  Motivated by recent experiments on Faraday waves in Bose-Einstein condensates
we investigate both analytically and numerically the dynamics of cigar-shaped
Bose-condensed gases subject to periodic modulation of the strength of the
transverse confinement. We offer a fully analytical explanation of the observed
parametric resonance, based on a Mathieu-type analysis of the non-polynomial
Schr{\"o}dinger equation. The theoretical prediction for the pattern
periodicity versus the driving frequency is directly compared with the
experimental data, yielding good qualitative and quantitative agreement between
the two. These results are corroborated by direct numerical simulations of both
the one-dimensional non-polynomial Schr{\"o}dinger equation and of the fully
three-dimensional Gross-Pitaevskii equation.
