A simple analytic approximation to the Rayleigh-Bénard stability threshold

The Rayleigh-Bénard linear stability problem is solved by means of a Fourier series expansion. It is found that truncating the series to just the first term gives an excellent explicit approximation to the marginal stability relation between the Rayleigh number and the wave number of the perturbation. Where the error can be compared with published exact results, it is found not to exceed a few percent over the entire wave number range. Several cases with no-slip boundaries of equal or unequal thermal conductivities are considered explicitly.