When dealing with wave propagation problems such as electromagnetic waves passing from one medium to another we set up boundary conditions to ensure the field is continuous and the flux is continuous as the waves pass from the first medium into the other.

Similarly in the case of acoustic waves passing from, say, a region of fluid into a region of gas. We set boundary conditions so that the field is continuous and the flux is continuous. For example, if we let $k_f$ and $k_g$ represent the wavenumbers in fluid and gas, respectively, we can model a transmission problem for a fluid-gas system with the Helmholtz equation, where the fluid occupies $\mathbb{R}^3\backslash \overline{\Omega}$ and the gas occupies $\Omega$, as follows:

\begin{align} \Delta p + k_f^2p = 0 & \quad x \in \mathbb{R}^3\backslash \overline{\Omega}\\ \Delta p + k_g^2p = 0 & \quad x \in \Omega\\ p_+ = p_- & \quad x \in \partial \Omega \\ \frac{1}{\rho_f}\frac{\partial p}{\partial \nu}\bigg|_+ = \frac{1}{\rho_g}\frac{\partial p}{\partial \nu}\bigg|_- & \quad x \in \partial \Omega \end{align} with the scattered wave $p^s = p - p^{inc}$ obeying the Sommerfeld radiation condition as $|x| \to \infty$, and $\rho$ represents the density in the fluid/gas.

Now what about the case where we have acoustic wave in a solid-fluid system? In this case we have two fundamentally different systems in each region as while the fluid region is governed by the Helmholtz equation like above, in the solid the wave propagation is governed by the linear elasticity while involves longitudinal waves as in the fluid case, but also shear waves.

So how do we model wave propagation in this fluid-solid case in which the governing equations are different in the fluid and solid regions? And in particular, how can we handle transmission boundary conditions?


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