Secondary bifurcations of under water sand-ripples under oscillatory flow in narrow channels.

Sand-ripples under oscillatory water flow form periodic patterns with wave lengths primarily controlled by the amplitude $d$ of the water motion. When $d$ is suddenly varied the sand-ripples undergo characteristic secondary bifurcations, which we study experimentally and compare to our proposed amplitude equation (previous lecture). In particular we focus on the so-called doubling transition where, initially, a new ripple is formed in each trough, and show that this transition is well reproduced theoretically for sufficiently large $\delta$ (asymmetry between trough and crest). We finally present experimental results showing that long range coupling is seen to a surprising degree in the initial details of the doubling transition: initially {\em two} new ripples form in every trough, but quickly either the left or the right one wins. And this choice is made collectively for the whole system.