exponential case (equation \[eq:fastest\_growing\_wavelength\_exp\]) $$\label{eq:fastest_growing_wavelength_sech2}
\lambda_{f_{\mathrm{sech^2}}} = A_{\mathrm{sech^2}} \; \lambda_{f_0},$$ with $A_{\mathrm{sech^2}} \simeq 1.926$. Expressed by the scaleheight it is $$\label{eq:fastest_growing_wavelength_sech2_z0}
\lambda_{f_{\mathrm{sech^2}}} = A_{\mathrm{sech^2}} \; 2 \upi z_0 \simeq  12.103 \;  z_0 .$$ The error for the exponential profile approximation (equation \[eq:fastest\_growing\_wavelength\_exp\]) is therefore $11.527 \%$ compared to the numerical solution (equation \[eq:fastest\_growing\_wavelength\_sech2\]), and for the thin disc approximation $48.079 \%$ (equation \[eq:fastest\_growing\_wavelength\_thin\_disk\]).\
Again, we get a constant reduction factor for $\lambda_{f_{\mathrm{sech^2}}}$ for all $z_0$ with (Fig. \[fig:reductionfactor\]) $$F_{\mathrm{sech^2}}(\lambda_{f_{\mathrm{sech^2}}}) \simeq 0.726,$$ with an error of $6.061 \%$ for $F_{\mathrm{exp}}(\lambda_{f_{\mathrm{exp}}})$ compared to the integral $F_{\mathrm{sech^2}}(\lambda_{f_{\mathrm{sech^2}}})$ and $37.741 \%$ for the thin disc approximation. Due to the steepness of the exponential profile in the innermost part, matter and self-gravity, respectively, is slightly underestimated in the mid-plane compared to the ${\, \mathrm{sech} }^2$ distribution.

Ring properties {#subsec:Ring properties}
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From the calculations above, we can derive properties of the possible growing structures if locally unstable. Due to axisymmetry, radial perturbations can grow within rings, with an overdense and an underdense part of the dominant growing wavelength. The initial radial thickness of the overdense region that will be investigated in detail in our disc simulation (see Section \[Results:subsec:Ring properties\]), is defined as $$\label{eq:half_wavelength_sech2}
L_{\mathrm{sech^2}} 