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.     

Time-domain calculations of sound interactions with outdoor ground surfaces

A time-domain formulation for sound propagation in rigid-frame porous media, including waveform attenuation and dispersion, is developed. The new formulation is based on inversion of the relaxation functions from a previous model [Wilson DK, Ostashev VE, Collier SL. J Acoust Soc Am 2004;116:1889-92], thereby casting the convolution integrals in a form amenable to numerical implementation. Numerical techniques are developed that accurately implement the relaxational equations and transparently reduce to previous results in low- and high-frequency limits. The techniques are demonstrated on calculations of outdoor sound propagation involving hills, barriers, and ground surfaces with various material properties. We also compare the relaxation formulation to a widely applied phenomenological model developed by Zwikker and Kosten. The two models can be made equivalent if the resistance constant, structure constant, and compression modulus in the ZK model are allowed to be weakly frequency dependent. But if the ZK parameters are taken to be constant, as is typically the case, the relaxation model provides more accurate calculations of attenuation by acoustically soft porous materials such as snow, gravel, and forest litter.     

Acoustic methods for measuring bullet velocity

This article describes two acoustic methods to measure bullet velocity with an accuracy of 1% or better. In one method, a microphone is placed within 0.1 m of the gun muzzle and a bullet is fired at a steel target 45 m away. The bullet’s flight time is the recorded time between the muzzle blast and sound of hitting the target minus the time for the sound to return from the target to the microphone. In the other method, the microphone is placed equidistant from both the gun muzzle and the steel target 91 m away. The time of flight is the recorded time between the muzzle blast and the sound of the bullet hitting the target. In both cases, the average bullet velocity is simply the flight distance divided by the flight time.     

Modelling of a city canyon problem in a turbulent atmosphere using an equivalent sources approach

The sound propagation into a courtyard shielded from direct exposure is predicted using an equivalent sources approach. The problem is simplified into that of a two-dimensional city canyon. A set of equivalent sources are used to couple the free half-space above the canyon to the cavity inside the canyon. Atmospheric turbulence causes an increase in the expected value of the sound pressure level compared to a homogeneous case. The level increase is estimated using a von Kármán turbulence model and the mutual coherences of all equivalent sources' contributions. For low frequencies the increase is negligible, but at 1.6 kHz it reaches 2-5 dB for the geometries and turbulence parameters used here. A comparison with a ray-based model shows reasonably good agreement.     

Prediction method for sound from passing vehicle transmitted through building façade

This study proposes a prediction method for assessing the sound of a passing vehicle that is transmitted through a glass plate, which employs the vibro-acoustic finite-difference time-domain (FDTD) method. The noise that is transmitted through the glass in a building façade, which is affected by the sound-insulation characteristics of its glass plates, can have a psychological influence such as a sleep disorder on the residents. In this study, a prediction method for the spectral characteristics of the transmitting sound through glass plates is proposed. The sound-insulation characteristics of glass plates are obtained using a vibro-acoustic FDTD method, and are then synthesized with the sound of passing vehicles obtained by in situ measurement. Firstly, the sound transmitted through several kinds of glass plates is simulated using the proposed method. Then, in order to confirm the validity of this method, the simulated results are compared to measured sounds transmitted by passing vehicles into a room near the street.     

Time-domain impedance formulation for transmission line matrix modelling of outdoor sound propagation

Outdoor sound propagation modelling has attracted considerable attention in the past and has lead to many analytical and numerical models. More recently, the increase of computational power has led to consider time-domain methods that enable to consider transient phenomena. Among these models, the transmission line matrix (TLM) method has been proposed, but the sound absorption at boundaries appears to be a somewhat underdeveloped aspect of this approach. In this paper, a specific implementation of impedance boundary condition is proposed. The method is based on the approximation of the impedance as a sum of linear systems, which allows the formulation of an equivalent impedance model in the time-domain. The proposed implementation is applied for two common impedance models of porous material. Numerical simulations have been carried out in the case of sound propagation over a flat ground with and without an impedance discontinuity, and for several values of specific airflow resistivity. TLM numerical predictions expressed in terms of excess attenuation relative to free field show a good agreement with analytical solutions.     

