Green’s function, lattice sums and rayleigh’s identity for a dynamic scattering problem
We have recently exhibited expressions for Green’s functions for dynamic scattering problems for gratings and arrays, expressed in terms of lattice sums. We have also discussed new ways to evaluate these sums, and how their use in Green’s function forms leads naturally to Rayleigh identities for scattering problems. These Rayleigh identities express connections between regular parts of wave solutions near a particular scatterer, and irregular parts of the solution summed over all other scatterers in a system. Here, we will discuss these ideas and techniques in the context of the problem of the scattering of a scalar wave by a singly-, doubly-and triply-periodic lattices of perfectly conducting obstacles. We will discuss expressions for lattice sums which can be integrated arbitrarily-many times to accelerate convergence, computationally-efficient Green’s function forms, and the appropriate Rayleigh identities for these problems. We will also discuss the long-wavelength limit, in which the dynamic identity tends to the static identity in a mathematically-interesting way.
KeywordsHelmholtz Equation Reciprocal Lattice Neumann Series Spherical Bessel Function Poisson Summation Formula
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