Hopf bifurcation in acoustically excited Faraday ripples on a bubble wall

Observations of surface waves on the wall of a bubble which is subjected to an acoustical field demonstrate differences in the transient processes, as well as a marked variation in steady state amplitudes, that depend on the insonification conditions. To clarify these observations, the stability analysis of the system has been carried out and the presence of the Hopf bifurcation was established. The appearance of the limiting circle corresponds to the presence of periodic variations in the amplitudes of interacting breathing (volume) and distortion (surface) modes. Comparison between observation and modeling indicate qualitative similarities, for example in that the oscillations will be seen for sufficiently high pressure amplitudes and will be absent at the bottom of the threshold curve for the excitation of the ripples.