Wave properties of a double one-dimensional periodic sheet grating

We show that the double one-dimensional periodic sheet gratings always have waveguide properties for acoustic waves. In general, there are two types of pass bands: i.e., the connected sets of frequencies for which there exist harmonic acoustic traveling waves propagating in the direction of periodicity and localized in the neighborhood of the grating. Using numerical-analytical methods, we describe the dispersion relations for these waves, pass bands, and their dependence on the geometric parameters of the problem. The phenomenon is discovered of bifurcation of waveguide frequencies with respect to the parameter of the distance between the gratings that decreases from infinity. Some estimates are obtained for the parameters of frequency splitting or fusion in dependence on the distance between the simple blade gratings forming the double grating. We show that near a double sheet grating there always exist standing waves (in-phase oscillations in the neighboring fundamental cells of the group of translations) localized near the grating. By numerical-analytical methods, the dependences of the standing wave frequencies on the geometric parameters of the grating are determined. The mechanics is described of traveling and standing waves localized in the neighborhood of the double gratings.