I am trying to understand what happens on the boundary between an elastic material and an acoustic one. I am trying to understand it in two dimensions:

Assume that we have a square separated made of two homogeneous materials one of them is elastic and the other one is acoustic and they are separated by a curve $\Gamma$.

To simplify the problem, we will assume the square $S$ mentioned above is the unit square and that $\Gamma$ is the line $y=1/2$. $S_a$ is the part above $\Gamma$ where the material is acoustic and $S_e$ is the lower part.

Now, I know that the elastic wave equation is $$\begin{cases}\frac{\partial E}{\partial t}&=\frac{1}{2}\left(\nabla v+\nabla v^{T} \right)\\ \rho \frac{\partial v}{\partial t} &= \nabla\cdot \left(\lambda \text{tr}(E)+2\mu E\right)+f\end{cases}$$ Where $(\lambda, \mu)$ are Lamé's parameters. and for the acoustic we just put $\mu=0$.

  1. Is those equations correct?
  2. how does the strain tensor $E$ relates to the pressure in the acoustic case.
  3. What conditions should we expect on $\Gamma$? for example $v$ will be continuous, how about $E$?

This is not a homework (someone just flagged it as homework), this is a part of a research I am doing on PDE solvers, I want to apply DG methods to such problems but I lack the necessary physics background.


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