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Mechanical metamaterials built up of compliant units can support the propagation of linear and nonlinear waves. A popular architecture consists of a one-dimensional chain of bistable elements connected by linear springs. This type of chain can support nonlinear transition waves that switch each element from one stable state to the other as they propagate along the chain. One way to manipulate the propagation of such waves is via introduction of a local inhomogeneity, i.e., a defect in the otherwise periodic chain. Recent analytical and numerical work has shown that based on its initial velocity, a transition wave may be reflected, transmitted, or captured upon interaction with the defect. In this thesis, we experimentally study the nonlinear wave-defect interaction in bistable mechanical metamaterials. We produce a physical bistable chain via fused deposition modeling (FDM) additive manufacturing using polylactic acid (PLA). The defects are introduced by modifying the local stiffness and/or mass at the defect site. This work contributes towards the development of a rational approach to defect engineering for manipulating nonlinear waves. Advisor: Piyush Grover

By calculating accurately the signals of wave sources following specific techniques, one can use their induced wave fields, in order to synthesize wave patterns with predefined spatial and temporal characteristics, inside various wave media. Using the previous idea, the first part of this thesis presents a linear wave field synthesis method which has potential applications in acoustic and electromagnetic media. Virtual sound reproduction mechanisms used in theaters and teleconference systems, as well as medical devices using ultrasound and electromagnetic radiation, could benefit from this method. The method is compared with traditional acoustic wave field synthesis techniques and simulations demonstrating its applicability on different source topologies with different radiation characteristics are also presented. Unlike the linear analysis of the first part, the second part of this thesis is focused on the theoretical and experimental study of certain nonlinear wave field synthesis phenomena which appear on two dimensional nonlinear LC lattices. More specifically, it is demonstrated how nonlinearity can help in synthesizing high frequency and high power wave pulses at the central points of these lattices, using many low power and low frequency sources at the boundaries. This idea has potential applications in ultra wide band communication and imaging systems and holds a promise of "closing the Terahertz window" formed by the power vs. frequency performance of electronic and optical devices.

The propagation of waves through nonlinear media is of interest here, namely as it pertains to two specific examples, a nonlinear dielectric and a hyperelastic solid. In both cases, we examine the propagation of two-dimensional, weakly nonlinear, quasi-planar waves. It is found that such systems will have a nonlinearity that is intrinsically cubic, and therefore, a classical Zabolotskaya-Khokhlov equation cannot give an accurate description of the wave evolution. To determine the general evolution equation in such systems, a multi-timing technique developed by Kluwick and Cox (1998) and Cramer and Webb (1998) will be employed. The resultant evolution equations are seen to involve only one new nonlinearity coefficient rather than the three coefficients found in other studies of cubically nonlinear systems. After determining the general evolution equation, inclusion of relaxation, dispersion and dissipation effects can be easily incorporated.