Numerical bisection applied to experiments

Imagine the situation where you have to detect threshold beyond which a phenomenon occurs. Below a critical value of the control parameter the phenomenon does not occur and beyond which you can observe it. This is the classic phase transition measurement problem in physics.

In our application, we had to detect when fluid jets emanating from a speaker excited at a particular frequency formed satisfying certain pre-defined criteria for a forest of jets due to highly nonlinear Faraday instability. This would involve analysis of the ‘run’ to decide whether the control parameter, in this case acceleration, crosses a threshold. The aim was to generate an acceleration-frequency phase plot to glean information on the physics behind the threshold curve. So, unlike a thermodynamic phase transition where the experimental run provides a continuous curve of the monitored property (with respect to temperature or some such parameter that is continuously varied), this would involve several individual runs. Since these are dynamic processes taking place over ms timescales, we used high-speed imaging. Here’s what one could do: change the control parameter slowly, take videos at each acceleration and analyze whether the ‘forest of jets’ criterion was satisfied, and continue by increasing the acceleration if not.

Is there a more eficient method than the blind search? We used successive bisection method from numerical analysis to make this task easier. You choose any acceleration and perform the analysis. Analyze video to check if the criteria are satisfied or not. Go to the other extreme where you expect the phenomenon to occur or not, opposite to the previous result. Somewhere between you have the threshold. To get to the threshold, perform the experiment at the mid-point of the two values of the acceleration. Keep bisecting at each iteration between the opposites of the phenomenon (in this case, occurrence of a forest of jets or not) until you find the threshold value for the occurrence of the phenomenon at the required precision or what is offered by the instrument.