# Circular membrane, PDE, separation of variables, coefficients.

1. Nov 20, 2012

### fluidistic

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 $\frac{\partial ^2 u }{\partial t^2 } - c^2 \triangle u =0$.
Separation of variables, $u(r, \theta , t ) = R(r)\Theta (\theta ) T(t)$.
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 $u_{mn}(r, \theta, t) = \left(A\cos c\lambda_{mn} t + B\sin c\lambda_{mn} t\right)J_m\left(\lambda_{mn} r\right)(C\cos m\theta + D \sin m\theta)$. 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 $\lambda _{mn}$ 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 $u(a ,\theta , t )=0$ should be an infinite linear combination of the eigenfunctions. The problem is that I am not sure of the following:
$u(a, \theta , t )=0 \Rightarrow \sum _{m=0}^\infty \sum _{n=1}^\infty \left(A_{mn}\cos c\lambda_{mn} t + B_{mn}\sin c\lambda_{mn} t\right)J_m\left(\lambda_{mn} a\right)(C_{mn}\cos m\theta + D_{mn} \sin m\theta)=0$, $\forall t$ and $\forall \theta$.
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 $A_{mn}$'s, $B_{mn}$'s, $C_{mn}$'s and $D_{mn}$'s. I do not see any trick to get them.

Edit: The infinite series result is pretty obvious because $J_m(\lambda _{mn} a)$ 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. Nov 21, 2012

### haruspex

Given that $\lambda _{mn}$ is the n-th root of Jm divided by the radius "a", what would the value of $J_m\left(\lambda_{mn} a\right)$ be?

3. Nov 21, 2012

### fluidistic

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?

4. Nov 21, 2012

### haruspex

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.

5. Nov 21, 2012

### fluidistic

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.

6. Nov 21, 2012

### haruspex

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