Oscillations, Waves, and Interactions - GWDG

The single bubble – a hot microlaboratory 145
influence in this region through forcing the sudden, but smooth, decay that occurs
at lower bubble radii. Without surface tension this decay would be missing and the
response would level off at a high expansion oscillation. To put the giant response
at small bubble radii into perspective, Fig. 4 gives the response for ˆpa = 130 kPa
and 70 kPa at 20 kHz for bubbles from 1 µm to 100 µm on a linear radius scale. The
by far greater amplitude of the giant response with respect to the main resonance is
easily noticed. The relation gets even more pronounced at higher driving amplitudes.
However, then the radial oscillation stability may be lost as is inevitable at sufficiently
strong driving.
2.4 Parameter space diagrams
The response curves already condense the information on the oscillation properties of
a bubble as not the full oscillation is retained but only the maximum elongation, or
a set of points in the case of a chaotic oscillation. An even more condensed survey of
information on the response of a bubble to a sound field can be given in the form of
a parameter space diagram, often also called phase diagram (Fig. 5). There, over a
space of parameters, best only two parameter variables, just the type of the resulting
oscillation is plotted into the respective volume (area). The areas are determined
by the boundaries of the bifurcation sets of the bubble oscillators. The type of
oscillation inside these boundaries may be colour coded for better vision or elsewise
made discernable. Oscillation types are, for instance, discerned by the period of the
resulting oscillation, that is period 1 or period 2, or generally period n, where n can
be any natural number up to infinity. Also, chaotic oscillations may be classified
Figure 5. Parameter space diagram obtained with the Keller-Miksis bubble model depicting
the periodicity properties of bubble oscillations in the (Rn, ˆpa)-parameter plane
(calculation by P. Koch).