Time-domain solution of the parabolic equation including nonlinearity

Abstract

This paper develops a two-dimensional (range and depth) formulation for small-angle propagation of nonlinear acoustic pulses and weak shocks in a refracting medium which can be range dependent. The formulation readily extends to three dimensions. The nonlinearity is most efficiently treated in the time domain, so the signal is not Fourier decomposed into component frequencies. Beginning with a second-order wave equation which includes the lowest-order nonlinearity, an approximate first-order nonlinear progressive wave equation (NPE) is derived. The NPE is the nonlinear time-domain counterpart of the linear frequency-domain parabolic equation (PE). The derivation is accomplished by transforming the wave equation to a pulse-following frame and perturbing about a unidirectional plane-wave signal. The NPE manifests a natural separation of terms governing refraction, diffraction, spreading and nonlinear steepening. Numerical methods are outlined for the solution of the time-domain problem. Calculations using the formulation developed here successfully follow the development of initially smooth pulses into N waves and the reflection of weak shocks from a caustic.