Dispersion diagram (Eq. (66)) plotting the non-dimensional frequency ω/(vlk) against the non-dimensional tube radius ka for the values of the parameters shown in the legend. Stable solutions are plotted in solid lines, while unstable solutions are plotted in dotted lines. The numerical solution of the system in Eq. (31) has been overplotted as diamonds. The dashed lines represent the external cut-off frequencies (Eq. (46)).
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