Linear Acoustic Wave Propagation Research Articles

Acoustic emissions based estimation of the temporal changes in microbubble radius during ultrasonic excitation

Our ability to study microbubble dynamics in vivo and link them to distinct mechano-biological effects hinges on our ability to accurately estimate the temporal changes in MB radius during ultrasonic excitation. Here, we hypothesize that real-time passive cavitation detection (PCD) monitoring combined with linear acoustic wave propagation theory can accurately estimate stable MB radius dynamics under in vivo conditions. To test our hypothesis, we employed numerical simulations, based on Rayleigh–Plesset modeling, followed by experimental validation, using calibrated PCDs with concurrent optical imaging of MB dynamics using high frame rate microscopy. Our method termed linear acoustic wave propagation and superposition (LAWPS) algorithm, combines Fourier series expansion with Euler’s relationship to estimate the acoustic emission (AE) from single MB, which is considered as a monopole source (i.e., R0 + ΔR < λ). LAWPS algorithm, which is independent of MB properties and, thus, can be linearly reversed to calculate MB oscillation radius from AE, was able to accurately capture the radiated pressure generated from a Rayleigh–Plesset MB modeling. Crucially, inverse LAWPS algorithm onto AE generated by Vokurka et al. resulted in the original MB model. Experimental observation using monodisperse MBs was able to accurately estimate the temporal changes in MB radius during 0.5 MHz ultrasonic excitation

An arbitrary high-order discontinuous Galerkin method with local time-stepping for linear acoustic wave propagation.

This paper presents a numerical scheme of arbitrary order of accuracy in both space and time, based on the arbitrary high-order derivatives methodology, for transient acoustic simulations. The scheme combines the nodal discontinuous Galerkin method for the spatial discretization and the Taylor series integrator (TSI) for the time integration. The main idea of the TSI is a temporal Taylor series expansion of all unknown acoustic variables in which the time derivatives are replaced by spatial derivatives via the Cauchy-Kovalewski procedure. The computational cost for the time integration is linearly proportional to the order of accuracy. To increase the computational efficiency for simulations involving strongly varying mesh sizes or material properties, a local time-stepping (LTS) algorithm accompanying the arbitrary high-order derivatives discontinuous Galerkin (ADER-DG) scheme, which ensures correct communications between domains with different time step sizes, is proposed. A numerical stability analysis in terms of the maximum allowable time step sizes is performed. Based on numerical convergence analysis, we demonstrate that for nonuniform meshes, a consistent high-order accuracy in space and time is achieved using ADER-DG with LTS. An application to the sound propagation across a transmissive noise barrier validates the potential of the proposed method for practical problems demanding high accuracy.

L^{p}-L^{q}-Maximal regularity of the Van Wijngaarden\u2013Eringen equation in a cylindrical domain

We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

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.

Numerical analysis of the lattice Boltzmann method for simulation of linear acoustic waves.

We analyze a linear lattice Boltzmann (LB) formulation for simulation of linear acoustic wave propagation in heterogeneous media. We employ the single-relaxation-time Bhatnagar-Gross-Krook as well as the general multirelaxation-time collision operators. By calculating the dispersion relation for various 2D lattices, we show that the D2Q5 lattice is the most suitable model for the linear acoustic problem. We also implement a grid-refinement algorithm for the LB scheme to simulate waves propagating in a heterogeneous medium with velocity contrasts. Our results show that the LB scheme performance is comparable to the classical second-order finite-difference schemes. Given its efficiency for parallel computation, the LB method can be a cost effective tool for the simulation of linear acoustic waves in complex geometries and multiphase media.

APSARA: A multi-dimensional unsplit fourth-order explicit Eulerian hydrodynamics code for arbitrary curvilinear grids

We present a new fourth-order, finite-volume hydrodynamics code named Apsara. The code employs a high-order, finite-volume method for mapped coordinates with extensions for nonlinear hyperbolic conservation laws. Apsara can handle arbitrary structured curvilinear meshes in three spatial dimensions. The code has successfully passed several hydrodynamic test problems, including the advection of a Gaussian density profile and a nonlinear vortex and the propagation of linear acoustic waves. For these test problems, Apsara produces fourth-order accurate results in case of smooth grid mappings. The order of accuracy is reduced to first-order when using the nonsmooth circular grid mapping. When applying the high-order method to simulations of low-Mach number flows, for example, the Gresho vortex and the Taylor-Green vortex, we discover that Apsara delivers superior results to codes based on the dimensionally split, piecewise parabolic method (PPM) widely used in astrophysics. Hence, Apsara is a suitable tool for simulating highly subsonic flows in astrophysics. In the first astrophysical application, we perform implicit large eddy simulations (ILES) of anisotropic turbulence in the context of core collapse supernova (CCSN) and obtain results similar to those previously reported.

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.

