  We investigate the combined effects of weak disorder and a two-dimensional
(2D) optical lattice on the collective excitations of a harmonically trapped
Bose-Einstein condensate (BEC) at zero temperature. Accordingly, we generalize
the hydrodynamic equations of superfluid for a weakly interacting Bose gas in a
2D optical lattice to include the effects of weak disorder. Our analytical
results for the collective frequencies beyond the mean-field approximation
reveal the peculiar role of disorder, interplaying with the 2D optical lattice
and interatomic interaction, on elementary excitations along the 3D to 1D
dimensional crossover. In particular, consequences of disorder on the phonon
propagation and surface modes are analyzed in detail. The experimental scenario
is also proposed.
