can derive a rotation-independent form for the fastest growing perturbations in the next section.

The fastest growing perturbation wavelength for the exponential profile {#subsec:Approximated fastest growing perturbation wavelength}
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The extremum of the dispersion relation as function of wavelength (equation \[eq:disp\_relation\_exp\]) is determined by $$\label{eq:calcfastestgrowingwavelength}
\frac{\upartial \omega_{\mathrm{exp}}^{2}} {\upartial \lambda} = 4 \upi^{2} \left( \frac{G \; \Sigma}{ (\lambda  + 2 \upi z_{0})^{2}} - \frac{2 c_{\mathrm{s}}^{2}}{\lambda^{3}}  \right)   = 0 .$$ We solve equation (\[eq:calcfastestgrowingwavelength\]) in Appendix \[App:Calculation of the fastest growing wavelength for the exp-profile approximation\] analytically and get the fastest growing wavelength $\lambda_{f_{\mathrm{exp}}}$. The minimum is ensured by $\frac{\upartial^{2} \omega_{\mathrm{exp}}^{2} (\lambda_{f_{\mathrm{exp}}})} {\upartial^{2} \lambda} > 0$. It marks the maximum growth rate for an unstable disc. For a disc in hydrostatic equilibrium with given $z_0$ (equation \[eq:scale-height\_\]) the solution simplifies and we can relate it to the classical solution in the razor-thin limit with $$\label{eq:fastest_growing_wavelength_thin_disk}
\lambda_{f_0}  = \frac{2 c_{\mathrm{s}}^{2}}{G \Sigma} .$$ For the exponential profile approximation, we get $$\lambda_{f_{\mathrm{exp}}} = A_{\mathrm{exp}} \; \lambda_{f_0} ,
\label{eq:fastest_growing_wavelength_exp}$$ with the constant factor $A_{\mathrm{exp}} \simeq 2.148$ and no additional dependences. Since the form of equation (\[eq:fastest\_growing\_wavelength\_thin\_disk\]) is similar to that of the scaleheight in equation (\[eq:scale-height\_\]), we can write $$\lambda_{f_{\mathrm{exp}}} = A_{\mathrm{exp}} \; 2 \upi z_{0} 