hear a 3D drum as a solid shape or as a surface

Question

Weyl's 'Can one hear the shape of a drum' is well explained in wiki. I have programmed the Laplacian for a given shape (2D), the "sound" is function of the eigenvalues of the Laplacian.

However, I cannot find any documents explaining the sound of a 3D shape: either as a solid shape or as a surface (void inside). Do researches on acoustics cover this topic?

Answer 1

It is the same idea, just solve the wave equation, but now in 3D:

$$ \nabla^2 u = \frac{1}{c^2}\frac{\partial ^2 u}{\partial t^2} $$

And this is a very active field. Here are some examples

1. Helioseismology: Propagation of acoustic waves in stars, allows you to figure out their internal structure

2. Medical imaging Use of acoustic waves to produce images of biological tissues

3. Reflection seismology Oil/water exploration by using the acoustic wave equation

The list is pretty large actually!