Sound velocity and attenuation in bubbly gels measured by transmission experiments

Linear pressure waves in mono- and poly-disperse bubbly liquids: Attenuation and propagation speed in slow and fast and evanescent modes

Using volumetric averaged equations from a two-fluid model, this study theoretically investigates linear pressure wave propagation in a quiescent liquid with many spherical gas bubbles. The speed and attenuation of sound are evaluated using the derived linear dispersion. Mono- and poly-disperse bubbly liquids are treated. To precisely describe the attenuation effect, some forms of bubble dynamics equations and temperature gradient models are employed. Focusing on the dissipative effect, we analyze the stop band that occurs in the linear dispersion relation. In the two-fluid model, even if the dissipation effect is considered, the inconvenience that the wavenumber diverges to infinity in the resonance frequency cannot be resolved. Additionally, the validity of terminating that wavenumber value in the middle of the frequency is demonstrated. To determine a linear dispersion relation that can exactly predict thermal conduction and acoustic radiation, wave propagation velocities and attenuation coefficients are compared with some experimental data and existing models. The results show that thermal conduction and acoustic radiation should be set appropriately to accurately predict the propagation velocity and attenuation except in the high frequency range, the phase velocity in the resonance frequency range, or the attenuation in the high frequency range.

Unconventional acoustic wave propagation transitions induced by resonant scatterers in the high-density limit

Experiments on ultrasound propagation through a gel doped with resonant encapsulated microbubbles provided evidence for a discontinuous transition between wave propagation regimes at a critical excitation frequency. Such behavior is unlike that observed for soft materials doped with non-resonant air or through liquid foams, and disagrees with a simple mixture model for the effective sound speed. Here, we study the discontinuous transition by measuring the transition as a function of encapsulated microbubble volume fraction. The results show the transition always occurs in the strong-scattering limit (l/λ < 1, l and λ are the mean free path and wavelength, respectively), that at the critical frequency the effective phase velocity changes discontinuously to a constant value with increasing microbubble volume fraction, and the measured critical frequency shows a power law dependence on microbubble volume fraction. The results cannot be explained by multiple scattering theory, viscous effects, mode decoupling, or a critical density of states. It is hypothesized the transition depends upon the microbubble on-resonance effective properties, and we discuss the results within the context of percolation theory. The results shed light on the discontinuous transition's physics, and suggest soft materials can be engineered in this manner to achieve a broad range of physical properties with potential application in ultrasonic actuators and switches.

Propagation of elastic waves in correlated dispersions of resonant scatterers

The propagation of coherent longitudinal and transverse waves in random distributions of spherical scatterers embedded in an elastic matrix is studied. The investigated frequency range is the vicinity of the resonance frequencies of the translational and rotational motion of the spheres forced by the waves, where strong dispersion and attenuation are predicted. A technique for making samples made of layers of carbide tungsten beads embedded in epoxy resin is presented, which allows control of the scatterers distribution, induce short-range positional correlations, and minimize the anisotropy of samples. Comparison between phase velocity and attenuation measurements and a model based on multiple scattering theory (MST) shows that bulk effective properties accurately described by MST are obtained from three beads layers. Besides, short-range correlations amplify the effect of mechanical resonances on the propagation of longitudinal and transverse coherent waves. As a practical consequence, the use of short-range positional correlations may be used to enhance the attenuation of elastic waves by disordered, locally resonant, elastic metamaterials, and MST globally correctly predicts the effect of short-range positional order on their effective properties.

Fractional derivative-based approximation of acoustic waveform dispersion measured in bubbly water beyond resonance frequency

Acoustic pulses transmitted across air bubbles in water are usually analyzed in terms of attenuation coefficient and phase velocity in the frequency domain. The present work expresses an analytical approximation of the acoustic waveform in the time domain. It is introduced by experiments performed with a Gaussian derivative source wavelet, S0(t), with a derivative order, β0 = 4, and a peak frequency, νp0, much larger than the bubble resonance frequency. The measurements highlight a significant shape variability of the waveform Bx(t), measured at x≤ 0.74 m and characterized by a peak frequency νpx≃νp0. The results are in good agreement with the approximation Bx(t)∝(dγx/dtγx)S0(δxt - T), where γx is an additional fractional-derivative order determined by an optimization procedure and T is related to the travel time. The time-scale parameter, δx=β0/(β0+γx), becomes a free parameter for more general source signals. The correlation coefficient between Bx(t) and the approximated waveform is used to identify the applicability of the method for a wide range of bubbly waters. The results may be of potential interest in characterizing gas bubbles in the ocean water column and, more generally, in modeling wave propagation in dispersive media with fractional-derivative orders in the time domain.

