The Nonlinear Pressure Acoustics, Time Explicit interface solves the nonlinear continuity equation and momentum equation that represent the physics described by the Westervelt equation (see Ref. 7). It is derived assuming that the cumulative nonlinear effects surpass the local nonlinear effects. That is, the contribution of the Lagrangian energy density is negligible:

where pt is the total acoustic pressure, ut is the total acoustic velocity fluctuations, and β is the coefficient of nonlinearity. Because the problem solved is nonlinear the superposition principle does not apply, which means that the “total” fields are always equal to the dependent variables.

Equation 7-10 is a nonlinear hyperbolic system written in the conservative form. The system allows solution discontinuities, and the conservative formulation is the only one that gives correct discontinuous (shock) solutions; see Ref. 8.