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.

Complex eigenrays algorithm for infrasound propagation in a windy range dependent atmosphere.

Ray tracing is a simple and efficient three-dimensional method which reduces the problem of infrasound propagation to a series of one-dimensional cases along acoustical rays. However, in relatively frequent cases, infrasound stations are located in geometrical shadow zones, where only diffracted waves are recorded. The corresponding arrivals cannot be predicted by ray theory. To simulate infrasound propagation in these zones, the ray tracing method is generalized to complex ray theory. The source, media, and ground parameters are all considered as complex numbers. For applications with realistic atmospheric data, including stratified temperature and wind as well as the range dependency of atmospheric profiles, an efficient algorithm determining complex eigenrays in the shadow zones is presented. It is illustrated by a two-dimensional case of a point source.

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.

A model for acoustic vaporization dynamics of a bubble/droplet system encapsulated within a hyperelastic shell.

Nanodroplets have great, promising medical applications such as contrast imaging, embolotherapy, or targeted drug delivery. Their functions can be mechanically activated by means of focused ultrasound inducing a phase change of the inner liquid known as the acoustic droplet vaporization (ADV) process. In this context, a four-phases (vapor + liquid + shell + surrounding environment) model of ADV is proposed. Attention is especially devoted to the mechanical properties of the encapsulating shell, incorporating the well-known strain-softening behavior of Mooney-Rivlin material adapted to very large deformations of soft, nearly incompressible materials. Various responses to ultrasound excitation are illustrated, depending on linear and nonlinear mechanical shell properties and acoustical excitation parameters. Different classes of ADV outcomes are exhibited, and a relevant threshold ensuring complete vaporization of the inner liquid layer is defined. The dependence of this threshold with acoustical, geometrical, and mechanical parameters is also provided.

Sound propagation in dilute suspensions of spheres: Analytical comparison between coupled phase model and multiple scattering theory.

Sound propagation in dilute suspensions of small spheres is studied using two models: a hydrodynamic model based on the coupled phase equations and an acoustic model based on the ECAH (ECAH: Epstein-Carhart-Allegra-Hawley) multiple scattering theory. The aim is to compare both models through the study of three fundamental kinds of particles: rigid particles, elastic spheres, and viscous droplets. The hydrodynamic model is based on a Rayleigh-Plesset-like equation generalized to elastic spheres and viscous droplets. The hydrodynamic forces for elastic spheres are introduced by analogy with those of droplets. The ECAH theory is also modified in order to take into account the velocity of rigid particles. Analytical calculations performed for long wavelength, low dilution, and weak absorption in the ambient fluid show that both models are strictly equivalent for the three kinds of particles studied. The analytical calculations show that dilatational and translational mechanisms are modeled in the same way by both models. The effective parameters of dilute suspensions are also calculated.

Properties of theranostic nanoparticles determined in suspension by ultrasonic spectroscopy.

In the context of growing use of nanoparticles, it is important to be able to characterize all their physical properties in order to understand their behavior, to optimize them, and to control their quality. We showed that ultrasonic spectroscopy provides many of the desired properties. To do so, we used as an example nanocapsules made of a polymer shell encaspulating a liquid perfluorocarbon core and designed them for theranostic applications. Frequency-dependent measurements of both ultrasound velocity and attenuation were performed on nanocapsule suspensions. Then the desired properties were extracted by analyzing the experimental data using a recently developed model that relates the speed of sound and attenuation of a suspension to the geometrical and viscoelastic properties of the nanocapsules.

A model for acoustic vaporization of encapsulated droplets.

