Prediction of Finite-Amplitude Waveform Distortion with Dissipation and Spreading Loss

Abstract

A new technique recently used for solution of the plane-wave Burgers' equation in a lossless medium has led to the development of an approximate analytical solution to Burgers' equation with dissipation for plane, cylindrical, and spherical sinusoidal waves. The solution is valid for the initial propagation zone prior to sawtooth formation for Gol'dberg numbers larger than 5 as indicated by comparison with a previous numerical solution of Burgers' equation for a plane wave. The approximate solution is expressed as a series that converges rapidly enough to permit calculations of harmonic levels by hand. For more viscous cases, where the approximate solution fails, the above technique leads easily to a numerical solution of Burgers' equation. By comparing the harmonic levels predicted by the numerical solution with empirical data taken on distorting spherical waves, it is concluded that numerical results tend to be below the measured values. It is felt that the numerical method can be of use in predicting distortion levels for practical underwater systems that exploit finite-amplitude effects.