Frequency-domain wave equation and its time-domain solutions in attenuating media

Our purpose in this paper is to describe the wave propagation in media whose attenuation obeys a frequency power law. To achieve this, a frequency-domain wave equation was developed using previously derived causal dispersion relations. An inverse space and time Fourier transform of the solution to this algebraic equation results in a time-domain solution. It is shown that this solution satisfies the convolutional time-domain wave equation proposed by Szabo [J. Acoust. Soc. Am. 96, 491-500 (1994)]. The form of the convolutional loss operator contained in this wave equation is obtained. Solutions representing the propagation of both plane sinusoidal and transient waves propagating in media with specific power law attenuation coefficients are investigated as special cases of our solution. Using our solution, comparisons are made for transient one-dimensional propagation in a medium whose attenuation is proportional to frequency with recently obtained numerical solutions of Szabo's equation. These show good agreement.     

Full wave modeling of therapeutic ultrasound: efficient time-domain implementation of the frequency power-law attenuation

For the simulation of therapeutic ultrasound applications, a method including frequency-dependent attenuation effects directly in the time domain is highly desirable. This paper describes an efficient numerical time-domain implementation of the power-law attenuation model presented by Szabo [Szabo, J. Acoust. Soc. Am. 96, 491-500 (1994)]. Simulations of therapeutic ultrasound applications are feasible in conjunction with a previously presented finite differences time-domain (FDTD) algorithm for nonlinear ultrasound propagation [Ginter et al., J. Acoust. Soc. Am. 111, 2049-2059 (2002)]. Szabo implemented the empirical frequency power-law attenuation using a causal convolutional operator directly in the time-domain equation. Though a variety of time-domain models has been published in recent years, no efficient numerical implementation has been presented so far for frequency power-law attenuation models. Solving a convolutional integral with standard time-domain techniques requires enormous computational effort and therefore often limits the application of such models to 1D problems. In contrast, the presented method is based on a recursive algorithm and requires only three time levels and a few auxiliary data to approximate the convolutional integral with high accuracy. The simulation results are validated by comparison with analytical solutions and measurements.     

Modified Szabo's wave equation models for lossy media obeying frequency power law

Szabo's models of acoustic attenuation [Szabo, J. Acoust. Soc. Am. 96(1), 491-500 (1994)] comply well with the empirical frequency power law involving noninteger and odd-integer exponent coefficients while guaranteeing causality, but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions. The purpose of this paper is to ease or remove these drawbacks of the Szabo's models via the Caputo fractional derivative concept. The positive time-fractional derivative is also introduced to include the positivity of the attenuation processes.     

A comparison of broadband models for sand sediments

Chotiros and Isakson [J. Acoust. Soc. Am. 116(4), 2011-2022 (2004)] recently proposed an extension of the Biot-Stoll model for poroelastic sediments that makes predictions for compressional wave speed and attenuation, which are in much better accord with the experimental measurements of these quantities extant in the literature than either those of the conventional Biot-Stoll model or the rival model of Buckingham [J. Acoust. Soc. Am. 108(6), 2796-2815 (2000)]. Using a local minimizer, the Nelder-Mead simplex method, it is shown that there are generally at least two choices of the Chotiros-Isakson parameters which produce good agreement with experimental measurements. Since one postulate of the Chotiros-Isakson model is that, due to the presence of air bubbles in the pore space, the pore fluid compressibility is greater than that of water, an alternative model based on a conjecture by Biot [J. Acoust. Soc. Am. 34(5), 1254-1264 (1962)], air bubble resonance, is considered. While this model does as well or better than the Chotiros-Isakson model in predicting measured values of wave speed and attenuation, the Rayleigh-Plesset theory of bubble oscillation casts doubt on its plausibility as a general explanation of large dispersion of velocity with respect to frequency.     

Modeling of nonlinear ultrasound propagation in tissue from array transducers

A computationally efficient model capable of simulating finite-amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described. It is based on a second-order operator splitting approach [Tavakkoli et al., J. Acoust. Soc. Am. 104, 2061-2072 (1998)], with a fractional step-marching scheme, whereby the effects of diffraction, attenuation, and nonlinearity can be computed independently over incremental steps. This approach is an extension to that of Christopher and Parker [J. Acoust. Soc. Am. 90, 507-521; 90, 488-499 (1991)], wherein linear and nonlinear effects are propagated separately over incremental steps, and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources. Results of the model are compared with published data. Predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented.     

