  This paper is concerned with the detailed behaviour of roll-waves undergoing
a low-frequency perturbation. We rst derive the so-called Whitham's averaged
modulation equations and relate the well-posedness of this set of equations to
the spectral stability problem in the small Floquet-number limit. We then fully
validate such a system and in particular, we are able to construct solutions to
the shallow water equations in the neighbourhood of modulated roll-waves proles
that exist for asymptotically large time.
