Metamaterials and metasurfaces for water waves and marine structures

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In application areas such as electromagnetics, optics, elasticity and acoustics, metamaterials are used to control waves in unusual ways. A metamaterial is a medium in which properties of a field can be propagated in a manner not normally found in naturally-occurring materials. They are most often comprised of microstructures much smaller than the natural lengthscales intrinsic to the underlying field variables in such a way that their macroscopic effect on the field allows complex phenomena to be exhibited. In the application areas listed above examples include "negative refraction" in which oblique waves bend backwards as they enter the metamaterial, "perfect lensing" and "invisibility cloaking", which both rely on negative refractivity. Likewise, metasurfaces are boundaries to domains which interact with a field in an unusual way. These are often comprised of microstructres that are man made rather than being found in naturally-occurring materials.The classical theory of linearised water waves is an example of a potential field theory sharing some common features with electromagnetic, acoustic and elastic field theories. In particular it is a wave-supporting environment. With a few notable exceptions, many of the ideas of metamaterials and metasurfaces developed in other application areas have not been extended or incorporated into water waves. Those that have have often taken advantage of an approximate "linearised shallow water theory" or "long wave theory" to transfer ideas and results across to water waves. This has led to some interesting ideas and developments particularly in the area of "invisibility cloaking" in water waves (an area which the PhD supervisor has made some contributions). However, the set of equations which define water waves have particular features which make them different to other field theories. The dependence on the field in the depth direction is different to the other two horizontal wave-bearing directions and the existence of lateral boundaries on the domain form an essential part of a water-wave problem in a way not shared by other field theories. Shallow water theory uses depth-averaging as a means of overcoming these difficulties but does so at the expense of introducing an approximation and restricting the range of parameters for which the approximation is justified.