Pythagorean triples as wave numbers in steady-state Rayleigh-Bénard convection

Abstract

In Rayleigh-Bénard convection, an eigenvalue equation for the vertical velocity in steady-state case admits a set of solutions in which the lateral dependence satisfies the wave equation. Two wave numbers are necessary for the general solution to this wave equation and are related to the critical wave number. Here we present explicit forms of the wave numbers using Pythagorean triples and the resulting velocity field is presented analytically and graphically. The plots for the first eigenfunction show superposition of two rolls perpendicular to another.