The focus of this dissertation is analyzing and manipulating acoustic wave propagation in metamaterials, which can be used to assist the design of acoustic devices. Metamaterials are artificial materials, which are arranged in certain patterns at a scale smaller than the wavelength and can exhibit properties beyond those naturally occurring materials. With metamaterials, novel phenomena, such as focusing, super absorption, cloaking and localization of ultrasound, are theoretically proposed and experimentally verified. In recent years, a planar version of metamaterials, often called meta-surfaces, has attracted a great deal of attention. Meta-surfaces can control and manipulate the amplitude, phase, and directions of waves. In this dissertation, we conducted a systematic study by deriving the effective medium theories (EMTs), and developing the theoretical and numerical models for our proposed designed metamaterial. Very recently, acoustic meta-surfaces have been used in the design of acoustic lenses, which can achieve various functionalities such as focusing and collimation. In the designs of acoustic lenses, impedance is an important issue because it is usually difficult to make the impedance of the lens equal to that of the environment, and mismatched impedance is detrimental to the performance of the acoustic lens. We developed an EMT based on a coupled-mode theory and transfer matrix method to characterize the propagation behavior and, based on these models, we report two designs of acoustic lenses in water and air, respectively. They are rigid thin plates decorated with periodically distributed sub-wavelength slits. The building block of the acoustic lens in water is constructed from coiling-up spaces, and that of the acoustic lens in air is made of layered structures. We demonstrate that the impedances of the lenses are indeed matched to those of the background media. With these impedance-matched acoustic lenses, we demonstrate acoustic focusing and collimation, and redirection of transmitted acoustic energy by finite-element simulations. In the framework of the hidden source of the volume principle, an EMT for a coupled resonator structure is derived, which shows that coupled resonators are characterized by a negative value of the effective bulk modulus near the resonance frequency and induce flat bands that give rise to the confinement of the incoming wave inside the resonators. The leakage of sound waves in a resonance-based rainbow trapping device prevents the sound wave from being trapped at a specific location. Based on our EMT, we report a sound trapping device design based on coupled Helmholtz resonators, loaded to an air waveguide, to effectively tackle the wave leakage issue. We show that a coupled resonators structure can generate dips in the transmission spectrum by an analytical model derived from Newton’s second law and a numerical analysis based on the finite-element method. We compute the transmission spectra and band diagram from the effective medium theory, which are consistent with the simulation results. Trapping and the high absorption of sound wave energy are demonstrated with our designed device.

Analytical solutions for the linear and nonlinear shallow-water wave equations are developed for evolution and runup of tsunamis &ndash / long waves&ndash / over one- and two-dimensional bathymetries. In one-dimensional case, the nonlinear equations are solved for a plane beach using the hodograph transformation with eigenfunction expansion or integral transform methods under different initial conditions, i.e., earthquake-generated waves, wind set-down relaxation, and landslide-generated waves. In two-dimensional case, the linear shallow-water wave equation is solved for a flat ocean bottom for initial waves having finite-crest length. Analytical verification of source focusing is presented. The role of focusing in unexpectedly high tsunami runup observations for the 17 July 1998 Papua New Guinea and 17 July 2006 Java Island, Indonesia tsunamis are investigated. Analytical models developed here can serve as benchmark solutions for numerical studies.