For a first book, I very much enjoyed Introduction to Wave Phenomena, By Hirose. This book uses very simple English to introduce waves at a level appropriate for a 2nd or 3rd year undergraduate. As a non-traditional physics graduate student, this book took me a week or so to master, and was very enjoyable.
For a second book, The Physics of Vibrations and Waves by Pain does a nice job of covering everything in an introductory matter with a strong rigor. After covering Hirose, this book took a few weeks to cover on my own.
The most advanced text on this topic, in my opinion, is Oscillations and Waves in Linear and Nonlinear Systems by Rabinovich. I think that if a person can understand half of what is written in this book, they will very well prepared indeed. Beware: In the Soviet Union, sometimes the book reads you.
A. Hirose and K.E. Lonngren, “Introduction to Wave Phenomena,” (Wiley Interscience, New York, 1985). Corrected edition (1995).
Pain, H. J., & Roelofs, L. (1976). The Physics of Vibrations and Waves, 5th ed.. The physics of vibrations and waves /. Wiley. (Newer Editions are available)
M. I. Rabinovich and D. I. Trubetskov, Oscillations and Waves in
Linear and Nonlinear Systems. Boston, MA: Kluwer, 1989.