Numerical solution of nonlinear acoustic wave problems employing a green's function approach

The design of phased array transducers for medical diagnostic ultrasound asks for an understanding of the nonlinear propagation of acoustic wavefields. Most existing numerical models are based on the linearized model equations, but in the recent decades several numerical models have been developed that incorporate weak nonlinear propagation as well. In this study, we present an approach that enables the computation of large-scale, three-dimensional, weakly nonlinear wavefields in the time domain. It is based on the Neumann iterative method, which enables the successive use of the solution of a linear wave problem, where the nonlinearity is treated as a contrast source. The linear wave problem is formulated as a convolution integral with a free space Green`s function kernel. If the Green`s function is adequately regularized, this method enables us to discretize the spatiotemporal domain with step sizes up to the limit given by the Nyquist-Shannon sampling theorem for a bandlimited signal. The remaining spatiotemporal derivatives are handled with high-order finite difference schemes. The proposed method is validated with a one-dimensional nonlinear wave problem. We evaluate the wavefield, including harmonic frequencies up to the fifth harmonic. The results are compared with a solution of the lossless Burgers` equation. It is observed that already after a small number of iterations the computed result shows perfect agreement with those of the Burgers` equation. In this paper, we make plausible that the proposed method can be straightforwardly extended to more complex problems. The volume integral equation approach opens the road to solving time domain nonlinear wave problems in three dimensions. Furthermore, the method can be adapted to media with attenuation and inhomogeneity.Applied Science