  Dissipation of magnetohydrodynamic (MHD) wave energy has been proposed as a
viable heating mechanism in the solar chromospheric plasma. Here, we use a
simplified one-dimensional model of the chromosphere to theoretically
investigate the physical processes and the spatial scales that are required for
the efficient dissipation of Alfv\'en waves and slow magnetoacoustic waves. We
consider the governing equations for a partially ionized hydrogen-helium plasma
in the single-fluid MHD approximation and include realistic wave damping
mechanisms that may operate in the chromosphere, namely Ohmic and ambipolar
magnetic diffusion, viscosity, thermal conduction, and radiative losses. We
perform an analytic local study in the limit of small amplitudes to
approximately derive the lengthscales for critical damping and efficient
dissipation of MHD wave energy. We find that the critical dissipation
lengthscale for Alfv\'en waves depends strongly on the magnetic field strength
and ranges from 10~m to 1~km for realistic field strengths. The damping of
Alfv\'en waves is dominated by Ohmic diffusion for weak magnetic field and low
heights in the chromosphere, and by ambipolar diffusion for strong magnetic
field and medium/large heights in the chromosphere. Conversely, the damping of
slow magnetoacoustic waves is less efficient, and spatial scales shorter than
10~m are required for critical damping. Thermal conduction and viscosity govern
the damping of slow magnetoacoustic waves and play an equally important role at
all heights. These results indicate that the spatial scales at which strong
wave heating may work in the chromosphere are currently unresolved by
observations.