Confined Phase Singularities Reveal the Spin-to-Orbital Angular Momentum Conversion of Sound Waves

We identify an acoustic process in which the conversion of angular momentum between its spin and orbital form takes place. The interaction between an evanescent wave propagating at the interface of two immiscible fluids and an isolated droplet is considered. The elliptical motion of the fluid supporting the incident wave is associated with a simple state of spin angular momentum, a quantity recently introduced for acoustic waves in the literature. We experimentally observe that this field predominantly forces a directional wave transport circling the droplet's interior, revealing the existence of confined phase singularities. The circulation of the phase, around a singular point, is characteristic of angular momentum in its orbital form, thereby demonstrating the conversion mechanism. The numerical and experimental observations presented in this Letter have implications for the fundamental understanding of the angular momentum of acoustic waves, and for applications such as particle manipulation with radiation forces or torques, acoustic sensing and imaging.

Probing the pressure dependence of sound speed and attenuation in bubbly media: Experimental observations, a theoretical model and numerical calculations

The problem of attenuation and sound speed of bubbly media has remained partially unsolved. Comprehensive data regarding pressure-dependent changes of the attenuation and sound speed of a bubbly medium are not available. Our theoretical understanding of the problem is limited to linear or semi-linear theoretical models, which are not accurate in the regime of large amplitude bubble oscillations. Here, by controlling the size of the lipid coated bubbles (mean diameter of ≈5.4μm), we report the first time observation and characterization of the simultaneous pressure dependence of sound speed and attenuation in bubbly water below, at and above microbubbles resonance (frequency range between 1-3 MHz). With increasing acoustic pressure (between 12.5-100 kPa), the frequency of the peak attenuation and sound speed decreases while maximum and minimum amplitudes of the sound speed increase. We propose a nonlinear model for the estimation of the pressure dependent sound speed and attenuation with good agreement with the experiments. The model calculations are validated by comparing with the linear and semi-linear models predictions. One of the major challenges of the previously developed models is the significant overestimation of the attenuation at the bubble resonance at higher void fractions (e.g. 0.005). We addressed this problem by incorporating bubble-bubble interactions and comparing the results to experiments. Influence of the bubble-bubble interactions increases with increasing pressure. Within the examined exposure parameters, we numerically show that, even for low void fractions (e.g. 5.1×10-6) with increasing pressure the sound speed may become 4 times higher than the sound speed in the non-bubbly medium.

Critical Role of a Nanometer-Scale Microballoon Shell on Bulk Acoustic Properties of Doped Soft Matter

A material's acoustic properties depend critically upon porosity. Doping a soft material with gas-filled microballoons permits a controlled variation of the porosity through a scalable fabrication process while generating well-tailored spherical cavities that are impermeable to liquids. However, evidence is lacking of how the nanometer-scale polymeric shell contributes to the overall effective material properties in the regime where the wavelength is comparable to the sample thickness. Here, we measure ultrasound transmission through a microballoon-doped soft material as a function of microballoon and impurity concentration, sample thickness, and frequency. The measured longitudinal wave speeds are an order of magnitude larger than those in similar systems where no shell is present, while the transverse wave speed is found to linearly increase with microballoon concentration, also in contrast to systems with no shell. Furthermore, we find the results are independent of the soft material's elastic moduli as well as a lesser contribution of the microballoon shell on material attenuation. The results are validated with a multiple scattering model and suggest the shell contributes significantly to the material's bulk acoustic properties despite its thickness being 4 orders of magnitude smaller than the acoustic wavelength. Our results demonstrate how a nanometer-scale interface between a gas cavity and a soft polymer can be used in the submicrometer design of acoustic materials, and are important for observations of such phenomena as strong interference effects in soft matter.

