\simeq 13.496 \;  z_{0},
\label{eq:fastest_growing_wavelength_z0_relation}$$ which is similar to the approximation found by @2012MNRAS.422..600G and @2014MNRAS.442.1230R with $\sim 4\upi z_0$.\
The reduction factor equation (\[eq:exp\_reductionfactor\]) then simplifies for $\lambda_{f_{\mathrm{exp}}}$ (equation \[eq:fastest\_growing\_wavelength\_z0\_relation\]) to a constant value $$\label{eq:F_exp_constant}
F_{\mathrm{exp}}(\lambda_{f_{\mathrm{exp}}})  = \left(  1 + \frac{1}{A_{\mathrm{exp}}}  \right)^{-1} \simeq 0.682,$$ independent of the scaleheight (Fig. \[fig:reductionfactor\]). According to the solution in equation (\[eq:F\_exp\_constant\]), the potential in the mid-plane of a razor-thin disc is therefore overestimated by $46.628 \%$ (see equation \[eq:phi\_tot\_reduction\_factor\]).

Fastest growing perturbation wavelength for the ${\, \mathrm{sech} }^2$-profile {#subsec:Exact fastest growing perturbation wavelength}
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We follow the steps described in Section \[subsec:Approximated fastest growing perturbation wavelength\] and determine the minimum of the dispersion relation for the ${\, \mathrm{sech} }^2$ -profile by using the reduction factor $F_{\mathrm{sech^2}}(\lambda , z_0)$ with $$\label{eq:sech2_dispersionrelation_peak}
\frac{\upartial \omega_{\mathrm{sech^2}}^{2}}{\upartial \lambda} = 4 \upi^{2} \left[  \frac{G \; \Sigma}{ \lambda  }  \left( \frac{F_{\mathrm{sech^2}}}{ \lambda } - \frac{\upartial F_{\mathrm{sech^2}}}{\upartial \lambda}   \right)    - \frac{2 c_{\mathrm{s}}^{2}}{\lambda^{3}}  \right]   = 0.$$ The wavelength $\lambda$ is varied iteratively for several fixed scaleheights $z_0$ until equation (\[eq:sech2\_dispersionrelation\_peak\]) reaches zero. The integral $F_{\mathrm{sech^2}}$ of equation (\[eq:integralsechhyperbolfunction\]) and its differentiation in equation (\[eq:sech2\_dispersionrelation\_peak\]) is solved numerically. We find a relation for the fastest growing perturbation wavelength that is similar to the 