Transformation elastodynamics and active exterior acoustic cloaking

TL;DR

This paper explores transformation elastodynamics and introduces active exterior acoustic cloaking techniques for elastic and acoustic waves, including new models, cloaking methods, and explicit solutions for larger objects.

Contribution

It presents novel cloaking methods based on active exterior devices, transformation elastodynamics models with torque springs, and explicit solutions for improved cloaking of larger objects.

Findings

Transformation of elastodynamic equations under coordinate changes.

Design of torque springs for realizing transformed networks.

Explicit cloaking solutions for larger objects at fixed distances.

Abstract

This chapter consists of three parts. In the first part we recall the elastodynamic equations under coordinate transformations. The idea is to use coordinate transformations to manipulate waves propagating in an elastic material. Then we study the effect of transformations on a mass-spring network model. The transformed networks can be realized with "torque springs", which are introduced here and are springs with a force proportional to the displacement in a direction other than the direction of the spring terminals. Possible homogenizations of the transformed networks are presented, with potential applications to cloaking. In the second and third parts we present cloaking methods that are based on cancelling an incident field using active devices which are exterior to the cloaked region and that do not generate significant fields far away from the devices. In the second part, the exterior cloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of analytic functions. An explicit solution is given that allows to cloak larger objects at a fixed distance from the cloaking device, compared to previous explicit solutions. In the third part we consider the active exterior cloaking problem for the Helmholtz equation in 3D. Our method uses the Green's formula and an addition theorem for spherical outgoing waves to design devices that mimic the effect of the single and double layer potentials in Green's formula.