Wave Stop Bands in a Double-Periodic Elastic Waveguide from Bodies Connected to Each Other by Thin Rods

A periodic isotropic and homogeneous elastic waveguide composed of identical massive bodies connected to each other by thin horizontal rods and attached to a rigid half-space by vertical rods was studied. Using the dimension reduction procedure on the rods and analyzing the interaction between rigid displacements and singular fields in a body, the asymptotics of eigenvalues depending on the Floquet parameters was constructed in the model problem on a periodicity cell. As a result, asymptotic formulas were found for the lengths and positions of spectral (wave pass) bands, and open spectral gaps (wave stop bands) were revealed.