  We propose an explanation for the quadratic dependence on the momentum $Q$,
of the broadening of the acoustic excitation peak recently found in the study
of the dynamic structure factor of many real and simulated glasses. We ascribe
the observed $Q^2$ law to the spatial fluctuations of the local wavelength of
the collective vibrational modes, in turn produced by the dishomegeneity of the
inter-particle elastic constants. This explanation is analitically shown to
hold for 1-dimensional disordered chains and satisfatorily numerically tested
in both 1 and 3 dimensions.
