Filtered Lebedev quadrature method for robust and efficient beam shape coefficient estimation in acoustic tweezers calibration.

Acoustic tweezers offer a contactless, three-dimensional, and selective approach to trapping objects by harnessing the acoustic radiation force. Precise control of this technique requires accurate calibration of the force, which depends on the object's properties and the spherical harmonics expansion of the incident field through the beam shape coefficients. Previous studies showed that these coefficients can be determined using either the Lebedev quadrature or the angular spectrum methods. However, the former is highly susceptible to noise, while the latter demands extensive implementation time due to the number of required measurement points. A filtered method with a reduced number of points is introduced to address these limitations. Initially, we emphasize the implicit filtering in the angular spectrum method, allowing relative noise insensitivity. Subsequently, we present its unfiltered version, enabling force estimation of a standing field. Finally, we develop a filtered method based on the Lebedev quadrature, requiring fewer points, and apply it to focused vortex beams. Numerical evaluation of the radiation force demonstrates the method's resilience to noise and a reduced need for points compared to previous methods. The filtered Lebedev method paves the way for characterizing high-frequency acoustic tweezers, where measurement constraints necessitate rapid and robust beam shape coefficient estimation techniques.

Calibration of the axial stiffness of a single-beam acoustic tweezers.

Single-beam acoustic tweezers have recently been demonstrated to be capable of selective three-dimensional trapping. This new contactless manipulation modality has great potential for many scientific applications. Its development as a scientific tool requires precise calibration of its radiation force, specifically its axial component. The lack of calibration for this force is mainly due to its weak magnitude compared to competing effects such as weight. We investigate an experimental method for the calibration of the axial stiffness of the radiation force by observing the axial oscillations of a trapped bead in a microgravity environment. The stiffness exhibits a linear relationship with the acoustic intensity and is of the mN/m order. Then, a predictive model, loaded with the experimental acoustic field, is compared to the measured stiffness with very good agreement, within a single amplitude coefficient. This study paves the way for the development of calibrated acoustic tweezers.

Propagation of classical and low booms through kinematic turbulence with uncertain parameters.

The propagation of sonic boom through kinematic turbulence is known to have an important impact on the noise perceived at the ground. In this work, a recent numerical method called FLHOWARD3D based on a one-way approach is used to simulate the propagation of classical and low-boom waveforms. Kinematic turbulence is synthesized following a von Kármán energy spectrum. Two- and three-dimensional (2D and 3D) simulations are compared to experimental measurements, and 2D simulations are found to be slightly less accurate than 3D ones but still consistent with experimental levels around 98% of the time. A stochastic study is carried out on the 2D simulation using the generalized polynomial chaos method with parameters of the von Kármán spectrum as uncertain parameters. Differences between the propagation of a classical N-wave and low booms are observed: the classical N-wave shows higher peak pressure and variations than low-boom signatures. The standard deviation for the peak pressure, the D-weighted sound exposure level (D-SEL), and the perceived level in dB (PLdB) metrics all show a linear increase with the distance, with a faster increase for the classical N-wave for the peak pressure and D-SEL and a similar increase between the different booms for PLdB. In general, it is found that low-boom waveforms show less sensitivity to turbulence.

Propagation of scalar waves in dense disordered media exhibiting short- and long-range correlations.

Correlated disorder is at the heart of numerous challenging problematics in physics. In this work we focus on the propagation of acoustic coherent waves in two-dimensional dense disordered media exhibiting long- and short-range structural correlations. The media are obtained by inserting elastic cylinders randomly in a stealth hyperuniform medium itself made up of cylinders. The properties of the coherent wave is studied using an original numerical software. In order to understand and discuss the complex physical phenomena occurring in the different media, we also make use of effective media models derived from the quasicrystalline approximation and the theory of Fikioris and Waterman that provides an explicit expression of the effective wave numbers. Our study shows a very good agreement between numerical and homogenization models up to very high concentrations of scatterers. This study shows that media with both short- and long-range correlations are of strong interest to design materials with original properties.

Impact of particle size and multiple scattering on the propagation of waves in stealthy-hyperuniform media.

