Quantum mechanical theory of nonlinear interaction of ultrasonic waves

Nonlinear interaction of ultrasonic waves is discussed from the very general quantum mechanical viewpoint. By using the concept of three-phonon interactions one is able to derive formulae for power and intensity of ultrasonic waves generated through nonlinear mixing of two ultrasonic waves, as well as the formula for parametric amplification of an ultrasonic wave through its nonlinear interaction with another ultrasonic wave. The effect of attenuation on nonlinear ultrasonic wave interactions also is discussed, and it is shown that in some situations attenuation strongly affects the generation of the second harmonic of an initially sinusoidal ultrasonic wave. It is shown that in the correspondence limit, the quantum mechanical viewpoint gives results which are in very good agreement with results obtained from classical physics.