Finite difference predictions of P-SV wave propagation inside submerged solids. I. Liquid–solid interface conditions

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Abstract
A finite difference scheme has been developed to analyze internal strains in submerged elastic solids of irregular geometry subjected to ultrasonic wave sources that simulate a clinical lithotripter. In part I of this paper, the finite difference formulation that accounts for arbitrary liquid-solid interfaces is presented and sample numerical results are discussed. Two different methods for discretizing the liquid-solid interface conditions are developed. The first treats the interface conditions explicitly. The second integrates the heterogeneous wave equations across the interface using the divergence theorem. Both schemes account for varying grid sizes and give similar results for a test problem consisting of a radially diverging source incident on the rectangular solid. The time sequence obtained numerically for strain at the center of a rectangular solid matches well with the experimental results [S. M. Gracewski et al., J. Acoust. Soc. Am. 94, 652-661 (1993)] in terms of the arrival times and the relative amplitudes of the peaks. In addition, strain contours are plotted to visualize the propagation of P (longitudinal) and S (shear vertical) waves inside a circular solid. The reflection from the concave back surface of the circular solid has a focusing effect with the subsequent formation of focal zones, known as caustics, where peak strains occur. In part II of this paper, the finite difference scheme is used to study the effects of geometry changes on the internal stresses and caustics predicted in model stones subjected to lithotripter pulses.