Projection of the Khokhlov–Zabolotskaya equation on the axis of wave beam as a model of nonlinear diffraction

Abstract

An equation is obtained that describes the nonlinear diffraction of a focused wave in a half-space x > 0 starting from the wave source, then through the focus region up to the far zone, where the wave becomes spherically divergent. In contrast to the Khokhlov–Zabolotskaya equation (KZ), which contains two spatial variables, the calculation of the field on the beam axis is reduced to a simpler one-dimensional problem. Integral relations that are useful for numerical calculation are indicated. The profiles of a periodic wave harmonic at the input to the medium are constructed. A comparison with the results of a numerical solution of problems based on KZ is made, which revealed a good accuracy of the approximate model. The passage of a wave through the focus region, accompanied by the formation of shock fronts, diffraction phase shifts and asymmetric distortion of regions of different polarity, is traced.