Ahmad Zareei Postdoctoral Fellow in Applied Mathematics, School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 
American Scientist Our research of ocean cloaking on American Scientist cover photo! 
Cloaking water waves in
shallow water A major obstacle in designing a perfect cloak for objects in shallowwater waves is that the linear transformation media scheme (also known as transformation optics) requires spatial variations of two independent medium properties. In the Maxwell’s equation and for the wellstudied problem of electromagnetic cloaking, these two properties are permittivity and permeability. Designing an anisotropic material with both variable permittivity and variable permeability, while challenging, is achievable. On the other hand, for long gravity waves, whose governing equation maps onetoone to the single polarization Maxwell’s equations, the two required spatially variable properties are the water depth and the gravitational acceleration; in this case changing the gravitational acceleration is simply impossible. Here we present a nonlinear transformation that only requires the change in one of the medium properties, which, in the case of shallowwater waves, is the water depth, while keeping the gravitational acceleration constant. This transformation keeps the governing equation perfectly intact and, if the cloak is large enough, asymptotically satisfies the necessary boundary conditions. [PDF]


Broadband Cloaking of
Flexural Waves The governing equation for elastic waves in flexural plates is not form invariant, and hence designing a cloak for such waves faces a major challenge. Here, we present the design of a perfect broadband cloak for flexural waves through the use of a nonlinear transformation in the region of the cloak and by matching term by term the original and transformed equations and also assuming a prestressed material with body forces. For a readily achievable flexural cloak in a physical setting, we further present an approximate adoption of our perfect cloak under more restrictive physical constraints. Through direct simulation of the governing equations, we show that this cloak, as well, maintains a consistently high cloaking efficiency over a broad range of frequencies. The methodology developed here may be used for steering waves and designing cloaks in other physical systems with nonforminvariant governing equations. [PDF]


Continuous Profile Flexural
GRIN Lens A significant challenge in flexural wave energy harvesting is the design of an aberrationfree lens capable of finely focusing waves over a broad frequency range. To date, flexural lenses have been created using discrete inclusions, voids, or stubs, often in a periodic arrangement, to focus waves via scattering. These structures are narrowband either because scattering is efficient over a small frequency range, or the arrangements exploit Bragg scattering bandgaps, which themselves are narrowband. In addition, current lens designs are based on a single frequency and approximate the necessary refractive index profile discretely, introducing aberrations and frequencydependent focal points. Here, we design a flexural GRIN lens in a thin plate by smoothly varying the plate’s rigidity, and thus its refractive index. Our lens (i) is broadband, since the design does not depend on frequency and does not require bandgaps, (ii) has a fixed focal point over a wide range of frequencies, and (iii) is theoretically capable of zeroaberration focusing. We numerically explore our Continuous Profile GRIN lens (CPGRIN lens) and then experimentally validate an implemented design. Furthermore, we use a piezoelectric energy harvester disk, located at the first focus of the CPGRIN, to document improvements in power gain. [PDF]


Cloaking
by a Floating Thin Plate An alternative way of cloaking is to use an engineered elastic buoyant carpet placed on water around the object. The carpet will effectively bend the wave rays around the object and shield the object from impinging waves. The method can potentially open up a new avenue in the protection of ocean objects, particularly offshore structures, from the action of oceanic waves hence reducing load on such structures. [PDF]

Interaction
between topographic features and internal
waves Interaction of stratified flows with the solid bottom boundary is a main source for generation of internal waves. This interaction can lead to generation of nearinertial internal waves which appear as a prominent peak in the internal wave spectrum. In some specific occasions, if the bottom topography wavenumber is an integer coefficient of the wavenumber of incident internal wave, the interaction could lead to generation of high wavenumer internal waves which are potentially susceptible to breaking. Here we show that the distribution of energy of internal gravity waves over a patch of seabed corrugations strongly depends on the distance of the patch to adjacent seafloor features located downstream of the patch.[PDF] 

Inherently
Unstable Internal Waves due to Resonant Harmonic
Generation Here we show that there exist internal gravity waves that are inherently unstable, that is, they cannot exist in nature for a long time. The instability mechanism is a oneway (irreversible) harmonicgeneration resonance that permanently transfers the energy of an internal wave to its higher harmonics. We show that, in fact, there are a countably infinite number of such unstable waves. For the harmonicgeneration resonance to take place, nonlinear terms in the free surface boundary condition play a pivotal role, and the instability does not occur in a linearlystratified fluid if a simplified boundary condition such as a rigid lid or a linearized boundary condition is employed. Harmonicgeneration resonance presented here also provides a mechanism for the transfer of internal wave energy to the higherfrequency part of the spectrum where internal waves are more prone to breaking, hence losing energy to turbulence and heat and contributing to oceanic mixing. [PDF]

EulerSchrodinger
Transformation Peculiar microscale quantum behaviors has been observed experimentally in the hydrodynamics of a bouncing droplet on a vibrant fluid bath. Supported by recent experiments, we have been recently working on a mathematical map between Schrodinger equation describing quantum systems and Euler’s equation describing classical small scale water waves. We are excited to investigate the implication, potentials, and limits of this fundamental transformation to (i) find a physical understanding of quantum behaviors observed in microscale fluid systems, (ii) potentially develop an equivalent hydrodynamic setups for quantum experiments and vice versa, and (iii) investigate possibility of a hydrodynamic quantum computer for solving complex problems. 
MicroSwimmers at Low
Reynolds number: Motion at low Reynolds number is impossible with periodic wave motions due to time symmetry of equations; an asymmetric wavelike motion can generate locomotion. We have created an artificial mechanisms that mimic the locomotory functions of nematodes with efficient viscous pumping. We show that the blade not only induces a flow structure similar to that of the worm, but also mixes the surrounding fluid by generating a circulatory flow. [PDF]
