Hydrodynamic spectrum of a superfluid in an elongated trap

--In this article we study the hydrodynamic spectrum of a superfluid confined in a cylindrical trap. We show that the dispersion relation $\omega$(q) of the phonon branch scales like \sqrt q at large q, leading to a vanishingly small superfluid critical velocity. In practice the critical velocity is set by the breakdown of the hydrodynamic approximation. For a broad class of superfluids, this entails a reduction of the critical velocity by a factor ($\omega$ $\perp$ /\"i1/2c) 1/3 with respect to the free-space prediction (here $\omega$ $\perp$ is the trapping frequency and \"i1/2c the chemical potential of the cloud).