T-Matrix Method of Elastic Wave Scattering on Imperfect Interface

In the scattering problem of elastic waves, the wave function extension method plays a basic and important role. When the scatterer is a circle cylinder, an elliptical cylinder, a sphere and a spheroid, the scattering problem of elastic waves can be solved by the wave function extension method directly. However, when the scatterer is of arbitrary shape, the wave function extension method meets a difficulty. Then the T-matrix method of elastic wave scattering was proposed. The advantage of T matrix method is that it can be used to deal with arbitrary shape. However, the traditional T-matrix method is applied to the perfect interface; namely, the displacement and traction vector keep continuous across the interface between the scatterer and the host material. For the imperfect interface, the displacement vector or traction vector jumps across the interface, how to compute the elements of T matrix? It is not discussed in the literature up to now. In this paper, three cases of imperfect interface are considered: 1) traction vector is continuous across interface, but displacement vector jumps; 2). displacement vector is continuous, but traction vector jumps; 3) both displacement and traction vectors jump. The modification and the specific calculation expression of T-matrix for these imperfect interfaces are discussed based on the T- matrix of perfect interface. In addition, the symmetry and the unitity of T matrix in case of imperfect interface are discussed. The outline of this paper is as following: in the section 2, the vector basis functions are defined and their orthogonality is discussed. In section 3, the T matrix in case of perfect interface is formulated in an alternative form in order to show easily and evidently the modification in case of imperfect interface. In section 4, the specific expressions of the modified T matrix for three cases of imperfect interface are given. Finally, the symmetry and unity of the modified T matrix are discussed in the section 5.