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.
Highlights
In [1], we applied a general theory to obtain spectral functional equations in electromagnetics and generalized Wiener–Hopf Equations (GWHEs) for scattering problems in angular regions filled by arbitrarily linear media, inspired by [2] and described in [3]
We demonstrate for validation that the GWHEs obtained from the proposed functional equations enforcing the boundary conditions and the functional equations obtained in [14] using the Gautesen (Kirchhoff) integral representations in the natural domain are identical, the applied notations are different from each other and not immediate in the comparison
We have introduced a general method for the deduction of spectral functional equations and GWHEs in angular regions filled by arbitrary linear isotropic homogeneous media in elasticity
Summary
In [1], we applied a general theory to obtain spectral functional equations in electromagnetics and generalized Wiener–Hopf Equations (GWHEs) for scattering problems in angular regions filled by arbitrarily linear media, inspired by [2] and described in [3]. Applying the Fourier transforms to the differential formulation of the elastic field and taking into account the boundary conditions, the authors obtain singular integral equations in terms of the spectral functions that are numerically solved by using the Galerkin collocation method. We state that the scope of our paper is to present algebraic spectral functional equations for arbitrary boundary conditions for threedimensional wave motion problems in angular regions that are useful for the examination of practical problems by imposing specific boundary conditions yielding GWHE formulations.
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