Propagation of waves in materials that exhibit stealthy-hyperuniform long-range correlations is investigated. By using a modal decomposition of the field that takes multiple scattering into account at all orders, we study the impact of the concentration of particles on the transparency of such materials at low frequency. An upper frequency limit for transparency is defined that include both the particle size and the degree of stealthiness. We show that the independent scattering approximation is not relevant to calculate elastic mean free paths when wavelength becomes comparable to the size of particles. We find that transparency is very robust with regard to the degree of heterogeneity of the host random medium and the polydispersity of particles. Finally, it is shown that resonances can be used as the frequency filter.

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.

Computation of the radiation force exerted by the acoustic tweezers using pressure field measurements.

Acoustic tweezers allow for manipulation of small objects like elastic spheres with a force generated by the radiation pressure which arises from the nonlinear interaction between the incident and scattered waves by the object. The accurate control of the object by acoustic tweezers requires the study of the components of the three-dimensional (3D) force. If the physical properties of the elastic sphere are known, then the 3D components of the force can be calculated thanks to a decomposition of the incident acoustic field in the spherical functions basis. This study proposes evaluating the expansion coefficients. Three methods are used and compared. The first one consists of measuring the acoustic field on a spherical surface centered on the theoretical position of the object and to calculate the spherical functions decomposition by Lebedev quadratures. The second method is based on the measurement of the acoustic field at random points in a spherical volume and on the resolution of the inverse problem by a sparse method called the orthogonal matching pursuit. In the third method, the incident beam is measured on a transverse plane, decomposed into a sum of plane waves, and then the expansion coefficients are calculated. The results of the three methods will be presented and compared.

Numerical simulation of two-dimensional multiple scattering of sound by a large number of circular cylinders.

The purpose of this article is to present an innovative resolution method for investigating problems of sound scattering by infinite cylinders immersed in a fluid medium. The study is based on the analytical solution of multiple scattering, where incident and scattered waves are expressed in cylindrical harmonics. This modeling leads to dense linear systems, which are made sparse by introducing a cutoff radius around each particle. This cutoff radius is deeply studied and quantified. Numerical resolution is performed using parallel computing methods designed to solve very large sparse linear systems. Comparisons with direct calculations made with another numerical software and homogenization techniques follow and show good agreement with the implemented method. The last part is dedicated to a comparison between the propagation of waves in a circular cluster made of a random distribution of cylinders and the propagation in the corresponding homogenized cluster where the multiple scattering formalism is combined with a statistical analysis to provide an effective medium.

Geometric calibration of very large microphone arrays in mismatched free field.

Large microphone arrays are an efficient means for source localization thanks to a wide aperture and a great number of sensors. When such arrays are deployed in situ, accurate geometric calibration becomes essential to obtain the microphone positions. In free field, the classic procedures rely on measured Times of Arrival (TOA) or Time Differences of Arrival (TDOA) between the microphones and several controlled sources. However, free field model mismatches, such as reflectors, generate outliers which severely deteriorate the positioning accuracy. This paper introduces a unified framework for robust calibration using TOA or TDOA by exploiting an outlier-aware noise model. Thanks to the largeness of the array, the existing outliers are sparse and can be identified by a Lasso regression. From this, three iterative robust solvers are proposed: (i) for TOA by Robust Multi Dimensional Unfolding, a particular variation of Robust Multi Dimensional Scaling, (ii) for TDOA by data predenoising based on sparse and low-rank matrix decomposition, and (iii) for TDOA by jointly identifying the outliers and the geometry. The relevance of outlier-aware approaches is asserted by numerical and experimental tests. Compared with the baseline least-square approaches, the proposed robust solvers significantly improve the positioning accuracy in a free field mismatched by reflectors.

Numerical observation of secondary Mach stem in weak acoustic shock reflection.

Recently Ram, Geva, and Sadot [J. Fluid Mech. 768, 219-235 (2015)] showed, experimentally, the formation of a secondary Mach stem generated from the reflection of the primary Mach stem in the aerodynamic regime. Such a phenomenon has never been observed, either experimentally or numerically, in the framework of weak acoustic shocks. In this work, the formation of a secondary Mach stem is observed from the reflection of acoustic shock waves on a convex-concave boundary giving rise to a complex five-shock pattern. This study is fully numerical and is based on the numerical solution of a nonlinear acoustic system of equations using a recently developed discontinuous Galerkin solver.

Orbital Angular Momentum Transfer to Stably Trapped Elastic Particles in Acoustical Vortex Beams.

