On the macroscopic scale, the wavelengths of sound waves in glasses are large enough that the details of the disordered microscopic structure are usually irrelevant, and the medium can be considered as a continuum. On decreasing the wavelength this approximation must of course fail at one point. We show here that this takes place unexpectedly on the mesoscopic scale characteristic of the medium range order of glasses, where it still works well for the corresponding crystalline phases. Specifically, we find that the acoustic excitations with nanometric wavelengths show the clear signature of being strongly scattered, indicating the existence of a cross-over between well-defined acoustic modes for larger wavelengths and ill-defined ones for smaller wavelengths. This cross-over region is accompanied by a softening of the sound velocity that quantitatively accounts for the excess observed in the vibrational density of states of glasses over the Debye level at energies of a few milli-electronvolts. These findings thus highlight the acoustic contribution to the well-known universal low-temperature anomalies found in the specific heat of glasses.disordered systems | elastic properties | inelastic X-ray scattering | vibrational density of states G lasses display a set of universal low-temperature properties (1). In particular, at a temperature of ∼10 K the specific heat is characterized by an excess over the level predicted by the continuum Debye model. This excess, absent in the corresponding crystalline phases, is related to the so-called boson peak, an excess over the Debye level that appears at energies of a few milli-electronvolts in the vibrational density of states. The physical origin of this universal property has been livelily discussed in the literature for many decades; however, an agreed on solution is still lacking. For example, one interpretation supports the idea that the boson peak is produced by soft vibrations that would be present in glasses in addition to the acoustic ones (2-4). Another interpretation is mainly based on models for the vibrational dynamics of glasses in terms of harmonic oscillators with disorder in the force constants: it supports the idea that the boson peak marks the transition between acoustic-like excitations and a disorder-dominated regime for the vibrational spectrum (5-7). Qualitatively similar results appear as well in recently developed theories for the vibrational spectrum of model systems with random spatial variations in the elastic moduli (8). In another interpretation, the boson peak is related to the characteristic vibrations of nanometric clusters (9, 10) that would exist in the glass as a consequence, for example, of a spatially inhomogeneous cohesion and that would hybridize with the acoustic modes (11). The idea of an inhomogeneous elastic response has been recently reconsidered by using a different approach: numerical studies show that the classical elasticity description breaks down in glasses on the mesoscopic length scale (12)(13)(14), and the boson peak woul...