On Budaev and Bogy's Approach to Diffraction by the 2D Traction-Free Elastic Wedge

Several semianalytical approaches are now available for describing diffraction of a plane wave by the 2D (two‐dimensional) traction free isotropic elastic wedge. In this paper we follow Budaev and Bogy, who reformulated the original diffraction problem as a singular integral one. This comprises two algebraic and two singular integral equations. Each integral equation involves two unknowns, a function and a constant. We discuss the underlying integral operators and develop a new semianalytical scheme for solving the integral equations. We investigate the properties of the solution obtained and argue that it is the solution of the original diffraction problem. We describe a comprehensive code verification and validation program.

[1]  A. Gautesen.  Scattering of an obliquely incident Rayleigh wave in an elastic quarterspace , 1986 .

[2]  The wedge subjected to tractions: a paradox re-examined , 1984 .

[3]  A. Gautesen.  Scattering of a Rayleigh wave by an elastic three-quarter space , 2002 .

[4]  G. Lebeau,et al.  Diffraction by an elastic wedge with stress-free boundary: existence and uniqueness , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  Excitation , 1964 .

[7]  M. Williams,et al.  Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension , 1952 .

[8]  Vladimir Maz’ya,et al.  Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations , 2000 .

[9]  R. Geroch.  Partial Differential Equations of Physics , 1996, gr-qc/9602055.

[10]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[11]  A. Maue.  Die Beugung elastischer Wellen an der Halbebene , 1952 .

[12]  B. Budaev.  Diffraction by Wedges , 1995 .

[13]  B. Plamenevskii,et al.  Elliptic Problems in Domains with Piecewise Smooth Boundaries , 1994 .

[14]  H. S. Carslaw..  Diffraction of Waves by a Wedge of any Angle , 1920 .

[15]  Diffraction of elastic waves by a rigid wedge , 1981, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[16]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[17]  A. Gautesen.  Scattering of a Rayleigh Wave by an Elastic Quarter Space , 1985 .

[18]  A. Gautesen.  Scattering of a Rayleigh Wave by an Elastic Wedge Whose Angle is Greater Than 180 Degrees , 2001 .

[19]  G. Lebeau,et al.  Diffraction by an Immersed Elastic Wedge , 2000 .

[20]  A. Gautesen.  Scattering of a Rayleigh wave by an elastic wedge whose angle is less than 180 , 2002 .

[21]  A. K. Gautesen,et al.  Scattering of a Rayleigh Wave by an Elastic Quarter Space-Revisited , 2002 .

[22]  A. Norris,et al.  The Malyuzhinets theory for scattering from wedge boundaries: a review , 1999 .

[23]  W. E. Williams.  Diffraction of an E-polarized plane wave by an imperfectly conducting wedge , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[24]  G. D. Maliuzhinets.  Excitation, Reflection and Emission of Surface Waves from a Wedge with Given Face Impedances , 1958 .

[25]  E. T. Copson,et al.  Asymptotic Expansions: The saddle-point method , 1965 .

[26]  RAYLEIGH WAVE SCATTERING BY A WEDGE: A BOUNDARY METHOD APPROACH , 1986 .

[27]  Kazunari Fujii.  Rayleigh-wave scattering at various wedge corners: Investigation in the wider range of wedge angles , 1994, Bulletin of the Seismological Society of America.

[28]  D. Bogy,et al.  Rayleigh wave scattering by two adhering elastic wedges , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.