Analytical solution of the nonlinear equations of acoustic in the form of Gaussian beam

The nonlinear acoustics equation for a dissipative medium is analytically solved. Continuous wave stimulation and an axisymmetric Gaussian spatial profile of the boundary conditions were assumed. The approximation of the D'Alambert operator by the wave diffusion operator was applied and justified. In this approximation and assuming classical absorption (dispersion), the equation to be solved is presented by the Khokhlov-Zabolotskaya-Kuznetsov model. A sequence of functions describing the spatial distribution of the harmonic components of the disturbance was determined. They are the form of spatially modulated Gauss functions for harmonic wave numbers (frequencies). For a lossless medium a universal numerical sequence describing non-linear interactions and harmonic generations was determined. In other cases, the description of the cooperation of dispersion and non-linear interactions in the harmonic generation process is given by a sequence of functions dependent on the dispersion coefficient and with boundary values given by the universal sequence mentioned above. It was unexpectedly discovered that the influence of geometrical parameters of the beam on nonlinear interactions depends on dispersion, and component of the dispersion, absorption may strengthen harmonic generation. In general, dispersion spatially modulates the amplitude and phase of nonlinear interactions. This is not against the law of conservation of energy. The energy exchange between the fundamental (initiating) component and other harmonics is described. The analytical solution was compared with the numerical one. The numerical solution was obtained in the scheme implementing the full Helmholtz operator (no axial - wave diffusion- approximation).