Scattering relations for point-generated dyadic fields in two-dimensional linear elasticity

Abstract

Abstract. The problem of scattering of elastic waves by a bounded obstacle in twodimensional linear elasticity is considered. The scattering problems are presented in a dyadic form. An incident dyadic field generated by a point source is disturbed by a rigid body, a cavity, or a penetrable obstacle. General scattering theorems are proved, relating the far-field patterns due to scattering of waves from a point source set up in either of two different locations. The most general reciprocity theorem is established, and mixed scattering relations are also proved. Finally, a relation between the incident and the scattered wave which refers to the mechanism of energy transfer of the scatterer, the so-called optical theorem, is established.

Point-Source Elastic Scattering by a Nested Piecewise Homogeneous Obstacle in an Elastic Environment

“…By letting the locations of the two point-sources coincide in the general scattering theorem, we obtain the optical theorem, relating a certain integral of the far-field pattern with the value of the scattered field at the point-source's location. All the derived scattering relations recover those of [11] and [12] for the 2D and 3D point-source excitation of a homogeneous elastic obstacle. However, contrary to [11] and [12], the different material parameters of the scatterer's layers impose different equations and boundary conditions in every layer, resulting in higher complexity for the analysis and derivation of the scattering relations.…”

“…In the present context the 2D and 3D scattered field must satisfy the Kupradze radiation conditions [20] In the radiation zone the 2D and the 3D scattered field has the asymptotic expression [11,12] 9 u sc a 8r3 3 9 g r a 81 r3 N 1 p e ik p10 r r N 4 9 g t a 81 r3 N1s e ik s10 r r N 4 38r 7 N 3 8r 9 3 (2.12) with 21 p 3 21s 3 1, 31 p 3 1 ik p10 , 31s 3 1 ik s10 , 2 3 1 2 , 3 3 1, 7 2 3 2 3 2 , 7 3 3 22a uniformly with respect to 1 r on the unit circle of 1 2 , and the unit sphere S 2 of 1 3 respectively. Furthermore, 9 g r a and 9 g t a are the radial (longitudinal) and tangential (transverse) dyadic far-field patterns [6].…”

“…Moreover, by extending and modifying the techniques of Lemma 3.1 in [11], one verifies the following useful relation: Now, by letting in (3.6) R 9 and 1 9 0, taking into account the radiation condition (2.11), and using (3.7) we find 9 u inc a 1 9 u sc b S 1 3 2 9 u sc b 8a3 81 a 1 a3 2 5 0 4 6 0 6 0 9 u sc b 8a3 8 9 I 2 1 a 1 a32 We note that, in view of (2.1) and (2.3), the reciprocity relation (3.2) holds for the incident as well as for the total fields in V 0 .…”

“…The 2D differential scattering cross-section due to a point source at a is given by [5,11,[15][16][17] where P inc a and P sc a are the power flux vectors of the incident and the scattered fields, and c is an arbitrary constant vector. Averaging the differential scattering cross-section over the unit circle we obtain the 2D scattering or total cross-section determining the amount of incident field's power, absorbed by the scatterer's core V n (since all the other layers have been assummed lossless).…”

“…For interior point-source excitation of piecewise homogeneous acoustic and electromagnetic obstacles such relations are established in [9,10]. Moreover, in elasticity, 2D and 3D scattering relations considering the incidence of exterior point-source fields on a homogeneous obstacle are established in [11,12].…”