Historical aspects of the Khokhlov–Zabolotskaya equation and its generalizations.

Abstract

Derivation of the Khokhlov–Zabolotskaya (KZ) equation provided a new approach to describing the combined effects of nonlinear propagation and diffraction in sound beams. In this paper, historical aspects of the KZ equation and its generalizations are presented. The interest in nonlinear acoustic beams of Academician Khokhlov and his colleagues at Moscow State University was inspired in the 1960s by emerging developments in laser physics and the corresponding models of nonlinear optical beams. The two cases, acoustical and optical, represent two limiting cases of nonlinear beams in weakly and strongly dispersive media, respectively, which required different theoretical approaches. The KZ equation and analogous nonlinear evolution equations of nonlinear wave physics are reviewed. It is illustrated how theoretical studies combined with numerical modeling resulted in predictions of new physical phenomena in nonlinear acoustic beams. Concurrently, newer applications of nonlinear acoustics such as parametric arrays, sonic booms, and medical acoustics stimulated the derivation of generalized KZ-type equations together with analytical and numerical methods to solve them. Modern applications and corresponding generalized KZ-type models that include effects such as frequency-dependent absorption, weak dispersion, scalar and vectorial inhomogeneities of the propagation medium, different orders of nonlinearity, and more accurate description of diffraction are presented.