Central equations and band structures of linear magnetohydrodynamic waves in a magneto-lattice

We investigate the band structures and propagation properties of linear ideal magnetohydrodynamic (MHD) waves in a plasma with a spatially periodic background magnetic field (a “magneto-lattice”). We develop a plane wave expansion approach in two equivalent forms: one written using the usual linearized MHD perturbation variables and another written in terms of the fluid displacement. We validate both formulations with numerical tests, including an empty-lattice limit that recovers the uniform-plasma dispersion. The method enables efficient computation of dispersion relations and reveals intrinsic frequency bandgaps and cutoff behavior caused by magnetic periodicity. We show that the bandgap width increases with the amplitude of the periodic magnetic-field modulation (relative to the uniform background field), leading to suppression of selected wave modes. In addition, the magnetic periodicity splits the Alfvén continuum into multiple branches, a feature absent in uniform plasmas. These results provide a framework for tailoring MHD-wave propagation in structured plasmas and may be useful for future studies of plasma metamaterials and topological plasma waves.