This book features a comprehensive discussion of the mathematical foundations of ultrasonic nondestructive testing of materials. The authors include a brief description of the theory of acoustic and electromagnetic fields to underline the similarities and differences with respect to elastodynamics. They also cover vector, elastic plane, and Rayleigh surface waves as well as ultrasonic beams, inverse scattering, and ultrasonic nondestructive imaging. A coordinate-free notation system is used that is easier to understand and navigate than standard index notation.