Relaxation of sound fields in rooms of diffusely reflecting boundaries and its application in acoustical radiosity simulation

The acoustical radiosity method is a computationally expensive acoustical simulation algorithm that assumes an enclosure with ideal diffuse reflecting boundaries. Miles observed that for such an enclosure, the sound energy decay of every point on the boundaries will gradually converge to exponential manner with a uniform decay rate. Therefore, the ratio of radiosity between every pair of points on the boundaries will converge to a constant, and the radiosity across the boundaries will approach a fixed distribution during the sound decay process, where radiosity is defined as the acoustic power per unit area leaving (or being received by) a point on a boundary. We call this phenomenon the "relaxation" of the sound field. In this paper, we study the relaxation in rooms of different shapes with different boundary absorptions. Criteria based on the relaxation of the sound field are proposed to terminate the costly and unnecessary radiosity computation in the later phase, which can then be replaced by a fast regression step to speed up the acoustical radiosity simulation.     

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.     

The prediction of speech intelligibility in underground stations of rectangular cross section

Long enclosures are spaces with nondiffuse sound fields, for which the classical theory of acoustics is not appropriate. Thus, the modeling of the sound field in a long enclosure is very different from the prediction of the behavior of sound in a diffuse space. Ray-tracing computer models have been developed for the prediction of the sound field in long enclosures, with particular reference to spaces such as underground stations which are generally long spaces of rectangular or curved cross section. This paper describes the development of a model for use in underground stations of rectangular cross section. The model predicts the sound-pressure level, early decay time, clarity index, and definition at receiver points along the enclosure. The model also calculates the value of the speech transmission index at individual points. Measurements of all parameters have been made in a station of rectangular cross section, and compared with the predicted values. The predictions of all parameters show good agreement with measurements at all frequencies, particularly in the far field of the sound source, and the trends in the behavior of the parameters along the enclosure have been correctly predicted.     

NUMERICAL MODELLING OF THE SOUND FIELDS IN URBAN STREETS WITH DIFFUSELY REFLECTING BOUNDARIES

A radiosity-based theoretical/computer model has been developed to study the fundamental characteristics of the sound fields in urban streets resulting from diffusely reflecting boundaries, and to investigate the effectiveness of architectural changes and urban design options on noise reduction. Comparison between the theoretical prediction and the measurement in a scale model of an urban street shows very good agreement. Computations using the model in hypothetical rectangular streets demonstrate that though the boundaries are diffusely reflective, the sound attenuation along the length is significant, typically at 20-30 dB/100 m. The sound distribution in a cross-section is generally even unless the cross-section is very close to the source. In terms of the effectiveness of architectural changes and urban design options, it has been shown that over 2-4 dB extra attenuation can be obtained either by increasing boundary absorption evenly or by adding absorbent patches on the façades or the ground. Reducing building height has a similar effect. A gap between buildings can provide about 2-3 dB extra sound attenuation, especially in the vicinity of the gap. The effectiveness of air absorption on increasing sound attenuation along the length could be 3-9 dB at high frequencies. If a treatment is effective with a single source, it is also effective with multiple sources. In addition, it has been demonstrated that if the façades in a street are diffusely reflective, the sound field of the street does not change significantly whether the ground is diffusely or geometrically reflective.     

Insertion loss of an acoustic enclosure

Acoustical enclosures are the common arrangements in reducing airborne noise from shipboard machinery such as engines and generators. In this paper the theoretical models, established based on statistical energy analysis, are presented for predicting the insertion loss of acoustical enclosures in different frequency ranges. In addition to the consideration of resonant modal coupling between internal sound field and enclosure structural vibration, the nonresonant transmission though and the interaction between enclosure walls in the models are also included. It is shown that the insertion loss of enclosures is mainly controlled by the nonresonant modes in the intermediate frequency range. At high frequencies, the insertion loss of enclosures can be improved by increasing the sound absorption at the internal boundaries of enclosures. Experiments were carried out on two enclosures made of different materials. The measured results are compared with the predicted values and the good agreement between them is the initial demonstration of the validity and feasibility of the theoretical models.     

