We consider dissipative Alfve´n waves in a viscous atmosphere with a magnetic field, so that the electrical conductivity is considered as a non-diagonal tensor. Under a vertical uniform magnetic field we consider transverse magnetic and velocity perturbations. We obtain two decoupled second-order systems to right and left polarized Alfve´n waves. The first electrical diffusivity&khgr;1and the viscous diffusivity &eegr; are associated with “real” damping; the second electrical diffusivity&khgr;2is associated with “imaginary damping”. Boundary conditions are imposed in order to obtain the velocity perturbation spectrum of an Alfve´n wave. The velocity perturbation is plotted versus altitude over a distance of five scale heights, for different values of the parameters. ©2000 American Institute of Physics.