Optimal focusing by spatio-temporal inverse filter. I. Basic principles

A focusing technique based on the inversion of the propagation operator relating an array of transducers to a set of control points inside a medium was proposed in previous work [Tanter et al., J. Acoust. Soc. Am. 108, 223-234 (2000)] and is extended here to the time domain. As the inversion of the propagation operator is achieved both in space and time, this technique allows calculation of the set of temporal signals to be emitted by each element of the array in order to optimally focus on a chosen control point. This broadband inversion process takes advantage of the singular-value decomposition of the propagation operator in the Fourier domain. The physical meaning of this decomposition is explained in a homogeneous medium. In particular, a definition of the number of degrees of freedom necessary to define the acoustic field generated by an array of limited aperture in a focal plane of limited extent is given. This number corresponds to the number of independent signals that can be created in the focal area both in space and time. In this paper, this broadband inverse-focusing technique is compared in homogeneous media with the classical focusing achieved by simple geometrical considerations but also with time-reversal focusing. It is shown that, even in a simple medium, slight differences appear between these three focusing strategies. In the companion paper [Aubry et al., J. Acoust. Soc. Am. 110, 48-58 (2001)] the three focusing techniques are compared in heterogeneous, absorbing, or complex media where classical focusing is strongly degraded. The strong improvement achieved by the spatio-temporal inverse-filter technique emphasizes the great potential of multiple-channel systems having the ability to apply completely different signal waveforms on each transducer of the array. The application of this focusing technique could be of great interest in various ultrasonic fields such as medical imaging, nondestructive testing, and underwater acoustics.     

An iterative effective medium approximation (IEMA) for wave dispersion and attenuation predictions in particulate composites, suspensions and emulsions

In the present work we deal with the scattering dispersion and attenuation of elastic waves in different types of nonhomogeneous media. The iterative effective medium approximation based on a single scattering consideration, for the estimation of wave dispersion and attenuation, proposed in Tsinopoulos et al., [Adv. Compos. Lett. 9, 193-200 (2000)] is examined herein not only for solid components but for liquid suspensions as well. The iterations are conducted by means of the classical relation of Waterman and Truell, while the self-consistent condition proposed by Kim et al. [J. Acoust. Soc. Am. 97, 1380-1388 (1995)] is used for the convergence of the iterative procedure. The single scattering problem is solved using the Ying and Truell formulation, which with a minor modification can accommodate the solution of scattering on inclusions in liquid. Theoretical results for several different systems of particulates and suspensions are presented being in excellent agreement with experimental data taken from the literature.     

Approximate expressions for viscous attenuation in marine sediments: relating Biot's "critical" and "peak" frequencies

Simple approximate relations are proposed for the viscous attenuation per cycle of the fast compressional and shear waves in the low-to-intermediate frequency range. Corresponding closed-form formulas are derived for frequencies at which maximum viscous attenuation per cycle occurs according to the Biot-Stoll theory of elastic wave propagation in marine sediments. In the new formulas, Biot's approximation [M. A. Biot, J. Acoust. Soc. Am. 34, 1254-1264 (1962)] for the frequency-dependent viscosity correction factor F(f) and the assumption of relatively low specific loss (Q(-1)<(0.2) [J. Geertsma and D. C. Smith, Geophysics 26(2), 169-181 (1962)] are used to provide an accurate representation of the fast compressional and shear wave attenuation from low frequencies through a transition region extending to two or three times Biot's critical frequency f(c). The approximate viscodynamic behavior of marine sediments for the fast compressional and shear waves shows similarities to that of a "homogeneous relaxation" process for an anelastic linear element [A. M. Freudenthal and H. Geiringer, Encyclopedia of Physics (Springer-Verlag. 1958), Vol. 6].     

