| Phononic crystals and quasicrystals are solid-solid, fluid-solid or fluid-fluid composites with periodic or quasiperiodic long-range order, respectively. They are of interest not only because of the profound effects of their periodic or quasiperiodic structure on wave propagation (e.g. the existence of elastic or acoustic band gaps), but also because of potential applications like sound filters, transducer design and acoustic mirrors, etc. In this thesis, I will present some research progress in the study of both phononic crystals and quasicrystals.; For phononic crystals, we have proposed a simple, systematic and efficient method to engineer elastic (acoustic) band gaps. A gap can be enhanced or created by altering the microstructure according to the field-energy distributions of the Bloch states at the band edges as well as their derivatives. Due to the structure of the acoustic wave equation, the engineering of acoustic band gaps is much more efficient than that of photonic band gaps. For elastic waves, a large absolute band gap can be enhanced or created by inserting air inclusions in a two-component elastic phononic crystal or quasicrystal with small density contrast and filling fraction. The positions of the insertion are chosen to suppress the shear potential energy of the acoustic branches and lower their frequencies. The validity of the proposed methods is supported by multiple-scattering calculations in various two-dimensional systems, such as steel cylinders in air, water cylinders in mercury and aluminum cylinders in epoxy. Our methods make the acoustic band gap "designable" and the realization of a light and effective sonic insulator possible.; The sonic band-gap structures of a 12-fold symmetric quasicrystal consisting of rigid cylinders in air are investigated by using the multiple scattering method. Large full gaps are found owing to its high symmetry. However, the gap structure evolves with the sample size. By using the SMCG method, we have observed a self-similar like behavior of states emerging inside original gaps and gaps emerging inside original bands when the sample size is increased. The long-range order has been proved important for the formation of gaps at low frequencies. We have also studied the properties of wave functions in large samples (up to 33919 cylinders). Various types of wave functions have been seen, including extended state, self-similar state with a power-law decay and confined state. |
