There are well-documented methods of calculating the impulse response of a room (ex. image method, ray-tracing method); however, I have not been able to find anything similar for the impulse response of an arbitrary solid given its size, density, etc. Can anyone point me in the right direction?
The exact answer can easily be computed by solving the elastic wave equation using the finite-difference time-domain method.
You mentioned the image and ray-tracing methods for room acoustics. Theses are geometric methods, which would not be useful for solids since there would need to be separate rays to describe the compressional and shear waves, and the transfer of energy between the two kinds of waves which is complicated.