If an MHD surface wave is supported by a true discontinuity, then the total pressure fluctuation, ΔPtot, is constant across the discontinuity. If the discontinuity is replaced by a thin transition layer, then ΔPtot will be approximately constant across the transition layer, with a value approximately the same as the value obtained for the case of a true discontinuity. We use this approximation to study the behavior of the plasma and fields in the transition layer. We regard ΔPtot as known, and write the relevant equations in forms in which ΔPtot appears as driving terms. Two resonances appear. The cusp resonance affects the density and pressure fluctuations, and the velocity and magnetic field components along the background magnetic field, B0. The Alfven resonance affects the velocity and magnetic field components normal to B0. We concentrate on the Alfven resonance, and show in a simple way how energy is pumped out of the surface wave into thin layers surrounding the resonant field lines. We consider also the effects of three types of viscosity on the Alfven resonance. Only classical shear viscosity is able to absorb the energy which is pumped into the thin resonant layer. In the steady state, the net viscous heating is independent of the viscosity coefficient, if the heating occurs in a sufficiently thin layer. We suggest that the large velocity shears which occur in the vicinity of the resonant field lines can lead to Kelvin-Helmholtz instabilities, which can in turn lead to an effective eddy viscosity, which we estimate to be large enough in the solar corona to distribute heat throughout large portions of coronal active region loops. We show also that coronal heating by the Alfven resonance is compatible with a variety of coronal data. ¿ American Geophysical Union 1988