Wavelength selection through boundaries in 1-D cellular structures
We have restated different results about wavelength selection in one dimensional cellular structures. The wavelength selection mechanism through boundaries, when boundary forcing is small or zero, was re viewed and the resulting bounds for the allowed wavenumbers were calculated again using the framework of amplitude theory at second order. Amplitude expansion at first order is sufficient to compute the selection effect when boundary forcing is strong. Wavenumber selection through boundaries is shown to disappear when end rolls have a larger amplitude than center ones. As inhomogeneous boundary conditions fix the phase of the amplitude at the boundaries, a periodic solution in the bulk must obey a phase matching condition. In some cases, this results in the appearance of “kinks“ is the slowly varying amplitude.
The case of a slowly varying environment was also discussed using second order amplitude expansion. When a subcritical region is related to a supercritical one by a slow variation of the parameters, selection of a unique wavenumber results from an adiabatic theory. If some non adiabatic effects are considered a band of wavenumbers is expected. The mean wavenumber selected through this process and the central one in the band of wavenumbers allowed by boundaries coincide, thus the two mechanisms can cooperate.
The above calculation exemplify simple methods based on the invariants of amplitude equations. The derivation of such an equation is not always possible in the two dimensional case. When possible, however, the use of adiabatic invariants again can lead to selection mechanisms as happens for instance in axisymmetric structures.
KeywordsPeriodic Solution Amplitude Equation Phase Match Condition Wavelength Selection Adiabatic Invariant
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