Gas bubbles play an important role in the generation, scattering, and absorption of sound in a liquid. The acoustical scattering cross section is usually employed to evaluate the scattering ability of the bubbles, and is defined as the square of the ratio between the amplitude of the radiated spherical wave by bubbles and the amplitude of the incident acoustic wave. Compared with the scattering behavior of incompressible objects, the acoustical scattering cross section of bubbles is quite significant due to the compressible nature of the gases inside the bubbles, and the scattering cross section of bubbles under acoustic excitation can be of the order of 1000 times of its own geometrical cross section.^[1,2]

The acoustical scattering cross section of gas bubbles has been studied over several decades. Linear models of bubble scattering cross sections are used to measure oceanic bubble size distributions and predict their acoustic effects, with applications involving sonar performance,^[3,4] and surf zone or subsurface acoustics.^[5,6] A paper by Ainslie^[1] has reviewed expressions and definitions in the literature for acoustical cross sections, resonance frequencies, and damping factors of a spherically pulsating gas bubble in an infinite liquid medium. The survey of bubble scattering models can be ordered into three threads, referred to as the “Wildt” thread, the “Devin” thread, and a third “nonlinear” thread. Zhang also proposed a generalized equation for the scattering cross section of spherical gas bubbles oscillating in liquids under acoustic excitation.^[7] Shi et al. studied the oscillation properties influenced by stimulated sound scattering processes.^[8] For the cases of bubble clouds, both the velocity and amplitude of the acoustic waves will be modulated due to the presence of bubbles. Similarly, the velocity and amplitude of the spherical waves generated by the oscillating bubbles will also be changed by the other bubbles.^[9] For the linear case, Commander and Prosperetti^[10] performed a systematic theoretical study. For a spherical bubble cloud, the acoustical scattering cross section could be theoretically calculated following the framework of d’Agostino and Brennen.^[11]

The previous investigations detailed in the literature mainly focused on the case of single-frequency acoustic excitation. However, to promote acoustic effects in many application areas, the multi-frequency approach (i.e., waves with two or three, even more acoustic components) could be an effective method.^[12] The multi-frequency approach has been employed to measure the bubble density in water,^[13–15] find the bubble size distributions with void fraction in the ocean,^[16,17] optimize the scattering from a micro bubble at a frequency different from the driving,^[18] and advance the biomedical ultrasound imaging techniques.^[19] Zhang et al. focused on the dual-frequency approach to obtain the influence of mass transfer on the oscillations of cavitation bubbles, and proposed a threshold of acoustic pressure amplitude dividing the bubble response into two regions.^[20] Furthermore, Zhang et al. studied the acoustical scattering cross section, and the combination and simultaneous resonances of gas bubbles under dual-frequency acoustic excitation in detail.^[21–23]

However, the problems of acoustical cross section of a bubble under multi-frequency excitation are still poorly investigated in the literature. The motivation of this paper is to achieve a much broader range of acoustical scattering cross section by the multi-frequency approach. Different from the work of Zhang et al.,^[21–24] our work improves the dual-frequency approach to the multi-frequency approach. The main aim of the paper is to obtain directly the analytical solution of the acoustical scattering cross section under multi-frequency excitation, and make a comprehensive analysis of the influence of several paramount parameters (e.g., bubble equilibrium radius, acoustic pressure amplitude, and acoustic frequencies) in the multi-frequency system.

The remainder of the article is organized as follows. In Section 2, the equations relating with the acoustical scattering cross section of gas bubbles under multi-frequency acoustic excitation are obtained by both analytical and numerical solutions. In Section 3, the nonlinear nature of the acoustical scattering cross section of gas bubbles under multi-frequency acoustic waves is revealed. The influences of several paramount parameters (e.g., bubble equilibrium radius, acoustic pressure amplitude, and acoustic frequencies) on the scattering cross section are shown. The conclusions are summarized in Section 4.

The dynamics of spherical gas bubbles in Newtonian, compressible, and viscous liquids are considered, and the equation of bubble motion can be written as the Keller–Miksis equation^[25,26]

For multi-frequency excitation, in Eq. (1) can be expressed as

Assume the amplitudes of the acoustic wave as (, respectively, are the non-dimensional amplitudes of the external acoustic excitation using the ambient pressure. The ratio of the pressure amplitudes of the multi-frequency acoustic waves is defined as , which reflects the power allocation between each high-frequency component and the lowest frequency component. We set . are the angular frequencies of the external acoustic excitation.

