  We present an explicit three-parameter class of $p=\gamma \varrho$,
($-1/3\leq\gamma<1$), cosmological models admitting a two-dimensional group
$G_{2}$ of isometries acting on spacelike surfaces. The family is self-similar
in the sense that it has a further homothetic vector field and it contains
subfamilies of both (previously unknown) tilted and non-tilted Bianchi models
with that equation of state. This is the first algebraically general class of
solutions of this kind including dust inhomogeneous solutions. The whole class
presents a universal spacelike big-bang singularity in the finite past. More
interestingly, the case $p=\varrho /3$ constitutes a new two-parameter
inhomogeneous subfamily which can be viewed as a Bianchi V background with a
gravitational wave travelling orthogonally to the surfaces of transitivity of
the $G_{2}$ group. This wave generates the {\it inhomogeneity} of the spacetime
and is related to the sound waves {\it tilting} the perfect fluid. It seems to
be the first explicit exact example of a gravitational wave travelling along a
homogeneous background that has a realistic equation of state $p=\varrho /3$.
