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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.

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I have no experience with gas/solid interactions, but I do have a couple thoughts that may or may not be correct.

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.

If you look at the definition of bulk modulus, you can see how to convert between strain (and the resulting changes in volume) and pressure. In fact, for isothermal compression of a gas, it looks like the bulk modulus is simply the gas pressure. It's a little weird to think of air being strained, but I don't think it causes any problems with the physics.

That said, the bulk modulus of air is going to be decidedly non-linear. (Think about how volume and pressure are related in the ideal gas law.) The non-linearity won't be a problem if the changes in pressure are relatively small (e.g. if you shout at the ground), but an explosion will cause big changes in pressure and temperature -- and thus also big changes in the bulk modulus.

Shock physics is quite complicated for exactly those reasons. Nevermind a gas-solid interface in an explosion, it's probably complicated to model the gas alone in an explosion because you're dealing with rapid and large changes in pressure and temperature -- not an ordinary sound wave. I doubt the usual wave equation for sound in air will work. If you want to model a realistic situation, you'll almost certainly need to do a computer simulation.

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