A cusp catastrophe in flow acoustics
Ingo Rehberg · The Journal of the Acoustical Society of America · 1983
The transonic flow in a duct following an abrupt enlargement of cross section leads to self-induced flow oscillations acting as a sound generator. It is shown that the frequency of the oscillation is influenced by the length of that duct: Increasing duct length will lead to a frequency decrease, but only up to a critical length lc of the duct. If this length is further increased, the frequency will jump to higher values. This jump is accompanied by hysteresis: decreasing the duct length again leads to a jump to the lower frequency at a length lc1 < lc. The above behavior is then simulated by two simple mathematical models. The transonic oscillator is represented by a nonlinear equation leading to oscillations, and the acoustic resonator, i.e., the duct, is represented by a term including a time lag. It is shown for both the experiment and the model that the hysteresis vanishes if the sound reflection at the end of the duct is decreased below a certain critical value. This behavior can be interpreted as being one of the rare examples for a cusp catastrophe of an oscillating system.