Rotating convection in the Earth's core produces columnar vortices of radius ~10 km or less near the inner core boundary. Small-scale motions in the core can travel as Alfvén waves in the face of Ohmic diffusion, provided the ratio of the magnetic diffusion time th to the Alfvén wave travel time tA (measured by the Lundquist number S0) is much greater than unity. These motions transfer angular momentum from the core to the mantle, a process that can help explain variations in length of day. Vortices subject to the combined influence of a magnetic field and background rotation give rise to fast and slow Magneto-Coriolis (MC) waves whose damping is not well understood. This thesis investigates the long-time evolution of magneto hydrodynamic (MHD) waves generated by an isolated, small-scale motion in an otherwise quiescent, electrically conducting fluid. The first part of the study focuses on the damping of small-scale Alfvén waves, which is independent of rotation. For a plausible magnetic field strength in the Earth's core, it is shown that flows of lengthscale ~ 5 km or larger can propagate across the core as damped Alfvén waves on sub-decadal timescales. The second part of the study looks at MC waves generated from an isolated blob under rotation and a uniform axial magnetic field. The decay laws for these waves are obtained by considering the decay of fast and slow waves individually. While the fast waves are subject to strongly anisotropic magnetic diffusion, the slow waves diffuse isotopically. New timescales are derived for the onset of damping and the transition from the wave-dominated to the diffusion-dominated (quasi-static) phase of decay. This study shows for the first time that MC waves originating from small-scale vortices of magnetic Reynolds number Rm ~ 1 can be long-lived. The results of this study are extendible to small-scale MHD turbulence under rotation, whose damped wave phase has not been adequately addressed in the literature. Furthermore, it is thought that this study would help place a lower bound on the poloidal magnetic field strength in the Earth’s core.