  A model that computes the secular evolution of a gravitating disk-planet
system is developed. The disk is treated as a set of gravitating rings, with
the rings'/planets' time-evolution governed by the classical Laplace-Lagrange
solution for secular evolution but modified to account for the disk's finite
thickness h. This system's Lagrange planetary equations yield a particular
class of spiral wave solutions, usually denoted as apsidal density waves and
nodal bending waves. There are two varieties of apsidal waves:long waves and
short waves. Planets typically launch long density waves at the disk's nearer
edge or else at a secular resonance in the disk, and these waves ultimately
reflect downstream at a more distant disk edge or else at a Q-barrier in the
disk, whereupon they return as short density waves. Planets also launch nodal
bending waves, and these have the property that they can stall in the disk,
that is, their group velocity plummets to zero upon approaching a disk region
too thick to support the continued propagation of bending waves. The rings
model is used to compute the secular evolution of a Kuiper Belt having a
variety of masses, and it is shown that the early massive Belt was very
susceptible to the propagation of low-amplitude apsidal and nodal waves
launched by the giant planets.
