Continuum limit of discrete Sommerfeld problems on square lattice
A low-frequency approximation of the discrete Sommerfeld diffraction problems, involving the scattering of a time harmonic lattice wave incident on square lattice by a discrete Dirichlet or a discrete Neumann half-plane, is investigated. It is established that the exact solution of the discrete model converges to the solution of the continuum model, i.e., the continuous Sommerfeld problem, in the discrete Sobolev space defined by Hackbusch. A proof of convergence has been provided for both types of boundary conditions when the imaginary part of incident wavenumber is positive.
KeywordsSommerfeld half-plane crack rigid ribbon continuum limit Wiener–Hopf Toeplitz operator
This work was partially supported by the Grant IITK/ME/20080334 and Project IITK/ME/20090027. The author thanks Anand and Munshi for suggestions. The author thanks an anonymous reviewer of  for a comment that prompted the author to prepare a rigorous proof of continuum limit, which had been deferred during the preparation of the papers [16, 17, 22, 23]. The author thanks the anonymous reviewers for several constructive comments and suggestions.
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