m = 2 power spectra for SimL in log  scale. The time window spans 269 Myr centred at t = 3.3 and t = 7.5 Gyr. The left vertical scale gives values of ω, the wave frequency in Myr^-1, whereas the right vertical scale is for the pattern speed Ω = ω/2 in kms^-1 kpc^-1. The horizontal white lines represent Ω_s and Ω_p, respectively, at the beginning of the time window (full line) and the end (dot-dashed one). Ω_s decreases from 146 to 139 at t = 3.3 Gyr and from 81 to 71 kms^-1 kpc^-1 at t = 7.5 Gyr, whereas Ω_p decreases by less than 1 kms^-1 kpc^-1 so that the two lines cannot be distinguished. The averaged curves 2Ω − κ (which allows the ILR to be identified) and 2Ω + κ (for the OLR) are drawn as white short dashed lines, 2Ω − κ/2 (for the UHR) as long dashed line, and 2Ω as a solid line (for the CR). A logarithmic scale has been chosen for the radius to emphasize the central kpc.