On the evolution of a spherical short pulse in nonlinear acoustics

Abstract

Planar wave propagation in nonlinear acoustics is modeled by the Burgers equation, which is exactly soluble. Spherical wave propagation is modeled by a generalized Burgers equation, in which the dissipative parameter of the plane wave Burgers equation is replaced by an exponentially growing function of the variable symbolizing the travelled length of the wave. A procedure previously used in 1998 by B.O. Enflo [1] on cylindrical short pulses is now used on spherical short pulses, which are originally N-waves. The procedure consists of the four steps: 1) A shock solution of the generalized Burgers equation is found by asymptotic matching. The shock fades in the region where the nonlinear term in the equation can be neglected. 2) The linear equation in step 1) is rescaled. It is identically solved by an integral representation containing an unknown function. 3) The integral representation found in step 2) is evaluated by the steepest descent method in the fading shock region introduced in step 1). The unknown function introduced in step 2) is determined by comparing the result of this evaluation with the fading shock solution found in step 1). 4) The integral representation with the unknown function determined is evaluated approximately asymptotically for large values of the original length (or time) variables in the original generalized Burgers equation (old-age regime). The result of this procedure is an old-age solution, controlled by numerical calculations. Curves of analytical and numerical solutions are given.