Linear Magnetohydrodynamic Waves in a Magneto-Lattice: A Unified Theoretical Framework and Numerical Validation

We present a systematic theoretical and numerical investigation of the propagation properties of linear magnetohydrodynamic (MHD) waves in a spatially periodic magnetic field, referred to as a magneto-lattice. Two types of central equations, expressed in terms of $\left(\rho,\boldsymbol{B},\boldsymbol{v}\right)$ (where $\rho$ is perturbed mass density, $\boldsymbol{B}$ is perturbed magnetic field, and $\boldsymbol{v}$ is perturbed velocity) and the perturbation displacement $\boldsymbol{\xi}$, are established using the plane wave expansion (PWE) method. The validity of both equations is demonstrated through two numerical examples. This framework enables the identification of intrinsic frequency bandgaps and cutoff phenomena within the system. Our numerical results show that the bandgap width increases with the magnetic modulation ratio $B_{m}$, leading to the suppression of specific MHD wave modes. Furthermore, the periodicity of the magnetic field induces the splitting of Alfv\'{e}n waves into multiple branches\textemdash a phenomenon absent in uniform plasmas. These findings provide new insights for manipulating MHD waves in a crystalline lattice framework of structured plasmas.