factor $F_{\mathrm{exp}}$ underestimates the numerical solution of $F_{\mathrm{sech^2}}$, with a maximum error of $13.7\%$ (Fig. \[fig:reductionfactor\_error\]), in agreement with the result found by e.g. @Kim:2002gl [@2006ApJ...649L..13K] and @Kim:2007ek. This maximum error is reached however at $\lambda / z_{0} \simeq 1.04 \upi$, which differs by roughly a factor of 2 from their result.

Modified dispersion relation {#subsec:Modified dispersion relation}
----------------------------

In this section, we introduce the modified local dispersion relation of @Lin:1964cx for geometrically thick gas discs, since it is important for our further calculations. It gives a relation between the angular frequency $\omega$ and density perturbations with wavelength $\lambda$ in radial direction. Here, it indicates the possibility for growing axisymmetric perturbations, depending on the local conditions.\
In general, to get the dispersion relation, linear perturbation theory is employed on the hydrodynamical equations and the Poisson equation (Section \[subsec:Reduction factor\]) for a self-gravitating and rotating gas disc in the razor-thin limit [@Binney:2011vb]. In our case, the reduction factor appears in addition and we find in the mid-plane for $\lambda > 0$ [e.g. @Wang:2010wr] $$\label{eq:modified_dispersion_relation_generalized}
\omega^{2} = \kappa^{2} - 2 \upi G \Sigma  \left(  \frac{2 \upi}{\lambda } \right)   F( \lambda, z_{0}) + c_{\mathrm{s}}^{2}  \left(\frac{2\upi}{\lambda} \right)^{2},$$ with the epicyclic frequency $\kappa$ and the 