unperturbed surface density $\Sigma$. For $\omega^2 > 0$ the disc is stable against axisymmetric instabilities. If $\omega^2 < 0$, a certain range of perturbations can grow exponentially in rings. The more negative the values, the faster they grow. All parameters in equation (\[eq:modified\_dispersion\_relation\_generalized\]) can change with radius for the case of an exponential surface density profile, which leads to a complex interplay between the three terms. However, in general we can say: the effect of self-gravity is destabilising, whereas the differential rotation, represented by $\kappa^2 > 0$, and the thermal pressure, with $c_{\mathrm{s}}^2$, stabilize the disc. The dispersion relation (equation \[eq:modified\_dispersion\_relation\_generalized\]) inherits the properties of the reduction factor $F$ (see Section \[subsec:Reduction factor\]). For larger perturbation wavelengths, $F$ approaches unity which gives the solution of $\omega$ for a razor-thin layer. At smaller wavelengths, the thickness of the vertical density structure plays a stronger role and self-gravity is reduced in the mid-plane [e.g. @Kim:2002gl]. With the reduction factor $F_{\mathrm{exp}}$, the relation leads, with all its dependences to the form [e.g. @2006ApJ...647..997S; @Wang:2010wr], $$\label{eq:disp_relation_exp} 
\omega_{\mathrm{exp}}^{2} = \kappa^{2} - \frac{4 \upi^{2} G \Sigma }{\lambda + 2 \upi  z_{0} } + c_{\mathrm{s}}^{2} \left(\frac{2\upi}{\lambda} \right)^{2} .$$ Since there is no wavelength dependence for $\kappa$, we 