  We investigate a classical phase-space approach of matter-wave propagation
based on the Truncated Wigner Equation (TWE). We show that such description is
suitable for ideal matter waves in quadratic time-dependent confinement as well
as for harmonically trapped Bose Einstein condensates in the Thomas-Fermi
regime. In arbitrary interacting regimes, the TWE combined with the moment
method yields the low-energy spectrum of a condensate as predicted by
independent variational methods. TWE also gives the right breathing mode
frequency for long-ranged interactions decaying as $1/r^2$ in 3D and for a
contact potential in 2D. Quantum signatures, beyond the TWE, may only be found
in the condensate dynamics beyond the regimes of classical phase-space
propagation identified here.
