affects the sizes of the growing structures. @2012MNRAS.422..600G estimated for an exponential vertical distribution a factor of $\sim 2$ larger wavelengths for the fastest growing perturbations. These implications on stability and structure properties demonstrate the importance of considering the thickness effects.\
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In this paper, we go beyond previous work by investigating in detail how axisymmetric instabilities grow in an idealized gas disc simulation. Section \[sec:Analytical model\] introduces the linear stability analysis for thick discs. From this we deduce the properties of the growing ring structures with respect to their sizes, masses and formation time-scales. Finally, we employ an idealized hydrodynamical simulation (Section \[sec:Numerical modelling\]) and compare the initial structure formation with the linear stability analysis (Section \[sec:Results\]).

Analytical model {#sec:Analytical model}
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We summarize the available literature on how perturbation theory has to be modified when moving from an infinitesimally thin disc to a more realistic disc with finite thickness. In general, this transition is expressed by a wavelength and scale-height dependent reduction factor of the perturbed potential. Based on the resulting modification of the dispersion relation, we explicitly present the analytical derivation of the fastest growing wavelength for a vertically exponential density distribution. The reduction factor is independent 