Nonlinear Acoustic Waves in Heterogeneous Materials
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This thesis concerns the partial differential equations governing acoustic wave propagation in heterogeneous materials. We start with an investigation of the standard Eulerian formulation of the equations, and point out why some of the underlying assumptions and approximations might give inaccurate results in some cases. We argue that a Lagrangian framework is better suited to accurately model wave propagation in materials with discontinuous material properties, and derive a momentum equation in Lagrangian coordinates without approximations. We continue by looking at conservation of energy and attenuation of the wave. To solve the Lagrangian equations on a computer, we propose a numerical scheme based on a Leapfrog on staggered grid-scheme that is second order both in space and time. We also do a numerical experiment and compare results from simulations with the Eulerian and Lagrangian equations. The simulations indicate that the Lagrangian equations are better suited model to certain phenomena.