Minnaert Resonance Research Articles

Minnaert resonance in an array of two-dimensional bubbles

Minnaert resonance in an array of two-dimensional bubbles

Modeling frequency shifts of collective bubble resonances with the boundary element method.

Increasing the number of closely packed air bubbles immersed in water changes the frequency of the Minnaert resonance. The collective interactions between bubbles in a small ensemble are primarily in the same phase, causing them to radiate a spherically symmetric field that peaks at a frequency lower than the Minnaert resonance for a single bubble. In contrast, large periodic arrays include bubbles that are further apart than half of the wavelength such that collective resonances have bubbles oscillating in opposite phases, ultimately creating a fundamental resonance at a frequency higher than the single-bubble Minnaert resonance. This work investigates the transition in resonance behavior using a modal analysis of a mass-spring system and a boundary element method. The computational complexity of the full-wave solver is significantly reduced to a linear dependence on the number of bubbles in a rectangular array. The simulated acoustic fields confirm the initial downshift in resonance frequency and the strong influence of collective resonances when the array has hundreds of bubbles covering more than half of the wavelength. These results are essential in understanding the low-frequency resonance characteristics of bubble ensembles, which have important applications in diverse fields such as underwater acoustics, quantum physics, and metamaterial design.

Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies

AbstractThis paper provides several contributions to the mathematical analysis of subwavelength resonances in a high‐contrast medium containing acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of an dimensional linear system, and characterizes resonant frequencies as the solutions to an ‐dimensional nonlinear eigenvalue problem. Under a simplicity assumptions on the eigenvalues of the capacitance matrix, we prove the analyticity of the scattering resonances with respect to the square root of the contrast parameter, and we provide a deterministic algorithm allowing to compute all terms of the corresponding Puiseux series. We then establish a nonlinear modal decomposition formula for the scattered field as well as point scatterer approximations for the far‐field pattern of the sound wave scattered by bodies. As a prerequisite to our analysis, a first part of the work establishes various novel results about the capacitance matrix and its symmetry properties, since qualitative properties of the resonances, such as the leading order of the scattering frequencies or of the corresponding far‐field pattern are closely related to its spectral decomposition.

Minnaert Resonances for Bubbles in Soft Elastic Materials

Minnaert resonance is a well known physical phenomenon and it has many important applications, in particular in the effective realisation of acoustic metamaterials using bubbly mediums in recent years. In this paper, motived by the Minnaert resonance, we consider the low-frequency resonance for acoustic bubbles embedded in soft elastic materials. This is a hybrid physical process that couples the acoustic and elastic wave propagations. By delicately and subtly balancing the acoustic and elastic parameters as well as the geometry of the bubble, we show that Minnaert resonance can (approximately) occur for rather general constructions. Our study poses a great potential for the effective realisation of negative elastic materials by using bubbly elastic mediums.

Three-dimensional acoustic lensing with a bubbly diamond metamaterial

A sound wave travelling in water is scattered by a periodic assembly of air bubbles. The local structure matters even in the low frequency regime. If the bubbles are arranged in a face-centered cubic (fcc) lattice, a total bandgap opens near the Minnaert resonance frequency. If they are arranged in the diamond structure, which one obtains by simply adding a second bubble to the unit cell, one finds an additional branch with a negative slope (optical branch). For a single specific frequency, the medium behaves as if its refractive index (relative to water) is exactly n=−1. We show that a slab of this material can be used to design a three-dimensional flat lens. We also report super-resolution focusing in the near field of the slab and illustrate its potential for imaging in three dimensions.

Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies

<p style='text-indent:20px;'>We analyze mathematically the acoustic imaging modality using bubbles as contrast agents. These bubbles are modeled by mass densities and bulk moduli enjoying contrasting scales. These contrasting scales allow them to resonate at certain incident frequencies. We consider two types of such contrasts. In the first one, the bubbles are light with small bulk modulus, as compared to the ones of the background, so that they generate the Minnaert resonance (corresponding to a local surface wave). In the second one, the bubbles have moderate mass density but still with small bulk modulus so that they generate a sequence of resonances (corresponding to local body waves). <p style='text-indent:20px;'>We propose to use as measurements the far-fields collected before and after injecting a bubble, set at a given location point in the target domain, generated at a band of incident frequencies and at a fixed <i>single backscattering direction</i>. Then, we scan the target domain with such bubbles and collect the corresponding far-fields. The goal is to reconstruct both the, variable, mass density and bulk modulus of the background in the target region. <p style='text-indent:20px;'>1.We show that, for each fixed used bubble, the contrasted far-fields reach their maximum value at, incident, frequencies close to the Minnaert resonance (or the body-wave resonances depending on the types of bubbles we use). Hence, we can reconstruct this resonance from our data. The explicit dependence of these resonances in terms of the background mass density of the background allows us to recover it, i.e. the mass density, in a straightforward way. <p style='text-indent:20px;'>2.In addition, this measured contrasted far-fields allow us to recover the total field at the location points of the bubbles (i.e. the total field in the absence of the bubbles). A numerical differentiation argument, for instance, allows us to recover the bulk modulus of the targeted region as well.

