An exact solution for finite-amplitude plane sound waves in a dissipative fluid

Abstract

The propagation of finite-amplitude plane sound waves in a dissipative fluid can be described by Burgers’ equation, and its exact solution obtained. In this paper, an exact solution for the sound pressure, which is suitable for direct numerical computation of the waveform in the time domain, is derived. Numerical results illustrate that this solution describes the entire propagation process, including shock formation and decay. Computation limits are determined in connection with the computational system. It is found that computation is possible for D0>0.0114 and any value of X with the present system, where D0 indicates the importance of dissipation relative to nonlinearity and X is distance normalized by the lossless shock formation distance. It is also found that the solution for the limiting value of D0 connects smoothly with the Fubini solution for X<1 and with the Fay solution for X>3.5. Since the present solution is exact and yields the waveform at any distance without introducing the Fourier series expansion of finite terms, it can serve as a standard solution for various approximate methods. As examples, changes in the energy density and the saturation phenomena are shown.