Plane sound waves of finite amplitude for intermediate Gol’dberg numbers

Abstract

Burgers equation provides a good description of the propagation of plane sound waves of finite amplitude for which nonlinear effects are important. Although an exact solution is known, simple asymptotic approximations are useful since they may provide starting points in the case of more general situations for which no exact solution is known. Approximation solutions of Burgers equation are sought by means of a “mixed asymptotic method” involving both the method of multiple scales and the method of strained coordinates. The main point is the introduction of a new slow variable binding the solution in the region before shock formation with the solution in the old age region. For an emitted monochromatic wave, the method yields a simple and complete description for Gol’dberg numbers up to 20, as it is shown by comparison to exact or other approximated solutions. For higher Gol’dberg numbers, the solution is valid up to the shock formation distance and it can be connected with Fay solution beyond.