The use of encapsulated liquid nanoparticles is currently largely investigated for medical applications, mainly because their reduced size allows them to enter targeted areas which cannot be reached by large microbubbles (contrast agents). Low-boiling point perfluorocarbon droplets can be vaporized on-site under the action of the ultrasonic field, in order to turn them into echogeneous-eventually cavitating-microbubbles. This paper presents a theoretical model describing this phenomenon, paying particular attention to the finite size of the droplet and its encapsulation by a thin viscoelastic layer. Numerical simulations are done for droplets of radii 1 and 10 µm and for frequencies of 1-5 MHz. Results reveal that droplet surface tension and shell rigidity are responsible for an increase of the acoustic droplet vaporization threshold. Furthermore, this threshold does not vary monotonically with frequency, and an optimal frequency can be found to minimize it. Finally, the role of some physical properties on the dynamics of the particle is analyzed, such as the contrast of inner and outer liquids densities and the mechanical properties of the shell.

Adaptive motion estimation of shear shock waves in soft solids and tissue with ultrasound.

Shear shock waves in soft solids, such as in tissue, have different regions of complex motion that can change rapidly across a single wave profile, especially at the shock front. Conventional tracking algorithms are not well adapted to the task of simultaneously tracking the discontinuous shock front and smooth regions away from the shock. An adaptive algorithm based on the normalized cross-correlation and a correlation-weighted median filter is presented. The proposed adaptive algorithm combines two features: first, it adapts the window size to optimize the correlation value based on the deformation, and second, it rejects inaccurate estimates with a median-weighted filter. For simulated ultrasound data, where the displacements are known, it is shown that the estimated velocity error for the adaptive algorithm is less than 1/3 of the error for non-adaptive normalized cross-correlation. The addition of the weighted median filter to the adaptive algorithm significantly improves the shock tracking performance. The shock position and rise-time error is almost an order of magnitude better with the median-weighted filter. This algorithm is then used to track shock wave propagation with data acquired by a high-frame-rate ultrasound scanner in a tissue-mimicking agar and gelatin phantom. The shock front is not resolved with conventional algorithms but it is clearly visible with the proposed adaptive median-weighted algorithm.

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.

Influence of shell compressibility on the ultrasonic properties of polydispersed suspensions of nanometric encapsulated droplets.

Liquid droplets of nanometric size encapsulated by a polymer shell are envisioned for targeted drug delivery in therapeutic applications. Unlike standard micrometric gas-filled contrast agents used for medical imaging, these particles present a thick shell and a weakly compressible core. Hence, their dynamical behavior may be out of the range of validity of the models available for the description of encapsulated bubbles. In the present paper, a model for the ultrasound dispersion and absorption in a suspension of nanodroplets is proposed, accounting for both dilatational and translational motions of the particle. The radial motion is modeled by a generalized Rayleigh-Plesset-like equation which takes into account the compressibility of the viscoelastic shell, as well as the one of the core. The effect of the polydispersity of particles in size and shell thickness is introduced in the coupled balance equations which govern the motion of the particles in the surrounding fluid. Both effects of shell compressibility and polydispersity are quantified through the dispersion and absorption curves obtained on a wide ultrasonic frequency range. Finally, some results for larger gas-filled particles are also provided, revealing the limit of the role of the shell compressibility.

A model for ultrasound absorption and dispersion in dilute suspensions of nanometric contrast agents.

Ultrasound dispersion and absorption are examined in dilute suspensions of contrast agents of nanometric size, with a typical radius around 100 nm. These kinds of contrast agents are designed for targeted delivery of drugs for cancer treatment. Compared to standard contrast agents used for imaging, particles are of smaller size to pass through the endothelial barrier, their shell, made up of biocompatible polymer, is stiffer to undergo a longer lifetime, and they have a liquid core instead of a gaseous one. Ultrasound propagation in dilute suspension is modeled by combining two modes for particle oscillations. The first one is a dilatational mode assuming an incompressible shell with a rheological behavior of Kelvin-Voigt or Maxwell type. The second one is a translational mode induced by visco-inertial interaction with the ambient fluid. The relative importance of these two modes of interaction on both dispersion and absorption is quantified and analyzed for a model system and for two radii (75 and 150 nm) and the two rheological models. The influence of shell parameters (Young modulus, viscosity, and relative thickness) is finally discussed.

