ASYMPTOTIC ESTIMATES TO SOLUTIONS OF THE TIME-DEPENDENT WAVE EQUATION (ACOUSTIC SCATTERING)
The short-time behavior of waves reflected from a transparent smooth surface is considered. Two cases are developed: first, when the surface is characterized by a jump in the gradient of propagation velocity (a gradient-type interface), and, second, when the surface is characterized by a jump in the velocity itself. In the first case, both one- and three-dimensional models are considered. The wave field is split into incident, reflected and transmitted waves, and relations among the amplitude coefficients are derived. One- and three-dimensional versions of an integral equation are derived, as well as reflection operators, in order to facilitate the analysis of error growth in time due to the asymptotic approximation. Various special cases are considered, and explicit time error bounds, as well as certain numerical results, are presented. In one dimension, the existence of an exact solution for a special velocity profile leads to a method of solution for an inverse problem: the reflection operator allows one to reconstruct a general velocity profile for a body with gradient-type boundaries using measurements of reflected data. In the second case, the waves are cast into the form of time-dependent single layer potentials, which are shown to match the geometrical optics approximation asymptotically. These results are of interest in themselves, as well as being valuable in the development of a general three-dimensional wave splitting, currently under investigation.
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