Dustybox and dustywave: two test problems for numerical simulations of two-fluid astrophysical dust-gas mixtures

In this paper we present the analytic solutions for two test problems involving two-fluid mixtures of dust and gas in an astrophysical context. The solutions provide a means of benchmarking numerical codes designed to simulate the non-linear dynamics of dusty gas. The first problem, dustybox, consists of two interpenetrating homogeneous fluids moving with relative velocity difference. We provide exact solutions to the full non-linear problem for a range of drag formulations appropriate to astrophysical fluids (i.e., various prescriptions for Epstein and Stokes drag in different regimes). The second problem, dustywave consists of the propagation of linear acoustic waves in a two-fluid gas-dust mixture. We provide the analytic solution for the case when the two fluids are interacting via a linear drag term. Both test problems are simple to set up in any numerical code and can be run with periodic boundary conditions. The solutions we derive are completely general with respect to both the dust-to-gas ratio and the amplitude of the drag coefficient. A stability analysis of waves in a gas-dust system is also presented, showing that sound waves in an astrophysical dust-gas mixture are linearly stable.

Analytical Solution for Acoustic Waves Propagation in Fluids

This paper presents a mathematical model of linear acoustic wave propagation in fluids. The benefits of a mathematical model over a normal mode analysis are first discussed, then the mathematical model for acoustic propagation in the test medium is developed using computer simulations. The approach is based on a analytical solution to the homogeneous wave equation for fluid medium. A good agreement between the computational presented results with published data.

A symmetric weak form of Biot's equations based on redundant variables representing the fluid, using a Helmholtz decomposition of the fluid displacement vector field

AbstractA novel symmetric weak formulation of Biot's equations for linear acoustic wave propagation in layered poroelastic media is presented. The primary variables used are the frame displacement, the acoustic pore pressure, the scalar potential and the vector potential obtained from a Helmholtz decomposition of the fluid displacement. Also a symmetric weak form based on the frame displacement, the pore pressure and the fluid displacement is obtained as an intermediate result. hp finite element simulations of a double leaf partition based on this new weak formulation is verified against simulation results from the classical frame displacement, fluid displacement formulation and a frame displacement pore pressure formulation. All three formulations simulated, displays the same rate of convergence with respect to finite element bases polynomial degree. The novel formulation also extends a previously published frame displacement, pore pressure, scalar fluid displacement potential formulation with an implicit irrotational fluid displacement assumption to a full representation of Biot's equations. Copyright © 2010 John Wiley & Sons, Ltd.

Global Acoustic Resonance in a Stratified Solar Atmosphere

The upward propagation of linear acoustic waves in a gravitationally stratified solar atmosphere is studied. The wave motion is governed by the Klein – Gordon equation, which contains a cutoff frequency introduced by stratification. The acoustic cutoff may act as a potential barrier when the temperature decreases with height. It is shown that waves trapped below the barrier could be subject to a resonance that extends into the entire unbounded atmosphere of the Sun. The parameter space characterizing the resonance is explored.

Resonant acoustic waves in a stratified atmosphere

AbstractThe upward propagation of linear acoustic waves in a gravitationally stratified atmosphere is studied. The wave motion is governed by the Klein-Gordon equation which contains a cut-off frequency introduced by stratification. The acoustic cut-off may act as a potential barrier when the temperature decreases with height. It is shown that waves trapped below the barrier could be subject to a resonance which extends into the entire unbounded atmosphere. The parameter space characterizing the resonance is explored.

Calcul de la propagation acoustique en milieux non homogènes infinis par la DRBEM

We present here an extension of the boundary element method, the DRBEM (Dual Reciprocity Boundary Element Method), in order to solve the Helmholtz equation governing the propagation of linear acoustic waves in an unbounded variable mean temperature medium at rest. The originality of the method is that it allows discretization and integration only on the boundary of the domain, without an inner grid. The method is used here to study the wave propagation inside a non-isothermal flanged cavity, and throughout a thermal flume without mean velocity expending in a medium of constant outer temperature.

Zonal Approach for Prediction of Jet Noise

A zonal approach for direct computation of sound generation and propagation from a supersonic jet is investigated. The present work splits the computational domain into a nonlinear, acoustic-source regime and a linear acoustic wave propagation regime. In the nonlinear regime, the unsteady flow is governed by the large-scale equations, which are the filtered compressible Navier-Stokes equations. In the linear acoustic regime, the sound wave propagation is described by the linearized Euler equations. Computational results are presented for a supersonic jet at M = 2. 1. It is demonstrated that no spurious modes are generated in the matching region and the computational expense is reduced substantially as opposed to fully large-scale simulation.

