Elastic wave scattering

Definition

Let $\R^3$ be filled with a homogeneous isotropic linear elastic medium with Lamé parameters $\lambda, \mu > 0$ and density $\rho > 0$. Let $D \subset \R^3$ be a bounded domain with smooth boundary such that $\R^3 \setminus \overline{D}$ is connected, and let $\omega > 0$ be a frequency. Put $c_p = \sqrt{(\lambda + 2\mu)/\rho}$, $c_s = \sqrt{\mu/\rho}$, $k_p = \omega / c_p$ and $k_s = \omega / c_s$. Let $u^{\mathrm{in}}$ be an incident displacement field that solves the time-harmonic Navier equation $$\mu \Delta u + (\lambda + \mu) \nabla (\nabla \cdot u) + \rho \omega^2 u = 0$$ in all of $\R^3$, for example a plane P-wave or S-wave. The scattering problem for the rigid obstacle $D$ asks for a scattered field $u^{\mathrm{sc}}$ such that:

1. $u^{\mathrm{sc}}$ solves the Navier equation in $\R^3 \setminus \overline{D}$.
2. The total field $u = u^{\mathrm{in}} + u^{\mathrm{sc}}$ vanishes on $\partial D$.
3. The parts $u_p = -k_p^{-2} \nabla (\nabla \cdot u^{\mathrm{sc}})$ and $u_s = u^{\mathrm{sc}} - u_p$ satisfy the Kupradze radiation condition $r (\partial_r u_p - i k_p u_p) \to 0$ and $r (\partial_r u_s - i k_s u_s) \to 0$ as $r = \abs{x} \to \infty$, uniformly in the direction.