Decay rates and wall absorption at low frequencies

This paper is concerned with evaluating the error of conventional estimates of the boundary absorption of rectangular enclosures, with particular reference to reverberation room sound power measurements. The reverberation process is examined theoretically; the relative contributions to the decay rate from different modes in a rectangular room are calculated from an ensemble average over rooms with nearly the same dimensions. It is shown that the traditional method of determining the absorption of the walls of reverberation rooms systematically underestimates the absorption at low frequencies; the error is computed from the ensemble average. Finally, an unbiased estimate of the sound power radiated by a source in a reverberation room is derived. This estimate involves measurement of the initial decay rates of the room and is, unlike the usual reverberation room sound power estimate, neither based on statistical diffuse field considerations nor on the normal mode theory.     

On the steady-state and the transient decay methods for the estimation of reverberation time

The discrepancy between reverberation times of an enclosed sound field measured by the steady-state method and by the transient decay method is well-known. So far, no clear explanation has been obtained. In this paper, the steady-state bandlimited energy in an enclosure and bandlimited power flow into modally reactive boundaries are derived to describe the energy balance relationship and thus the reverberation time in a frequency band. This reverberation time is then compared to that obtained from the transient decay of the sound field based on the modal analysis. The comparison provides an understanding of the discrepancy mentioned above as well as the physical interpretations of the reverberation times estimated by both methods.     

Prediction of reverberation time and speech transmission index in long enclosures

It is known that the sound field in a long space is not diffuse, and that the classic theory of room acoustics is not applicable. A theoretical model is developed for the prediction of reverberation time and speech transmission index in rectangular long enclosures, such as corridors and train stations, where the acoustic quality is important for speech. The model is based on an image-source method, and both acoustically hard and impedance boundaries are investigated. An approximate analytical solution is used to predict the frequency response of the sound field. The reverberation time is determined from the decay curve which is computed by a reverse-time integration of the squared impulse response. The angle-dependence of reflection coefficients of the boundaries and the change of phase upon reflection are incorporated in this model. Due to the relatively long distance of sound propagation, the effect of atmospheric absorption is also considered. Measurements of reverberation time and speech transmission index taken from a real tunnel, a corridor, and a model tunnel are presented. The theoretical predictions are found to agree well with the experimental data. An application of the proposed model has been suggested.     

Sound propagation in street canyons: comparison between diffusely and geometrically reflecting boundaries

This paper systematically compares the sound fields in street canyons with diffusely and geometrically reflecting boundaries. For diffuse boundaries, a radiosity-based theoretical/computer model has been developed. For geometrical boundaries, the image source method has been used. Computations using the models show that there are considerable differences between the sound fields resulting from the two kinds of boundaries. By replacing diffuse boundaries with geometrical boundaries, the sound attenuation along the length becomes significantly less; the RT30 is considerably longer; and the extra attenuation caused by air or vegetation absorption is reduced. There are also some similarities between the sound fields under the two boundary conditions. For example, in both cases the sound attenuation along the length with a given amount of absorption is the highest if the absorbers are arranged on one boundary and the lowest if they are evenly distributed on all boundaries. Overall, the results suggest that, from the viewpoint of urban noise reduction, it is better to design the street boundaries as diffusely reflective rather than acoustically smooth.     

Monotonic curvature of low frequency decay records in reverberation chambers

In the papers by Larsen [1] and Brüel [2], two interesting problems connected with reverberation room measurements are pointed out and discussed. The first problem is that the ensemble averaged decay curve reveals a monotonic curvature at low frequencies. The second phenomenon is that often systematically larger sound power output values are reported at low frequencies according to the free field method than according to the reverberation room method. In searching for an explanation of these anomalies some measurements and a classical normal mode theory analysis have been made. It is shown that it is not possible to explain fully the curvature of the low frequency decay curves by means of the normal mode theory. The measured curves are more bent than the respective theoretical ones. Most probably, it should be possible to explain this lack of agreement by the fact that the absorption characteristics of normal reverberation chambers significantly deviate from the situation of uniform wall admittance which has been assumed in the theoretical deductions. The theoretical analysis and the comparison between theory and practice indicate that the damping characteristics of the individual waves vary much more than is predicted for a uniform wall admittance. This reasoning is supported by the observation that the monotonic curvature increases when a plane concentrated absorbent is added to one of the walls. One way to decrease the curvature has also been identified. When the room surfaces are provided with randomly placed small samples of low frequency absorbents the resulting decay curves turn out to be almost perfectly linear. Furthermore, it is found that the normal mode theory does not imply significantly different sound power output values than the ISO 3741 model. This fact has been verified with a comparative test. According to the normal mode theory the average sound power output as measured in the reverberant room should equal the free field output. Therefore, one is forced to conclude that the analysis of the classical normal mode theory fails in explaining the anomalies observed.     

