  The speed of sound of a Bose-Einstein condensate in an optical lattice is
studied both analytically and numerically in all three dimensions. Our
investigation shows that the sound speed depends strongly on the strength of
the lattice. In the one-dimensional case, the speed of sound falls
monotonically with increasing lattice strength. The dependence on lattice
strength becomes much richer in two and three dimensions. In the
two-dimensional case, when the interaction is weak, the sound speed first
increases then decreases as the lattice strength increases. For the three
dimensional lattice, the sound speed can even oscillate with the lattice
strength. These rich behaviors can be understood in terms of compressibility
and effective mass. Our analytical results at the limit of weak lattices also
offer an interesting perspective to the understanding: they show the lattice
component perpendicular to the sound propagation increases the sound speed
while the lattice components parallel to the propagation decreases the sound
speed. The various dependence of the sound speed on the lattice strength is the
result of this competition.
