In this chapter we study the band gap structures as well as the effect of inhomogeneities within the perfect one-dimensional (1D) phononic crystal (PnC) such as a free surface, a PnC/substrate interface, and a defect layer embedded in these systems. Such inhomogeneities are usually present in actual device structures as a support (substrate) or as a protection (cap layer) for the PnC; the defect layers offer the possibility of wave filtering and sometimes they can be introduced as an imperfection during the epitaxial growth process. The PnCs are considered as semiinfinite or finite structures. A general rule about the existence of localized surface modes in elastic semiinfinite PnCs with a free surface is presented. For a finite PnC, due to the interaction between the surface, interface, and bulk waves, different localized and resonant modes are obtained and their properties are investigated. These results are obtained in the frame of a Green’s function formalism that enables to deduce the dispersion curves, local and total densities of states, transmission and reflection coefficients as well as the scattering of light by phonons. In particular, an exact relation between the density of states and the phase times is pointed out. The application of 1D PnC as acoustic mirrors that exhibit total reflection of waves for all incident angles and polarizations in a given frequency range is indicated. These structures may also be used as acoustic filters when a defect layer is inserted within the finite PnC. A discussion is also included about some spectroscopic techniques used to probe the acoustic waves such as Raman and Brillouin light scattering and other acoustic techniques as the surface acoustic waves and the picosecond laser techniques among others. A comparison of the theoretical results with experimental data available in literature is also presented.