  Recently Eisenstein and collaborators introduced a method to `reconstruct'
the linear power spectrum from a non-linearly evolved galaxy distribution in
order to improve precision in measurements of baryon acoustic oscillations. We
reformulate this method within the Lagrangian picture of structure formation,
to better understand what such a method does, and what the resulting power
spectra are. We show that reconstruction does not reproduce the linear density
field, at second order. We however show that it does reduce the damping of the
oscillations due to non-linear structure formation, explaining the improvements
seen in simulations. Our results suggest that the reconstructed power spectrum
is potentially better modeled as the sum of three different power spectra, each
dominating over different wavelength ranges and with different non-linear
damping terms. Finally, we also show that reconstruction reduces the
mode-coupling term in the power spectrum, explaining why mis-calibrations of
the acoustic scale are reduced when one considers the reconstructed power
spectrum.
