# Wave equation on Air- Solid interaction

Suppose that there is, due to an explosion $h$ meters above the ground, a wave in the air with high density, velocity and pressure, capable of inducing an elastic wave on the earth's surface. How does this air wave produce a wave in the ground, that is, how can we represent the wave equation on the solid, as a function of displacement ($\rho \vec{\ddot{u}}=(\lambda +\mu)\nabla\nabla\vec{u}+\mu\nabla^2\vec{u}$)?

First, air is a gas and earth is an elastic solid and the interaction occurs only on the earth's surface, does that mean that the stress tensor on the surface of the solid only has the $\tau_{z,i}$ non-zero components? However, the pressure in the air can also be represented as a tensor, whose only non-zero components are $\sigma_i$.

Second, it is common to describe a wave in the air, as a function of its pressure. But, on the solid, it may be more appropriate to write the wave as a function of displacement (whose stress tensor can be easily discovered by the relation $\tau_{i,j}=\partial_ju_i$). Is there a way to reach a relationship between these two concepts of wave?

Any help would be much appreciated, specially with bibliographic references.

I'm guessing that you can model air as an elastic material with a non-zero bulk modulus but zero shear modulus. If that's so, you can use the same wave equation (with position-dependent $\mu$ and $\lambda$) for the air and elastic material, and you don't have to worry about matching two different wave equations and the air/solid interface.