The Eckhaus and Benjamin-Feir resonance mechanisms

A unified treatment is given of the Eckhaus mechanism of stability or instability of two-dimensional flows, which are periodic in one spatial dimension, and the Benjamin—Feir instability mechanism of the two-dimensional Stokes water wave. The method of the amplitude equation is used, following the lead of Newell in a related context. This method easily allows the analysis of the so-called side-band perturbations, which are a crucial feature of the Eckhaus and Benjamin—Feir resonance mechanisms. In particular, it is shown that Eckhaus’s result, that a periodic flow is stable only within a particular band of wavenumbers narrower than the span of the neutral curve of linearized theory, is only valid when the eigenvalues and other parameters are real. A corrected and extended form of the result is given for the general case of complex eigenvalues and coefficients. It is noted, however, that Eckhaus’s result is valid for the important examples of Taylor vortices and Bénard cells.