Oscillations, Waves, and Interactions - GWDG

142 W. Lauterborn et al.
Figure 2. Synopsis of the steady-state radial oscillations of a bubble in water as a function
of acoustic driving pressure. The bubble’s equilibrium radius is 7 µm, the driving frequency,
25 kHz. The maximum radius strongly increases for pressures above 1 bar, which is the
static pressure in the liquid.
Closed analytical solutions to the above bubble models are not known. Numerical
solutions, on the other hand, are easily obtained today. When starting at some initial
condition the bubble oscillator responds with transients until it settles to a steady
state determined by the damping. Figure 2 gives a sequence of steady-state radiustime
curves for a bubble of 7 µm radius at rest driven by a sound field of 25 kHz at
various sound pressure amplitudes from 0.7 bar to 1.7 bar. The variation from an
almost sinusoidal oscillation of small amplitude to one with strong elongation, fast
collapse and afterbounces at compression is demonstrated.
All oscillations in the figure repeat after one oscillation of the sound field. This
need not be the case [6]. The steady state oscillation may only be stably reached after
two periods of the sound field, or four or eight. Indeed, the period of the steady state
oscillation may go to infinity despite periodic driving of the bubble. The oscillation
is then said to be chaotic, as it never repeats, albeit everything is deterministic.
Acoustically driven bubbles follow a period-doubling route to chaos, where upon
altering a parameter (either bubble radius at rest or sound field frequency or sound
pressure amplitude) the bubble oscillation doubles its period at ever finer alteration,
so that chaotic oscillations are reached at a finite parameter value. Examples of the
period-doubling route to chaos are given in Lauterborn and Parlitz [12] and Parlitz
et al. [10], for instance. Indications of this behaviour can already be found in an
early paper by Lauterborn [6].