travels with the angular velocity $\omega_R/m$. The factor $e^{\omega_I t}$ shows the exponential growth/decay of the wave amplitude. I normalize all length, velocity, and time variables to $R_C$, $v_0$ and $R_C/v_0$, respectively, and set $G=R_C=v_0=1$.

I begin my case studies in §\[sec:maximal-disk\] with a near maximal disk of $(N,\lambda,\alpha)$=$(6,1,0.42)$ and compute its eigenfrequency spectra for the wavenumbers $0\le m\le 5$. I then classify unstable $m=2$ modes of this model and investigate their evolution as the parameter $\alpha$ is varied. In §\[sec:variations-of-lambda\] and §\[sec:variation-of-N\], I study the behavior of unstable $m=2$ waves as the parameters $\lambda$ and $N$ are changed. The eigenfrequency spectrum of a model with an inner cutout is also computed and discussed in §\[sec:cutout-models\].

A Near Maximal Disk {#sec:maximal-disk}
-------------------

I pick up the first model from Table 4 of JH with $(N,\lambda,\alpha)=(6,1,0.42)$ and start solving the eigensystem (\[eq:eigenvalue-problem-two\]) with $(l_{\rm max},j_{\rm max})=(2,4)$ and increase these limits until complex eigenfrequencies converge. For an error threshold of $1\%$ the program terminates when $(l_{\rm max},j_{\rm max})=(10,15)$, which gives a size of $336\times 336$ for the matrix $\textbf{\textit{A}}$. In such a circumstance, out of $336$ eigenfrequencies of $\textbf{\textit{A}}$ (for each wavenumber $m$), less than 15 pair have non-zero growth rates ($\omega_I\not 