Waves in layered media are governed by the one dimensional wave equation with variable coefficient. Piecewise constant coefficients are a natural approximation to the general case of arbitrary variable coefficient. There are two paradigms within the piecewise constant context, depending on the spacing of jump points. Recent work on the 1D wave equation has revealed that disk polynomials underpin a closed form representation of the reflection Green's function when jump points of the coefficient are irregularly spaced. But when jump points are regularly spaced the representation in terms of disk polynomials collapses. Remarkably, orthogonal polynomials on the unit circle (OPUC) enter the picture in the case of a coefficient with regularly spaced jump points, and the recurrence relation for OPUC provides a simple solution to the inverse problem of computing the (unknown) coefficient from time limited reflection data.