Time domain formulation for pulse propagation including nonlinear behavior at a caustic

This paper derives a simple three-dimensional time domain formulation for small angle propagation of finite amplitude acoustic pulses and weak shocks in a medium that can be range dependent. The formulation is closely related to a full wave model, and does not contain the artificial singularities of ray-based models at caustics. The simplicity of the formulation suggests that it may be of value for broadband linear as well as nonlinear propagation. Two derivations are given: one heuristic and the other a formal series expansion from the fluid equations. The result is a first-order nonlinear progressive wave equation (NPE) cast in a wave-following frame of reference.The NPE is shown to be the nonlinear time domain counterpart of the frequency domain parabolic wave equation (PE). The NPE gives a natural separation of terms governing refraction, spreading, and nonlinear steepening. Numerical calculations (not involving Fourier synthesis) are presented using the NPE to simulate the following cases: (a) broadband linear pulse propagation in a waveguide; (b) the development of an initially smooth nonlinear pulse into an N wave; and (c) the behavior of a weak shock at a caustic. Cases (a) and (b) are compared with analytic solutions.