# An Inverse Problem in Elastodynamics

## Abstract

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.

## Keywords

Inverse Problem Surface Measurement Geodesic Segment Integral Geometry Inverse Boundary## Preview

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## References

- [Ba-Sy]Bao, G. and W.W. Symes.
*On the sensitivity of solutions of hyperbolic equations to the coefficients*, Inst. Math. Appl. preprint series, Vol. 1249, submitted to Comm. in PDE.Google Scholar - [B-K]Belishev, M. and Y. Kurylev. (1991).
*Boundary control, wave field continuation and inverse problems for the wave equation*, Computers Math. Applic., Vol. 22(4/5), (pages 27–52).MathSciNetCrossRefGoogle Scholar - [C]Croke, C. (1991).
*Rigidity and the distance between boundary points*, Journal of Differential Geometry, Vol. 33(2), (pages 445–464).MathSciNetMATHGoogle Scholar - [De]Dencker, N. (1982).
*On the propagation of singularities for pseudodifferential operators of principal type*, Arkiv for Matematik, Vol. 20(1), (pages 23–60).MathSciNetMATHCrossRefGoogle Scholar - [D-L]G. Duvaut and J. L. Lions.
*Inequalities in mechanics and physics*, Springer-Verlag, Berlin, 1976.MATHCrossRefGoogle Scholar - [Hö]Hörmander, L. (1985).
*The Analysis of Linear Partial Differential Oper-**ators*, Vol. III, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo.Google Scholar - [H-K-M]Hughes, T.J.R., T. Kato, and J. E. Marsden. (1977).
*Well-posed, quasi-linear, second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity*, Arch. for Rat. Mech. Anal., Vol. 63(3), (pages 273–294).Google Scholar - [K-V]Kohn, R. and M. Vogelius. (1984).
*Identification of an unknown conductivity by means of measurements at the boundary*, SIAM-AMS Proceedings, Vol. 14, (pages 113–123 ).Google Scholar - [Ku]Kupradze, V. D. (editor). (1979).
*Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity*, North-Holland, Amsterdam.MATHGoogle Scholar - [Lu]Ludwig, D. (1960).
*Exact and asymptotic solutions of the Cauchy problem*, Comm. Pure Appl. Math., Vol. 13, (pages 473–508 ).Google Scholar - [La]Lax, P.D. (1957).
*Asymptotic solutions of oscillatory initial value problems*, Duke Math. J., Vol. 24, (pages 627–646 ).Google Scholar - [M-R]R. G. Mukhometov and V. G. Romanov. On the problem of finding an isotropic Riemannian metric in an n-dimensional space,
*Dokl. Akad. Nauk*. SSSR 243 (1), 1978, 41–44.MathSciNetGoogle Scholar - [Na]A. Nachman. Global uniqueness for a two-dimensional inverse boundary value problem, Ann.
*of Math*. (2) 143 (1), 1996, 71–96.MathSciNetMATHCrossRefGoogle Scholar - [N-U ‘83]Nakamura, G. and G. Uhlmann. (1993).
*Identification of Lamé parameters by boundary measurements*, Amer. J. Math., Vol. 115(5), (pages 1161–1187).Google Scholar - [N-U ‘84]Nakamura, G. and G. Uhlmann (1994).
*Global uniqueness for an inverse boundary problem arising in elasticity, Invent*. Math., Vol. 118(3), (pages 457–474).Google Scholar - [N-U ‘85]Nakamura, G. and G. Uhlmann. (1995).
*Inverse problems at the boundary for an elastic medium*, SIAM J. Math. Anal., Vol. 26, (pages 263–279 ).Google Scholar - [R I]Rachele, L.
*Boundary determination for an inverse problem in elastodynamics*, Preprint: See http://www.math.purdue.edu/-lrachele. Google Scholar - [R II]Rachele, L.
*An Inverse Problem in Elastodynamics: Determination of the wave speeds in the interior*. Preprint: See http://www.math.purdue.edu/~lrachele. - [Ra-Sy]Rakesh and W. W. Symes. (1988).
*Uniqueness for an inverse problem for the wave equation*. Communications in Partial Differential Equations, Vol. 13(1), (pages 87–96).Google Scholar - [Sa-Sy]Sacks, P.E. and W.W. Symes. (1990).
*The inverse problem for a fluid over a layered elastic half-space*, Inverse Problems, Vol. 6(6), (pages 1031–1054).MathSciNetMATHCrossRefGoogle Scholar - [S-U ‘87]Sylvester, J. and G. Uhlmann. (1987).
*A global uniqueness theorem for an inverse boundary value problem*, Annals of Mathematics, Vol. 125, (pages 153–169 ).MathSciNetMATHCrossRefGoogle Scholar - [S-U ‘88]Sylvester, J. and G. Uhlmann. (1988).
*Inverse boundary value problems at the boundary — continuous dependence*. Comm.of Pure and Applied Math., Vol. 41(2), (pages 197–219).Google Scholar - [Sy-U]Sylvester, J. and G. Uhlmann. (1991).
*Inverse problems in anisotropic media*, Contemporary Mathematics, Vol. 122, (pages 105–117 ).MathSciNetCrossRefGoogle Scholar - [U]Uhlmann, G. (1992).
*Inverse boundary value problems and applications*, Methodes semi-classiques, Vol. 1 (Nantes, 1991),*Asterisque*, Vol. 207 (6), (pages 153–211 ).MathSciNetGoogle Scholar