Here, I have some acoustic waves generated in a compressible fluid by small oscillations of a cylinder with boundary at $r=a$.
In this problem, I am only interested in the solution outside of the cylinder.
I have worked out that the PDE to be solved is
$$\frac{\partial^2 \phi}{\partial t^2} = c_0^2 \nabla^2 \phi \\ \frac{\partial \phi}{\partial r}=-\varepsilon\omega aie^{-i\omega t} \qquad \qquad \text{ on } r = a$$
where $\phi = \phi(r,\theta,t)$ is the velocity potential and $(r,\theta)$ are plane polar coordinates.
The general solution I have found is
$$\phi(r,\theta,t) = \bigg[AJ_0\bigg(\frac{\omega}{c_0}r \bigg) + BY_0\bigg(\frac{\omega}{c_0}r \bigg)\bigg]e^{-i\omega t}$$
where $J_0$ and $Y_0$ are Bessel functions of the first and second kind (please assume that this general solution is correct).
However, I only have the one boundary condition at $r=a$.
Imposing a boundedness condition as $r\rightarrow \infty$ is useless because $J_0$ and $Y_0$ are already bounded.
Any hints as to what the other boundary condition should be?
What boundary condition is typically imposed when solving the wave equation outside a circle?
Any help would be much appreciated. Thanks!