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In this paper, we study the nonlinear periodic Westervelt equation with excitations located within a bounded domain in $\mathbb{R}^d$, where $d \in \{2,3\}$, subject to Robin boundary conditions. This problem is of particular interest for advancing imaging techniques %in invasive ultrasound, such as intravascular ultrasound imaging. that exploit nonlinearity of the acoustic propagation. We establish the existence and uniqueness of solutions in both the linear and the nonlinear setting, thereby allowing for spatially varying coefficients as relevant in quantitative imaging. %, highlighting the crucial role of small excitations in ensuring existence and uniqueness in the nonlinear setting. Derivation of a multiharmonic formulation enables us to show the generation of higher harmonics (that is, responses at multiples of the fundamental frequency) due nonlinear wave propagation. An iterative scheme for solving the resulting system is proposed that relies on successive resolution of these higher harmonics, and its convergence under smallness conditions on the excitation is proven. Furthermore, we investigate the numerical solution of the resulting system of Helmholtz equations, employin

Author

Rainer, Benjamin; Kaltenbacher, Barbara

Published

2024

Language

EN