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.     

Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation

Finite-difference, time-domain (FDTD) calculations are typically performed with partial differential equations that are first order in time. Equation sets appropriate for FDTD calculations in a moving inhomogeneous medium (with an emphasis on the atmosphere) are derived and discussed in this paper. Two candidate equation sets, both derived from linearized equations of fluid dynamics, are proposed. The first, which contains three coupled equations for the sound pressure, vector acoustic velocity, and acoustic density, is obtained without any approximations. The second, which contains two coupled equations for the sound pressure and vector acoustic velocity, is derived by ignoring terms proportional to the divergence of the medium velocity and the gradient of the ambient pressure. It is shown that the second set has the same or a wider range of applicability than equations for the sound pressure that have been previously used for analytical and numerical studies of sound propagation in a moving atmosphere. Practical FDTD implementation of the second set of equations is discussed. Results show good agreement with theoretical predictions of the sound pressure due to a point monochromatic source in a uniform, high Mach number flow and with Fast Field Program calculations of sound propagation in a stratified moving atmosphere.     

Efficient calculation of two-dimensional periodic and waveguide acoustic Green's functions

New representations and efficient calculation methods are derived for the problem of propagation from an infinite regularly spaced array of coherent line sources above a homogeneous impedance plane, and for the Green's function for sound propagation in the canyon formed by two infinitely high, parallel rigid or sound soft walls and an impedance ground surface. The infinite sum of source contributions is replaced by a finite sum and the remainder is expressed as a Laplace-type integral. A pole subtraction technique is used to remove poles in the integrand which lie near the path of integration, obtaining a smooth integrand, more suitable for numerical integration, and a specific numerical integration method is proposed. Numerical experiments show highly accurate results across the frequency spectrum for a range of ground surface types. It is expected that the methods proposed will prove useful in boundary element modeling of noise propagation in canyon streets and in ducts, and for problems of scattering by periodic surfaces.     

水平变化双层波导声传播问题的耦合简正波解

为了验证现有模型的精度,导出了全反射下边界双层波导中简正波耦合矩阵的解析表达式,并将其应用到全局矩阵耦合简正波模型(Direct Global Matrix Coupled-Mode)中,使得该模型可以提供水平变化双层波导问题的标准解。文中首先利用COUPLE的简正波及耦合矩阵数值解验证了该简正波及耦合矩阵解析表达式的正确性;其次,采用改进的DGMCM模型求解了双层波导海山声传播损失,结果表明,改进后的DGMCM模型可以非常精确地求解水平变化双层波导问题,可作为求解此类问题的标准模型使用。     

Simulation of outdoor sound propagation with a transmission line matrix method

The present paper focusses on the application of the transmission line matrix method (TLM) for outdoor sound propagation simulations. It is shown how the reflection at the ground and vertical sound speed gradients can be taken into account. The comparison with calculations using finite differences in the time domain (FDTD), with calculations using parabolic equation and with analytical solutions showed all very good agreement. Thereby the computational effort of TLM is significantly lower compared to common FDTD calculation schemes.     

Traffic noise prediction with the parabolic equation method: validation of a split-step Padé approach in complex environments

This study deals with sound propagation in typical traffic noise conditions. The numerical results are obtained through the split-step Padé method and the discrete random Fourier modes technique. These are first evaluated qualitatively, by color contour maps showing noise propagation, diffraction by an impedance discontinuity or a screen edge, and scattering by atmospheric turbulence. Next, our numerical results are quantitatively validated by comparison with analytical models and other parabolic equation models. For all the atmospheric conditions and geometrical configurations available in literature, the agreement between the different methods is very good, except for some cases involving the atmospheric turbulence. However, in those particular cases, the split-step Padé results are shown to be more consistent with physical theory. Finally, our method seems to be very powerful and reliable for traffic noise prediction.     

