  We propose a new formulation for atomic side mode dynamics from super-radiant
light scattering of trapped atoms. A detailed analysis of the recently observed
super-radiant light scattering from trapped bose gases [S. Inouye {\it et al.},
Science {\bf 285}, 571 (1999)] is presented. We find that scattered light
intensity can exhibit both oscillatory and exponential growth behaviors
depending on densities, pump pulse characteristics, temperatures, and geometric
shapes of trapped gas samples. The total photon scattering rate as well as the
accompanied matter wave amplification depends explicitly on atom number
fluctuations in the condensate. Our formulation allows for natural and
transparent interpretations of subtle features in the MIT data, and provides
numerical simulations in good agreement with all aspects of the experimental
observations.

Condensed Matter;Optics;Quantum Physics