Sound field of thermoacoustic tomography based on a modified finite-difference time-domain method

A modified finite-difference time-domain （FDTD） method is proposed for the sound field simulation of the thermoacoustic tomography （TAT） in the acoustic speed inhomogeneous medium. First, the basic equations of the TAT are discretized to difference ones by the FDTD. Then the electromagnetic pulse, the excitation source of the TAT, is modified twice to eliminate the error introduced by high frequency electromagnetic waves. Computer simulations are carried out to validate this method. It is shown that the FDTD method has a better accuracy than the commonly used time-of-flight （TOF） method in the TAT with the inhomogeneous acoustic speed. The error of the FDTD is ten times smaller than that of the TOF in the simulation for the acoustic speed difference larger than 50%. So this FDTD method is an efficient one for the sound field simulation of the TAT and can provide the theoretical basis for the study of reconstruction algorithms of the TAT in the acoustic heterogeneous medium.     

有机分子第一超极化率色散效应和双光子共振增强理论研究

采用含时Hartree-Fock和多态求和方法计算了半花菁衍生物生色团的第一超极化率色散效应. 采用二能级模型研究了第一超极化率的双光子共振增强特征. 研究发现，第一激发态对半花菁非线性光学性质起决定性作用，随着入射光波长向短波方向移动，二次谐波产生β(-2ω;ω,ω)不断增大并且在950nm附近出现双光子共振效应，理论结果与实验结果较好相符. 此外，共振条件下的二能级模型需要考虑激发态弛豫效应. 研究结果为实验测量和实际应用提供了理论参考. 关键词： 分子非线性光学 超极化率 色散效应 共振增强     

阻尼受击板的时域建模及声音合成

为满足声源辨识中对合成冲击声的迫切需求,建立了球-板撞击的时域模型,提出一种时域快速求解方法,并进行了实验验证.首先给出一种将时域有限差分法(FDTD)和模态展开法(MEM)相结合的时域混合方法,求解板的振动方程,并解决了混合方法中MEM的模态截断和初值问题,及两种方法中阻尼的一致性问题;随后,给出了简支矩形板的冲击声计算结果,通过与FDTD方法的运算量进行对比,验证了混合方法的高效性;最后,针对自由边界下的L形板进行了实验验证.结果表明,与传统FDTD方法相比,时域混合方法在保证合成冲击声精度的前提下可将计算效率提高100至1200倍。      

Quasi-wavelet calculations of sound scattering behind barriers

Quasi-wavelets (QWs) are a representation of turbulence consisting of self-similar, eddy-like structures with random orientations and positions in space. They are used in this paper to calculate the scattering, due to turbulent velocity fluctuations, of sound behind noise barriers as a function of the size and spatial location of the eddies. The sound scattering cross-section for QWs of an individual size class (eddy size) is derived and shown to reproduce results for the von Kármán spectrum when the scattered energies from a continuous distribution of QW sizes are combined. A Bragg resonance condition is derived for the eddy size that scatters most strongly for a given acoustic wavenumber and scattering angle. Results for scattering over barriers show that, for typical barrier conditions, most of the scattered energy originates from eddies in the size range of approximately one-half to twice the size of the eddies responsible for maximum scattering. The results also suggest that scattering over the barrier due to eddies with a line of sight to both the source and receiver is generally significant only for frequencies above several kilohertz, for sources and receivers no more than a few meters below the top of the barrier, and for very turbulent atmospheric conditions.     

Numerical simulation of partially coherent broadband optical imaging using the finite-difference time-domain method

Rigorous numerical modeling of optical systems has attracted interest in diverse research areas ranging from biophotonics to photolithography. We report the full-vector electromagnetic numerical simulation of a broadband optical imaging system with partially coherent and unpolarized illumination. The scattering of light from the sample is calculated using the finite-difference time-domain (FDTD) numerical method. Geometrical optics principles are applied to the scattered light to obtain the intensity distribution at the image plane. Multilayered object spaces are also supported by our algorithm. For the first time, numerical FDTD calculations are directly compared to and shown to agree well with broadband experimental microscopy results.     