The radiation pressure at radial coordinate r from the origin of the bubble center can be given as^[27] 5Prad(r,t)=ρRr(2R˙2+RR¨), where is the radiation pressure, and r is the radial coordinate with the origin at the bubble center. The solution of the bubble motion can be obtained by solving Eqs. (1)–(4), i.e., the dynamic oscillations of R, , and . Substituting the above solutions into Eq. (5), the radiation pressure generated by the bubble oscillations can be obtained. The amplitude of the radiation pressure can be written as

Here, B/r is the amplitude of the divergent spherical scattered wave generated by the bubble oscillations. Following the definition proposed by Wildt,^[2] the acoustical scattering cross section ( of an oscillating gas bubble in the liquid, which reflects the square of the ratio between scattered waves and incident waves, can be defined as

2.1. Analytical analysis

The analytical solution for the acoustical scattering cross section under multi-frequency acoustic excitation is obtained in this section. The solution of the Keller–Miksis equation under multi-frequency acoustic field can be expressed as

Thermal damping is ignored. Substituting Eq. (8) to Eq. (1), we obtain the following equation by considering the terms up to the first order of :

The sum of all terms with order in the above equation should be equal to zero respectively. The result can be expressed as

where

The solution of Eq. (10) can be written as

where

Here, is the natural frequency of the gas bubbles oscillating in the liquids; , , and are the total, viscous, and acoustic damping constants, respectively. Therefore, the acoustical scattering cross section under multi-frequency excitation is

The typical parameters for a bubble in water, as used in the calculations with the Keller–Miksis model, are , , , , , kPa, , kPa, , and . For simplicity, the acoustical scattering cross sections of individual bubbles are normalized by their respective geometrical cross sections, i.e., .

For comparison, the total input power should be a constant during simulations of different cases, so we can obtain the acoustic pressure amplitude during different cases

Figure 1 shows the acoustical scattering cross section (i.e., curves of against equilibrium bubble radius) predicted by Eq. (17) under single-frequency excitation, dual-frequency excitation (different combinations of frequencies), and tri-frequency excitation ( kHz, kHz, and kHz), respectively. However, the analytical solution can give the main resonances clearly, but cannot predict complex nonlinear phenomena when is big enough.

Fig. 1.

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	Fig. 1. (color online) Prediction of acoustical scattering cross section against equilibrium bubble radius under single-frequency, dual-frequency and tri-frequency acoustic excitations by analytical solution: (a) acoustic excitation versus different single-frequency, (b) acoustic excitation versus different dual-frequency, and (c) tri-frequency acoustic excitation. , ().

2.2. Numerical analysis

Next, the numerical solution is employed to study the acoustical scattering cross section generated by bubble oscillations. The Keller–Miksis equation can be reformulated as a set of autonomous differential equations of first order

20. where the variable is introduced for eliminating variable t. This equation can be directly solved using an explicit Runge–Kutta formula. For this bubble oscillation model, the main parameters of interest are the equilibrium bubble radius , acoustic frequency , and acoustic pressure amplitude . The other parameters are connected with the properties of the liquid outside the bubble and the gas and vapor of the liquid inside the bubble. In the present paper, the origin of the coordinate is placed at the center of the bubble. In numerical simulations, the acoustical scattering cross section is evaluated at the point with distance from the origin.

In this section, demonstrating examples of radiation pressure and acoustical scattering cross section generated by an oscillating bubble under multi-frequency acoustic excitation within a wide range of parameters are shown. The nonlinear characteristics and influential parameters of the acoustical cross section are discussed in detail.

3.1. The instantaneous bubble radius and corresponding radiation pressure

Figure 2 demonstrates the variations of non-dimensional instantaneous bubble radius and radiation pressure versus time for 10 μm, 20 μm, 50 μm, and 100 μm bubbles oscillating under multi-frequency excitation respectively. For a certain , both the bubble oscillations and radiation pressure curves under single-, dual-, and tri-frequency excitations show obvious periodical oscillations. However, the bubble oscillations and the radiation pressure curves under dual-frequency and tri-frequency excitations exhibit much more complicated patterns owing to the involvement of extra frequency components. Additionally, the addition of the extra acoustic waves leads to great increases of the amplitude of the bubble oscillations and the radiation pressure when the equilibrium bubble radius is not too big.

Fig. 2.

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	Fig. 2. (color online) The instantaneous bubble radius under the single-frequency acoustic excitation (solid line), dual-frequency acoustic excitation (dashed line), tri-frequency acoustic excitation (dash-dotted line) and the corresponding radiation pressure versus time for (a) 10 μm, (b) 20 μm, (c) 50 μm, and (d) 100 μm bubbles. kHz, kHz, kHz. , .

3.2. The characteristic of the scattering cross section under multi-frequency excitation

3.2.1. Influences of the pressure amplitude and the equilibrium bubble radius

Figure 3 shows the acoustical scattering cross section under single-frequency, dual-frequency, and tri-frequency acoustic excitations versus both equilibrium bubble radius and acoustic pressure amplitude by numerical solution. A parameter plane, where only two parameters are varied, is best suited for viewing the details of the acoustical scattering cross section. This parameter space diagram (, is color coded, in which the driving pressure amplitude has been increased from 0.01 to 0.4, and the bubble radius has been increased from 1 μm to 150 μm. The scale of acoustical scattering cross section is limited from −40 dB to 40 dB for different cases.

Fig. 3.

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	Fig. 3. (color online) Parameter space diagram spanned by and giving the acoustical scattering cross section under single-frequency, dual-frequency, and tri-frequency acoustic excitations: (a) kHz; (b) kHz; (c) kHz, kHz; (d) kHz, kHz, kHz. The logarithmic bubble radius scale stretches the acoustical scattering cross section for better viewing.

