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.