An Inverse Problem in Elastodynamics
We show that the p-wave and s-wave speeds of an isotropic elastic object are determined in the interior by surface measurements, and that the density and elastic properties are determined to infinite order at the boundary. The material properties of the bounded, fully 3-dimensional object, that is, the density and elastic properties, are represented by the (nonconstant, leading) coefficients of the system of linear differential equations for elastodynamics. Surface measurements are modelled by the Dirichlet-to-Neumann map on a finite time interval.
The proof of these results makes use of high frequency asymptotic expansions, Hamilton-Jacobi theory, microlocal analysis, propagation of singularities results for systems of real principal type, and a result in integral geometry. Here we announce the results of [R I] and [R II].
We consider an inverse problem in elastodynamics. The physical setting for the problem is a bounded, 3-dimensional, isotropic elastic object with smooth boundary. The inverse problem can be formulated as the study of whether measurements made at the surface of the object determine the material properties (that is, the density and elastic properties) of the object. Surface measurements consist of the following pairs of surface data: all possible forces applied normal to the surface and the resulting displacements of the surface.
KeywordsInverse Problem Surface Measurement Geodesic Segment Integral Geometry Inverse Boundary
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