Numerical simulation of nonlinear propagation of sound waves in a finite horn

Nonlinear acoustic propagation generated by a piston in a finite horn is numerically studied.A quasi-one-dimensional nonlinear model with varying cross-section uses high-order low-dispersion numerical schemes to solve the governing equation.Because of the nonlinear wave distortion and reflected sound waves at the mouth,broadband time-domain impedance boundary conditions are employed.The impedance approximation can be optimized to identify the complex-conjugate pole-residue pairs of the impedance functions,which can be calculated by fast and efficient recursive convolution.The numerical results agree very well with experimental data in the situations of weak nonlinear wave propagation in an exponential horn,it is shown that the model can describe the broadband characteristics caused by nonlinear distortion.Moreover,finite-amplitude acoustic propagation in types of horns is simulated,including hyperbolic,conical,exponential and sinusoidal horns.It is found that sound pressure levels at the horn mouth are strongly affected by the horn profiles,the driving velocity and frequency of the piston.The paper also discusses the influence of the horn geometry on nonlinear waveform distortion.     

An improved acoustical wave propagator method and its application to a duct structure

The pseudospectral time-domain method has long been used to describe the acoustical wave propagation. However, due to the limitation and difficulties of the fast Fourier transform (FFT) in dealing with nonperiodic problems, the dispersion error is inevitable and the numerical accuracy greatly decreases after the waves arrive at the boundary. To resolve this problem, the Lagrange-Chebyshev interpolation polynomials were used to replace the previous FFT, which, however, brings in an additional restriction on the time step. In this paper, a mapped Chebyshev method is introduced, providing the dual benefit of preserving the spectral accuracy and overcoming the time step restriction at the same time. Three main issues are addressed to assess the proposed technique: (a) Spatial derivatives in the system operator and the boundary treatment; (b) parameter selections; and (c) the maximum time step in the temporal operator. Furthermore, a numerical example involving the time-domain evolution of wave propagation in a duct structure is carried out, with comparisons to those obtained by Euler method, the fourth-order Runge-Kutta method, and the exact analytical solution, to demonstrate the numerical performance of the proposed technique.     

Diffraction of sound from a dipole source near to a barrier or an impedance discontinuity

Pierce's formulation for the diffraction of spherical waves by a hard wedge has been extended to the case of the sound field due to a dipole source. The same approach is also used to extend a semiempirical model for sound propagation above an impedance discontinuity due to a dipole source. The resulting formulas have been validated by comparing their numerical solutions with that computed by summing the sound fields due to two closely spaced monopole sources of equal magnitude but opposite in phase. These new formulations are then used to develop a simple model for calculating the dipole sound field diffracted by a barrier above an impedance ground. Applications of these models relate to transportation noise prediction, particularly railway noise abatement, for which dipole sources are commonly used. The numerical predictions have been found to compare reasonably well with indoor measurements using piezoceramic transducers as dipole sources.     

A Chebyshev–Lagrangian method for acoustic analysis of a rectangular cavity with arbitrary impedance walls

A general Chebyshev–Lagrangian method is proposed to obtain the analytical solution for a rectangular acoustic cavity with arbitrary impedance boundary conditions. The originality of the present paper is the successful attempt of applying orthogonal polynomials, such as Chebyshev polynomials of the first kind, to the analysis of a rectangular sound field with general wall impedance. The sound pressure is uniformly expressed as triplicate Chebyshev polynomial series which is independent in each direction. The Chebyshev polynomial series solution is obtained using the Rayleigh–Ritz procedure after considering the influence of boundary impedance on the cavity as the work done by the impedance surfaces in the Lagrangian function. The accuracy and reliability of the proposed method are validated against the analytical solutions and some numerical results available in the literature. Excellent orthogonality and complete properties of the Chebyshev polynomials ensure the rapid convergence, numerical stability, high accuracy of the current solution. The simplicity and low computational cost of the present approach make it preferable to obtain the results of complex models even in the relative high frequency range by choosing enough truncated terms in the sound pressure expression. Numerous cases with various uniform or non-uniform impedance boundary conditions are analyzed numerically and some of the results can be used as benchmark. It is shown that the impedance boundary condition can effectively influence or modify the acoustic characteristics and response of a cavity.     

The extended Fourier pseudospectral time-domain method for atmospheric sound propagation

An extended Fourier pseudospectral time-domain (PSTD) method is presented to model atmospheric sound propagation by solving the linearized Euler equations. In this method, evaluation of spatial derivatives is based on an eigenfunction expansion. Evaluation on a spatial grid requires only two spatial points per wavelength. Time iteration is done using a low-storage optimized six-stage Runge-Kutta method. This method is applied to two-dimensional non-moving media models, one with screens and one for an urban canyon, with generally high accuracy in both amplitude and phase. For a moving atmosphere, accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen. The method has also been validated for three-dimensional sound propagation over a screen. For that application, the developed method is in the order of 100 times faster than the second-order-accurate FDTD solution to the linearized Euler equations. The method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated.     

