The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on August 4, 2008
The Quarterly Journal of Mechanics and Applied Mathematics 2008 61(4):549-580; doi:10.1093/qjmam/hbn017
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Point source excitation of a layered sphere: direct and far-field inverse scattering problems
( Department of Mathematics, University of Athens, Panepistemiopolis, GR 15784 Athens, Greece )
ntsitsas{at}math.uoa.gr
cathan{at}math.uoa.gr
Received 8 January 2008.
Revise 23 May 2008.
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Abstract |
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A layered sphere is excited by a time-harmonic spherical acoustic wave, generated by a point source located either in the interior or in the exterior of the sphere. The sphere's core may be acoustically soft, hard, resistive or penetrable. Significant applications such as radiation from the neuron currents lying inside the human brain and localization and shape reconstruction of buried objects in layered media motivate the investigation of direct and inverse scattering problems involving such types of scatterers and excitations. The exact Green's function is determined by solving the corresponding boundary-value problem, by applying a combination of Sommerfeld's and T-matrix methods. We then introduce the low-frequency assumption and extract from the determined exact Green's function the low-frequency far-field results for a small layered sphere. The spherical wave low-frequency far-fields obtained reduce to those due to plane wave incidence on a layered sphere and also recover as special cases several classic results of the literature concerning the exterior spherical wave excitation of homogeneous spheres. The derived low-frequency far-field expansions are then utilized in order to establish inverse scattering algorithms for the determinations of (i) the sphere's center and the layers’ radii, (ii) the layers’ physical parameters and (iii) the location of the point source. The distance of the point source from the sphere's centre plays a significant role in the development of these algorithms. Several numerical results are included concerning the far-field interactions between the point source and the layered sphere.