Noise abatement schemes for shielded canyons

Access to quiet areas in cities is important to avoid adverse health effects due to road traffic noise. Most urban areas which are or can become quiet (L_A,eq < 45 dB) are shielded from direct road traffic noise. By transfer paths over roof level, many road traffic noise sources contribute to the level in these shielded areas and noise abatement schemes may be necessary to make these areas quiet. Two real life shielded courtyards in Göteborg have been selected as reference cases for a numerical investigation of noise abatement schemes. The selected areas are modelled as canyons with a road traffic noise source modelled outside the canyon by a finite incoherent line source, which is more realistic than both a coherent and an incoherent line source of infinite length. The equivalent sources method has been used for the calculations. For all studied noise abatement schemes in the shielded canyon, the reductions are largest for the lower canyon observer positions. Façade absorption is the most effective when placed in the upper part of the canyon and can typically yield a reduction of 4 dB(A). Constructing 1 m wide walkways with ceiling absorption reduces the level typically by 3 dB(A). These effects are most effective for narrower canyons. For treatments at the canyon roof, reductions are independent of the canyon observer position and amount to 4 dB(A) for a 1 m tall screen and 2 dB(A) for a grass covering of a saddle roof. Downward refracting conditions increase the levels for the lower canyon observer positions and higher frequencies. For sources located in canyons, abatement schemes therein are more effective for noise reduction in the shielded canyon than similar abatement schemes in the shielded canyon itself, given that all contributing source canyons are treated.     

阶梯状声扩散体形体优化方法

应用时域有限差分法计算阶梯状声扩散体扩散系数,结合免疫遗传算法分别对周期排布的六阶和十二阶阶梯状声扩散体的形体进行优化。首先,验证了时域有限差分法计算声场的有效性,之后采用两种方式进行优化设计:第一种是直接对多周期重复排布的扩散体形体进行优化;第二种首先对单周期扩散体形体进行优化,再将该扩散体重复周期排布。结果显示:第一种优化方式能获得扩散性能更优的周期排布的扩散体。单周期扩散性能最优的扩散体重复一定周期后,其性能不会保持最优,相反,扩散性能并非最优的单周期扩散体重复一定周期后的扩散性能却优于前者;扩散体总长度一定条件下,增加扩散体的阶次、降低重复周期数,可有效提升扩散体的扩散性能。因此,直接对多周期重复排布的高阶次扩散体形体进行优化,能够获得扩散性能最优的周期重复排布的扩散体。      

Calculation of B/A for n-alkane liquids using the Tait equation

The B/A parameter of acoustic nonlinearity was calculated for a series of n-alkane liquids using the Tait PVT equation of state supplemented with specific heat data. The calculations of sound speed, sound speed derivatives, the two components of B/A, and the value of B/A itself were compared with experimental data taken from the literature and with earlier calculations using a different equation of state. In addition, a comparison of the results with Ballou's rule (linear relation of B/A and reciprocal sound speed) was made. It is concluded that B/A can be calculated from the Tait equation of state with about the same accuracy as direct measurements of sound speed versus pressure and temperature, though the the temperature derivatives of the sound speed are calculated with much lower accuracy than pressure derivatives. The calculations made using the Tait equation are about the same accuracy as calculations made using our equation of state. Also, Ballou's rule does not hold for these liquids.     

Calculation of Dispersion Characteristics of a Circular Waveguide with Deep Helical Corrugation by the FDTD Method

We study the dispersion characteristics of a circular waveguide with helical corrugation, which are used in microwave amplifiers and oscillators. The results of calculations by the finite difference time domain (FDTD) method are compared with the results obtained using the perturbation theory and the experimental results. The problems of improving the calculation accuracy are considered     

Finite difference time domain modelling of sound scattering by the dynamically rough surface of a turbulent open channel flow

