The physical problem I'm trying to solve is this: I would like to find the "reflection" of a harmonic plane sound wave in a liquid, from a spherical air bubble. I'm modeling the problem as follows: The sound (pressure) wave is a solution to the wave equation $$ (\partial_t^2 - c^2\nabla^2) p = J$$ (where $J$ is a possible source term) on $\mathbb{R}^3 \setminus B(0, R)$, i.e. on all of space minus a ball with radius $R$. The boundary condition is $p(x) = 0$ where $|x| = R$. The harmonic sound wave is traveling along the $x_3$-axis and may be written $$p_0(x, t) = e^{i (k x_3 - \omega t)}, \qquad k := \frac{\omega}{c}.$$
I'm not sure where to go from here. My dilemma is whether the incoming plane wave is an initial condition or whether it comes in as the source term $J$.
I know the Green's function for the wave equation on $\mathbb{R}^3$, and I believe I could use the method of images to find a Green's function for my boundary conditions. Then, if I knew the source $J$ for the plane wave, I could find the solution via a convolution. So my question is, what kind of source $J$ corresponds the plane wave above?
Secondary question: Is there a smarter way to solve this problem?
The correct way to formulate problem is to solve the differential equation $$(\partial_t^2 - c^2 \nabla^2) p = 0$$ with the boundary condition $p(\mathbf{x}, t) = 0$ for $|x| = R$ and in addition the following boundary condition at infinity: $$ p(\mathbf{x}, t) \rightarrow p_0(\mathbf{x},t) \qquad \text{as } |\mathbf{x}| \rightarrow \infty $$
We see that the solution will be of the form $p(\mathbf{x}, t) = \psi(\mathbf{x}) e^{i\omega t}$, where $\psi$ satisfies the Helmholtz equation $(\nabla^2 + k^2) \psi = 0$. Write $\psi(\mathbf{x}) = \phi(\mathbf{x}) + e^{ik x_3}$. Then we want $\phi$ to vanish at infinity and satisfy $\phi(\mathbf{x}) = - e^{ikx_3}$ at the surface of the sphere. The general solution which describes an outgoing wave is $$ \phi(\mathbf{x}) = \sum_{l, m} c_{lm} h_l^{(1)}(kr) Y_{lm}(\theta, \phi).$$ where $h_{l}^{(1)}$ is a spherical Hankel function of the first kind and $Y_{lm}$ is a spherical Harmonic. This function vanishes as $|\mathbf{x}| \rightarrow \infty$ because $\lim_{z \rightarrow \infty} h_l^{1}(z) = 0$. Thus we only need to choose the coefficients $c_{lm}$ to make the solution $\psi = \phi + e^{ikx_3}$ vanish at the surface of the sphere. This is done by expanding the plane wave as follows: $$e^{ikx_3} = \sum_{l = 0}^{\infty} i^l (2l+1) j_l(kr) P_l(\cos\theta)$$ where $j_l$ is a spherical Bessel function and $P_l$ is a Legendre polynomial. By using the orthogonality of the spherical harmonics, we find that $c_{lm} = 0$ for $m > 0$ and $$c_{l0} = - \frac{(2l+1) i^l j_l(ka)}{h_l^{(1)}(ka) \sqrt{\frac{2l+1}{4 \pi}}}.$$
I hope this helps someone.