of the scale-height when the disc is in hydrostatic equilibrium. The fastest growing wavelength for the vertical ${\, \mathrm{sech} }^2$ density profile is numerically approximated. At this wavelength, which is the dominant growing perturbation in a disc, the reduction factor is a constant and allows us to simplify the deduction of the physical properties of the growing ring structures and the local instability parameter.

Reduction factor {#subsec:Reduction factor}
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In classical linear perturbation theory, matter is assumed to be concentrated in a razor-thin layer with a surface density $\Sigma$. This overestimates the potential as in real discs matter is vertically arranged with the density having its maximum in the mid-plane where the initial structures form first. This softens the potential and requires a correction factor $F$ in addition to modify the classical theory.\
Density perturbations correlate with variations in the gravitational field and are usually specified by the perturbed local gravitational potential $\Phi_{\mathrm{0}}$ of an infinitesimal thin disc, which is given by the solution of the Poisson equation in the stationary form (see Appendix \[App:Derivation of the reduction factor of the potential due to the disc thicknes\]). The total potential at the mid-plane, $z=0$, of a disc with finite thickness 