  Layered media have been studied extensively both for their importance in
imaging technologies and as an example of a hyperbolic PDE with discontinuous
coefficients. From the perspective of acoustic imaging, the time limited
impulse response at the boundary, or boundary Green's function, represents
measured data, and the objective is to determine coefficients, which encode
physical parameters, from the data. The present paper resolves two fundamental
open problems for layered media: (1) how to compute the time limited Green's
function in the presence of discontinuous coefficients; and (2) to determine
precisely how data depends on coefficients. We show that there exists a single
system of equations in $3n$-dimensional space that governs the parameterized
family of all $n$-layered media simultaneously. The alternate system has smooth
coefficients, can be solved directly by separation of variables, and recovers
the impulse response at the boundary for the original equations. The analysis
brings to light an exotic laplacian---hybrid between the euclidean and
hyperbolic laplacians---that plays a central role in the scattering process.
Its eigenfunctions comprise a new family of orthogonal polynomials on the disk.
These serve as building blocks for a universal wavefield in terms of which the
dependence of data on coefficients has a simple description: reflection data is
obtained by sampling a translate of the wavefield on the integer lattice and
then pushing forward by a linear functional, where the translate and
pushforward correspond to reflectivity and layer depth vectors, respectively.