Propagation-invariant classification of sounds in channels with dispersion and absorption

In a previous paper [G. Okopal et al., J. Acoust. Soc. Am. 123, 832-841 (2008)], a method to obtain features of a wave that are unaffected by dispersion, per mode, was developed for improving classification of underwater sounds (e.g., sonar backscatter). The current paper builds on this work and presents additional contributions. First, it is shown that the dispersion-invariant moments developed previously are not invariant to frequency-dependent attenuation (absorption); consequently, their classification performance degrades in such channels. Second, a feature extraction method is developed to obtain features that are invariant to dispersion, and to two forms of absorption (known a priori): namely, absorption that yields spectral magnitude attenuation (in dB) that is linear with frequency, and linear with log-frequency. Third, the relationship of these absorption- and dispersion-invariant moment (ADIM) features to the cepstrum of the wave is examined, and it is shown that cepstral moments are also invariant to dispersion, and to the first form of absorption for odd-order moments. Finally, simulations are conducted to illustrate the performance of the ADIMs and cepstral moments on classifying the backscatter from steel shells in a dispersive channel with absorption. Receiver operator characteristic curves quantify the superior discriminability of the ADIMs and cepstral moments compared to ordinary moments.     

A new formalism for time-dependent wave scattering from a bounded obstacle

A time-dependent three-dimensional acoustic scattering problem is considered. An incoming wave packet is scattered by a bounded, simply connected obstacle with locally Lipschitz boundary. The obstacle is assumed to have a constant boundary acoustic impedance. The limit cases of acoustically soft and acoustically hard obstacles are considered. The scattered acoustic field is the solution of an exterior problem for the wave equation. A new numerical method to compute the scattered acoustic field is proposed. This numerical method obtains the time-dependent scattered field as a superposition of time-harmonic acoustic waves and computes the time-harmonic acoustic waves by a new "operator expansion method." That is, the time-harmonic acoustic waves are solutions of an exterior boundary value problem for the Helmholtz equation. The method used to compute the time-harmonic waves improves on the method proposed by Misici, Pacelli, and Zirilli [J. Acoust. Soc. Am. 103, 106-113 (1998)] and is based on a "perturbative series" of the type of the one proposed in the operator expansion method by Milder [J. Acoust. Soc. Am. 89, 529-541 (1991)]. Computationally, the method is highly parallelizable with respect to time and space variables. Some numerical experiments on test problems obtained with a parallel implementation of the numerical method proposed are shown and discussed from the numerical and the physical point of view. The website: http://www.econ.unian.it/recchioni/w1 shows four animations relative to the numerical experiments.     

On boundary conditions for the diffusion equation in room-acoustic prediction: Theory, simulations, and experiments

This paper proposes a modified boundary condition to improve the room-acoustic prediction accuracy of a diffusion equation model. Previous boundary conditions for the diffusion equation model have certain limitations which restrict its application to a certain number of room types. The boundary condition employing the Sabine absorption coefficient [V. Valeau et al., J. Acoust. Soc. Am. 119, 1504-1513 (2006)] cannot predict the sound field well when the absorption coefficient is high, while the boundary condition employing the Eyring absorption coefficient [Y. Jing and N. Xiang, J. Acoust. Soc. Am. 121, 3284-3287 (2007); A. Billon et al., Appl. Acoust. 69, (2008)] has a singularity whenever any surface material has an absorption coefficient of 1.0. The modified boundary condition is derived based on an analogy between sound propagation and light propagation. Simulated and experimental data are compared to verify the modified boundary condition in terms of room-acoustic parameter prediction. The results of this comparison suggest that the modified boundary condition is valid for a range of absorption coefficient values and successfully eliminates the singularity problem.     

Compilation of empirical ultrasonic properties of mammalian tissues. II

The compilation of the literature on ultrasonic propagation properties of mammalian tissues [J. Acoust. Soc. Am. 64, 423 (1978)] is continued with the addition of 45 papers yielding over 700 lines of parametric data.     

