Book recommendation for wave motion

Question

What is a good mathematical treatment of wave motion (i.e. the wave equation and SHM)?

Answer 1

For introduction to wave motion I recommend book by J. Billingham Wave Motion. It's introductory, with a lot of good examples and different applications. It covers acoustics, elastic waves, water waves, electromagnetics...

Second is Linear and Nonlinear Waves by G. B. Whitham. Besides "ordinary waves" there you can find traffic flow, kinetic theory, shock and some advanced topics about wave theory.

Radiation of electromagnetic waves is covered in Theory of Electromagnetic Wave Propagation by C. H. Papas.

This books are more about physics of waves but with fair amount of mathematics involved in it. For more mathematical approach you can always consult Colton & Kress for Integral equation methods in scattering theory (BIE for time-harmonic acoustic and electromagnetic waves) or some similar books from the authors.

Answer 2

Depending on what exactly you are interested in, there are of course many different books that cover these areas.

For the physical background of acoustic wave motion, most authors refer to Theoretical Acoustics by Morse and Ingard. It covers the modeling of acoustic wave motion in various medie ((in-)compressible, (non-)viscous)), leading to equations rather different to treat mathematically.

A good introduction in the mathematics of the wave equation is provided in the PDE book by Evans. In a more or less self-contained way, you get an idea of explicit solutions via spherical means as well as a more qualitative analysis by energy methods, also covering the case of variable speed and a wide class of hyperbolic operators.

An important aspect of hyperbolic PDE (and so of the wave equation) is the propagation of singularities. To dive deeper into this topic, a lot of mathematics (pseudodifferential operators ($\Psi$Dos), wave front sets, microlocal analysis) has to be introduced. This is done in the PDE book series by Hörmander, where book I and II cover the constant coefficient case, whereas III and IV use $\Psi$Dos and Fourier integral operators to investigate more general situations.

Finally, if you are interested in time-harmonic problems, I suggest that you take a look at Colton&Kress' standard book about "Inverse Acoustic and Electromagnetic Scattering Theory". While aiming at the inverse scattering problem, a lot of theory regarding the forward problem (=wave motion) is treated. There are also some section about modeling in the homogeneous and heterogeneous media case.