Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids. II. Second-order asymptotic boundary conditions

In this paper, using a transfer matrix method, second-order asymptotic boundary conditions (B.C.) are introduced to model an anisotropic interfacial layer when the layer is thin compared to the wavelength. Compared with the first-order asymptotic B.C. which have previously been introduced by Rokhlin and Huang [J. Acoust. Soc. Am. 92, 1729–1742 (1992)], the second-order B.C. greatly improve the accuracy and consistency of approximation and at the same time preserve the same order of simplicity. Other attractive features are that unlike the first-order approximation the wave solutions for the second-order B.C. exactly satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. Furthermore, for decoupled symmetric and antisymmetric problems, the second-order B.C. can be simplified through lower-order stiffness matrices so that the rank of the system of equations is reduced by half. Here the second-order B.C. are presented for a thin orthotropic layer between two solids which is rotated about one of its axes of symmetry with respect to the incident plane. Such boundary conditions couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. The results are illustrated by numerical examples of ultrasonic wave scattering from an imperfect interface between anisotropic solids by comparison with exact solutions.