4.4. Asymmetric Encounters
The perturbation caused by any collision in which the center of the companion passes through the primary disk at a large impact angle relative to the disk will have a strong radial component. Thus, a ring-like wave will result even if the impact point is well out from the disk center. The simple kinematic, impulse models above can be extended to model these asymmetric waves (see Appendix 2). A wealth of new caustic waveforms, i.e., combinations of the elementary caustics, are possible (see Struck-Marcell 1990; Donner et al. 1991; Gerber and Lamb 1994; Gerber 1993). The type and structure of these waves as a function of the KIA parameters have not been completely analyzed. Gerber and Lamb's identification of several of the most important dimensionless parameters is an important step. However, as noted above, they assumed specific structural forms for both primary and companion galaxies, so additional parameters could be added.
Both the general theory and the work that has been carried out on modeling specific systems (see Section 5, and Appendix 2) show that the wave morphology is very sensitive to how far off-center the impact is. The wave morphology is also sensitive to a number of other parameters. Thus, there are a myriad of morphologies possible within a limited range of parameter values. Yet the number of disk galaxies containing asymmetric waves with a strong radial component in the driving perturbation ("ring relatives") is probably only a few times greater than the number of nearly symmetric rings. This means that many possible morphologies will not be found in any reasonably nearby sample of galaxies.
It is possible that each asymmetric wave morphology is unique, and is the result of a specific set of collision and structural parameters. If so, then a successful model of the morphology of an individual system would uniquely specify these parameter values. On the other hand, we have seen in the case of symmetrical collisions that there are several ways to generate broad rings. As a second example, the ripples produced in retrograde tidal interactions (e.g., Hernquist and Quinn 1987) can appear similar to the late-time ringing resulting from off-center collisions in early-type disk galaxies (Wallin and Struck-Marcell 1988). However, it is possible to tell the difference in these cases if sufficient morphological or kinematic detail can be observed.