ONE-DIMENSIONAL SPATIAL STRUCTURE NEAR A HOPF-BIFURCATION AT FINITE WAVE-NUMBER

The theory of the spatial structure in a finite geometry in the weakly non-linear regime of an instability to a traveling wave state with x --> -x symmetry is described. Very close to the threshold a direct expansion in the unstable modes yields a quantitative description of the structure in terms of the reflection coefficient of the ends. The calculation of the reflection coefficient and its dependence on fluid and end wall properties is illustrated using the model of binary fluid convection between free slip boundaries. Further away from threshold the phase diagram of the spatial structure, involving four distinct phases, is calculated using the amplitude equation approach. The validity of this method is judged by comparison with the free slip convection model. The transition between a state where the disturbance fills the system and one where only a portion is filled ("filling" to "confined") is analyzed in detail. Our theory is compared with experiments on the oscillatory instability in pure fluid convection.