Transformation method and wave control

Abstract

Transformation method provides an efficient way to control wave propagation by materials. The transformed relations for field and material during a transformation are essential to fulfill this method. We propose a systematic method to derive the transformed relations for a general physic process, the constraint conditions are obtained by considering geometrical and physical constraint during a mapping. The proposed method is applied to Navier's equation for elastodynamics, Helmholtz's equation for acoustic wave and Maxwell's equation for electromagnetic wave, the corresponding transformed relations are derived, which can be used in the framework of transformation method for wave control. We show that contrary to electromagnetic wave, the transformed relations are not uniquely determined for elastic wave and acoustic wave, so we have a freedom to choose them differently. Using the obtained transformed relations, we also provide some examples for device design, a concentrator for elastic wave, devices for illusion acoustic and illusion optics are conceived and validated by numerical simulations.

Fluid Flow Control with Transformation Media

“…The key idea that enabled progress in those areas was the combination of coordinate transformations with coordinate-dependent, and often extremely exotic [5,[13][14][15], material properties. Transformation optics [3][4][5]11,16] and transformation acoustics [6-8] offer solutions to some inverse scattering problems [17] by reducing them to an elementary one-for example, scattering off a point object in free space.There is no reason why this conceptual approach cannot be used in other areas of physics [18], and, in fact, it has already been applied to conductive heat transfer [19], linear elastodynamics [5,8,[20][21][22][23][24], surface wave [25], and quantum-mechanical matter wave [26] dynamics. The fundamental requirement for the applicability of this concept is the presence of a medium with sufficiently flexible properties, which enables manipulation of the coefficients in the equations describing the dynamical process.…”

“…There is no reason why this conceptual approach cannot be used in other areas of physics [18], and, in fact, it has already been applied to conductive heat transfer [19], linear elastodynamics [5,8,[20][21][22][23][24], surface wave [25], and quantum-mechanical matter wave [26] dynamics. The fundamental requirement for the applicability of this concept is the presence of a medium with sufficiently flexible properties, which enables manipulation of the coefficients in the equations describing the dynamical process.…”

Fractional Fourier analysis of elastic wave scattering in inhomogeneous materials

“…[1][2][3][4] Due to zero-and first-order approximation method itself and the impedance mismatch in the materials, the scattering phenomena, which refers to the frequency changes as waves pass through the media, is caused in the material design of wave propagation control. 5 Generally, the operating mechanism of elastic waves is more complicated and difficult to control than electromagnetic waves.…”

“…5 Generally, the operating mechanism of elastic waves is more complicated and difficult to control than electromagnetic waves. 3 Furthermore, according to the existing control theory of wave propagation, some assumptions about the characteristics of the material, such as the anisotropy of the medium parameters, the continuity in the radial direction, and lossless dielectric are necessary but difficult to implement in practice. It is common that scattering exists in material designs of wave propagation control, including the directional cloak design of elastic waves.…”

Non-singular acoustic cloak derived by the ray tracing method with rotationally symmetric transformations

“…The method solved the problem to some extent, however, it should be emphasized that with a small regularization parameter the material parameters near the inner boundaries are still extremely large and are impossible for practical fabrication. Inspired by the deformation description in the continuum mechanics, Chang et al [6] regarded the geometry transformation as deformation in the language of finite elasticity and gained an intrinsic comprehension of the transformation method for electromagnetics, acoustics and elastodynamics. Based on the deformation perspective, Hu et al [7] revealed the underlying mechanism of the non-uniqueness of transformation acoustics and derived several sets of transformation relations by considering energy conservation and form invariance of governing equations under the local affine transformation.…”