  We present a study of the spiral responses in a stable disc galaxy model to
co-orbiting perturbing masses that are evenly spaced around rings. The
amplitudes of the responses, or wakes, are proportional to the masses of the
perturbations, and we find that the response to a low-mass ring disperses when
it is removed -- behaviour that is predicted by linear theory. Higher mass
rings cause nonlinear changes through scattering at the major resonances,
provoking instabilities that were absent before the scattering took place. The
separate wake patterns from two rings orbiting at differing frequencies,
produce a net response that is an apparently shearing spiral. When the rings
have low mass, the evolution of the simulation is both qualitatively and
quantitatively reproduced by linear superposition of the two separate
responses. We argue that apparently shearing transient spirals in simulations
result from the superposition of two or more steadily rotating patterns, each
of which is best accounted for as a normal mode of the non-smooth disc.