Influence of the microstructure of two-dimensional random heterogeneous media on propagation of acoustic coherent waves

Multiple scattering of waves arises in all fields of physics in either periodic or random media. For random media the organization of the microstructure (uniform or nonuniform statistical distribution of scatterers) has effects on the propagation of coherent waves. Using a recent exact resolution method and different homogenization theories, the effects of the microstructure on the effective wave number are investigated over a large frequency range (ka between 0.1 and 13.4) and high concentrations. For uniform random media, increasing the configurational constraint makes the media more transparent for low frequencies and less for high frequencies. As a side but important result, we show that two of the homogenization models considered here appear to be very efficient at high frequency up to a concentration of 60% in the case of uniform media. For nonuniform media, for which clustered and periodic aggregates appear, the main effect is to reduce the magnitude of resonances and to make network effects appear. In this case, homogenization theories are not relevant to make a detailed analysis.

Low frequency propagation through random polydisperse assemblies of cylindrical or spherical poroelastic obstacles

The effective wavenumbers, moduli, and mass densities are found for polydisperse assemblies of poroelastic obstacles (considering fluid flow and solid deformation in the porous medium). The obstacles are infinite length cylinders and spheres. To achieve this, recent formulas for the effective wavenumbers, given by Linton and Martin [SIAM J. Appl. Math. 66(5), 1649-1668 (2006)] and Norris and Conoir [J. Acoust. Soc. Am. 129(1), 104-113 (2011)] in the dilute monodisperse case (obstacles of identical sizes in a fluid matrix), have been modified. Given the uncertainty in predicting the distribution in size of the obstacles, three quite different probability density functions are studied and compared: uniform, Schulz, and lognormal. Specifically, the Rayleigh approximation (low frequency regime) is considered, in which the wavelengths can be assumed very large compared to the size of the obstacles. Within this limit, simplified formulas are provided for the concentrations depending on the parameter characterizing the size dispersion.

The inferred determination of the phase speed of a bubbly liquid of less than 2 m/s by using a transfer function technique

The phase speed of a bubbly liquid has been inferred at the single bubble resonance. The difficulty in this measurement is the fact that acoustic absorption is very strong at the single bubble resonance making it impossible to measure with standard techniques, e.g., time-of-flight or the use of standing waves. Instead, the strong absorption was exploited by first measuring the change in the reflection coefficient and relating it to the acoustic impedance and the sound speed. A sound speed of less than 2 m/s was determined possibly making it one of the slowest sound speeds ever recorded.

Bubbly flow velocity measurement in multiple scattering regime

We propose a technique to measure the velocity of a bubble cloud based on the coda correlation. The method is founded on successive recordings of multiple scattered waves from a bubble cloud. Our model predicts the dependence between the correlation coefficient of these coda waves and the velocity of the bubble cloud under diffusion approximation. The Acoustic experiments are validated by simultaneous optical measurements in a water tank, with a good agreement between the acoustical and the optical methods (relative difference smaller than 7%). This technique can be transposed to any particle flow velocity problems involving multiple scattering effects in acoustics.

The effect of speech training experiences on speechreading skills of Chinese children with hearing impairment

Speechreading contributes significantly to effective communication, and persons with hearing impairment (HI) may need to rely more on speechreading. However, whether they may benefit from training/practice to improve their speechreading performance remains unclear. In this study, we examine the effect of speech training on speechreading performance of children with HI in China, and how such effect, if any, may be influenced by age. Fifty-nine HI children with speech training experiences, and fifty-eight HI children without speech training experiences completed tasks measuring their ability to speechread Chinese at the linguistic levels of words, phrases, and sentences. The children ranged from 7 to 14 years old, and were divided into four age groups by two-year age intervals. Both accuracy rate and response time data were collected. Results revealed three findings: (1) HI children with speech training experiences speechread more accurately but more slowly than those without speech training experiences; (2) while speechreading performance generally improved with age, age didn't alter the relative performance between the two groups; and (3) speechreading performance was best for phrases, and worst for sentences. These findings suggest that HI children benefit from speech training to improve their speechreading performance, and their speechreading performance is influenced by age, linguistic level, and the specific measure used.