Predicted attenuation of sound in a rigid-porous ground from an airborne source

An approximate analytical formula has been derived for the prediction of sound fields in a semi-infinite rigid-porous ground due to an airborne source. The method starts by expressing the sound fields in an integral form, which can subsequently be evaluated by the method of steepest descents. The concept of effective impedance has been introduced by using a physically plausible assumption. The integral can then be simplified and evaluated analytically. The analytical solution can be expressed in a closed form analogous to the classical Weyl-Van der Pol formula that has been used for predicting sound fields above a rigid-porous ground. Extensive comparisons with the wave-based numerical solutions according to the fast field formulation and the direct evaluation of the integral have been conducted. It has been demonstrated that the analytical formula is sufficiently accurate to predict the penetration of sound into a wide range of outdoor ground surfaces.     

Nonlinear TLM modelling of high-frequency varactor multipliers and halverstm

A numerical modelling procedure in the time-domain of microstrip circuits containing non-linear devices such as varactor multipliers and halvers ^tm , is described. The two-dimensional Transmission Line Matrix (TLM) method has been extended to model elements with voltage-dependent capacitance (varactors). Waveforms and spectra computed with the TLM model compare well with measured results.     

Air-ground interaction in long range propagation of low frequency sound and vibration—field tests and model verification

An extensive program of intermediate and long range impulsive sound propagation field tests have been conducted. The test program and the performed measurements are presented. Particular focus is given on the air-ground interaction and its effect on low frequency sound and vibration propagation. It is found that the pressure wave interaction with the viscoelastic Rayleigh wave in the ground may have a significant effect on the ground impedance and the sound and vibration propagation. This introduces an important mechanism not covered in commonly used ground impedance models. Numerical simulation models have been developed and verified against the test data. The ground impedance does not only effect the sound pressure propagation. If either acoustically induced ground vibration, or ground to building transmitted vibration, is to be considered, the acousto-seismic impedance has a dramatic effect on the level of ground vibration induced by a given sound pressure. For a site where Rayleigh wave interaction appears at the dominant frequencies of the sound pressure, the ground vibration may be greater than a factor 100 (40 dB) than at a site with ground conditions not making the interaction happen.     

The diffraction of sound by an impedance sphere in the vicinity of a ground surface

The problem of sound diffraction by an absorbing sphere due to a monopole point source was investigated. The theoretical models were extended to consider the case of sound diffraction by an absorbing sphere with a locally reacting boundary or an extended reaction boundary placed above an outdoor ground surface of finite impedance. The separation of variables techniques and appropriate wave field expansions were used to derive the analytical solutions. By adopting an image method, the solutions could be formulated to account for the multiple scattering of sound between the sphere and its image near a flat acoustically hard or an impedance ground. The effect of ground on the reflected sound fields was incorporated in the theoretical model by employing an approximate analytical solution known as the Weyl-van der Pol formula. An approximation solution was suggested to determine the scattering coefficients from a set of linearly coupled complex equations for an absorbing sphere not too close to the ground. The approximate method substantially reduced the computational time for calculating the sound field. Preliminary measurements were conducted to characterize the acoustical properties of an absorbing sphere made of open cell polyurethane foam. Subsequent experiments were carried out to demonstrate the validity of the proposed theoretical models for various source/receiver configurations around the sphere above an acoustically hard ground and an impedance ground. Satisfactory comparative results were obtained between the theoretical predictions and experimental data. It was found that the theoretical predictions derived from the approximate solution agreed well with the results obtained by using the exact solutions.     

