Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media

We consider scalar waves in periodic media through the lens of a second-order effective, i.e., macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coeficients in the governing mean field equation|including both the leading order (i.e., quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit cell solutions used to formulate the unperturbed macroscopic model; (ii) two adoint-field solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of nonperiodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) subwavelength sensing of periodic defects or perturbations.