This research is focused upon applying mathematical techniques to the modelling of modern materials, so-called metamaterials, in alignment to research in continuum mechanics. The ideas originate from electromagnetism, but have recently begun to be implemented in acoustics and elasticity, where there are nuances and changes in this new context. The objectives are to bring the power of mathematical ideas to bear upon the engineering and physics problems of current interest and to develop new and efficient methodologies that enable physical interpretation, and thus motivate the design, of such materials. Applications include many areas where wave energy and transport are important, since such materials can ultimately allow the control of wave propagation. Specific areas include underwater acoustics and re-direction of ground vibration.