  The origin of spiral patterns in galaxies is still not fully understood.
Similar features also develop readily in N-body simulations of isolated cool,
collisionless disks, yet even here the mechanism has yet to be explained. In
this series of papers, I present a detailed study of the origin of spiral
activity in simulations in the hope that the mechanism that causes the patterns
is also responsible for some of these features galaxies. In this first paper, I
use a suite of highly idealized simulations of a linearly stable disk that
employ increasing numbers of particles. While the amplitudes of initial
non-axisymmetric features scale as the inverse square-root of the number of
particles employed, the final amplitude of the patterns is independent of the
particle number. I find that the amplitudes of non-axisymmetric disturbances
grow in two distinct phases: slow growth occurs when the relative overdensity
is below ~2%, but above this level the amplitude rises more rapidly. I show
that all features, even of very low amplitude, scatter particles at the inner
Lindblad resonance, changing the distribution of particles in the disk in such
a way as to foster continued growth. Stronger scattering by larger amplitude
waves provokes a vigorous instability that is a true linear mode of the
modified disk.
