  The behavior of the momentum transferred to a trapped Bose-Einstein
condensate by a two-photon Bragg pulse reflects the structure of the underlying
Bogoliubov spectrum. In elongated condensates, axial phonons with different
number of radial nodes give rise to a multibranch spectrum which can be
resolved in Bragg spectroscopy, as shown by Steinhauer {\it et al.} [Phys. Rev.
Lett. {\bf 90}, 060404 (2003)]. Here we present a detailed theoretical analysis
of this process. We calculate the momentum transferred by numerically solving
the time dependent Gross-Pitaevskii equation. In the case of a cylindrical
condensate, we compare the results with those obtained by linearizing the
Gross-Pitaevskii equation and using a quasiparticle projection method. This
analysis shows how the axial-phonon branches affect the momentum transfer, in
agreement with our previous interpretation of the observed data. We also
discuss the applicability of this type of spectroscopy to typical available
condensates, as well as the role of nonlinear effects.