The controlled rotation of solid particles trapped in a liquid by an ultrasonic vortex beam is observed. Single polystyrene beads, or clusters, can be trapped against gravity while simultaneously rotated. The induced rotation of a single particle is compared to a torque balance model accounting for the acoustic response of the particle. The measured torque (∼10 pN m for a driving acoustic power ∼40 W/cm^{2}) suggests two dominating dissipation mechanisms of the acoustic orbital angular momentum responsible for the observed rotation. The first takes place in the bulk of the absorbing particle, while the second arises as dissipation in the viscous boundary layer in the surrounding fluid. Importantly, the dissipation processes affect both the dipolar and quadrupolar particle vibration modes suggesting that the restriction to the well-known Rayleigh scattering regime is invalid to model the total torque even for spheres much smaller than the sound wavelength. The findings show that a precise knowledge of the probe elastic absorption properties is crucial to perform rheological measurements with maneuverable trapped spheres in viscous liquids. Further results suggest that the external rotational steady flow must be included in the balance and can play an important role in other liquids.

A robust and passive method for geometric calibration of large arrays.

This paper presents a complete strategy for the geometry estimation of large microphone arrays of arbitrary shape. Largeness is intended here in both number of microphones (hundreds) and size (few meters). Such arrays can be used for various applications in open or confined spaces like acoustical imaging, source identification, or speech processing. For so large array systems, measuring the geometry by hand is impractical. Therefore a blind passive method is proposed. It is based on the analysis of the background acoustic noise, supposed to be a diffuse field. The proposed strategy is a two-step process. First the pairwise microphone distances are identified by matching their measured coherence function to the one predicted by the diffuse field theory. Second, a robust multidimensional scaling (MDS) algorithm is adapted and implemented. It takes advantage of local characteristics to reduce the set of distances and infer the geometry of the array. This work is an extension of previous studies, and it overcomes unsolved drawbacks. In particular it deals efficiently with the outliers known to ruin standard MDS algorithms. Experimental proofs of this ability are presented by treating the case of two arrays. They show that the proposed improvements manage large spatial arrays.

Nonlinear acoustic propagation in bubbly liquids: Multiple scattering, softening and hardening phenomena.

The weakly nonlinear propagation of acoustic waves in monodisperse bubbly liquids is investigated numerically. A hydrodynamic model based on the averaged two-phase fluid equations is coupled with the Rayleigh-Plesset equation to model the dynamics of bubbles at the local scale. The present model is validated in the linear regime by comparing with the Foldy approximation. The analysis of the pressure signals in the linear regime highlights two resonance frequencies: the Minnaert frequency and a multiple scattering resonance that strongly depends on the bubble concentration. For weakly nonlinear regimes, the generation of higher harmonics is observed only for the Minnaert frequency. Linear combinations between the Minnaert harmonics and the multiple scattering resonance are also observed. However, the most significant effect observed is the appearance of softening-hardening effects that share some similarities with those observed for sandstones or cracked materials. These effects are related to the multiple scattering resonance. Downward or upward resonance frequency shifts can be observed depending on the characteristic of the incident wave when increasing the excitation amplitude. It is shown that the frequency shift can be explained assuming that the acoustic wave velocity depends on a law different from those usually encountered for sandstones or cracked materials.

Numerical Simulation of Transit-Time Ultrasonic Flowmeters by a Direct Approach.

This paper deals with the development of a computational code for the numerical simulation of wave propagation through domains with a complex geometry consisting in both solids and moving fluids. The emphasis is on the numerical simulation of ultrasonic flowmeters (UFMs) by modeling the wave propagation in solids with the equations of linear elasticity (ELE) and in fluids with the linearized Euler equations (LEEs). This approach requires high performance computing because of the high number of degrees of freedom and the long propagation distances. Therefore, the numerical method should be chosen with care. In order to minimize the numerical dissipation which may occur in this kind of configuration, the numerical method employed here is the nodal discontinuous Galerkin (DG) method. Also, this method is well suited for parallel computing. To speed up the code, almost all the computational stages have been implemented to run on graphical processing unit (GPU) by using the compute unified device architecture (CUDA) programming model from NVIDIA. This approach has been validated and then used for the two-dimensional simulation of gas UFMs. The large contrast of acoustic impedance characteristic to gas UFMs makes their simulation a real challenge.

Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers.

