  We present a comprehensive study of MHD waves and instabilities in a weakly
ionised system, e.g., an interstellar molecular cloud. We determine all the
critical wavelengths of perturbations across which the sustainable wave modes
can change radically (and so can their decay rates), and various instabilities
are present or absent. These critical wavelengths are essential for
understanding the effects of MHD waves (or turbulence) on the structure and
evolution of molecular clouds. Depending on the angle of propagation relative
to the magnetic field and the physical parameters of a cloud, there are
wavelength ranges in which no wave can be sustained as such. Yet, for other
directions of propagation or different properties of a model cloud, there may
exist some wave mode(s) at all wavelengths. For a typical cloud,
magnetically-driven ambipolar diffusion leads to removal of any support against
gravity that most short-wavelength waves (or turbulence) may have had, and
gravitationally-driven ambipolar diffusion sets in and leads to cloud
fragmentation into stellar-size masses, as first suggested by Mouschovias three
decades ago -- a single-stage fragmentation theory of star formation, distinct
from the hierarchical fragmentation picture. Phase velocities, decay times, and
eigenvectors (the densities and velocities of neutral particles and the plasma,
and components of the magnetic field) are determined as functions of the
wavelength of the disturbances and are explained physically. Comparison of the
results with nonlinear analytical or numerical calculations is also presented,
excellent agreement is found, and confidence in the analytical, linear approach
is gained to explore phenomena difficult to study through numerical
simulations. Mode splitting and mode merging, which are impossible in
single-fluid systems for linear perturbations, occur naturally in multifluid
systems.
