We have derived the dispersion relation of the gravitational instability in the rotating thin disk as
Plotting the wavenumber against the normalized frequency of equation (3.39) as Figure 3.11(left), it is shown that, in the case of , the wavenumber exists for all . Since , 0, and +1 correspond to the points of ILR, CR, and OLR and these three resonance points appear in accordance with the radial distance, the -axis of Figure 3.11(left) seems to correspond to the radial distance from the center. In the case of , it is shown that a forbidden region appears around the co-rotation resonance point. Waves cannot propagate into the region. Figure 3.11(left) shows that the has two possible wavenumbers in the permitted region. The waves with larger and smaller are called short waves and long waves, respectively.
Consider a wave expressed by . If , moving from a point in the direction and the phase difference between the two points can be equal to zero. That is, in the case of the wave is leading. On the other hand, if , moving in the direction and the phase will be unchanged. In this case, the wave pattern is trailing. Since the dispersion relation is symmetric for and , there are two waves, trailing waves and leading waves. Therefore there are four waves: a short trailing wave, a long trailing wave, a short leading wave, and a long leading wave.