Small-wavelength thermal approximation for a solidification front ($\sigma =-V^2/\kappa +V\alpha -2\kappa \Gamma {\alpha }^3/\Delta T$)

Approximating perturbation temperatures by harmonic fields decaying as $e^{\pm \alpha \xi }$, while retaining displacement through the base gradient, produces this reduced dispersion relation. The Gibbs--Thomson relation supplies $\Gamma =\gamma T_m/(\rho L)$. This short-wave reduction suppresses perturbation advection and time dependence; its constant term and a threshold at $\alpha \kappa /V=O(1)$ are not controlled next-order results of the full diffusion problem.

The reduced dimensionless growth rate $g(q)=-1+q-2q^3/R_c$ has a positive maximum precisely when $R_c>27/2$. Its maximizing $q=\sqrt{R_c/6}$ follows by differentiation. At equality $q=3/2$, outside a strict $q\gg 1$ short-wave regime. The inequality is an exact criterion for the reduced cubic, not a quantitative onset theorem for full thermal stability of a travelling solidification front.