  Exact space-time propagator for the wave (second-order in time) equation in a
layered system, made up of a layer sandwiched between two other different
semi-infinite layers, is obtained by means of the multiple scattering theory
(MST) approach. The wave equation for this layered system is characterized by
the spatial dependence of the wave phase velocity which changes in the
perpendicular-to-interfaces direction when crossing the interfaces and thus
having different constant values in different layers of the threelayer. The MST
approach is made possible due to obtained effective potentials localized at
interfaces and responsible for reflection from and propagation through them.
The obtained space-time propagator exactly solves the considered wave equation
for any initial value and allows for following in time the processes of
reflection and transmission. The solution for an initial Gaussian wave packet
is obtained, analyzed and numerically visualized for a simple case of the
perpendicular-to-interfaces incoming wave. In particular, the contribution of
the backward-moving components of the wave packet is revealed.