The problem of scattering of airborne sound by a dynamically rough surface of a turbulent, open channel flow is poorly understood. In this work, a laser-induced fluorescence (LIF) technique is used to capture accurately a representative number of the instantaneous elevations of the dynamically rough surface of 6 turbulent, subcritical flows in a rectangular flume with Reynolds numbers of 10,800?Re?47,300$10\text{,}800?\text{Re}?47\text{,}300$ and Froude numbers of 0.36?Fr?0.69$0.36?\mathit{Fr}?0.69$. The surface elevation data were then used in a finite difference time domain (FDTD) model to predict the directivity pattern of the airborne sound pressure scattered by the dynamically rough flow surface. The predictions obtained with the FDTD model were compared against the sound pressure data measured in the flume and against that obtained with the Kirchhoff approximation. It is shown that the FDTD model agrees with the measured data within 22.3%. The agreement between the FDTD model and stationary phase approximation based on Kirchhoff integral is within 3%. The novelty of this work is in the direct use of the LIF data and FDTD model to predict the directivity pattern of the airborne sound pressure scattered by the flow surface. This work is aimed to inform the design of acoustic instrumentation for non-invasive measurements of hydraulic processes in rivers and in partially filled pipes.     

Study of the sound-vortex interaction: direct numerical simulations and experimental results

We present a study of sound propagation through a single vortex by direct numerical simulations (DNS) compared to experimental measurements. We analyse the amplitude and the phase shift of the sound wave when it interacts with the vortical flow and we display the focusing effects produced by the vortex. We show that the turbulent fluctuations have a little effect on the sound phase shift whereas they induce a strong defocusing effect on the sound amplitude. Received 9 October 2002 / Received in final form 20 January 2003 Published online 1st April 2003 RID="a" ID="a"e-mail: rberthet@lps.ens.fr RID="b" ID="b"UMR CNRS 8550     

Prediction-step staggered-in-time FDTD: An efficient numerical scheme to solve the linearised equations of fluid dynamics in outdoor sound propagation

The finite-difference time-domain (FDTD) method to solve the linearised equations of fluid dynamics has shown to be very powerful and useful in outdoor sound propagation. Practical applications are however limited due to the large need for computational resources. The numerical discretisation influences computational efficiency to an important degree. In this paper, some possible ways to discretise temporal derivatives are studied. Two obvious ways of time-discretisation namely staggered-in-time (SIT) and a simple collocated-in-time (CIT) scheme are compared to the prediction-step staggered-in-time (PSIT) scheme. The latter is intended to be used for the calculation of sound propagation in the typical low wind speeds encountered in the outdoor environment at low heights above the earth’s surface. It was shown that the PSIT scheme is more stable than the SIT scheme, so practical calculations are possible. Computational efficiency is increased to an important degree compared to the CIT scheme. The numerical accuracy (more precisely the amplitude error) of the PSIT scheme is an important improvement upon SIT. The CIT scheme on the other hand conserves amplitude better. The amplitude error becomes larger with increasing wind speed because of some simplifications during the numerical discretisation. In low wind speeds, the PSIT algorithm can serve as an interesting compromise between numerical accuracy and the required amount of computing power.     

收发分置目标强度的计算及前向散射信号的分离

研究时域有限差分(FDTD)法计算收发分置弹性目标散射声场的有效性,和实验测量中从直达与散射干涉信号中分离目标前向散射信号的方法。将FDTD法应用于三维声场计算,分别计算了平顶圆柱、球顶圆柱、球锥顶圆柱和扁长回转椭球等几种不同形状的弹性体散射近场空间分布,再用近远场变换方法获取远场目标强度空间分布。模拟结果表明:目标强度的空间指向性是受声波入射方向、无量纲波数kL(k-波数,L-目标特征尺度)、目标形状、材料等影响;在本文所计算的几种不同形状目标的尺度和kL取值范围内,计算得到的收发分置目标强度空间分布均有显著的前向散射与镜反射特征。计算结果得到解析解和实验结果的印证,从而表明了FDTD法计算收发分置目标散射场的有效性。对接收信号采用相关处理方法,能有效地分离出目标前向散射信号,表明对于收发分置目标散射声强的测量,这是一种简单有效的分离目标前向散射信号的方法。