Transformation Elastodynamics and Active Exterior Acoustic Cloaking

This chapterAcoustics consistsCloaking of three parts. In the first part we recall the elastodynamicElastodynamic equations under coordinate transformations. The idea is to use coordinate transformations to manipulate wavesWaves propagating in an elasticElastic material. Then we study the effect of transformations on a massMass-spring network model. The transformed networks can be realized with “torque springsTorque 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 homogenizationsHomogenization of the transformed networks are presented, with potential applications to cloakingCloaking. In the second and third parts we present cloakingCloaking 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 cloakingCloaking problem for the Laplace equation is reformulated as the problem of polynomial approximation of functions. An explicit solution is given that allows cloakingCloaking of larger objects at a fixed distance from the cloakingCloaking device, compared to previous explicit solutions. In the third part we consider the active exterior cloakingCloaking problem for the HelmholtzHelmholtz equation in 3D. Our method uses the Green’s formulaGreen’s formula and an addition theorem for spherical outgoing wavesWaves to design devices that mimic the effect of the single and double layer potentials in Green’s formulaGreen’s formula.