  Stability of self-similar solutions for gravitational collapse is an
important problem to be investigated from the perspectives of their nature as
an attractor, critical phenomena and instability of a naked singularity. In
this paper we study spherically symmetric non-self-similar perturbations of
matter and metrics in spherically symmetric self-similar backgrounds. The
collapsing matter is assumed to be a perfect fluid with the equation of state
$P=\alpha\rho$. We construct a single wave equation governing the
perturbations, which makes their time evolution in arbitrary self-similar
backgrounds analytically tractable. Further we propose an analytical
application of this master wave equation to the stability problem by means of
the normal mode analysis for the perturbations having the time dependence given
by $\exp{(i\omega\log|t|)}$, and present some sufficient conditions for the
absence of non-oscillatory unstable normal modes with purely imaginary
$\omega$.
