Iterative numerical solution of nonlinear wave problems

Nowadays, the design of phased array transducers for medical diagnostic ultrasound asks for an understanding of the nonlinear propagation of acoustic wavefields. In the last decade an imaging modality called Tissue Harmonic Imaging (THI) has become the standard for many echography investigations. THI specifically benefits from the nonlinear distortion of ultrasound propagation in tissue. Since most existing numerical models are based on a linear approximation of the underlying nonlinear physical reality, they cannot account for this kind of disortion. Several numerical models have been developed in recent years that incorporate weak nonlinear propagation. However, as yet no model has enabled the computation of large scale, full-wave nonlinear wavefields in the time domain. In this study, we present an approach that handles weak nonlinear propagation by means of an iterative Neumann scheme. This approach enables the successive use of the solution of a linear wave problem, where the nonlinearity is treated as a contrast source. Thus, we can employ well-known linear methods for large scale wave problems to obtain the desired nonlinear wavefield. The general formalism is outlined and applied to a one-dimensional nonlinear wave problem to obtain the desired nonlinear wavefield. The general formalism is outlined and applied to a one-dimensional nonlinear wave problem. The wavefield is evaluated, including harmonic frequencies up to the fifth harmonic. For each successive linear step a Green's function approach is employed. The results that already after a small number of iterations the results are in very good agreement with this exact result. The proposed method can easily be extended can easily be extended to more complex problems. The Green's function approach enables us to discretize the spatiotemporal domain very efficiently, which opens the road to solving time domain nonlinear wave problems in three dimensions. Furthermore, media with attenuation and inhomogeneity can be inlcuded straightforwardly in the algorithm.