In the recent years, much attention has been devoted to the inhomogeneous
nature of the mechanical response at the nano-scale in disordered solids.
Clearly, the elastic heterogeneities that have been characterized in this
context are expected to strongly impact the nature of the sound waves which, in
contrast to the case of perfect crystals, cannot be completely rationalized in
terms of phonons. Building on previous work on a toy model showing an
amorphisation transition [Mizuno H, Mossa S, Barrat JL (2013) EPL {\bf
104}:56001], we investigate the relationship between sound waves and elastic
heterogeneities in a unified framework, by continuously interpolating from the
perfect crystal, through increasingly defective phases, to fully developed
glasses. We provide strong evidence of a direct correlation between sound waves
features and the extent of the heterogeneous mechanical response at the
nano-scale
