= \frac{\lambda_{f_{\mathrm{sech^2}}}}{2},$$ and related to the scale-height $$\label{eq:l_ring_z0}
L_{\mathrm{sech^2}} = A_{\mathrm{sech^2}} \; \upi z_0.$$ For the thin disc approximation, we have $L_0 = \lambda_{f_0} / 2$ and get with equations (\[eq:fastest\_growing\_wavelength\_sech2\]) and (\[eq:fastest\_growing\_wavelength\_sech2\_z0\]) the relation $$L_{\mathrm{sech^2}} = A_{\mathrm{sech^2}} \;  L_{0} = A_{\mathrm{sech^2}} \; \frac{ c_{\mathrm{s}}^{2}}{G \Sigma}.
\label{eq:clump_radius_S}$$ From this we can conclude that in thick discs, the initial radial ring widths are much larger than the scale-height and than in the razor-thin limit assumed.\
The total mass of a ring can be estimated to first order as $$M_{\mathrm{Ring}} = \upi \Sigma(R) \; \left[ \left( R + \frac{ \lambda_{f_{\mathrm{sech^2}}}}{2}\right)^{2} - \left( R - \frac{\lambda_{f_{\mathrm{sech^2}}}}{2}\right)^{2} \right],$$ which gives $$M_{\mathrm{Ring}} = 2 \upi \Sigma (R) \; R \; \lambda_{f_{\mathrm{sech^2}}} ,$$ where $R$ is the location of the maximum of the density perturbation.

Local disc instability parameter {#subsec:Local disc instability parameter}
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Here we derive a local dimensionless stability parameter for a disc with finite thickness, following the example of @1964ApJ...139.1217T for an infinitesimal-thin gas disc with $$\label{eq:Toomre_Q}
 Q_0 = \frac{\kappa \; c_{\mathrm{s}}}{\upi G \Sigma}.$$ In contrast to other derivations we, apply the fastest growing wavelength on the dispersion relation to determine the parameter $Q$.\
The disc is locally unstable for axisymmetric instabilities if 