We consider the general problem of linear Alfvèn waves propagating in a dissipative atmosphere Cl], and obtain an exact solution of the wave equation with double diffusion, i.e., by fluid viscosity and electrical resistance. This solution includes, as particular cases, nondissipative Alfvèn waves in an isothermal atmosphere, with a vertical [2,3] or oblique [4] magnetic field, and the case when resistive dissipation alone is present C5,6]. This solution may be relevant to a variety of solar atmospheric problems in which dissipation is an essential element, e.g., atmospheric heating C7] or resonances [81, which have been modelled using the undamped solutions of the Alfvèn wave equation. The present, exact solution, can be used to assess the domain of validity of the RLC-analogy C9],and of the phase mixing approximation [10-11]. An exhamination of the effects, on wave amplitude and phase, of changing wave frequency, horizontal wavenumber, magnetic field inclination and viscous and resistive damping scales (Figures 2 to 6), shows that intense, localized dissipation can occur generally at an intermediate altitude; this mechanism of atmospheric heating by propagation-diffuse coupling is a spatial analogue of some properties [12] unsteady magnetic fields.