=a.
    2)That the membrane is free.



    2. Relevant equations

    The wave equation [itex]\frac{\partial ^2 u }{\partial t^2 } - c^2 \triangle u =0[/itex].
    Separation of variables, [itex]u(r, \theta , t ) = R(r)\Theta (\theta ) T(t)[/itex].
    3. The attempt at a solution
    I've basically followed wikipedia's article http://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane#The_general_case and reached exactly the same solution, namely [itex]u_{mn}(r, \theta, t) = \left(A\cos c\lambda_{mn} t + B\sin c\lambda_{mn} t\right)J_m\left(\lambda_{mn} r\right)(C\cos m\theta + D \sin m\theta)[/itex]. This would be the eigenfunctions. m goes from 0 to infinity (and is an integer) and n goes from 1 to infinity and is an integer too. Also [itex]\lambda _{mn}[/itex] is the n-th root of the Bessel function of the first kind of order m divided by the radius "a".
    Now the solution that satisfies the boundary condition [itex]u(a ,\theta , t )=0[/itex] should be an infinite linear combination of the eigenfunctions. The problem is that I am not sure of the following:
    [itex]u(a, \theta , t )=0 \Rightarrow \sum _{m=0}^\infty \sum _{n=1}^\infty \left(A_{mn}\cos c\lambda_{mn} t + B_{mn}\sin c\lambda_{mn} t\right)J_m\left(\lambda_{mn} a\right)(C_{mn}\cos m\theta + D_{mn} \sin m\theta)=0[/itex], [itex]\forall t[/itex] and [itex]\forall \theta[/itex].
    In other words I am not sure whether there are so many constants and if there are 2 infinite series as I believe.
    If this is right, I'd like some tip to get all those [itex]A_{mn}[/itex]'s, [itex]B_{mn}[/itex]'s, [itex]C_{mn}[/itex]'s and [itex]D_{mn}[/itex]'s. I do not see any trick to get them.

    Edit: The infinite series result is pretty obvious because [itex]J_m(\lambda _{mn} a)[/itex] is the Bessel function evaluated in its zero, which gives zero. And this term appears in every term of the infinite series. Thus I do not know how to get the constants I'm looking for.
    I've no idea how to find them.
     
  2. jcsd
  3. Nov 21, 2012 #2

    haruspex

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    Given that [itex]\lambda _{mn}[/itex] is the n-th root of Jm divided by the radius "a", what would the value of [itex]J_m\left(\lambda_{mn} a\right)[/itex] be?
     
  4. Nov 21, 2012 #3

    fluidistic

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    0 like I wrote in the edit part of my 1st post? This would imply that any value for A_{mn}, B_{mn}, etc. would work. Is this right?
     
  5. Nov 21, 2012 #4

    haruspex

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    Yes. (When did you do that? I don't think it was there when I made my post.)
    Yes, all the possible values of the constants are solutions.
     
  6. Nov 21, 2012 #5

    fluidistic

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    Probably 2 minutes after writing my post.
    Oh wow, I'm surprised. I guess I should not and this basically mean that the waves can have any amplitude or something like that.
    Thanks for the comment, I did not know.
     
  7. Nov 21, 2012 #6

    haruspex

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    Ok - I must have skimmed over it in reading. Sorry about that.
     
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