Nonlinear effects in focused sound beams

Abstract

Nonlinear effects in focused sound beams are investigated with numerical solutions of the Khokhlov–Zabolotskaya–Kuznetsov (KZK) parabolic wave equation. The KZK equation accounts consistently for the combined effects of nonlinearity, diffraction, and absorption. To improve the efficiency of the calculations, a coordinate transformation is introduced that follows the convergent geometry of the focused beam. The transformation is similar to one introduced earlier to accommodate the divergent geometry of unfocused beams [Hamilton et al., J. Acoust. Soc. Am. 78, 202–216 (1985)]. A Fourier series expansion of the acoustic pressure is used to reduce the transformed KZK equation to a set of coupled parabolic equations that are solved using a finite difference method. Attention is devoted to radiation from circular sources with uniform amplitude distributions and linear focusing gains of order 50. The combined effects of nonlinearity, diffraction, and absorption in the focal region are clearly revealed in both the time and frequency domains. For the range of parameters considered, beam patterns in the focal region are found to be less dependent on source amplitude and absorption than are beam patterns in the farfield of unfocused sources.