Asymptotically unified exact solutions of the Khokhlov–Zabolotskaya equation

Abstract

For the large distance traversed by the sonic beam the exact analytical and physically realistic solutions of the Khokhlov–Zabolotskaya (KZ) equation were found for the first time. The procedure of finding these solutions as the self-similar traveling waves identifies their asymptotical generality with respect to the initial arbitrary wave profiles. The resulting solutions consist of the periodically repeating parts of the second-order curvers such as hyperbolas, ellipses, and, in particular, parabolas. The boundaries between these parts are defined with the use of the preserved integrals of the KZ equation. The influence of nonlinearity, diffraction, which is seen in many-dimensional cases, and initial curvature of the phase front of the wave beam on the forming of the typical form of a specific asymptotical wave profile (N shaped or U shaped) was discussed. A comparison was made between analytical solutions with the numerical results for the KZ equation and the experimental data and a full agreement was discovered. These solutions play the same role among all the variations of solutions of the KZ equation, such as the well-known sawtoothlike wave for the equation of the simple waves (Riemann’s equation). [Work supported by CRDF.]