Time-varying resonators in acoustic waveguides: a transfer matrix formalism for space–time-modulated metamaterials

The transfer matrix method remains a simple yet powerful tool for modelling acoustic systems, particularly in a closed waveguide geometry. Here, we present a generalization of this method based on the theory of mode-matching, that incorporates the effect of ultra-fast temporal variations in the geometry, applying it to a system of side-branching resonant cavities (quarter wavelength resonators) fixed to an acoustic waveguide, modulated through alteration of the cavity length. We calculate propagation in a waveguide containing both a single resonator and a periodic array. In particular, we predict the generation of additional Doppler-like terms in the reflected and transmitted fields that leads to the modification of the band structure, comparing our results to finite element simulations of space–time-modulated acoustic crystals.