Application of the Transmission Line Matrix method for outdoor sound propagation modelling – Part 2: Experimental validation using meteorological data derived from the meso-scale model Meso-NH

Outdoor sound prediction is both a societal concern and a scientific issue. This paper deals with numerical simulations of micrometeorological (temperature and wind) fields for environmental acoustics. These simulations are carried out using the reference meso-scale meteorological model at the Meteo-France weather agency (Meso-NH). Meso-NH predictions at very fine scales (up to 3 m), including new developments (drag force approach), are validated both numerically and experimentally under stable, unstable and neutral conditions. Then, this information can be used as input data for the acoustic propagation model. The time-domain acoustic model is based on the Transmission Line Matrix method. Its development has also been promoted for application to outdoor sound propagation, i.e. to take into account topography, ground impedance, meteorological conditions, etc. In part 1, the presentation and evaluation of the Transmission Line Matrix method showed the relevance of this method’s use in the context of environmental acoustics. Finally, simulated noise levels under different propagation conditions were compared to in situ measurements. Satisfactory results were obtained regarding the variability of the observed phenomena.     

Benchmark solutions for sound propagation in an ideal wedge

Sound propagation in a wedge-shaped waveguide with perfectly reflecting boundaries is one of the few range-dependent problems with an analytical solution, and hence provides an ideal benchmark for a full two-way solution to the wave equation. An analytical solution for the sound propagation in an ideal wedge with a pressure-release bottom was presented by Buckingham and Tolstoy [Buckingham and Tolstoy 1990 J. Acoust. Soc. Am. 87 1511]. The ideal wedge problem with a rigid bottom is also of great importance in underwater acoustics. We present an analytical solution to the ideal wedge problem with a perfectly reflecting bottom, either rigid or pressure-release, which may be used to provide a means for investigating the sound field in depth-varying channels, and to establish the accuracy of numerical propagation models. Closed-form expressions for coupling matrices are also provided for the ideal waveguides characterized by a homogeneous water column bounded by perfectly reflecting boundaries. A comparison between the analytical solution and the numerical solution recently proposed by Luo et al. [Luo W Y, Yang C M and Zhang R H 2012 Chin. Phys. Lett. 29 014302] is also presented, through which the accuracy of this numerical model is illustrated.     

Counterpropagation of waves with shock fronts in a nonlinear tissue-like medium

A numerical model for describing the counterpropagation of one-dimensional waves in a nonlinear medium with an arbitrary power law absorption and corresponding dispersion is developed. The model is based on general one-dimensional Navier-Stokes equations with absorption in the form of a time-domain convolution operator in the equation of state. The developed algorithm makes it possible to describe wave interactions in the presence of shock fronts in media like biological tissue. Numerical modeling is conducted by the finite difference method on a staggered grid; absorption and sound speed dispersion are taken into account using the causal memory function. The developed model is used for numerical calculations, which demonstrate the absorption and dispersion effects on nonlinear propagation of differently shaped pulses, as well as their reflection from impedance acoustic boundaries.     

非线性波动方程的新数值迭代方法

提出了一种新的求解非线性波动方程的数值迭代法,它是一种半解析的方法.与完全的数值计算方法扰法相比,它能够考虑各阶谐波的相互作用,且能够满足能量守恒定律.用它研究了非线性声波在液体中的传播性质,结果表明,在微扰法适用的声强范围内迭代法也适用,在微扰法不适用的一个较宽的声强范围内迭代法依然适用.     

A coherent model for predicting noise reduction in long enclosures with impedance discontinuities

A theoretical model has been developed for the prediction of sound propagation in a rectangular long enclosure with impedance discontinuities. Based on the image-source method, the boundaries are assumed to be geometrically reflective. An infinite number of image sources are generated by multiple reflections. The sound pressure of each image is obtained by an approximate analytical solution, known as the Weyl-van der Pol formula. The total sound field is then calculated by summation of the contribution from all images. The phase information of each image and the phase change upon reflection are included in the model. A single change of impedance in a two-dimensional duct is focused on as the fundamental problem of the current study. The diffraction effect at the impedance discontinuity is proved to be insignificant, and it is ignored in the formulation. On the assumption that the diffraction effect is not important, the investigation is moved on to a rectangular long enclosure. Measurements are conducted in two model tunnels to validate the proposed prediction model. The predictions are found to give good approximations of the experimental results. The theoretical model serves as the first attempt to optimize the position and pattern of sound absorption materials in a long enclosure, such as an underground railway station or a building corridor, for the reduction of noise and improvement of sound quality.