We demonstrate the trapping of elastic particles by the large gradient force of a single acoustical beam in three dimensions. Acoustical tweezers can push, pull and accurately control both the position and the forces exerted on a unique particle. Forces in excess of 1 micronewton were exerted on polystyrene beads in the submillimeter range. A beam intensity less than 50 W/cm^{2} was required, ensuring damage-free trapping conditions. The large spectrum of frequencies covered by coherent ultrasonic sources provides a wide variety of manipulation possibilities from macroscopic to microscopic length scales. Our observations could open the way to important applications, in particular, in biology and biophysics at the cellular scale and for the design of acoustical machines in microfluidic environments.

One-way approximation for the simulation of weak shock wave propagation in atmospheric flows.

A numerical scheme is developed to simulate the propagation of weak acoustic shock waves in the atmosphere with no absorption. It generalizes the method previously developed for a heterogeneous medium [Dagrau, Rénier, Marchiano, and Coulouvrat, J. Acoust. Soc. Am. 130, 20-32 (2011)] to the case of a moving medium. It is based on an approximate scalar wave equation for potential, rewritten in a moving time frame, and separated into three parts: (i) the linear wave equation in a homogeneous and quiescent medium, (ii) the effects of atmospheric winds and of density and speed of sound heterogeneities, and (iii) nonlinearities. Each effect is then solved separately by an adapted method: angular spectrum for the wave equation, finite differences for the flow and heterogeneity corrections, and analytical method in time domain for nonlinearities. To keep a one-way formulation, only forward propagating waves are kept in the angular spectrum part, while a wide-angle parabolic approximation is performed on the correction terms. The numerical process is validated in the case of guided modal propagation with a shear flow. It is then applied to the case of blast wave propagation within a boundary layer flow over a flat and rigid ground.

Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere.

This work aims to model the acoustic radiation forces acting on an elastic sphere placed in an inviscid fluid. An expression of the axial and transverse forces exerted on the sphere is derived. The analysis is based on the scattering of an arbitrary acoustic field expanded in the spherical coordinate system centered on the spherical scatterer. The sphere is allowed to be arbitrarily located. The special case of high order Bessel beams, acoustical vortices, are considered. These types of beams have a helicoidal wave front, i.e., a screw-type phase singularity and hence, the beam has a central dark core of zero amplitude surrounded by an intense ring. Depending on the sphere's radius, different radial equilibrium positions may exist and the sphere can be set in rotation around the beam axis by an azimuthal force. This confirms the pseudo-angular moment transfer from the beam to the sphere. Cases where the axial force is directed opposite to the direction of the beam propagation are investigated and the potential use of Bessel beams as tractor beams is demonstrated. Numerical results provide an impetus for further designing acoustical tweezers for potential applications in particle entrapment and remote controlled manipulation.

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation.

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.

Nonlinear waves and shocks in a rigid acoustical guide.

A model is developed for the propagation of finite amplitude acoustical waves and weak shocks in a straight duct of arbitrary cross section. It generalizes the linear modal solution, assuming mode amplitudes slowly vary along the guide axis under the influence of nonlinearities. Using orthogonality properties, the model finally reduces to a set of ordinary differential equations for each mode at each of the harmonics of the input frequency. The theory is then applied to a two-dimensional waveguide. Dispersion relations indicate that there can be two types of nonlinear interactions either called "resonant" or "non-resonant." Resonant interactions occur dominantly for modes propagating at a rather large angle with respect to the axis and involve mostly modes propagating with the same phase velocity. In this case, guided propagation is similar to nonlinear plane wave propagation, with the progressive steepening up to shock formation of the two waves that constitute the mode and reflect onto the guide walls. Non-resonant interactions can be observed as the input modes propagate at a small angle, in which case, nonlinear interactions involve many adjacent modes having close phase velocities. Grazing propagation can also lead to more complex phenomena such as wavefront curvature and irregular reflection.

Transverse shift of helical beams and subdiffraction imaging.

An imaging technique is here proposed to overcome the classical "diffraction limit" by using helical beams. This technique and the analysis presented are valid for all kinds of waves (either optical or acoustical) as long as the field can be considered as scalar. We show that the stable structure of such phase singularities turns out to be appropriate to measure both the position and the diameter of subdiffraction circular apertures. The property used is a shift of the scattered vortex. Its location is obtained with a very high resolution thanks to a nonclassical correlation method exploiting the superoscillating property of a vortex near its axis. This theoretical analysis is supported by acoustic experiments performed underwater evidencing subdiffraction imaging.