Figure 15.

Real and imaginary parts (top and bottom, respectively) of modes q_n in eqn. (40) in terms of the circulation ratio $r\stackrel{\text{def}}{=}{F}_{HF}/F_A$. Note that for r greater than about 30, the real part of $q_n,n=-1,0,1,2,\dots$ suggests that the decay of all $\exp \left(-\frac{F_A}{V}{q}_{n}t\right),n\ne 0$ is so much faster than the decay of $\exp \left(-\frac{F_A}{V}{q}_{0}t\right)\approx \exp \left(-\frac{F_A}{V}t\right)$ as to be negligible. Furthermore, for r greater than about 30, the imaginary part of $q_n,n=-1,1,2,\dots$ suggests that no appreciable oscillations are going to appear in $\exp \left(-\frac{F_A}{V}{q}_{n}t\right),n\ne 0$, as all oscillation frequencies $\frac{\left|Im\left[q_n\right]\right|}{2\pi },n=-1,1,2,\mathrm{...}$ are comparable to the exponential decay rates $Re\left[q_n\right]$.