Propagation of acoustic wave in viscoelastic medium permeated with air bubbles

Based on the modification of the radial pulsation equation of an individual bubble, an effective medium method (EMM) is presented for studying propagation of linear and nonlinear longitudinal acoustic waves in viscoelastic medium permeated with air bubbles. A classical theory developed previously by Gaunaurd (Gaunaurd GC and \"{U}berall H, {\em J. Acoust. Soc. Am}., 1978; 63: 1699--1711) is employed to verify the EMM under linear approximation by comparing the dynamic (i.e. frequency-dependent) effective parameters, and an excellent agreement is obtained. The propagation of longitudinal waves is hereby studied in detail. The results illustrate that the nonlinear pulsation of bubbles serves as the source of second harmonic wave and the sound energy has the tendency to be transferred to second harmonic wave. Therefore the sound attenuation and acoustic nonlinearity of the viscoelastic matrix are remarkably enhanced due to the system's resonance induced by the existence of bubbles.     

Stability of self-similar plane shocks with Hertzian nonlinearity

The nonlinear progressive wave equation [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406-1417 (1987)] is expressed in a form to accommodate Hertzian nonlinearity of order n=3/2 typical of granular media. Stability of self-similar plane weak shocks against perturbations in three dimensions is demonstrated for arbitrary order nonlinearity with 1 相似文献   

Nonlinear surface waves in soft, weakly compressible elastic media

Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.     

Unification and extension of monolithic state space and iterative cochlear models

Time domain cochlear models have primarily followed a method introduced by Allen and Sondhi [J. Acoust. Soc. Am. 66, 123-132 (1979)]. Recently the "state space formalism" proposed by Elliott et al. [J. Acoust. Soc. Am. 122, 2759-2771 (2007)] has been used to simulate a wide range of nonlinear cochlear models. It used a one-dimensional approach that is extended to two dimensions in this paper, using the finite element method. The recently developed "state space formalism" in fact shares a close relationship to the earlier approach. Working from Diependaal et al. [J. Acoust. Soc. Am. 82, 1655-1666 (1987)] the two approaches are compared and the relationship formalized. Understanding this relationship allows models to be converted from one to the other in order to utilize each of their strengths. A second method to derive the state space matrices required for the "state space formalism" is also presented. This method offers improved numerical properties because it uses the information available about the model more effectively. Numerical results support the claims regarding fluid dimension and the underlying similarity of the two approaches. Finally, the recent advances in the state space formalism [Bertaccini and Sisto, J. Comp. Phys. 230, 2575-2587 (2011)] are discussed in terms of this relationship.     

On the interpretability of speech/nonspeech comparisons: a reply to Fowler

Fowler [J. Acoust. Soc. Am. 88, 1236-1249 (1990)] makes a set of claims on the basis of which she denies the general interpretability of experiments that compare the perception of speech sounds to the perception of acoustically analogous nonspeech sound. She also challenges a specific auditory hypothesis offered by Diehl and Walsh [J. Acoust. Soc. Am. 85, 2154-2164 (1989)] to explain the stimulus-length effect in the perception of stops and glides. It will be argued that her conclusions are unwarranted.     

Rapid computation of acoustic impulse scattering for rough penetrable surfaces

Chu’s theory for the impulse response of a point source to an isovelocity density contrast wedge [Chu D. Impulse response of density contrast wedge using normal coordinates. J Acoust Soc Am 1989;86:1883-96] enables wedge-assemblage rough surface scattering models to be extended to a broad range of penetrable seafloors, but is computationally intensive since it necessitates finding the multifold roots of a characteristic eigenvalue equation, and summing a power series, for each wedge apex. This present work considers the properties and relationships of the direct, reflected, and diffracted field components of a density contrast wedge. In particular, an analysis of the physical origin and behavior of diffractions associated with specular reflections of the source in the wedge faces leads to a simple extension of the Biot-Tolstoy theory [Biot MA, Tolstoy I. Formulation of wave propagation in infinite media by normal coordinates with an application to diffraction. J Acoust Soc Am 1957;29:381-91] to density contrast wedges with reflectivity ∣R∣ < 1, for wedge angles within the range 150 ? θ_w ? 210°, where the diffractions are predominantly associated with a single reflection in each wedge face. This facilitates rapid time domain calculations of acoustic bottom scattering and penetration for complex multilayered seafloors.