Abstract
In this paper the problem of scattering of time-harmonic elastic waves by a non-penetrable partially coated obstacle buried in a piecewise homogeneous medium is considered. We study the direct scattering problem as well as the inverse one. The direct problem for the Navier equation is formulated in a dyadic form and the issues of solvability due to uniqueness, existence and regularity are discussed. Uniqueness results for the corresponding inverse scattering problem are proved. In particular, the unique determination of the non-penetrable partially coated obstacle with its boundary condition as well as of the penetrable interface(s) between the layered media are established via the knowledge of the far-field pattern for elastic waves. The proof of the uniqueness is based on a mixed reciprocity relation and its connection between plane-waves and point-sources. The paper at hand deals with two-dimensional problems for a body consisting of two layers, but the obtained results also hold for multi-layered media; they are valid as well for the three-dimensional case.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Athanasiadis, C. E., Natroshvili, D., Sevroglou, V. Stratis, I. G.: Mixed impedance transmission problems for vibrating layered elastic problems. Math. Methods Appl. Sc. (accepted 2014)
Athanasiadis, C. E., Natroshvili, D., Sevroglou V., Stratis, I. G.: A boundary integral equations approach for direct mixed impedance problems in elasticity. J. Integral Eqns. Appl. 23, 183–222 (2011)
Athanasiadis, C. E., Natroshvili, D., Sevroglou V., Stratis, I. G.: An application of the reciprocity gap functional to inverse mixed impedance problems in elasticity. Inverse Problems. 26, 085011 19pp (2010)
Athanasiadis, C. E., Sevroglou V., Stratis, I. G.: Scattering relations for point-generated dyadic fields in two-dimensional linear elasticity. Quart. Appl. Math. 4, 695–710 (2006)
Cakoni, F., Colton, D.: Qualitative Methods in Inverse Electromagnetic Scattering Theory. Springer-Verlag (2005)
Xiaodong L., Bo, Z.: Direct and inverse obstacle scattering problems in a piecewise homogeneous medium. SIAM J. Appl. Math. 70, No 8 (2010)
Natrosvilli, D., Z. Tediashvili, Z.: Mixed type direct and inverse scattering problems. In: Elschner, J., Gohberg, I., Silbermann, B. (eds.) Operator Theory: Advances and Applications, 121, 366–389 Birkhuser, Basel (2001)
Pelekanos G., Sevroglou, V.: Inverse scattering by penetrable objects in two-dimensional elastodynamics. J. Comp. Appl. Math. 151, 129–140 (2003)
Sevroglou, V.: The far-field operator for penetrable and absorbing obstacles in 2D inverse elastic scattering. Inverse Problems. 17, 717–738 (2005)
Tai, C. T.: Dyadic Greens Functions in Electromagnetic Theory. IEEE, New York (1994)
Twersky, V.: Multiple scattering of electromagnetic waves by arbitrary configurations. J. Math. Phys. 8, 589–610 (1967)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Athanasiadis, C.E., Natroshvili, D., Sevroglou, V., Stratis, I.G. (2015). A Mixed Impedance Scattering Problem for Partially Coated Obstacles in Two-Dimensional Linear Elasticity. In: Constanda, C., Kirsch, A. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16727-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-16727-5_3
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-16726-8
Online ISBN: 978-3-319-16727-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)