I am having trouble solving my first wave equation on the unit disk. The problem that we are posed is to consider a membrane covering the unit disk: $$ D =\{(x,y) \in \mathbb{R}^2: x^2+y^2\leq 1 \} $$ A vibration across the membrane solves the following problem: $$ u_{tt}-\Delta u=0 \;\;\;\;\;\;\forall (t,x,y)\in(0,\infty)\times D\\u(0,x,y)=g(x,y)=g(\sqrt{x^2+y^2}), \;\;u_t(0,x,y)=0 \;\;\;\forall(x,y)\in D\\u(t,x,y)=0\;\;\;\;\forall(t,x,y)\in (0,\infty)\times\partial D $$
We are meant to use separation of variables and due to the nature of the domain, I also attempted using polar coordinates. So first I supposed $u(t,x,y)=w(t,r,\theta)$ where $w$ is just the solution in polar. I then supposed that $w$ was separable and therefore of the form $w(t,r,\theta) = R(r)T(t)\Lambda(\theta)\\$.
Substituting this in to the PDE, I got the following ODEs:
$$ \frac{T''}{T}=-k\\ \frac{\Lambda ''}{\Lambda}=-\gamma\\ \frac{R''}{R}+\frac{R'}{rR}+k-\frac{\gamma}{r^2}=0 $$ where $\gamma, k$ are constants. I then got that$$ \Lambda(\theta)=\sum_n A_n \cos(n\theta) + B_n \sin(n\theta) $$ where $n^2 = \gamma$ and using this as well as the change of variables $\rho = \sqrt{k}r$, I derived a Bessel equation (something we have not covered in class so not sure if this is the right route) of the form $$ R'' + \frac{1}{\rho}R' + (1-\frac{n^2}{\rho^2})R =0 $$ I tried to solve this using the power series method that I found online but no luck. I did find out that the solution to this equation is the Bessel function of the first kind $J_n(\sqrt{k}r)$ which has form $$ \sum_{j=0}^{\infty}\frac{(-1)^j(\frac{1}{2}\sqrt{k}r)^{n+j}}{j!(n+j)!} $$ The I solved the final ODE which gave $$ T(t) = A_m\cos(\sqrt{k_{mn}}t)+B_m\sin(\sqrt{k_{mn}}t) $$ Combining all of this together you get the very ugly (and most probably incorrect solution) $$ w(t,r,\theta) = \sum_{n=0}^{\infty}\sum_{m=1}^{\infty} J_n(r \sqrt{k_{mn}})\cos(\sqrt{k_{mn}}t)(A_{nm}\cos(n\theta)+B_{nm}\sin(n\theta))+\sum_{n=0}^{\infty}\sum_{m=1}^{\infty} J_n(r \sqrt{k_{mn}})\sin(\sqrt{k_{mn}}t)(C_{nm}\cos(n\theta)+D_{nm}\sin(n\theta)) $$ But now I have a LOT of constants to derive the value of. So I tried to sub in my initial conditions. Firstly, at time $t=0$ we know the solution depends only on $r$ so I subbed in $t=0$ and I get $$ g(r) =\sum_{n=0}^{\infty}J_n(r\sqrt{k_{mn}})A_{nm} $$ However, I see no way to get the $A_{mn}$ in terms of $g$ from this. Also using the other initial conditions didn't seem to help either.
Have I done anything right, am I down the wrong route entirely? How does one actually get the solution to the Bessel equation and what should the final solution look like for this question? Any help or hints would be very much appreciate. Thank you :)