基于四水听器的充水弹性管声速测量方法

实验室中水声材料声学参数的测量主要在水声声管中进行。管内平面波声速是正确测量这些参数的基础。该文提出一种基于四水听器结合不同边界的测量充水弹性管中声速的新方法。该方法利用4个固定位置处的水听器,采用最小二乘的方法,使得两组水听器分别得到的声管末端入射波声压差值的平方最小的声速即为管内平面波声速。该方法利用单频信号,在每一频率点均可测得声速,可以在任一种声管末端边界下进行测量,同时无需知道各水听器到边界的精确距离,在文中的3种边界下声速测量结果具有很好的一致性,实验操作简单、误差很小。该方法的仿真结果与管内声速的理论值吻合得很好,同时实验测量结果与仿真值之间的误差很小,证明了方法的准确性以及鲁棒性,为声管声速测量提供一个很好的思路。     

基于联合波叠加法的浅海信道下圆柱壳声辐射研究

针对浅海信道下弹性结构声辐射预报尚无高效可靠的研究方法,提出了一种浅海信道下弹性结构声辐射快速预报的联合波叠加法.该方法结合了浅海信道传输函数、多物理场耦合数值计算法和波叠加法理论,运用该方法可对浅海信道下弹性结构辐射声场进行快速预报.经数值法和解析解法验证后,从信道下辐射源、环境影响和辐射声场测量的角度研究分析了浅海信道下弹性圆柱壳的声辐射特性,阐释了进行浅海信道下结构声辐射研究的必要性.研究结果表明,仅在低频浅海信道下弹性结构可近似等效为点源,信道上下边界对声场产生显著的耦合影响,高频段的空间声场指向性分布尤为明显,垂直线列阵进行信道下结构辐射声功率测量时,测量结果受到信道环境边界和潜深的影响较大.     

两端开口空间的声场研究

研究了两端开口空间中声压分布同封闭空间和长空间中的不同,比较传统赛宾公式、改进赛宾公式和声线法对这种空间的估计准确度,同时对其中出现非指数衰变曲线进行讨论,分析其产生的原因、影响因素和规律。文中使用声线法进行数值计算,并且通过实验来和数值结果比较,验证结论。研究表明:两端开口空间中的稳态声压分布与封闭空间和长空间明显不同,混响时间的计算不能再使用传统赛宾公式,声线追踪法无法完全准确地估计稳态声压;其中出现非指数衰变曲线的原因经分析为空间内各不同活跃度的小区域之间的声能量耦合造成,而能否被观察到取决于能否满足耦合空间理论中对于双斜率衰变曲线出现的要求,直观上只要观察点和声源距离相差不远,并且不都在空间中央便可观察到明显的双斜率衰变曲线。      

非规则封闭空间声场建模的Chebyshev-变分法及其固有声学特性分析

针对非规则封闭空间声场建模问题,提出了一种基于Chebyshev-变分原理的声场建模方法。该方法首先选取包络非规则声场的矩形空间并将此矩形空间内的声压函数展开成三重Chebyshev级数形式,然后通过坐标变换得到定义域空间中的声场势能和声场动能,最后按照里茨方法对声场的拉格朗日泛函进行求解,得到声场的特征方程并求得声场固有频率和模态。通过与曲面声场的数值结果对比,验证了本建模方法的正确性和有效性。在此基础上研究具有不同倾角的梯形声场固有特性,分析内部凹槽深度对"凹"型声场频率和模态的影响。结果表明,梯形声场模态会随着倾角增大而逐步演变;"凹"型声场低阶频率随凹槽深度的增加而逐渐减小,但第一阶频率却呈现先减小后增大的特点。      

