A fundamental Lagrangian approach to transformation acoustics and spherical spacetime cloaking

Abstract

Transformation acoustics centers on the construction of advanced acoustic devices by combining mathematical transformation techniques with the engineering of acoustic metamaterials. We show how differential-geometric methods together with a variational principle form the basis of a powerful framework to control acoustic waves as desired. This formalism is required to leave the acoustic wave equation invariant under coordinate transformations and is shown to consist of a proposed acoustic Lagrangian function on a smooth spacetime manifold. As an immediate consequence, we can derive the general constitutive relations between the acoustic parameters (bulk modulus and mass-density tensor) of the physical and virtual spaces under consideration. We conclude with a practical application of this theory by presenting acoustic spherical cloaking with time dilation.