Appropriate boundary conditions of the wave equation

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!

• If you are considering viscous fluid, then you need to impose condition on the tangential component of the velocity field. Feb 12, 2019 at 1:59
• But here my velocity potential $\phi$ is independent of $\theta$, meaning that the velocity field has no tangential component? Feb 12, 2019 at 3:01
• I am not familiar with compressible fluid, but $\phi$ is not necessarily independent of $\theta$ at hindsight. I assume you take no-penetration on $r=a$, which means the radial component of the fluid velocity is 0; this doesn't necessarily imply that $\phi$ is independent of \$\theta. Anyway, at this point I'm not even sure what slip condition (or if we even have slip?) one should impose on the problem, I was thinking of the Milne-Thompson Circle Theorem before but wasn't sure if the theorem is applicable to compressible flow too. It might be worth checking that out. Feb 12, 2019 at 7:26
• I should also probably mentioned that my suggestions are from the fluid dynamics perspective (: Feb 12, 2019 at 10:18