Quasi-one-dimensional character of sound propagation in elongated Bose-Einstein condensed clouds

We show that under rather general conditions the propagation of sound pulses along the long axis of a Bose-Einstein condensed cloud in a very anisotropic trap is essentially one dimensional. In the linear regime, we demonstrate that the square of the velocity of propagation is given by the square of the local sound velocity $c^2{=nU}_0/m$ averaged over the cross section of the cloud. We also carry out calculations in the nonlinear regime and determine how the speed of the pulse depends on its amplitude.