Resonant acoustic scattering by two spherical bubbles

The mutual interaction between two close bubbles in an acoustic field is studied. This interaction is modeled in the linear framework of the multiple scattering theory using spherical harmonics expansions and the addition theorem. In order to deal with small as well as large bubbles, viscous dissipation in the liquid, thermal dissipation in the gas, and surface tension are taken into account in the calculations of the scattering coefficients of a unique bubble. Under the assumption of the long wavelengths, the scattering coefficient of the monopolar mode is linked to the one obtained by using the Rayleigh-Plesset equation. The exact characteristic equation providing the symmetric and antisymmetric resonances of the two bubbles is established. Numerical results show that a great number of modes of vibration is required to describe the acoustic field around the bubbles. Moreover, whatever the spacing between two identical bubbles, the scattering cross section has a maximum value at the frequency of the symmetric mode while the antisymmetric mode is not detected. However, the strengthening of the scattering observed close to the symmetric resonance frequency is clearly due to the presence of the antisymmetric mode.

Influence of bubble distributions on the propagation of linear waves in polydisperse bubbly liquids

The influence of the spatial distributions of bubbles on the propagation of linear acoustic waves in polydisperse bubbly liquids is studied. Using the diagrammatic approach, the effective wavenumber, which includes both spatial information and higher orders of multiple scattering, is presented. The phase speed and attenuation coefficient of acoustic waves in bubbly liquids are calculated from the effective wavenumber. A three-dimensional random model, the Generalized Matérn's hard-core point process, is used to close the model. Numerical simulations reveal that as the bubble volume fraction becomes larger so does the effect of the bubble distributions on the attenuation and phase speed. The irregular discrepancy between previously reported experimental results and the classical theory is attributed to the influence of bubble clustering on the propagation of linear waves. The comparison between the present model and the experimental measurements [Leroy, Strybulevych, Page, and Scanlon. (2011). Phys. Rev. E 83, 046605] reveals that the proposed correction term significantly improves the theoretical predictions.

Gravitational collapse of Mount Etna's southeastern flank

The southeastern flank of Etna volcano slides into the Ionian Sea at rates of centimeters per year. The prevailing understanding is that pressurization of the magmatic system, and not gravitational forces, controls flank movement, although this has also been proposed. So far, it has not been possible to separate between these processes, because no data on offshore deformation were available until we conducted the first long-term seafloor displacement monitoring campaign from April 2016 until July 2017. Unprecedented seafloor geodetic data reveal a >4-cm slip along the offshore extension of a fault related to flank kinematics during one 8-day-long event in May 2017, while displacement on land peaked at ~4 cm at the coast. As deformation increases away from the magmatic system, the bulk of Mount Etna's present continuous deformation must be driven by gravity while being further destabilized by magma dynamics. We cannot exclude flank movement to evolve into catastrophic collapse, implying that Etna's flank movement poses a much greater hazard than previously thought. The hazard of flank collapse might be underestimated at other coastal and ocean island volcanoes, where the dynamics of submerged flanks are unknown.

A strict formulation of a nonlinear Helmholtz equation for the propagation of sound in bubbly liquids. Part I: Theory and validation at low acoustic pressure amplitudes

This paper strictly demonstrated a nonlinear Helmholtz equation, with its corresponding new expressions for the wave number of the mixture, for the propagation of sound trough a bubbly liquid. The demonstration was conducted under the assumption of periodicity of volume fluctuations, the acoustic approximation and considering only mono-harmonic pressure oscillations. The model revealed a beautiful symmetry between the average acoustic energy density and the average energy dissipation, as well as between the time average of the first and second derivatives of such fluctuations. The nonlinear model was validated with available experimental data at very low pressure amplitudes yielding the same results as the linear model. However, unlike the linear model, the advantage of the nonlinear model is that the wave number of the mixture is function of the pressure amplitude, which has great implications to model the sound propagation on cavitating bubbly liquids where the linear theory greatly under-predicts.

