Reddit reviews An Introduction to the Mathematical Theory of Waves (Student Mathematical Library, V. 3)

If you're going for a traditional old school textbook on vibrations and waves, maybe you want the book by A. P. French, or some of the books suggested below.

But if you want a more interesting look on the matter there was a book I loved to read as a physics undergrad called An Introduction to the Mathematical Theory of Waves. It goes on for a while trying to define what is a wave, and what kinds of waves are there, then goes on with a mathematical description of a huge number of wave-like phenomena - from the good old waves on a string and d'Alembert's equation to traffic jams, solitons and non-linear waves.

It's a little heavy on partial differential equations, but it is kind of self contained.

I haven't read this book in a while, but I really loved it when I was an undergrad. Sometimes this means the book is really good, sometimes it just means I wasn't mature enough to judge the book. :P

This will give you some solid theory on ODEs (less so on PDEs), and a bunch of great methods of solving both ODEs and PDEs. I work a lot with differential equations and this is one of my principal reference books.

This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.

This covers PDEs from a very basic level. It assumes no previous knowledge of PDEs, explains some of the theory, and then goes into a bunch of elementary methods of solving the equations. It's a small book and a fairly easy read. It also has a lot of examples and exercises.

This is THE book on PDEs. It assumes quite a bit of knowledge about them though, so if you're not feeling too confident, I suggest you start with the previous link. It's something great to have around either way, just for reference.

Hope this helped, and good luck with your postgrad!