Exponential Convergence For Multipole And Local Expansions And Their Translations For Sources In Layered Media: Two-Dimensional Acoustic Wave

In this paper, we first derive the multipole expansion (ME) and local expansion (LE) for far fields from wave sources in two-dimensional (2-D) layered media as well as the multipole-to-local translation (M2L) operator, by using the generating function of Bessel functions and Sommerfeld integral representations of Hankel functions. Then, we give a rigorous proof of the exponential convergence of the ME, LE, and M2L. It is shown that the convergence of ME, LE, and M2L for the reaction field components of the 2-D Helmholtz Green's function in layered media depends on the distance between the target charge and an equivalent polarization source. The polarization sources can be used in the implementation of fast multipole methods for wave sources embedded in layered media.