Coronal hole regions are well-known sources of high-speed solar wind; however, to account for the observed properties of the solar wind, a source of energy must be included in addition to heat conduction. Alfv¿n waves were suggested as the possible source of heating that accelerates the solar wind. We investigate the heating and propagation of the fast and shear Alfv¿n waves in coronal holes via numerical solution of the time-dependent, linearized, resistive, low-&bgr;, two-dimensional MHD equations in slab geometry. The waves are driven at the lower boundary of the coronal hole and propagate into the corona. We find that fast waves are partially reflected at the coronal hole boundary and significant part of the wave energy leaks out of the coronal hole. We compare the calculated wavelengths and the attenuation rate of the fast waves in the leaky waveguide formed by the coronal hole with the analytical ideal MHD solutions for ky=0, where ky is the perpendicular wavenumber, and find an excellent agreement. When ky≠0 the fast waves couple to the shear Alfv¿n waves and transfer energy across field lines.

Resonance heating layers are found to occur when shear Alfv¿n waves are driven and a continuous density profile is assumed for the coronal hole. When resonance absorption is considered, the leakage is small compared to the heating rate. The heating is enhanced by phase mixing when coronal hole inhomogeneities (i.e., plumes) are included. We investigate the dependence of the heating rate on the driver frequency and the Lundquist number S and find a good agreement with the analytical S1/3 scaling of the dissipation length. We find that when S=104 the low-frequency Alfv¿n waves can be a significant source of heating of coronal holes at several solar radii. At larger values of S, nonlinear effects might reduce the effective dissipation length. We discuss the relation of our results to the observed properties of high-speed solar wind and coronal holes. ¿ American Geophysical Union 1995