The fluctuating elasticity (FE) model, introduced phenomenologically and developed by Schirmacher [J. Non-Cryst. Solids 357, 518 (2011)], is today the only theoretical framework available to analyze low-temperature elastic acoustic scattering in glasses. Its existing formulations, which neglect the tensorial nature of elasticity and exclude long-range disorder correlations, predict that the acoustic damping coefficients obey the standard Rayleigh scaling law: Γ ∼ k dþ1 , with k the acoustic wave vector, in dimension d. However, recent numerical data, supported by the analysis of existing experimental results, show that Γ does not obey this scaling law but Γ ∼ −k dþ1 ln k. Here we analyze in detail how a fully tensorial FE model can be constructed as a long wavelength approximation of the elastic response of the discrete, atomistic, problem. We show that, although it incorporates all long-range correlations, it fails to capture the observed damping in two respects: (i) it misses the anomalous scaling, and predicts the standard Rayleigh law; (ii) it grossly underestimates the amplitude of scattering by about 2 orders of magnitude. This brings clear evidence that the small scale nonaffine displacement fields, although not simply reducible to local defects, play a crucial role in acoustic wave scattering and hence cannot be ignored.