In this Appendix the epicyclic KIA model is used to further examine the effects of a second symmetric collision described in Section 4.3. Specifically, immediately after the second collision, Equation (4.5) can be replaced with

(A1.1) |

where *t*_{2} is the time of the second collision (with
*t* = 0 at the first
collision). The amplitude *A*_{2} is defined through the
velocity equation at time *t*_{2},

(A1.2) |

where
*v*_{2}
is the velocity impulse in the second collision, and the
second equality defines *A*_{2}. Equations (A1.1) and
(A1.2) can be combined
to eliminate the phase ,

(A1.3) |

with
*r*_{2} = *r*(*t*_{2}), the radius of the
given star at the time of the second
impact. This can be simplified using the relations

(A1.4) |

Thus, we obtain

(A1.5) |

The orbit equation then becomes

(A1.6) |

In zones where the last term in the square brackets is small after the
second impact, i.e.,
sin(*t*_{2}
+ )
0, Equation
(A1.6) is **essentially
the same as Equation (4.5), but with a position-dependent amplitude.**

In zones where the first term in the square brackets in Equation
(A1.6) is small, i.e.,
cos(*t*_{2}
+ )
0, that
equation is similar to
Equation (4.5), but the perturbation term is off by a phase of
/ 2. As a result we expect a
change in the morphology and rate of
propagation at phases where there is a change of dominance from one
term to the other. The models confirm-that the rings do not propagate
as rapidly outward following the second impact as they did during the
first.

When
cos(*t*_{2})
> 1, all of the terms in Equation (A1.5) add,
and the wave amplitude will be relatively large (unless
cos(*t*_{2}
+ ) is small). When
cos(*t*_{2})
< 1 the last two terms on the rights-hand-side of
Equation (A1.5) will tend to cancel, resulting in weak waves in that
region. Indeed if the phase
*t*_{2} is
such that
sin(*t*_{2})
0, then
*A*_{2}
0, i.e., the second perturbation can essentially cancel the
first. Such trajectories are seen in the models. If the amplitude of
nearby trajectories is large enough they can cross the
*A*_{2} 0
trajectory. The result is a ring that does not propagate outward, but
its width varies with time. Though it will be affected by stellar
diffusion, such a stationary ring could be long-lived. Because their
locations depend on the details of the collisions as well as the disk
structure, they will in general have no relation to classical Lindblad
resonances.