Transformation Based Wave Control
The method of Transformation Optics, which is based on transformation solutions, was originally developed in optics community for creating passive invisibility cloaks. This method uses the subtle idea of coordinate transformation and form-invariance of Maxwell equations and offers an unprecedented way of steering optical waves through architecting the constitutive parameters (i.e. permittivity and permeability) of a dielectric.
In this thesis, we present the method transformation media, inspired by transformation optics, as a general mathematical framework for an arbitrary steering of mechanical waves including surface gravity waves in fluids and also flexural waves in thin plates. Particularly, we prove that surface gravity wave equation in shallow water (long-wave limit) is form-invariant and therefore permits transformation solutions. We further formulate a general coordinate free equation that determines spatial variations required for water depth and gravitational acceleration as constitutive parameters for an arbitrary steering of long surface gravity waves. This general equation provides transformation media scheme for water waves. Additionally, since one of the constitutive parameters is gravitational acceleration and is actually a physical constant, we formulate a nonlinear transformation to keep the gravitational acceleration intact and with only sea-bed variations, we design and numerically validate a broadband invisibility cloak for surface gravity waves as the benchmark of transformation media method. In order to fully enable various applications of transformation media for surface gravity waves, a variable gravitational acceleration is required. We show that an altered effective gravitational acceleration is possible through a visco-elastic sea-bed, where the effective gravity depends on viscosity and elasticity of the bottom carpet. The visco-elastic sea bed itself provides a new mechanism to control oceanic wave energy through variations in the effective gravitational acceleration. Based on this idea, we present a theoretical design for an equivalent of graded-index optical fibers for water-waves which defies the spreading loss of water waves and can transmit wave energy over long distances in a broad range of frequencies and without a need for side-walls. Since the effect of sea-bed variation decreases exponentially as the water depth increases, a whole new approach is required for controlling waves in finite/deep water. We propose a flexural buoyant carpet, with an engineered rigidity and mass, floating on the free surface to affect and control waves. We further design a cloaking carpet for water waves in finite/deep water waves to prove the efficacy of the proposed method.
In the case of flexural waves, the governing equation for elastic waves in thin plates is not form invariant, and hence transformation based solution does not apply to such waves. Applying a transformation to the flexural wave equation, most of the terms in the transformed equation do not match with the exact orthotropic and inhomogeneous flexural wave equation, and therefore the governing equation is not form-invariant. We show that by carefully designing the transformation and also assuming a pre-stressed material with body forces, all of the terms in the transformed equation match with the exact inhomogeneous and orthotropic equation, and a perfect broadband flexural cloak can be designed. 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 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 non form-invariant governing equations. Lastly, we show the relation between flexural rigidity in thin plates and the equivalent of refractive index for flexural waves. We design a flexural GRIN lens in a thin plate by smoothly varying the plate’s rigidity, and thus its refractive index. We show that the proposed lens is broadband, has a fixed focal point over a wide range of frequencies, and is theoretically capable of zero-aberration focusing. We numerically explore our Continuous Profile GRIN lens (CP-GRIN lens) and further experimentally validate an implemented design.