This paper presents the first full band-structure calculations for periodic, elastic composites. The authors investigate the acoustic band structure of such composites, focusing on transverse polarization of vibrations, which results in a "phononic" band gap that extends throughout the Brillouin zone. This band gap has significant implications for the suppression of zero-point motion and the localization of phonons, potentially leading to improvements in transducers and the creation of vibrationless environments. The study is inspired by developments in photonic band structures for periodic structures made of two different dielectric materials. The authors compare their findings with those of photonic band structures, noting that complete band gaps in photonic systems are achieved when the dielectric constant ratio of the two constituents is sufficiently large. In contrast, elastic composites have limited band-structure calculations, with most studies focusing on a single direction of the wave vector. The authors consider a system composed of an array of straight, infinite cylinders made of an isotropic solid "a" embedded in an elastic background "b." They derive the wave equation for inhomogeneous solids and use periodicity to expand the density and sound speed functions in two-dimensional Fourier series. The resulting equations are solved to determine the band structure, showing the presence of a phononic band gap for vibrations parallel to the cylinders. The study highlights the importance of understanding complete acoustic or phononic band gaps for applications in mechanical systems and transducers. It also discusses the potential for Anderson localization of phonons in strongly scattering dielectric structures, a phenomenon observed in photonic systems. The authors conclude that the study of complete gaps in classical-wave band structures could provide insights into the conditions for strong localization of these waves. The paper also notes that the interest in periodic elastic composites remains valid even in the presence of a low density of states rather than a full gap.This paper presents the first full band-structure calculations for periodic, elastic composites. The authors investigate the acoustic band structure of such composites, focusing on transverse polarization of vibrations, which results in a "phononic" band gap that extends throughout the Brillouin zone. This band gap has significant implications for the suppression of zero-point motion and the localization of phonons, potentially leading to improvements in transducers and the creation of vibrationless environments. The study is inspired by developments in photonic band structures for periodic structures made of two different dielectric materials. The authors compare their findings with those of photonic band structures, noting that complete band gaps in photonic systems are achieved when the dielectric constant ratio of the two constituents is sufficiently large. In contrast, elastic composites have limited band-structure calculations, with most studies focusing on a single direction of the wave vector. The authors consider a system composed of an array of straight, infinite cylinders made of an isotropic solid "a" embedded in an elastic background "b." They derive the wave equation for inhomogeneous solids and use periodicity to expand the density and sound speed functions in two-dimensional Fourier series. The resulting equations are solved to determine the band structure, showing the presence of a phononic band gap for vibrations parallel to the cylinders. The study highlights the importance of understanding complete acoustic or phononic band gaps for applications in mechanical systems and transducers. It also discusses the potential for Anderson localization of phonons in strongly scattering dielectric structures, a phenomenon observed in photonic systems. The authors conclude that the study of complete gaps in classical-wave band structures could provide insights into the conditions for strong localization of these waves. The paper also notes that the interest in periodic elastic composites remains valid even in the presence of a low density of states rather than a full gap.