Wave propagation in two physical structures is described and analyzed in this dissertation. In the first problem, the propagation of a normally incident plane acoustic wave through a three dimensional rigid slab with periodically placed holes is modeled and analyzed. The spacing of the holes A and B, the wavelength λ and the thickness of the slab L are order one parameters compared to the characteristic size D of the holes, which is a small quantity. Scattering matrix techniques are used to derive expressions for the transmission and reflection coefficients of the lowest mode. These expressions depend only on the transmission coefficient, r₀ of an infinitely long slab with the same configuration. The determination of r₀ requires the solution of an infinite set of algebraic equations. These equations are approximately solved by exploiting the small parameter D/√AB. Remarkably, this structure is transparent at certain frequencies which could prove useful in narrow band filters and resonators.

In the second problem, a systematic mathematical approach is given to find the solutions of microstrip transmission lines. Specifically, we employ an asymptotic method to determine an approximation to the field components and propagation constant when the wavelength is much bigger than the thickness of the substrate. It is found that the transverse electrical and magnetic fields can be expressed in terms of two potential functions which are elliptic in character and are coupled through the longitudinal electrical field boundary conditions. The solvability conditions for the longitudinal magnetic field yield an approximation to the propagation constant. Transmission line equations are also obtained for coupled microstrip transmission lines and single microstrips with smoothly changing widths by using the same techniques.