Transient wave propagation through bubbly layers via the Foldy–Twersky integral equation

This work considers transient linear acoustic wave propagation through a layer of small gas bubbles in a liquid. The analysis is based on multiple scattering theory and the Foldy–Twersky integral equation is used. The transmitted coherent wave for a distribution of uncorrelated scattering bubbles is studied. The case of a weak scattering density is investigated for the regimes of single scattering and Foldy–Twersky multiple scattering. A condition for the column density for the single scattering regime is derived and found to be more restrictive than earlier heuristically derived results. The influence of the low-frequency region of the monopole part of the scattering amplitude of a spherical gas bubble is analyzed. It is shown that a combination of the expression for the wave number for the transmitted coherent wave and the formula used for the low-frequency region of the monopole coefficient yield a well-posed initial-value problem. The effects due to single scattering are shown to be negligible so a pulse propagates essentially undisturbed. For the case of Foldy–Twersky scattering, however, the influence of the bubbles can be substantial. The main characteristics of linear transient wave propagation in bubbly liquids experimentally observed by Kuznetsov et al. are qualitatively predicted by the theory used in this paper.

Acoustic pulse propagation in Maxwell fluids

Transient linear acoustic wave propagation in fluids exhibiting Maxwell relaxation processes is investigated theoretically. The case of a single relaxation proces is studied in detail. The inverse problem concerning the estimation of the governing parameters of Maxwell fluids on the basis of measured transients is analyzed. First, a method based on numerical Fourier transformation is discussed. Then, it is shown how the governing parameters can be estimated via certain moments of a distorted pulse. The above two methods involve formally the entire pressure–time history of a distorted pulse. Since part of it can also be influenced by other effects than dissipative it is of interest to establish complementary methods. The full solution to the distortion problem is derived and the governing parameters can be estimated by means of, for example, least‐mean‐square approximation. This method requires only that a finite part of a pulse be solely influenced by dissipation. However, it results in two overdetermined nonlinear equations and a first‐order approximation is needed. By evaluation of the time derivatives of the pressure signature at the wave front it is shown that a first‐order approximation can be obtained.

Resonances in periodic media

AbstractWe study the propagation of linear acoustic waves (a) in an infinite string with a periodic material distribution, (b) in an infinite cylinder with a meterial distribution that is periodic in the longitudinal direction and does not depend on the transverse coordinates. We assume that the wave field is generated by a time‐harmonic force distribution of frequency ω acting in a compact set. We show in both cases that resonances of order t1/2 occur for a discrete set of frequencies and that the solution is bounded as t→∞ for the remaining frequencies. In case (a) ω is a resonance frequency if and only if ω2 is a boundary point of one of the spectral bands of the corresponding spatial differential operator of Hill's type. A similar characterization of the resonance frequencies is given in case (b).

On the fundamental solution of an equation describing the propagation of longitudinal waves in a dispersive medium

We study the fundamental solution of the equation ∂ ∂t ( ∂ 2ρ ∂t 2 − Δρ) + ∂ 2ρ ∂t 2 − α Δρ=0 , 0< α<1, which describes the propagation of linear acoustic waves in a medium with dispersion. The fundamental solution is written in integral form. Its asymptotic expansion is obtained as t→+∞, δ⩽¦x¦/ t⩽1−gd, where δ>0 are small but fixed numbers.

Generation of acoustic waves by an impulsive point source in a fluid/porous‐medium configuration with a plane boundary

The space‐time acoustic wave motion generated by an impulsive monopole point source in a fluid/porous‐medium configuration with a plane boundary is calculated with the aid of the modified Cagniard technique. To describe the macroscopic propagation of linear acoustic waves through the fluid‐saturated porous solid, four coupled partial differential equations are used that follow upon applying a volume averaging procedure to the two constituents, a perfectly elastic solid and an ideal fluid of the porous medium. The source is located in the fluid part of the configuration. Numerical results are presented for the reflected‐wave acoustic pressure, especially in those regions of space where head waves occur. There is a marked difference in time response in the different regimes that exist for the wave speed in the fluid in relation to the different wave speeds in the porous medium (fast compressional, slow compressional, shear). These differences are of importance to the situation where the reflected wave in the fluid is used to determine experimentally the acoustic properties of the porous medium.

Numerical calculations of ultrasonic fields I: Transducer near fields

A computer code for the calculation of linear acoustic wave propagation in homogeneous fluid and solid materials has been derived from the thermal-hydraulics code STEALTH. The code uses finite-difference techniques in a two-dimensional mesh made up of arbitrarily shaped quadrilaterals. Problems with two-dimensional plane strain or two-dimensional axial symmetries can be solved. Free, fixed, or stressed boundaries can be used. Transducers can be modeled by time dependent boundary conditions or by moving pistons. This paper gives a brief description of the method and shows the results of the calculation of the near fields of circular flat and focused transducers. These results agree with analytic theory along the axis of symmetry and with other codes that use a Huygens reconstruction technique off-axis.