Sound wave inside a closed cylinder - Bessel function

[ArkadyK]

Homework Statement

The question is as follows, there is a cylinder with length L and radius R, there is a sound wave with a phase velocity v, they ask for the normal modes and the 5 lowest frequencies when L=R

Homework Equations

Wave equation for 3D, (d^2/dt^2)ψ=v^2*(∇^2)ψ

The Attempt at a Solution

I have used the wave equation and derived an expression for ψ as function of t,ϑ,z,ρ using bessel formula.
the expression I derived is: Ψ=e^(-iωt)e^(inϑ)e^(ikz)*J(κρ)
I wouldn't like to post all the derivation here (it will take hours to type everything), but the idea was clear, assume that ψ is a function of R,Θ,Z,T which are all independent of one another, and insert into the wave equation.
I`m not sure how to continue, what initial conditions should I use?
Hopefully that was clear enough, that`s my first post, so I hope i did it well :)

 

[diazona]

Normal modes themselves are only the patterns in which the system naturally oscillates. You don't use initial conditions to determine the normal modes themselves; instead, initial conditions would tell you how the system's energy is distributed among the normal modes.

As far as how to continue, it seems that you haven't actually used the 3D wave equation yet. You can tell that you have to use that at some point because (for one thing) the answer has to depend on the phase velocity of the wave somehow, and the only place that comes in is the wave equation. Right now the coefficients $\omega$, $n$, $k$, and $\kappa$ are arbitrary; using the wave equation should give you a relation between those coefficients for a given normal mode.