Sound, infrasound, and sonic boom absorption by atmospheric clouds.

This study quantifies the influence of atmospheric clouds on propagation of sound and infrasound, based on an existing model [Gubaidulin and Nigmatulin, Int. J. Multiphase Flow 26, 207-228 (2000)]. Clouds are considered as a dilute and polydisperse suspension of liquid water droplets within a mixture of dry air and water vapor, both considered as perfect gases. The model is limited to low and medium altitude clouds, with a small ice content. Four physical mechanisms are taken into account: viscoinertial effects, heat transfer, water phase changes (evaporation and condensation), and vapor diffusion. Physical properties of atmospheric clouds (altitude, thickness, water content and droplet size distribution) are collected, along with values of the thermodynamical coefficients. Different types of clouds have been selected. Quantitative evaluation shows that, for low audible and infrasound frequencies, absorption within clouds is several orders of magnitude larger than classical absorption. The importance of phase changes and vapor diffusion is outlined. Finally, numerical simulations for nonlinear propagation of sonic booms indicate that, for thick clouds, attenuation can lead to a very large decay of the boom at the ground level.

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.

Scattering of ultrasonic shock waves in suspensions of silica nanoparticles.

Experiments are carried out to assess, for the first time, the validity of a generalized Burgers' equation, introduced first by Davidson [J. Acoust. Soc. Am. 54, 1331-1342 (1973)] to compute the nonlinear propagation of finite amplitude acoustical waves in suspensions of "rigid" particles. Silica nanoparticles of two sizes (33 and 69 nm) have been synthesized in a water-ethanol mixture and precisely characterized via electron microscopy. An acoustical beam of high amplitude is generated at 1 MHz inside a water tank, leading to the formation of acoustical shock waves through nonlinear steepening. The signal is then measured after propagation in a cylinder containing either a reference solution or suspensions of nanoparticles. In this way, a "nonlinear attenuation" is obtained and compared to the numerical solution of a generalized Burgers' equation adapted to the case of hydrosols. An excellent agreement (corresponding to an error on the particles size estimation of 3 nm) is achieved in the frequency range from 1 to 40 MHz. Both visco-inertial and thermal scattering are significant in the present case, whereas thermal effects can generally be neglected for most hydrosols. This is due to the value of the specific heat ratio of water-ethanol mixture which significantly differs from unity.

Generalization of helicoidal beams for short pulses.

A generalization to the transient regime is developed for waves with a phase singularity of the screw type. These singular waves are commonly called vortices for all kind of waves as, for instance, optical vortex or acoustical vortex. We generalize the definition of vortices to get an azimuthal velocity invariant for all the frequency components contained in the broad spectrum of a short pulse. This generalization leads to a modification of the orbital angular momentum definition. Another generalization is introduced by considering helicoidal waves with a finite number of turns. We demonstrate that, in this last case, the topological charge is no longer an integer. This provides a physical interpretation to vortices of fractional charge that are involved here to take into account the diffraction occurring at both tips of the now finite helical wave front. We show that shortening the pulse implies an angular localization of the wave energy and, as a consequence, a spreading of the angular momentum amplitude due to the uncertainty principle.

Nonlinear reflection of shock shear waves in soft elastic media.

For fluids, the theoretical investigation of shock wave reflection has a good agreement with experiments when the incident shock Mach number is large. But when it is small, theory predicts that Mach reflections are physically unrealistic, which contradicts experimental evidence. This von Neumann paradox is investigated for shear shock waves in soft elastic solids with theory and simulations. The nonlinear elastic wave equation is approximated by a paraxial wave equation with a cubic nonlinear term. This equation is solved numerically with finite differences and the Godunov scheme. Three reflection regimes are observed. Theory is developed for shock propagation by applying the Rankine-Hugoniot relations and entropic constraints. A characteristic parameter relating diffraction and non-linearity is introduced and its theoretical values are shown to match numerical observations. The numerical solution is then applied to von Neumann reflection, where curved reflected and Mach shocks are observed. Finally, the case of weak von Neumann reflection, where there is no reflected shock, is examined. The smooth but non-monotonic transition between these three reflection regimes, from linear Snell-Descartes to perfect grazing case, provides a solution to the acoustical von Neumann paradox for the shear wave equation. This transition is similar to the quadratic non-linearity in fluids.

