Proof of two theorems related to the energy of acoustic Bloch waves in periodically inhomogeneous media

Abstract

Two theorems which are applicable to plane waves in homogeneous elastic media are generalized to the case of periodic inhomogeneity. The first theorem is known as Rayleigh's principle and essentially says that, on average, kinetic energy equals potential energy. The second theorem states that group velocity and velocity of energy transport ("energy velocity") are the same. Both equalities are verified below for periodically inhomogeneous media. As distinguished from the homogeneous case one has to deal with Bloch waves instead of plane waves and has to perform spatial averages over an elementary cell of the lattice. By means of plausible arguments the theorems can also be applied to the modes of infinitely extended plates and beams.