The propagation of magnetohydrodynamic waves above the E region of the ionosphere is treated with the aim of developing a computer program to predict the space-time history of magnetohydrodynamic waves generated by a given source. Changes in ionization density as a function of altitude and the geomagnetic field in the form of a magnetic dipole are incorporated. The Carovillano-McClay method of solution (expansion in vector spherical harmonics) was used. This leads to a set of coupled, second order, ordinary, differential equations to solve. Boundary conditions, for the case where the ionospheric plasma is treated as a cavity bounded by vacuum, are derived. Some preliminary numerical calculations, for a source consisting of an axisymmetric ring current located over the magnetic pole, are presented. The coupling of wave modes due to the Hall Effect is discussed. The physics of positive ion-neutral particle collisions, important in the propagation of magnetohydrodynamic waves in the E region, has been considered from the point of view of possible modifications to the hydrodynamic equations. Corrections to the hydrodynamic moment equations due to space-time correlation effects have been investigated by the theory of Green and found to be small. Corrections to the adiabatic pressure relations due to the interaction of the positive ions with the neutrals is considered and found to be significant for the propagation of magnetohydrodynamic waves through the E layer. Various forms for the correction terms as they effect the momentum equations have been derived but no detailed calculations are carried out. (Author).