Scattering Problem of Sound Waves
Explanation1
The canonical problem of scattering theory is, given the incident field $u^{i}$, to find the scattered field $u^{s}$ in the situation where the total field is as per $u = u^{i} + u^{s}$. For acoustic waves, it is assumed that the incident field is given as the following time-harmonic plane wave.
$$ u^{i} (x,t) = e^{i(k x\cdot d - \omega t)},\quad x\in \mathbb{R}^{3} $$
Here, $k = \dfrac{\omega}{c_{0}}$ denotes the wavenumber, $\omega$ denotes the frequency, $c_{0}$ is the speed of the sound wave, and $d \in \mathbb{R}^{3}$ is a vector representing the direction of propagation of the sound wave. Furthermore, for simplicity, let’s deal with the following Helmholtz equation where the time term is removed, instead of the wave equation.
$$ \Delta u (x) + k^{2}u (x) = 0 $$
The following problems are the simplest examples among the physically realistic problems in scattering theory. Despite this, they have unresolved aspects (especially in the numerical part) and are currently a topic of research.2
In the Absence of Obstacles
The simplest problem for an inhomogeneous medium is to find the total field $u$ that satisfies the following.
$$ \begin{align} \Delta u + k^{2} n(x) u = 0 \quad \text{in } \mathbb{R}^{3} \\[1em] u(x) = e^{i k x \cdot d} + u^{s}(x) \\[1em] \lim \limits_{r \to \infty} r \left( \dfrac{\partial u^{s}}{\partial r} - ik u^{s} \right) = 0 \end{align} $$
Here, $r = \left| x \right|$, and $n(x) = c_{0}^{2}(x) / c^{2}$ is the refractive index, $c$ is the speed of sound in air. $(3)$ is the Sommerfeld radiation condition, a condition that must be satisfied for the solution to be physically meaningful.
$n$ is natural to assume that the air is $\dfrac{c}{c}=1$ and the medium is finite. Therefore, $1-n(x)$ is $0$ in the air, and a positive value in the medium. Hence, $1-n$ has a compact support.
In the Presence of Obstacles
In the case of scattering by an impenetrable obstacle $D$, the simplest problem is to find the total field $u$ that satisfies the following.
$$ \begin{align} \Delta u + k^{2}u = 0 \quad \text{in } \mathbb{R}^{3} \setminus \overline{D} \\[1em] u(x) = e^{i k x \cdot d} + u^{s}(x) \\[1em] u = 0 \quad \text{on } \partial D \\[1em] \lim \limits_{r \to \infty} r \left( \dfrac{\partial u^{s}}{\partial r} - ik u^{s} \right) = 0 \end{align} $$
$(6)$ is the Dirichlet boundary condition for a sound-soft obstacle. For a sound-hard obstacle, one could consider the Neumann boundary condition or the Robin boundary condition.
$$ \dfrac{\partial u}{\partial \nu_{}} = 0 \quad \text{on } \partial D $$
$$ \dfrac{\partial u}{\partial \nu} + i k \lambda u = 0 \quad \text{on } \partial D, \quad \lambda \gt 0 $$