The generalized Wiener–Hopf equations for the elastic wave motion in angular regions

Abstract

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener–Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions.

A New Method to Estimate the Contribution of Geometrical Optics in Arbitrary Linear Stratified Planar Structures

“…The Wiener-Hopf technique in its generalized form has been applied effectively in electromagnetic wave scattering problems for angular regions (wedge problems); see [1,2] and references therein. Following the procedure first proposed in [3], we aim to extend the Wiener-Hopf technique in angular regions for arbitrary linear wave scattering problems [3][4][5][6]. This technique can be also extended to geometries containing angular regions or stratified planar regions; see, for instance, [7].…”

“…This technique can be also extended to geometries containing angular regions or stratified planar regions; see, for instance, [7]. We start our formulation from electromagnetic applications [3][4][5] and extend the procedure to elasticity as reported in [6]. The method is based on two steps: the deduction of the generalized Wiener-Hopf equations for angular-region problems [3][4][5][6] and the solution of the equations using the semianalytical factorization procedure known as Fredholm factorization; see, for instance, [8,9].…”

“…We start our formulation from electromagnetic applications [3][4][5] and extend the procedure to elasticity as reported in [6]. The method is based on two steps: the deduction of the generalized Wiener-Hopf equations for angular-region problems [3][4][5][6] and the solution of the equations using the semianalytical factorization procedure known as Fredholm factorization; see, for instance, [8,9]. A key point in implementing the solution of generalized Wiener-Hopf equations for arbitrary linear stratified media via Fredholm factorization is the extraction of the source term that is related to geometrical-optics components.…”