  The translational symmetry of solids gives rise to the existence of
low-frequency phonons. In ordered systems, some phonons characterized by
different wavevectors are degenerate, i.e. they share the same frequency
$\omega$; in finite-size systems, phonons form a discrete set of bands with
$n_q(\omega)$-fold degeneracy. Here we focus on understanding how this
degeneracy is lifted in the presence of disorder, and its physical
implications. Using standard degenerate perturbation theory and simple
statistical considerations, we predict the dependence of the disorder-induced
frequency width of phonon bands to be
$\Delta\omega\!\sim\!\sigma\,\omega\sqrt{n_q}/\sqrt{N}$, where $\sigma$ is the
strength of disorder and $N$ is the total number of particles. This theoretical
prediction is supported by extensive numerical calculations for disordered
lattices characterized by topological, mass, stiffness and positional disorder,
and for computer glasses, where disorder is self-generated, thus establishing
its universal nature. The predicted scaling leads to the identification of a
crossover frequency $\omega_\dagger\!\sim\!L^{-2/(d+2)}$ in systems of linear
size $L$ in $d\!>\!2$ dimensions, where the disorder-induced width of phonon
bands becomes comparable to the frequency gap between neighboring bands.
Consequently, phonons continuously cover the frequency range
$\omega\!>\!\omega_\dagger$, where the notion of discrete phonon bands becomes
ill-defined. Two basic applications of the theory are presented; first, we show
that the phonon scattering lifetime is proportional to $(\Delta\omega)^{-1}$
for $\omega\!<\!\omega_\dagger$. Second, the theory is applied to the basic
physics of glasses, allowing to determine the range of frequencies in which the
recently established universal density of states of non-phononic excitations
can be directly probed for different system sizes.
