  General formalism for describing dynamics of modulated beams along linear
beamlines is developed. We describe modulated beams with spectral distribution
function which represents Fourier transform of the conventional beam
distribution function in the 6-dimensional phase space. The introduced spectral
distribution function is localized in some region of the spectral domain for
nearly monochromatic modulations and it can be characterized with a small
number of typical parameters which we choose to be the lowest order moments of
the spectral distribution. We study evolution of modulated beams in linear
beamlines and find that characteristic spectral parameters transform linearly.
The developed approach significantly simplifies analysis of various schemes
proposed for seeding X-ray free electron lasers. We use this approach to study
several recently proposed schemes and find the bandwidth of the output bunching
in each case.
