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Scattering Problem of Sound Waves

Scattering Problem of Sound Waves

Explanation1

The canonical problem of scattering theory is, given the incident field uiu^{i}, to find the scattered field usu^{s} in the situation where the total field is as per u=ui+usu = u^{i} + u^{s}. For acoustic waves, it is assumed that the incident field is given as the following time-harmonic plane wave.

ui(x,t)=ei(kxdωt),xR3 u^{i} (x,t) = e^{i(k x\cdot d - \omega t)},\quad x\in \mathbb{R}^{3}

Here, k=ωc0k = \dfrac{\omega}{c_{0}} denotes the wavenumber, ω\omega denotes the frequency, c0c_{0} is the speed of the sound wave, and dR3d \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.

Δu(x)+k2u(x)=0 \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 research2.

In the Absence of Obstacles

The simplest problem for an inhomogeneous medium is to find the total field uu that satisfies the following.

Δu+k2n(x)u=0in R3u(x)=eikxd+us(x)limrr(usrikus)=0 \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=xr = \left| x \right|, and n(x)=c02(x)/c2n(x) = c_{0}^{2}(x) / c^{2} is the refractive index, cc is the speed of sound in air. (3)(3) is the Sommerfeld radiation condition, a condition that must be satisfied for the solution to be physically meaningful.

nn is natural to assume that the air is cc=1\dfrac{c}{c}=1 and the medium is finite. Therefore, 1n(x)1-n(x) is 00 in the air, and a positive value in the medium. Hence, 1n1-n has a compact support.

In the Presence of Obstacles

In the case of scattering by an impenetrable obstacle DD, the simplest problem is to find the total field uu that satisfies the following.

Δu+k2u=0in R3Du(x)=eikxd+us(x)u=0on Dlimrr(usrikus)=0 \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)(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.

uν=0on D \dfrac{\partial u}{\partial \nu_{}} = 0 \quad \text{on } \partial D

uν+ikλu=0on D,λ>0 \dfrac{\partial u}{\partial \nu} + i k \lambda u = 0 \quad \text{on } \partial D, \quad \lambda \gt 0


  1. David Colton and Rainer Kress, Inverse Acoustic and Electromagnetic Scattering Theory (4th Edition, 2019), p2-3 ↩︎

  2. Based on the latest edition of the reference textbook (4th Edition, 2019). ↩︎