  Gravitational instabilities play an important role in structure formation of
gas-rich high-redshift disc galaxies. In this paper, we revisit the
axisymmetric perturbation theory and the resulting growth of structure by
taking the realistic thickness of the disc into account. In the unstable
regime, which corresponds for thick discs to a Toomre parameter below the
critical value Q_0,crit = 0.696, we find a fastest growing perturbation
wavelength that is always a factor 1.93 times larger than in the classical
razor-thin disc approximation. This result is independent of the adopted disc
scaleheight and by this independent of temperature and surface density. In
order to test the analytical theory, we compare it with a high-resolution
hydrodynamical simulation of an isothermal gravitationally unstable gas disc
with the typical vertical sech^2 density profile and study its break up into
rings that subsequently fragment into dense clumps. In the first phase, rings
form, that organize themselves discretely, with distances corresponding to the
local fastest growing perturbation wavelength. We find that the disc
scaleheight has to be resolved initially with five or more grid cells in order
to guarantee proper growth of the ring structures, which follow the analytical
prediction. These rings later on contract to a thin and dense line, while at
the same time accreting more gas from the inter-ring region. It is these dense,
circular filaments, that subsequently fragment into a large number of clumps.
Contrary to what is typically assumed, the clump sizes are therefore not
directly determined by the fastest growing wavelength.
