Longitudinal sound mode of a Bose-Einstein condensate in an optical lattice

We examine the effect of the transverse breathing mode on the longitudinal sound propagation in a Bose-Einstein condensate in a one-dimensional optical lattice. In particular, we discuss how the coupling with the transverse breathing mode influences the sound velocity in an optical lattice. Using a variational approach we calculate the dispersion relations for the longitudinal sound mode and the transverse breathing mode analytically, and find that the shift in the sound velocity from the uncoupled result can be large enough to be experimentally relevant. We also find that the effective mass of the transverse breathing mode is affected considerably by the coupling to longitudinal sound.