Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

The Ewald method is applied to accelerate the evaluation of the Green’s function (GF) of an infinite equispaced linear array of point sources with linear phasing. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. It is shown analytically that the choice of the standard “optimal” Ewald splitting parameter E0E0 causes overflow errors at high frequencies (period large compared to the wavelength), and convergence rates are analyzed. A recipe for selecting the Ewald splitting parameter is provided.