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We investigate the combined effects of weak disorder and a two-dimensional(2D) optical lattice on the collective excitations of a harmonically trappedBose-Einstein condensate (BEC) at zero temperature. Accordingly, we generalizethe hydrodynamic equations of superfluid for a weakly interacting Bose gas in a2D optical lattice to include the effects of weak disorder. Our analyticalresults for the collective frequencies beyond the mean-field approximationreveal the peculiar role of disorder, interplaying with the 2D optical latticeand interatomic interaction, on elementary excitations along the 3D to 1Ddimensional crossover. In particular, consequences of disorder on the phononpropagation and surface modes are analyzed in detail. The experimental scenariois also proposed.

Collective excitations of a trapped Bose-Einstein condensate in the presence of weak disorder and a two-dimensional optical lattice