  This paper proposes a novel method to establish the wellposedness and
convergence theory of the uniaxial-perfectly-matched-layer (UPML) method in
solving a two-dimensional acoustic scattering problem due to a compactly
supported source, where the medium consists of two layers separated by the
horizontal axis. When perfectly matched layer (PML) is used to truncate the
vertical variable only, the medium structure becomes a closed waveguide. The
Green function due to a primary source point in this waveguide can be
constructed explicitly based on variable separations and Fourier
transformations. In the horizontal direction, by properly placing periodical
PMLs and locating periodic source points imaged by the primary source point,
the exciting waveguide Green functions by those imaging points can be assembled
to construct the Green function due to the primary source point for the
two-layer medium truncated by a UPML. Incorporated with Green's identities,
this UPML Green function directly leads to the wellposedness of the acoustic
scattering problem in a UPML truncation with no constraints about wavenumbers
or UPML absorbing strength. Consequently, we firstly prove that such a UPML
truncating problem is unconditionally resonance free. Moreover, we show, under
quite general conditions, that this UPML Green function converges to the exact
layered Green function exponentially fast as absorbing strength of the UPML
increases, which in turn gives rise to the exponential convergence of the
solution of the UPML problem towards the original solution.
