Let's consider the one-dimensional string first. Standard text-book physics considers the three usual boundary conditions here, namely Dirichlet (endpoints of the string are fixed), Neumann (endpoints are free) and Robin conditions (obtained e.g. when the endpoints are attached to a spring).
Now, for real sonic propagation, those boundary conditions won't ...
I would suggest Introduction to Numerical Methods for Variational Problems by Langtangen and Mardal.
Besides that, you could check:
I would also suggest our own course
Introductory Finite Elements. It has Notebooks with the material and lecture notes;
and our own package SolidsPy.