# Simulation of Acoustic Wave Propagation in Anisotropic Media Using Dynamic Programming Techniques

Nikolai Botkin and Varvara Turova^{*}

The paper concerns the development of methods for modeling the propagation of acoustic waves in anisotropic media. This investigation is very important for many applications such as acoustic sensors whose operating principle is based on the excitation and detection of acoustic waves of very high frequency in piezoelectric crystals. \newline For anisotropic media, the WKB approximation yields eikonal equations whose Hamiltonians are neither convex nor concave in the impulse variable. Therefore, the well-known Fermat principle of wave propagation fails in this case. Moreover, the propagation occurs in such a way, as if an antagonistic opponent aims to slow down the movement of wave fronts. Thus, we come to the idea to adopt methods of differential games to the analysis of wave propagation. If the Hamiltonian of a differential game approximates the Hamiltonian of the eikonal equation, then the value function of the game approximates the phase function satisfying the eikonal equation. Utilizing dynamic programming approach, the authors have developed very effective algorithms for solving Hamilton-Jacoby equations arising from differential games, so that very accurate computing of approximate value functions and optimal trajectories in differential games is possible. \newline The level sets of the value function of the corresponding game represent the wave fronts, and optimal trajectories are associated with the propagation of rays. Thus, it makes possible to describe very complicated behavior of rays using game theoretical classification of the so-called singular surfaces that can attract, repulse, and break the trajectories. For example, the caustic-like behavior of rays can be interpreted as the attraction of neighboring optimal trajectories to a singular surface. \newline Using this approach, methods of modeling the propagation of bulk and surface acoustic waves in anisotropic monocrystals and multi-layered structures used in surface acoustic wave sensors are developed. With such algorithms, the propagation fronts can be found very precisely even in the case of very complicated geometry of the excitation source (transducer). Numerical results are presented for the case of non-convex bulk and surface wave slowness surfaces typical for anisotropic quartz crystals.$\\$ \newline This investigation is inspired by the cooperation with professor A.$\,$A$\,$Melikyan (deceased) from the Institute for Problems in Mechanics, Moscow, Russia.

Mathematics Subject Classification: 74J20 49N90 49L25 49M25

Keywords: Anisotropic media; eikonal equation; conflict control problems; grid methods

Minisymposion: Dynamic Programming Approach to Optimal Control Methods and Applications