interactions are beyond the scope of this study and will be investigated in a subsequent paper.

Results {#sec:Results}
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Perturbation theory {#Results:subsec:Perturbation theory}
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From the modified dispersion relation (equation \[eq:modified\_dispersion\_relation\_generalized\]) for the ${\, \mathrm{sech} }^2$ profile, we can derive the possible growing wavelengths for our disc model (Fig. \[Results:fig:theory:growing\_wavlength\]) in the unstable regime $R_{\mathrm{u}} = 0- 10.517$ kpc. The fastest growing perturbation wavelength is different for every radius and increases outwards. For the ${\, \mathrm{sech} }^2$ profile the range lies between $\lambda_{f_{\mathrm{sech^2}}} = [0.393, 2.862]$ kpc, while the exponential profile approximation would indicate $\lambda_{f_{\mathrm{exp}}} = [0.438, 3.191]$ kpc and in the razor-thin disc approximation we have $\lambda_{f_{0}} = [0.204, 1.486]$ kpc. For axisymmetric disturbances, the dispersion relation holds so long as $| k R | \gg 1$ [@Binney:2011vb], which is $| 2 \upi R | \gg \lambda_{\mathrm{f_{sech^2}}}$ and here fulfilled for $R \gg 63$ pc and $\lambda_{\mathrm{f_{sech^2}}} \gg 392.7$. The growth rate is decreasing the larger the difference between $\lambda$ and $\lambda_{f_{\mathrm{sech^2}}}$ (Fig. \[Results:fig:growing\_wavlength\_\_and\_\_growth\_rates\]) and reaches zero at the maximum range. At every radius there are two wavelengths with the same growth rate $p_{\mathrm{sech^2}} = \sqrt{- \omega^2_{\mathrm{sech^2}}}$, a smaller and a larger one, relative to $\lambda_{f_{\mathrm{sech^2}}}$. With decreasing growth 