Oscillations, Waves, and Interactions - GWDG

2.3 Response curves
The single bubble – a hot microlaboratory 143
Oscillators are often described by response curves. In linear systems the amplitude of
the (then sinusoidal) steady state oscillation is plotted versus the driving frequency
to give the amplitude response curve or resonance curve. It shows just one peak,
the resonance, when the damping is sufficiently small. With nonlinear oscillators
this main resonance starts to lean over to higher or lower frequencies of the driving,
eventually developing hysteresis, where there exist two steady states at the same
frequency of the driving each with its own basin of attraction (set of initial conditions)
leading to the respective steady state (attractor). Such basins can be found in
Parlitz et al. [10]. Moreover, additional resonances appear at or near rational numbers
of the ratio of driving frequency and linear resonance frequency of the bubble,
called harmonic resonances, subharmonics resonances and ultraharmonic (or ultrasubharmonic)
resonances [6]. They also may show hysteresis. And even more, period
doubling sets in along with the resonances with incomplete (“period bubbling”) and
complete routes to chaos.
Figure 3 shows a set of response curves with the bubble radius at rest as the control
parameter instead of the driving sound field frequency. The maximum relative
response, (Rmax − Rn)/Rn, for a driving frequency of 20 kHz and for sound pressure
amplitudes from 10 to 70 kPa is plotted versus Rn. The emerging peaks are
resonances that can be labelled with two numbers, the torsion number and the period
number (both natural numbers) [6,10]. Two sets of resonances are to be seen:
the large peaks that fall off towards lower radii, labeled with a ‘1’ in the denomi-
(Rmax - Rn) / Rn
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
70 kPa
3/1
4/1
5/1
6/1
0
10 20 50 100 200 500
Rn [ µm ]
Figure 3. Response curves for bubbles of different sizes subject to driving at a fixed
frequency of 20 kHz. Driving amplitudes are 10 (bottom curve), 30, 50, and 70 kPa (upper
curve). Initial condition is the respective bubble at rest. The logarithmic radius scale
stretches the harmonic resonances for better viewing.
5/2
2/1
3/2
1/1
1/2