The Equivalent Media Generated by Bubbles of High Contrasts: Volumetric Metamaterials and Metasurfaces

We deal with the point-interaction approximations for the acoustic wave fields generated by a cluster of highly contrasted bubbles for a wide range of densities and bulk moduli contrasts. We derive the equivalent fields when the cluster of bubbles is appropriately distributed (but not necessarily periodically) in a bounded domain $\Omega$ of $\mathbb{R}^3$. We handle two situations. 1. In the first one, we distribute the bubbles to occupy a $3$ dimensional domain. For this case, we show that the equivalent speed of propagation changes sign when the medium is excited with frequencies smaller or larger than (but not necessarily close to) the Minnaert resonance. As a consequence, this medium behaves as a reflective or absorbing depending on whether the used frequency is smaller or larger than this resonance. In addition, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above a certain threshold, then the medium behaves as a 'wall', i.e. allowing no incident sound to penetrate. 2. In the second one, we distribute the bubbles to occupy a $2$ dimensional (open or closed) surface, not necessarily flat. For this case, we show that the equivalent medium is modeled by a Dirac potential supported on that surface. The sign of the surface potential changes for frequencies smaller or larger than the Minnaert resonance, i.e. it behaves as a smart metasurface reducing or amplifying the transmitted sound across it. As in the $3$D case, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above an appropriate threshold, then the surface allows no incident sound to be transmitted across the surface, i.e. it behaves as a white screen.

СОУДАРЕНИЕ КАПЛИ СО СВОБОДНОЙ ПОВЕРХНОСТЬЮ: АКУСТИЧЕСКОЕ ИЗЛУЧЕНИЕ В ТОЛЩЕ ЖИДКОСТИ

An experimental study of the emission of sound by droplets falling into an unperturbed liquid was performed using wideband hydrophones and a high-speed video camera. Collision of a drop with a water surface is accompanied by air entrainment and the formation of underwater gas cavities, which leads to the appearance of surface forces and fast flows, generating extreme accelerations, which are sufficient for resonant acoustic excitation of cavities and emission of sound packets. The sequence of emitted signals contains a shock pulse, as well as one or more sound packets generated by resonant bubbles detached from underwater cavities. The number of resonating bubbles varies from experience to experience, depending on the collision scenario and further behavior, including, in particular, the process of fragmentation of the primary cavities. The dimensions of the bubbles, measured on video frames, correspond to the values calculated in accordance with the Minnaert resonance frequency.

Double-negative acoustic metamaterials

The aim of this paper is to provide a mathematical theory for understanding the mechanism behind the double-negative refractive index phenomenon in bubbly fluids. The design of double-negative metamaterials generally requires the use of two different kinds of subwavelength resonators, which may limits the applicability of double-negative metamaterials. Herein we rely on media that consists of only a single type of resonant element, and show how to turn the acoustic metamaterial with a single negative effective property obtained in [H. Ammari and H. Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency. SIAM J. Math. Anal., 49 (2017), 3252--3276.] into a negative refractive index metamaterial, which refracts waves negatively, hence acting as a superlens. Using bubble dimers made of two identical bubbles, it is proved that both the effective mass density and the bulk modulus of the bubbly fluid can be negative near the anti-resonance of the two hybridized Minnaert resonances for a single constituent bubble dimer. A rigorous justification of the Minnaert resonance hybridization, in the case of a bubble dimer in a homogeneous medium, is established. The acoustic properties of a single bubble dimer are analyzed. Asymptotic formulas for the two hybridized Minnaert resonances are derived. Moreover, it is proved that the bubble dimer can be approximated by a point scatterer with monopole and dipole modes. For an appropriate volume fraction of bubble dimers with certain conditions on their configuration, a double-negative effective medium when the frequency is near the anti-resonance of the hybridized Minnaert resonances can be obtained.

The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies

We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of M small bubbles, distributed in a bounded domain Ω, with relative densities having contrasts of the order aβ,β>0, where a models their relative maximum diameter, a≪1. We provide useful and natural conditions on the number M, the minimum distance and the contrasts parameter β of the small bubbles under which the point interaction approximation (called also the Foldy-Lax approximation) is valid.With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order aβ, β∈(32,2) and in this case we can handle any fixed frequency. In the particular and important case β=2, we can handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.

Minnaert resonances for acoustic waves in bubbly media

Through the application of layer potential techniques and Gohberg–Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.

Subwavelength Localized Modes for Acoustic Waves in Bubbly Crystals with a Defect

The ability to control wave propagation is of fundamental interest in many areas of physics. Photonic and phononic crystals have proved very useful for this purpose but, because they are based on Bragg interference, these artificial media require structures with large dimensions. In [Ammari et al., Subwavelength phononic bandgap opening in bubbly media, J. Diff. Eq., 263 (2017), 5610--5629], it has been proved that a subwavelength bandgap opening occurs in bubbly phononic crystals. To demonstrate the opening of a subwavelength phononic bandgap, a periodic arrangement of bubbles is considered and their subwavelength Minnaert resonance is exploited. In this paper, this subwavelength bandgap is used to demonstrate cavities, very similar to those obtained in photonic and phononic crystals, albeit of deeply subwavelength dimensions. The key idea is to perturb the size of a single bubble inside the crystal, thus creating a defect. The goal is then to analytically and numerically show that this crystal has a localized eigenmode close to the defect bubble.

