  This paper addresses the fine-scale axisymmetric structure exhibited in
Saturn's A and B-rings. We aim to explain both the periodic microstructure on
150-220m, revealed by the Cassini UVIS and RSS instruments, and the irregular
variations in brightness on 1-10km, reported by the Cassini ISS. We propose
that the former structures correspond to the peaks and troughs of the nonlinear
wavetrains that form naturally in a viscously overstable disk. The latter
variations on longer scales may correspond to modulations and defects in the
wavetrains' amplitudes and wavelength. We explore these ideas using a simple
hydrodynamical model which captures the correct qualitative behaviour of a disk
of inelastically colliding particles, while also permitting us to make progress
with analytic and semi-analytic techniques. Specifically, we calculate a family
of travelling nonlinear density waves and determine their stability properties.
Detailed numerical simulations that confirm our basic results will appear in a
following paper.
