A time domain computer algorithm for solving the KZK equation

Abstract

A time domain computer algorithm for solving the KZK (Khokhlov-Zabolotskaya-Kuznetsov) nonlinear parabolic wave equation is being developed for pulsed, axisymmetric, finite-amplitude sound beams. The sound beams may be weakly focused. A coordinate transformation is introduced in the KZK equation [Hamilton et al., J. Acoust. Soc. Am. 78, 202–216 (1985)], and the resulting parabolic equation is integrated once analytically with respect to time. The diffraction and thermoviscous dissipation terms are integrated numerically with implicit backward finite-difference methods, and the Earnshaw solution [Pestorius and Blackstock, J. Acoust. Soc. Am. 53, 383 (1973)] is used to account for the nonlinearity. Weak shock theory may be incorporated in the algorithm. Numerical results for several limiting cases are compared with analytical solutions. Comparisons are also made with numerical results obtained with a frequency domain computer algorithm for solving the KZK equation [see, e.g., Berntsen, in Frontiers of Nonlinear Acoustics—12th ISNA, edited by Hamilton and Blackstock (Elsevier, London, 1990), pp. 191– 196]. [Work supported by the Office of Naval Research and the Texas Advanced Research Program.]