Sub-wavelength focusing of acoustic waves in bubbly media.

The purpose of this paper is to investigate acoustic wave scattering by a large number of bubbles in a liquid at frequencies near the Minnaert resonance frequency. This bubbly media has been exploited in practice to obtain super-focusing of acoustic waves. Using layer potential techniques, we derive the scattering function for a single spherical bubble excited by an incident wave in the low frequency regime. We then propose a point scatterer approximation for N bubbles, and describe several numerical simulations based on this approximation, that demonstrate the possibility of achieving super-focusing using bubbly media.

Subwavelength phononic bandgap opening in bubbly media

The aim of this paper is to show both analytically and numerically the existence of a subwavelength phononic bandgap in bubble phononic crystals. The key result is an original formula for the quasi-periodic Minnaert resonance frequencies of an arbitrarily shaped bubble. The main findings in this paper are illustrated with a variety of numerical experiments.

Random acoustic metamaterial with a subwavelength dipolar resonance.

The effective velocity and attenuation of longitudinal waves through random dispersions of rigid, tungsten-carbide beads in an elastic matrix made of epoxy resin in the range of beads volume fraction 2%-10% are determined experimentally. The multiple scattering model proposed by Luppé, Conoir, and Norris [J. Acoust. Soc. Am. 131(2), 1113-1120 (2012)], which fully takes into account the elastic nature of the matrix and the associated mode conversions, accurately describes the measurements. Theoretical calculations show that the rigid particles display a local, dipolar resonance which shares several features with Minnaert resonance of bubbly liquids and with the dipolar resonance of core-shell particles. Moreover, for the samples under study, the main cause of smoothing of the dipolar resonance of the scatterers and the associated variations of the effective mass density of the dispersions is elastic relaxation, i.e., the finite time required for the shear stresses associated to the translational motion of the scatterers to propagate through the matrix. It is shown that its influence is governed solely by the value of the particle to matrix mass density contrast.

Multiple scattering in bubbly media: Highlighting the role played by interactions between neighboring bubbles

A bubble in water exhibits a low-frequency monopolar acoustic resonance, the so-called Minnaert resonance, which makes an assembly of such bubbles an ideal system to study multiple scattering of ultrasound. Following Foldy’s seminal work, various approaches, such as the ones by Keller or Waterman and Truell, have been developed to infer the effective acoustic properties of a bubble cloud. Here, we confront the predictions of these different approaches with numerical results obtained with a multiple scattering theory (MST) code that fully incorporates the multiple scattering effects. Based on this study, we show the importance to take into account the interactions between neighboring bubbles to predict the behavior of a bubble sample. Finally, we demonstrate how the introduction of a local order can affect the effective parameters and allows interesting transmission properties.

Propagation of ultrasound in aqueous foams: bubble size dependence and resonance effects

We report experimental results on the propagation of ultrasonic waves (at frequencies in the range of 40 kHz) in aqueous foams. Monitoring the acoustics of the foams as they age, i.e. as the mean bubble radius increases by coarsening, we recover at short times some trends that are already known: decrease of the speed of sound and increase of attenuation. At long times, we have identified, for the first time, robust non-monotonic behaviors of the speed of sound and attenuation, associated with a critical bubble size, which decreases at increasing frequency. The experimental features appear to be surprisingly reminiscent of the Minnaert resonance known for a single isolated bubble in a fluid. Transposing the Minnaert theoretical framework to the limit of a dense packing of bubbles gives some qualitative agreement with the data, but still cannot explain quantitatively the measured properties.

Single bubble oscillations in viscoelastic media

We report a new ultrasonic technique for determining the viscoelasticity of soft materials based on the oscillations of single bubbles injected into the material of interest. It is known that bubbles in a liquid act as strong acoustic scatterers that exhibit a low frequency resonance known as the Minnaert resonance. In the case of viscoelastic media, the complex frequency‐dependent shear modulus causes the Minnaert frequency to be shifted to a higher value and leads to additional ultrasonic absorption. Therefore, both storage and loss shear moduli can be determined from the resonance frequency and the damping rate of the acoustic oscillations of a single bubble that has been injected into the sample. Experiments were performed on optically transparent commercial hair gel, agar gel, and PDMS rubber, allowing independent measurements of the bubble sizes to be made by an optical imaging technique. The acoustical properties of the samples were measured by sweeping the frequency of a continuous sinusoidal sign...

Acoustic transmission through one plane of bubbles

We measured the transmission of ultrasonic waves through one layer of bubbles, for frequencies ranging from 30 to 250 kHz. The layer was a true 2D structure obtained by injecting very monodisperse bubbles (with radius r=100 μm) into a yield‐stress polymer gel. Even for layer with a low concentration of bubbles (areal fraction, nπr2, of 10‐20%, where n is the number of bubbles per unit area), the 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 individual Minnaert resonance of the bubbles, but depended on the concentration, which we interpret as 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.