APPLICATION OF DROPLET DETECTORS TO ALPHA RADIATION DETECTION

Superheated droplet detectors (SDDs) are traditionally employed in the detection of neutrons. In this work the focus is on the detection of alpha particles using C2ClF5 as the target liquid. The alpha-droplet interaction is examined via computational studies, and a geometric model developed to describe the expected detector response. Experiments with alpha-emitting uranium- and samarium-doped SDDs at temperatures of 5-12°C confirm that the event rate is related to the size of the droplets, and are in model agreement for temperatures below 8°C; above this temperature, the acoustic sensitivity is reduced by signal attenuation as a result of the increasing bubble population, for which the addition of an attenuation coefficient restores the agreement with experiment. The results suggest the viability of a SDD-based alpha spectrometer using mono-sized droplets.

Multiple scattering of an ultrasonic shock wave in bubbly media

This experimental study deals with the propagation of an ultrasonic shock wave in a random heterogeneous medium, constituted of identical 75μm radius bubbles, trapped in a yield-stress fluid. The fundamental frequency of the incident wave (in the MHz range) was much larger than the resonance frequency of bubbles (38kHz). A well-expanded coda, resulting from the multiple scattering of the incident shock wave through the heterogeneous medium, was experimentally measured in transmission. Despite the significant amplitude of the shock wave (90kPa), no sign of nonlinear response of the bubbles was detected. Both the coherent and incoherent fields were successfully described by linear theories. Using a shock wave presents the advantage of characterizing the medium over a large frequency range (1.5-15MHz).

An iterative fullwave simulation approach to multiple scattering in media with randomly distributed microbubbles

Ultrasound contrast agents (UCA), such as microbubbles, enhance the scattering properties of blood, which is otherwise hypoechoic. The multiple scattering interactions of the acoustic field with UCA are poorly understood due to the complexity of the multiple scattering theories and the nonlinear microbubble response. The majority of bubble models describe the behavior of UCA as single, isolated microbubbles suspended in infinite medium. Multiple scattering models such as the independent scattering approximation can approximate phase velocity and attenuation for low scatterer volume fractions. However, all current models and simulation approaches only describe multiple scattering and nonlinear bubble dynamics separately. Here we present an approach that combines two existing models: (1) a full-wave model that describes nonlinear propagation and scattering interactions in a heterogeneous attenuating medium and (2) a Paul-Sarkar model that describes the nonlinear interactions between an acoustic field and microbubbles. These two models were solved numerically and combined with an iterative approach. The convergence of this combined model was explored in silico for 0.5 × 10⁶ microbubbles ml^-1, 1% and 2% bubble concentration by volume. The backscattering predicted by our modeling approach was verified experimentally with water tank measurements performed with a 128-element linear array transducer. An excellent agreement in terms of the fundamental and harmonic acoustic fields is shown. Additionally, our model correctly predicts the phase velocity and attenuation measured using through transmission and predicted by the independent scattering approximation.

Multiple scattering in random dispersions of spherical scatterers: Effects of shear-acoustic interactions

The propagation of acoustic waves through a suspension of spherical particles in a viscous liquid is investigated, through application of a multiple scattering model. The model is based on the multiple scattering formulation of Luppé, Conoir, and Norris [J. Acoust. Soc. Am. 131, 1113-1120 (2012)] which incorporated the effects of thermal and shear wave modes on propagation of the acoustic wave mode. Here, the model is simplified for the case of solid particles in a liquid, in which shear waves make a significant contribution to the effective properties. The relevant scattering coefficients and effective wavenumber are derived in analytical form. The results of calculations are presented for a system of silica particles in water, illustrating the dependence of the scattering coefficients, effective wavenumber, speed, attenuation on particle size and frequency. The results demonstrate what has already been shown experimentally; that the shear-mediated processes have a very significant effect on the effective attenuation of acoustic waves, especially as the concentration of particles increases.

Investigating the Existence of Bulk Nanobubbles with Ultrasound

Nanobubbles are expected to dissolve in milliseconds. Experimental evidence of nanobubbles that were stable for days had thus been first received with circumspection. If the large number of experimental confirmations has now made clear that surface nanobubbles could exist, bulk nanobubbles are still subject to debate. When observations are reported, the main problem is to make sure the observed particles are really made of gas. We show that ultrasound is an ideal tool for investigating the existence of bulk nanobubbles: 1) it is sensitive to minute quantities of gas, 2) it allows one to determine the bubble size distribution, 3) it discriminates unambiguously between gaseous and solid/liquid inclusions. To illustrate the efficiency of ultrasonic detection, we performed size measurements of bubbles produced by a commercial nano-/microbubble generator. No nanobubble was detected with this device. It would be insightful to use ultrasonic detection in experimental situations for which stable nanobubbles were reported.

