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Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener–Hopf factorization

  • Gaurav Maurya
  • Basant Lal SharmaEmail author
Article

Abstract

Scattering of time-harmonic plane wave by two parallel semi-infinite rows, but with staggered edges, is considered on square lattice. The condition imposed on the semi-infinite rows is a discrete analogue of Neumann boundary condition. A physical interpretation assuming an out-of-plane displacement for the particles arranged in the form of a square lattice and interacting with nearest neighbours associates the scattering problem with lattice wave scattering due to the presence of two staggered but parallel crack tips. The discrete scattering problem is reduced to the study of a pair of Wiener–Hopf equation on an annulus in complex plane, using Fourier transforms. Due to the offset between the crack edges, the Wiener–Hopf kernel, a \(2\times 2\) matrix, is not amenable to factorization in a desirable form and an asymptotic method is adapted. Further, an approximation in the far field is carried out using the stationary phase method. A graphical comparison between the far-field approximation based on asymptotic Wiener–Hopf method and that obtained by a numerical solution is provided. Also included is a graphical illustration of the low-frequency approximation, where it has been found that the numerical solution of the scattering problem coincides with the well-known formidable solution in the continuum framework.

Keywords

Out-of-plane motion Neumann Crack Discrete Wiener–Hopf Edge diffraction Lattice wave 

Mathematics Subject Classification

Primary 78A45 Secondary 39A14 47B35 

Notes

Acknowledgements

GM acknowledges MHRD (India) and IITK for providing financial assistance in the form of Senior Research Fellowship. BLS acknowledges the partial support of SERB MATRICS Grant MTR/2017/000013.

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Copyright information

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology KanpurKanpurIndia

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