  We consider the radial wave equation in similarity coordinates within the
semigroup formalism. It is known that the generator of the semigroup exhibits a
continuum of eigenvalues and embedded in this continuum there exists a discrete
set of eigenvalues with analytic eigenfunctions. Our results show that, for
sufficiently regular data, the long time behaviour of the solution is governed
by the analytic eigenfunctions. The same techniques are applied to the linear
stability problem for the fundamental self--similar solution $\chi_T$ of the
wave equation with a focusing power nonlinearity. Analogous to the free wave
equation, we show that the long time behaviour (in similarity coordinates) of
linear perturbations around $\chi_T$ is governed by analytic mode solutions. In
particular, this yields a rigorous proof for the linear stability of $\chi_T$
with the sharp decay rate for the perturbations.
