Analytical method for describing the paraxial region of finite amplitude sound beams

A special analytical method, which combines the parabolic approximation (KZ equation) with nonlinear geometrical acoustics, is developed to model nonlinear and diffraction effects near the axis of a finite amplitude sound beam. The corresponding system of nonlinear equations describing waveform evolution is derived. For the case of an initially sinusoidal wave radiated by a Gaussian source, an analytic solution of the coupled equations is obtained for the paraxial region of the beam. The axial solution is expressed in both the time and frequency domains, and is analyzed in detail for both unfocused and focused beams in their preshock regions. Harmonic propagation curves are compared with finite difference solutions of the KZ equation, and good agreement is obtained for a variety of parameter values.