Focusing of Finite-Amplitude Cylindrical and Spherical Sound Waves in a Viscous and Heat-Conducting Medium

Abstract

The focusing of cylindrical and spherical pulses of finite amplitude in a medium of small viscosity and heat conductivity has been studied by dividing the region of interest into three parts: converging, interaction, and diverging regions. In the converging region, the flow field is governed by the radial Burgers equation with a small parameter multiplying the term with second derivatives. The method of matched asymptotic expansion is found applicable to this equation with an N wave as an initial condition; a composite solution obtained describes a converging N wave with increasing amplitude and wavelength. The front and rear shocks of the N wave are locally described by Taylor's shock structure. In the interaction region, no small perturbation solution exists for the shocks. However, the flow field between the front and rear shocks satisfies to first order the inviscid linear wave equation, which is solved by the Fourier transform technique. The treatment of the diverging region is similar to that of the converging region, except for trivial changes in the analysis. It is shown that the effect of entropy increase on the failure of the solution at the focus is inconsequential if certain restrictions on the initial strength and wavelength are satisfied.