It is shown that the harmonics and subharmonics increase when driving the bubbles stronger in the case of single-frequency acoustic excitation. With the increase of , both the harmonic and subharmonic resonances grow significantly, and the peaks of these resonances bend toward smaller bubble radius. This bending phenomenon is a nonlinear feature determined by the Keller–Miksis equation.^[24,25] In the case of multi-frequency acoustic excitation, the combination resonances increase with the increase of the acoustic pressure amplitude and the number of the excitation frequency. It means that the acoustical scattering cross section is promoted significantly within a much broader range of bubble sizes and acoustic frequencies due to the generation of more resonances.

Figure 4 shows a slice of Fig. 3 in the case of . Comparing the scattering cross section curves under single-frequency and multi-frequency excitations with the acoustic pressure amplitude , the amplitudes of the main resonances are not changed, but the harmonics, subharmonics, and the combination resonances increase with the increase of the number of the excitation frequency.

Fig. 4.

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Fig. 4. (color online) The acoustical scattering cross section under single-frequency (dashed line kHz and dotted line kHz), dual-frequency (dash-dotted line kHz, kHz), and tri-frequency (solid line kHz, kHz, kHz) acoustic excitations versus equilibrium bubble radius by numerical solution. . The logarithmic bubble radius scale stretches the acoustical scattering cross section cruves for better viewing.

3.2.2. Influence of the frequency and the equilibrium bubble radius

Figure 5 shows the predicted scattering cross sections under single-frequency (first column), dual-frequency (second column), and tri-frequency (third column) excitations versus both excitation frequency f and equilibrium bubble radius . In this parameter space diagram (f, (first column), the driving frequency f has been increased from 0 to 300 kHz, and the bubble radius has been increased from 1 μm to 150 μm. In the cases of dual-frequency and tri-frequency excitions, the driving frequency has been increased from 0 to 300 kHz, and , .

Fig. 5.

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Fig. 5. (color online) The acoustical scattering cross section under single-frequency (first column), dual-frequency (second column), and tri-frequency (third column) acoustic excitations versus both excitation frequency f and equilibrium bubble radius by numerical solution. (a) , (b) , (c) , (d) , (e) . The logarithmic bubble radius scale and frequency f scale stretch the acoustical scattering cross section for better viewing.

In the case of single-frequency excitation, for a certain acoustic pressure amplitude , the scattering cross section reaches maximum when the bubble radius is equal to the resonant bubble radius corresponding to the driving frequency, i.e., at the main resonance. The scattering cross section curves show nonlinear features, such as harmonic, subharmonic, and ultraharmonic resonances. Compared to the single-frequency approach, the multi-frequency approach can generate much more resonance regions involving all the main, harmonic, and subharmonic resonances corresponding to the multiple component frequencies and a lot of combination resonances.

3.2.3. Influence of the frequency and the pressure amplitude

Figure 6 shows the predicted scattering cross sections under single-frequency (first column), dual-frequency (second column) and tri-frequency (third column) excitations versus both excitation frequency f and acoustic pressure amplitude . In this parameter space diagram (f, (first column), the driving frequency f has been increased from 0 to 300 kHz, and the acoustic pressure amplitude has been increased from 0.01 to 0.4. In the cases of dual-frequency and tri-frequency excitions, the driving frequency has been increased from 0 to 300 kHz, and , .

Fig. 6.

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	Fig. 6. (color online) The acoustical scattering cross section under single-frequency, dual-frequency, and tri-frequency acoustic excitations versus both excitation frequency f and by numerical solution. (a) , (b) , (c) , (d) . The logarithmic frequency f scale stretches the acoustical scattering cross section for better viewing.

Compared to the single-frequency approach, for a certain bubble radius , harmonic, subharmonic, and combination resonances become more and more complex with the increase of under the multi-frequency excitation.

Both the analytical and numerical solutions of the acoustical scattering cross section of a bubble under multi-frequency excitation are obtained. The prediction of the scattering cross section by the analytical method can give the main resonances clearly, but cannot predict complex nonlinear phenomena when is big enough because many nonlinear features will present.

The nonlinear characteristics of the scattering cross section under the multi-frequency approach are investigated numerically by using parameter space diagrams (, , (f, , and (f, in detail. Generally, the scattering cross section curves show typical nonlinear features such as harmonics, subharmonics, and ultraharmonics. Furthermore, the multi-frequency approach displays more resonances termed as combination resonances. Compared to the single-frequency approach, the multi-frequency approach could promote the acoustical scattering cross section significantly within a much broader range of bubble sizes due to the generation of more resonances.

The influence of several paramount parameters in the multi-frequency system on the scattering cross section has been discussed. Due to the vast parameter space, the properties of the scattering cross section are still not fully explored. Meanwhile, a large parameter zone is involved, leading to great difficulties for the design. So optimization of the multi-frequency approach is still needed and worth studying.