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

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.

Finite difference predictions of P-SV wave propagation inside submerged solids. II. Effect of geometry.

To understand better direct stress wave contributions to stone fragmentation during extracorporeal shock wave lithotripsy (ESWL), the numerical formulation developed in part I is applied to study the time evolution of stress wave fields produced inside submerged isotropic elastic solids having irregular geometries. Cut spheres are used to model stones that have already had an initial fracture. Ellipses are used to approximate other deviations from a spherical geometry. The propagation and focusing of the longitudinal (P) and shear (S) wave fronts are visualized by presenting internal strain contours. Internal strain measurements are obtained from strain gauges embedded inside plaster specimens to confirm the focusing effect obtained from the concave back surfaces of the stones. Fragmentation experiments indicate damage caused by spalling and direct stress wave focusing as well as a front surface pit presumably created by cavitation activity.

Internal stress wave measurements in solids subjected to lithotripter pulses.

Semiconductor strain gauges were used to measure the internal strain along the axes of spherical and disk plaster specimens when subjected to lithotripter shock pulses. The pulses were produced by one of two lithotripters. The first source generates spherically diverging shock waves of peak pressure approximately 1 MPa at the surface of the specimen. For this source, the incident and first reflected pressure (P) waves in both sphere and disk specimens were identified. In addition, waves reflected by the disk circumference were found to contribute significantly to the strain fields along the disk axis. Experimental results compared favorably to a ray theory analysis of a spherically diverging shock wave striking either concretion. For the sphere, pressure contours for the incident P wave and caustic lines were determined theoretically for an incident spherical shock wave. These caustic lines indicate the location of the highest stresses within the sphere and therefore the areas where damage may occur. Results were also presented for a second source that uses an ellipsoidal reflector to generate a 30-MPa focused shock wave, more closely approximating the wave fields of a clinical extracorporeal lithotripter.