Comment on "Attenuation and dispersion of sound in dilute suspensions of spherical particles" [J. Acoust. Soc. Am. 108(1), 126-146 (2000)]

Comment is made on an article by Temkin [J. Acoust. Soc. Am. 108(1), 126-146 (2000)]. This work illustrates the impact of particle size distribution on the dispersion and attenuation of sound waves propagating through particle suspensions. Temkin's propagation theory is extended for poly-disperse distributions of spherical particles, applicable to real suspensions. Significant changes in dispersion and attenuation appear near the particle resonance for both bubble and solid particle suspensions. Existing experimental data for bubble distributions also shows a good agreement with the poly-disperse modifications proposed.

Soft 3D acoustic metamaterial with negative index

Many efforts have been devoted to the design and achievement of negative-refractive-index metamaterials since the 2000s. One of the challenges at present is to extend that field beyond electromagnetism by realizing three-dimensional (3D) media with negative acoustic indices. We report a new class of locally resonant ultrasonic metafluids consisting of a concentrated suspension of macroporous microbeads engineered using soft-matter techniques. The propagation of Gaussian pulses within these random distributions of 'ultra-slow' Mie resonators is investigated through in situ ultrasonic experiments. The real part of the acoustic index is shown to be negative (up to almost - 1) over broad frequency bandwidths, depending on the volume fraction of the microbeads as predicted by multiple-scattering calculations. These soft 3D acoustic metamaterials open the way for key applications such as sub-wavelength imaging and transformation acoustics, which require the production of acoustic devices with negative or zero-valued indices.

Effect of direct bubble-bubble interactions on linear-wave propagation in bubbly liquids

We study the influence of bubble-bubble interactions on the propagation of linear acoustic waves in bubbly liquids. Using the full model proposed by Fuster and Colonius [J. Fluid Mech. 688, 253 (2011)], numerical simulations reveal that direct bubble-bubble interactions have an appreciable effect for frequencies above the natural resonance frequency of the average size bubble. Based on the new results, a modification of the classical wave propagation theory is proposed. The results obtained are in good agreement with previously reported experimental data where the classical linear theory systematically overpredicts the effective attenuation and phase velocity.

Resonant acoustic propagation and negative density in liquid foams

We measured the dispersion relation for acoustic longitudinal waves in liquid foams, over a broad frequency range (60-600 kHz). Strong dispersion was found, with two nondispersive behaviors, separated by a negative density regime. A new model, based on the coupled displacements of films, liquid channels, and gas in the foam, rationalizes all the experimental findings.

Numerical simulation of effective phase velocity and attenuation of shear elastic wave propagation in unidirectional composite materials

In this paper, a simple simulation approach is presented for calculating the effective phase velocity and attenuation coefficient of elastic shear waves propagating in composite materials with randomly distributed unidirectional inclusions. As an application of the developed numerical approach, the phase velocities and attenuation coefficients of the coherent waves in four different types of composite material are simulated for various incident frequencies up to π. Numerical results are compared with theoretical predictions obtained from three representative theoretical models. While all theoretical results agree very well with numerical values at low incident frequencies, the discrepancies increase with the increased incident frequency and volume fraction of inclusions. It has been found that within the frequency and volume fraction ranges considered in this work, the generalized self-consistent model by Kanaun and Levin [18] seems to provide the most accurate estimations.

Impact of polydispersity on multipolar resonant scattering in emulsions

The influence of size polydispersity on the resonant acoustic properties of dilute emulsions, made of fluorinated-oil droplets, is quantitatively investigated. Ultrasound attenuation and dispersion measurements on various samples with controlled size polydispersities, ranging from 1% to 13%, are found to be in excellent agreement with predictions based on the independent scattering approximation. By relating the particle-size distribution of the synthesized emulsions to the quality factor of the predicted multipolar resonances, the number of observable acoustic resonances is shown to be imposed by the sample polydispersity. These results are briefly discussed into the context of metamaterials for which scattering resonances are central to their effective properties.