Evidence of wave front folding of sonic booms by a laboratory-scale deterministic experiment of shock waves in a heterogeneous medium.

The influence of the planetary boundary layer on the sonic boom received at the ground level is known since the 1960s to be of major importance. Sonic boom propagation in a turbulent medium is characterized by an increase of the mean rise time and a huge variability. An experiment is conducted at a 1:100,000 scale in water to investigate ultrasonic shock wave interaction with a single heterogeneity. The experiment shows a very good scaling with sonic boom, concerning the size of the heterogeneities, the wave amplitude, and the rise time of the incident wave. The wave front folding associated with local focusing, and its link to the increase of the rise time, are evidenced by the experiment. The observed amplification of the peak pressure (by a factor up to 2), and increase of the rise time (by up to about one magnitude order), are in qualitative agreement with sonic boom observations. A nonlinear parabolic model is compared favorably to the experiment on axis, though the paraxial approximation turns out less precise off axis. Simulations are finally used to discriminate between nonlinear and linear propagations, showing nonlinearities affect mostly the higher harmonics that are in the audible range for sonic booms.

On the influence of spatial correlations on sound propagation in concentrated solutions of rigid particles.

In a previous paper [J. Acoust. Soc. Am. 121, 3386-3387 (2007)], a self-consistent effective medium theory has been used to account for hydrodynamic interactions between neighboring rigid particles, which considerably affect the sound propagation in concentrated solutions. However, spatial correlations were completely left out in this model. They correspond to the fact that the presence of one particle at a given position locally affects the location of the other ones. In the present work, the importance of such correlations is demonstrated within a certain frequency range and particle concentration. For that purpose, spatial correlations are integrated in our two-phase formulation by using a closure scheme similar to the one introduced by Spelt et al. [''Attenuation of sound in concentrated suspensions theory and experiments," J. Fluid Mech. 430, 51-86 (2001)]. Then, the effect is shown through a careful comparison of the results obtained with this model, the ones obtained with different self-consistent approximations and the experiments performed by Hipp et al. ["Acoustical characterization of concentrated suspensions and emulsions. 2. Experimental validation," Langmuir, 18, 391-404 (2002)]. With the present formulation, an excellent agreement is reached for all frequencies (within the limit of the long wavelength regime) and for concentrations up to 30% without any adjustable parameter.

Numerical investigation of the properties of nonlinear acoustical vortices through weakly heterogeneous media.

Acoustical vortices (AV) are the acoustical equivalent of optical vortices (OV) that are a key feature in the discipline of singular optics. For linear waves, OV and AV possess the same properties. But as nonlinearities are different in optics and acoustics, the nonlinear behavior of these structures has to be different. In this paper, a numerical investigation of the three-dimensional (3D) nonlinear propagation of acoustical vortices through homogeneous or heterogeneous media is reported. First, an original numerical method is described and compared to existing ones. Then, it is used to study the dynamics of AV in a nonlinear regime. The nonlinear properties of acoustical vortices in a homogeneous medium are investigated. It is shown that shock waves can be produced during propagation, leading to an interesting spatiotemporal wave field with an azimuthal shock. The dynamics of the topological charge, intrinsic property of AV or OV, is studied in the nonlinear regime through different focusing lenses. Inversion of the topological charge is observed if the AV propagates through a 1D focusing medium (cylindrical lens), while the charge remains constant if the medium is 2D (spherical lens). These last results already observed in linear optics are generalized here to the nonlinear behavior through the investigation of harmonics which show the same behavior as the fundamental with respect to inversion.