Certain regions of the solar atmosphere are at sufficiently low temperatures to be only partially ionised. The lower chromosphere contains neutral atoms, the existence of which greatly increases the efficiency of the damping of waves due to collisional friction momentum transfer. More specifically the Cowling conductivity can be up to 12 orders of magnitude smaller than the Spitzer value, so that the main damping mechanism in this region is due to the collisions between neutrals and positive ions (Khudachenko et al., A&A, 422, 1073). Using values for the gas density and temperature as functions of height taken from the VAL C model of the quiet Sun (Vernazza et al., ApJS, 45, 635), an estimate is made for the dependance of the Cowling conductivity on height and strength of magnetic field. Using both analytic and numerical approaches the passage of Alfvén waves over a wide spectrum through this partially ionised region is investigated. Estimates of the efficiency of this region in the damping of Alfvén waves are made and compared for both approaches. We find that Alfvén waves with frequencies above 0.6 Hz are completely damped and frequencies below 0.01 Hz unaffected.