  In this paper, a fast multipole method (FMM) is proposed to compute
long-range interactions of wave sources embedded in 3-D layered media. The
layered media Green's function for the Helmholtz equation, which satisfies the
transmission conditions at material interfaces, is decomposed into a free space
component and four types of reaction field components arising from wave
reflections and transmissions through the layered media. The proposed algorithm
is a combination of the classic FMM for the free space component and FMMs
specifically designed for the four types reaction components, made possible by
new multipole expansions (MEs) and local expansions (LEs) as well as the
multipole-to-local translation (M2L) operators for the reaction field
components. { Moreover, equivalent polarization source can be defined for each
reaction component based on the convergence analysis of its ME. The FMMs for
the reaction components, implemented with the target particles and equivalent
polarization sources, are found to be much more efficient than the classic FMM
for the free space component due to the fact that the equivalent polarization
sources and the target particles are always separated by a material interface.}
As a result, the FMM algorithm developed for layered media has a similar
computational cost as that for the free space. Numerical results validate the
fast convergence of the MEs and the $O(N)$ complexity of the FMM for
interactions of low-frequency wave sources in 3-D layered media.