A study of barriers in enclosures by a ray-tracing computer model

The theoretical study of the behaviour of barriers in enclosures is much more complicated than that of barriers in the free field. As a matter of fact, in the latter case one has to take into account the sound energy reflected at the boundary surface of the enclosure. The barrier-absorption joint effect has to be considered in defining the barrier sound attenuation. From this point of view, the barrier attenuation also depends on the enclosure shape and proportions. In this paper we suggest a numerical method, based on a ray-tracing technique, in which the concepts inherent in Keller's theory of diffraction are included. The results of some simulation examples are reported, together with some qualitative considerations which allow emphasis of the problems connected with the optimization of barrier design in enclosures.     

适用于任意形状封闭空间的声场预测相干模型

针对现有几何声学的方法对封闭空间内声场进行预测时在中低频段出现较大误差的问题,该文提出一种近似圆锥声束追踪法和相干反射场理论相结合的声场预测新模型。在近似圆锥声束追踪法基础上,考虑声束轴线在边界多次反射时声压和相位的改变,最后计算不同声波之间的干涉效应,建立一种适用于任意形状封闭空间的声场预测相干模型。利用该模型对某一矩形封闭空间进行声场预测,通过对边界元法、Raynoise软件相干和非相干算法的预测结果和本模型的数值模拟结果对比。结果表明,文中提出的方法和边界元法的计算结果在中低频段非常吻合,两者的计算结果平均绝对误差为1.1 d B。本模型在中低频率下与同样考虑了相位的Raynoise相干算法相比有更好的准确性,在较高频率上,本模型计算结果与Raynoise相干算法计算结果非常吻合。     

An analytical model to estimate the performance of an indoor barrier at low-medium frequencies

Indoor barriers are now widely used for sound insulation. This paper examines the performance of indoor barriers in the low-medium frequency range and analyses the interaction between different natural modes of a room-barrier-room system. Morse proposed a theoretical model to calculate the sound field in a coupled-room, but this model neglects the surface integral of the boundary values of sound pressure. To estimate the performance of a barrier in an indoor environment, an analytical model is proposed that modifies the Green’s function for a non-rigid boundary enclosure and approximates the surface integral by a pre-estimated sound pressure based on Morse’s model. An additional approximation has been made in the proposed model to neglect the coupling area in the calculation of the surface integral. The proposed model used to predict the insertion loss of the barrier is verified by the experimental results using a 1:5 scale model. The predicted results agree well with the measured results at lower frequencies.     

Modeling and acoustic analysis of irregular sound enclosure by using Chebyshev-variational method

A modeling method for irregular sound enclosures was proposed based on the Chebyshev-variational theory. A rectangular space was first assumed to bound the irregular sound space and the sound pressure in the rectangular space expressed as a triple-Chebyshev series. Next, a coordinate transformation was performed and the Lagrangian functional of the irregular sound space obtained. Finally, the Lagrangian functional was solved under the Ritz method framework, and the enclosure's acoustic characteristic equation deduced and the eigenpairs obtained. The accuracy of the present method was validated according to agreement between the present results and finite element results for an enclosure with a curved surface.Furthermore, the acoustic characteristics of a trapezoidal enclosure and an enclosure with an inner groove were investigated. The results showed that the mode shapes of the trapezoidal sound space changed with increased inclination angle and the natural frequencies, except the first order, of the sound space with a rectangular inner groove decreased with increased groove depth.     

Prediction of sound fields in rooms of arbitrary shape: Validity of the image sources method

In order to find the sound pressure field in a rectangular room, the classical modal method may be applied. However, it does not apply to polyhedric rooms of any shape, to which, however, the image sources method may be applied. The two methods are compared for several geometric shapes: two parallel planes (a layer), a rectangular wave guide and a rectangular enclosure. It is shown under what conditions the image sources method can give accurate enough results. A comparison of theoretical and experimental results is presented.