  In this work we consider perturbations of homogeneous and hypersurface
orthogonal cosmological backgrounds with local rotational symmetry (LRS), using
a method based on the 1 + 1 + 2 covariant split of spacetime. The backgrounds,
of LRS class II, are characterised by that the vorticity, the twist of the
2-sheets, and the magnetic part of the Weyl tensor all vanish. They include the
flat Friedmann universe as a special case. The matter contents of the perturbed
spacetimes are given by vorticity-free perfect fluids, but otherwise the
perturbations are arbitrary and describe gravitational, shear, and density
waves. All the perturbation variables can be given in terms of the time
evolution of a set of six harmonic coefficients. This set decouples into one
set of four coefficients with the density perturbations acting as source terms,
and another set of two coefficients describing damped source-free gravitational
waves with odd parity. We also consider the flat Friedmann universe, which~has
been considered by several others using the 1 + 3 covariant split, as a check
of the isotropic limit. In agreement with earlier results we find a
second-order wavelike equation for the magnetic part of the Weyl tensor which
decouples from the density gradient for the flat Friedmann universes. Assuming
vanishing vector perturbations, including the density gradient, we find a
similar equation for the electric part of the Weyl tensor, which was previously
unnoticed.
