3D elastic scattering theorems for point‐generated dyadic fields

Abstract

SUMMARYThe problem of scattering of elastic spherical waves by a rigid body, cavity or a penetrable obstacle in 3D linear elasticity is considered. For two point sources, dyadic far-field pattern generators are defined, which are used for the formulation of a general scattering theorem. The main reciprocity principle and mixed scattering relations are also established. We provide the necessary energy considerations, presenting relative energy functionals and expressions for the differential and the scattering cross section due to point-source dyadic incidence. Finally, an application of the general scattering theorem for appropriate locations of the point source leads to an optical theorem.

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].…”

Retrieval of Green’s function and generalized optical theorem for the scattering of complete dyadic fields

“…We adopt the similar method as that used by Athanasiadis et al (2008) to calculate the integral in Eq. (23).…”

“…For a large a ¼ |a|, we have |a À r 0 | ¼ a À â Á r 0 þ O(a À1 ) and 1/|a2 r 0 | ¼ a À1 þ O(r À2 ). Neglecting the items with higher order, in the far field,G 0 ða; r 0 Þ can be then expressed as the dyadic form in spherical coordinate system (Dassios et al, 1995;Dassios and Kleinman, 2000;Snieder, 2002;Athanasiadis et al, 2008) G 0 ðâ; r 0 Þ ¼ U pââ e ik p a a e Àik pâ Ár 0 þ U s ðĨ ÀââÞ e ik s a a e Àik sâ Ár 0 ; (10)…”

“…(16) (17) where the dagger † denotes the Hermitian conjugate (transpose and conjugate). T r is the surface stress operator and it is defined as (Dassios et al, 1995;Dassios and Kleinman, 2000;Athanasiadis et al, 2008) T r ¼ 2l 0n Á r þ k 0n r Á þl 0n  r  :…”