An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation

Abstract

An explicit time domain algorithm is developed which is capable of numerically simulating pulsed pressure waves propagating through media whose attenuation increases with frequency according to a power law dependency. Because of possible noninteger exponents in the power law formulation, standard temporal differential operators cannot be defined and traditional finite difference approximations are therefore inappropriate. We derive a method that is consistent with the fact that the complex wavenumber often must include a nonlinear phase if system causality is to be ensured. This phase is derived from the power law attenuation and, due to their interdependency, the two terms in the wave equation corresponding to attenuation and phase can be combined into a single factor. This so-called dispersion wave equation is mapped into complex discrete-time frequency. In this domain, noninteger exponents can be eliminated via a power series expansion, and the resulting equations transform naturally to discrete time operators. The algorithm is tested by comparing numerically evaluated attenuation with the exact power law form, and the issues of stability in relation to the convergence of the power series and the accuracy of the mapping are investigated.