Animation redrawn from Sellwood & Carlberg (2021)

The top left panel of the animation, which loops back after a short pause, shows the time evolution of the net overdensity of the disk response to two rings at R=2.5R0 and R=4R0 each of three very light disturbance masses that are driven at the local corotation speed. These rings of particles, marked by large dots and each having a mass of 10-4R0V02/G or almost 1000 times the mass of a background disk particle, were added into a simulation of the stable half-mass Mestel disk in which Q=1.5.

The disk response in the simulation was fitted by two steadily rotating m=3 waves of fixed shape and amplitude that are illustrated in the top panel. The fitted pattern speeds of each of the responses are Ωp,1=0.404 and Ωp,2=0.252, and the principal resonances are marked by circles. As in the animation, distances are in units of R0 and times are in units of R0/V0.

The top right panel of the animation shows the time evolution of the superposed fitted disturbances, each turning at the fitted rate, that appear as a shearing transient pattern. From time to time, the pattern breaks into two separate spiral segments that quickly rejoin into a more open spiral that winds up again. The similarity of the more intense parts of the disturbances in the top left and top right panels is evident.

The bottom two panels illustrate the evolution of the logarithmic spiral spectrum in the simulation on the left and the superposed two waves on the right. The abscissa in these panels is tan γ, where γ is the angle between the radius vector and the tangent to the spiral; it is therefore the complement to the pitch angle α. Here one can see a repeating pattern of a peak moving to the right from leading to trailing as the amplitude rises to a maximum near tan γ = +2, followed by a decrease as the spiral begins to wind more tightly.

This is a clear demonstration that the superposition of two steady disturbances can create the appearance of swing amplification.

Reference: Sellwood, J. A. & Carlberg, R. G. 2021, MNRAS, 500, 5043