  An analytic theory of the waves in colliding ring galaxies was presented some
years ago, but the observations where not of sufficient quality then to make
quantitative comparisons. Well-resolved observations of a few systems are now
available to make such comparisons, and structure imaged in several dozen
systems, derived from the recent compilation of Madore, Nelson and Petrillo and
the Galaxy Zoo project, can further constrain the theory. Systems with two
rings are especially useful for deriving such constraints. After examining the
implications of recent observations of ring sizes and structure, I extend the
analytic theory, investigate limiting cases, and present several levels of
approximation. The theory is especially simple in the case of nearly flat
rotation curves. I present observational comparisons for a few systems,
including: Arp 10, the Cartwheel and AM2136-492. The fit is quite good over a
large range of cases. For the Cartwheel there are discrepancies, but the areas
of disagreement are suggestive of additional factors, such as multiple
collisions. A specific prediction of the theory in the case of nearly flat
rotation curves is that the ratio of the outward velocity of successive rings
approximately equals the ratio of ring sizes. Ring velocities are also shown to
scale simply with local circular velocities in this limit.