Review of scattering and extinction cross-sections, damping factors, and resonance frequencies of a spherical gas bubble

Perhaps the most familiar concepts when discussing acoustic scattering by bubbles are the resonance frequency for bubble pulsation, the bubbles' damping, and their scattering and extinction cross-sections, all of which are used routinely in oceanography, sonochemistry, and biomedicine. The apparent simplicity of these concepts is illusory: there exist multiple, sometimes contradictory definitions for their components. This paper reviews 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, deriving two expressions for "resonance frequency" that are compared and reconciled with two others from the reviewed literature. In order to prevent errors, care is needed by researchers when combining results from different publications that might have used internally correct but mutually inconsistent definitions. Expressions are presented for acoustical cross-sections associated with forced pulsations damped by liquid shear and (oft-neglected) bulk or dilatational viscosities, gas thermal diffusivity, and acoustic re-radiation. The concept of a dimensionless "damping coefficient" is unsuitable for radiation damping because different cross-sections would require different functional forms for this parameter. Instead, terms based on the ratio of bubble radius to acoustic wavelength are included explicitly in the cross-sections where needed.

Influence of positional correlations on the propagation of waves in a complex medium with polydisperse resonant scatterers

We present experimental results on a model system for studying wave propagation in a complex medium exhibiting low-frequency resonances. These experiments enable us to investigate a fundamental question that is relevant for many materials, such as metamaterials, where low-frequency scattering resonances strongly influence the effective medium properties. This question concerns the effect of correlations in the positions of the scatterers on the coupling between their resonances, and hence on wave transport through the medium. To examine this question experimentally, we measure the effective medium wavenumber of acoustic waves in a sample made of bubbles embedded in an elastic matrix over a frequency range that includes the resonance frequency of the bubbles. The effective medium is highly dispersive, showing peaks in the attenuation and the phase velocity as functions of the frequency, which cannot be accurately described using the independent scattering approximation (ISA). This discrepancy may be explained by the effects of the positional correlations of the scatterers, which we show to be dependent on the size of the scatterers. We propose a self-consistent approach for taking this "polydisperse correlation" into account and show that our model better describes the experimental results than the ISA.

Sound propagation in a monodisperse bubble cloud: from the crystal to the glass

We present a theoretical study of the propagation of a monochromatic pressure wave in an unbounded monodisperse bubbly liquid. We begin with the case of a regular bubble array--a bubble crystal--for which we derive a dispersion relation. In order to interpret the different branches of this relation, we introduce a formalism, the radiative picture, which is the adaptation to acoustics of the standard splitting of the electric field in an electrostatic and a radiative part in Coulomb gauge. In the case of an irregular or completely random array--a bubble glass--and at wavelengths large compared to the size of the bubble array spatial inhomogeneities, the difference between order and disorder is not felt by the pressure wave: a dispersion relation still holds, coinciding with that of a bubble crystal with the same bubble size and air volume fraction at the centre of its first Brillouin zone. This relation is discussed and compared to that obtained by Foldy in the framework of his multiscattering approach.

Transmission of ultrasound through a single layer of bubbles

We investigate, both experimentally and theoretically, the effect of coupling between resonant scatterers on the transmission coefficient of a model system of isotropic scatterers. The model system consists of a monodisperse layer of bubbles, which exhibit a strong monopole scattering resonance at low ultrasonic frequencies. The layer was a true 2D structure obtained by injecting very monodisperse bubbles (with radius a approximately 100 microm) into a yield-stress polymer gel. Even for a layer with a low concentration of bubbles (areal fraction, n pi a(2), of 10-20%, where n is the number of bubbles per unit area), the ultrasonic transmission was found to be significantly reduced by the presence of bubbles (-20 to -50 dB) and showed a sharp minimum at a particular frequency. Interestingly, this frequency did not correspond to the resonance frequency of the individual, isolated bubbles, but depended markedly on the concentration. This frequency shift is an indication of strong coupling between the bubbles. We propose a simple model, based on a self-consistent relation, which takes into account the coupling between the bubbles and gives good agreement with the measured transmission coefficient.