Two alternative expressions for the spherical wave expansion of the time domain scalar free-space Green’s function and an application: Scattering by a soft sphere

Abstract

The importance of expanding Green's functions, particularly free-space Green's functions, in terms of orthogonal wave functions is practically self-evident when frequency domain scattering problems are of interest. With the relatively recent and widespread interest in time domain scattering problems, similar expansions of Green's functions are expected to be useful in the time domain. In this paper, two alternative expressions, expanded in terms of orthogonal spherical wave functions, for the free-space time domain scalar Green's functions are presented. Although the two expressions are equivalent, one of them is seen to be more convenient for the calculation of the scattered field for a known equivalent source density, whereas the second expression is more suitable for setting up an integral equation for the equivalent source density. Such an integral equation may be setup, for example, by the application of a time domain equivalent of the T-matrix ͑extended boundary condition͒ method.

Fast Evaluation of Three-Dimensional Transient Wave Fields Using Diagonal Translation Operators

“…However, the use of the interpolation function defined in Eq. (24) implies that knowledge of the field at the edge of the observation sphere requires knowledge of samples from incoming rays that reside p t samples exterior to the sphere in all directions. Therefore, when working with sampled field representations, constraints (21) and (22) should be satisfied in terms of T s and R s = R s + p t c t, instead of T s and R s .…”

“…This is in spite of the fact that the last decade has witnessed significant speedup of frequency domain integral equation solvers with the advent of the fast multipole method (FMM) [16][17][18][19], the impedance matrix localization technique [20], the multilevel matrix decomposition algorithm [21], etc. Although the structure of transient wave fields has been well studied [2,6,[22][23][24], to our knowledge no TDIE algorithms with reduced computational complexity have been reported. Recently, preliminary research has indicated that fast methods, similar in spirit to the frequency domain algorithms, can also be developed in the time domain [25].…”

Time-Dependent Lorentz-Mie-Debye Formulation for Electromagnetic Scattering From Dielectric Spheres (Invited Paper)

“…Both methods have been extended to electromagnetics [44,45]. Compared to [39], the main advantages of TDIE based method lie in the stability and flexibility, because no explicit time domain wave functions is used. Though stable and self-consistent, Laplace based methods involves finding the roots of special functions (recursion rules have to be used).…”

“…Considering the fact that mode matching concept is used to extract the time-domain signatures, this type of approach is direct time-domain variant of conventional LorentzMie-Debye approach. However, the method involves explicit convolutions between inverse Fourier transform (IFT) of spherical Hankel and Bessel functions (the IFT of spherical Hankel function only exists in convolution sense according to [39]), which introduces instability especially for high order modes. Recently, for acoustic scattering, Laplace transform based approach [41,42] and TDIE based approach [43] have been proposed to solve the time-dependent scattering problem.…”

“…None of those methods provides a time-domain Lorentz-Mie-Debye solution for electromagnetic scattering from spheres. In [39], timedomain acoustic scattering from a sphere is studied with the help of time-dependent wave functions (inverse Fourier transform of the frequency domain spherical wave functions). The same idea is also extended to two-sphere case [40].…”