  We consider a trapped repulsive one-dimensional (1D) Bose gas at very low
temperature. In order to study the collective modes of this strongly
interacting system, we use a hydrodynamic approach, where the gas is locally
described by the Lieb-Liniger model of bosons interacting via a repulsive delta
potential. Solving the corresponding linearized hydrodynamic equations, we
obtain the collective modes and concentrate more specifically on the lowest
compressional mode. This is done by finding models, approaching very closely
the exact equation of stae of the gas, for which the linearized hydrodynamic
equations are exactly solvable. Results are in excellent agreement with those
of the sum rule approach of Menotti and Stringari.
