  In this paper, we present a powerful method (Atomistic Green's Function, AGF)
for calculating the effective Hamiltonian of acoustic and elastic
wave-scatterers. The ability to calculate the effective Hamiltonian allows for
the study of scattering problems in infinite systems without the introduction
of any artificial truncating boundaries such as perfectly matched layers or
Dirichlet to Neumann (DtN) maps. Furthermore, the AGF formalism also allows for
the efficient calculation of the Green's function of the scatterer as well as
all relevant scattering metrics including reflection and transmission ratios.
The formalism presented here is especially suited to scattering problems
involving waveguides, phononic crystals, metamaterials, and metasurfaces. We
show the application of the method to three scattering problems: scattering
from a slab (1D), scattering from a finite phononic crystal (1D), and
scattering from defects in a waveguide (2D).
