Diffraction of Plane SH Waves by a Cylindrical Cavity in an Infinite Wedge

Abstract

Diffraction of plane harmonic SH waves by a cylindrical cavity in a homogeneous, isotropic and linear elastic infinite wedge is investigated. Analytical closed form solution is obtained by utilizing Wave Function Expansion Method and Image Technique. Governing equation of this two dimensional steady-state wave motion problem is the Helmholtz equation. Helmholtz equation for polar coordinates is solved by using Separation of Variables Method. Displacement fields are expressed in terms of Fourier-Bessel series in complex form. Complex coefficients of Fourier-Bessel series are to be determined from boundary conditions. Zero-stress boundary condition at cavity inner surface is satisfied directly since polar coordinate systems are used. Free stress conditions at flat surfaces of the infinite wedge are satisfied in closed form via imaging method. In this technique, imaginary cavities symmetric with respect to flat surfaces are considered. With this consideration, scattered waves from imaginary cavities would represent reflections from flat surfaces so that stress free boundary condition on flat surfaces would be automatically satisfied. This technique would generate two additional displacement fields with two additional set of unknown complex Fourier-Bessel constants. These unknowns could be determined from boundary conditions of imaginary cavities. The cylindrical scattered waves from each cavity are defined in different polar coordinate systems. Addition Theorems are used for adequate coordinate transformation. Since Image Technique is used, to maintain angular symmetry, analytical solutions are obtained for a specific wedge whose internal angle is 120 degrees. Also, direction of incident waves has to be either horizontal or on the symmetry line of the wedge and cavity center has to be on the symmetry line. In numerical results, parameters in length dimension are normalized with respect to cavity radius.