have been developed and reviewed by Plesset and Prosperetti (1977),
Prosperetti (1984a; 1984b), Feng and Leal (1997), Brenner et al. (2002)
and Lauterborn and Kurz (2010).
Lauterborn (1976) gave a thorough investigation of the basic properties of
nonlinear oscillations of gas bubbles in liquids numerically. The response
of a bubble to a single-frequency excitation was calculated and displayed
in the form of frequency response curves, i.e., the maximum bubble radius
in steady-state oscillation (non-dimensionalized by the equilibrium bubble
radius) versus driving frequency. Figure 1.4 shows typical response curves
with special features of nonlinear oscillations. The expression n m (here,
m and n are two integers) above the peaks represents the order of the resonance. Cases with m=1 and n=2, 3… correspond to harmonics; cases withm=2, 3… andn=1 correspond to subharmonics; cases with m=2, 3… and n=2, 3… correspond to ultraharmonics. There exist thresholds for subharmonics and ultraharmonics. Figure 1.5 illustrates the threshold for
the subharmonic of the order 1 2 varying with bubble radii. For detailed
definition and the descriptions of the resonances and nonlinear phenomena
(e.g., jump phenomenon, hysteresis) mentioned above, readers are referred
Figure 1.4 Frequency response curves for a bubble in water with a radius
at rest of Rn 10 μm for different sound pressure amplitudes PA of (a) 0.4, (b) 0.5, (c) 0.6, (d) 0.7, and (e) 0.8 bar. is the frequency of the driving sound field. 0 is the natural frequency of the bubble oscillation.
max
R is the maximum radius of the bubble during its steady-state oscillation. The numbers marked above the peaks are the orders of the
resonances, represented as n m. The dots and the arrows belong to curve (e). The arrows indicate that the corresponding stationary solution is out of
the range of the diagram or that no stationary solution could be found. In
this case the values of the amplitudes were also very high oscillating
around some value outside the diagram. This figure was adapted from Fig.
Figure 1.5 Threshold for the occurrence of the first subharmonic
oscillation (of order 1 2 at 0 2) versus the equilibrium bubble radius (solid line). is the frequency of the driving sound field. 0 is the natural frequency of the bubble oscillation. PA is the amplitude of sound pressure. This figure was adapted from Fig. 13 of Lauterborn
(1976).
The power spectrum can be used to describe the property of bubble
oscillators. By solving the bubble motion equations, the variations of
bubble radius with time could be obtained (as shown in Figure 1.6). Then
through the Fourier transform, the “time domain” diagram could be
transformed to the “frequency domain” diagram, i.e., the power spectrum
[as shown in Figure 1.7(a)], the corresponding frequencies of the bands
resonance and its harmonics. In particular conditions, as shown in Figure
1.7(b), there are bands at 2 , 3 2 and 5 2 which represent subharmonic and ultraharmonics respectively. These lines are also typical
bands which usually appear in the scattered signals (i.e., acoustical echo of
bubble oscillations) in experiments.
Figure 1.6 Non-dimensionized bubble radius versus time. Equilibrium
bubble radius Rn is 10 μm. Sound pressure amplitude is 90 kPa. Driving
frequencyvis 207 kHz for upper diagram and 197 kHz for lower diagram. This figure was adapted from Fig. 13 of Lauterborn (1988).
Figure 1.7 Power spectrum of bubble oscillator. Equilibrium bubble radius
n
R is 10 μm. Sound pressure amplitude
207 kHz; (b) 197 kHz.
and Parlitz (1988).
Newhouse and Shanka
oscillators under a dual
1
f and f2 would contain the bands at in single-frequency
the radiated pressure at frequencies
1 2
f f . They (Newhouse and Shankar, 1984; Shanka proved that the resonance at
at f2, which means obtained through dual
(a)
(b)
1.7 Power spectrum of bubble oscillator. Equilibrium bubble radius
ound pressure amplitude is 90 kPa. Driving frequency
kHz. This figure was adapted from Fig. 13 of Lauterborn
Newhouse and Shankar (1984) pointed out that the echo of bubble
a dual-frequency acoustical excitation with frequencies
would contain the bands at f1 f2 besides the typical bands frequency approach. They further gave the analytical solutions of
pressure at frequencies f1, f2 , 2f1, 2f2, f1 f2
They (Newhouse and Shankar, 1984; Shankar et al., 1986) also
the resonance at f1 f2 is much sharper than the resonance , which means more accurate measurements of bubble size can be
dual-frequency approach.
1.7 Power spectrum of bubble oscillator. Equilibrium bubble radius
kPa. Driving frequencyv: (a) This figure was adapted from Fig. 13 of Lauterborn
r (1984) pointed out that the echo of bubble
with frequencies
besides the typical bands
gave the analytical solutions of
1 2
f f , and , 1986) also
the resonance
However, in multi-
bands rather than those corresponding to
driving frequencies. Fig
scattered signal of bubbles ex
frequencies f1 and
corresponding to main resonances (marked by
by □), subharmarnics (marked by
△). But besides these, there are other peaks (marked by magnitudes of which
Figure 1.8 Experimental spectr
excited by a dual-frequency ultrasound field. The frequencies of the two
sound waves are
1 2 12
f f MHz. of Ma (2010).
-frequency systems, the scattered echo contains
those corresponding to the difference and sum
driving frequencies. Figure 1.8 is an experimental spectrum
scattered signal of bubbles excited by a dual-frequency field (with
and f2 respectively). In this figure, there are peaks corresponding to main resonances (marked by■), their harmonics (marked ), subharmarnics (marked by▲) and sum and difference (marked by
besides these, there are other peaks (marked by
of which are of the same order of the harmonics.
1.8 Experimental spectrum of the scattered signal from bubbles
frequency ultrasound field. The frequencies of the two
sound waves are f15.5 MHz and f2 6.5 MHz, respectively. 2 1 1
f f MHz. This figure was adapted from Fig. contains more
the difference and sum of the
um of the
frequency field (with
In this figure, there are peaks
), their harmonics (marked
sum and difference (marked by
besides these, there are other peaks (marked by ◆), the
of the scattered signal from bubbles
frequency ultrasound field. The frequencies of the two
respectively.
So far as we know, most published papers mainly focused on the
application of the sum and difference of the driving frequencies in
multi-frequency systems (Wyczalkowski and Szeri, 2003; Phelps and
Leighton, 1994; Wu et al., 2005). The fundamental properties of bubble
dynamics (for instance, the special bands marked by ◆ shown in Figure 1.8) under multi-frequency acoustical excitation have not been studied
systematically. However, these features are essential for understanding the
bubble behaviour under multi-frequency excitation as well as expanding
their applications.