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Cloaking in shallow-water waves via nonlinear medium transformation

Published online by Cambridge University Press:  30 July 2015

Ahmad Zareei  and
Mohammad-Reza Alam
Ahmad Zareei
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Mohammad-Reza Alam*
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
*
†Email address for correspondence: reza.alam@berkeley.edu
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Abstract

A major obstacle in designing a perfect cloak for objects in shallow-water 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 well-studied 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 one-to-one 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 shallow-water 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. We show that with this nonlinear transformation an object can be cloaked from any wave that merely satisfies the long-wave assumption. The presented transformation can be applied as well for the design of non-magnetic optical cloaks for electromagnetic waves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 273 - 287
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Copyright
© 2015 Cambridge University Press 

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