Approximate reconstruction of sound fields close to the source surface using spherical nearfield acoustical holography

This paper presents an investigation of the reconstruction of sound field parameters close to the surface of arbitrarily shaped sound sources. The field is reconstructed using nearfield acoustical holography (NAH) in spherical coordinates. Of particular interest are source shapes where the Rayleigh hypothesis is violated. To overcome the limitation of the minimal sphere given by the validity restriction of the Rayleigh hypothesis an algorithm is proposed for extracting local information from the nonconvergent NAH solution. For the assessment of the results an appropriate virtual test rig is developed employing the Kirchhoff-Helmholtz integral theorem.     

Study of the comparison of the methods of equivalent sources and boundary element methods for near-field acoustic holography

Boundary element methods (BEM) based near-field acoustic holography (NAH) has been used successfully in order to reconstruct the normal velocity on an arbitrarily shaped structure surface from measurements of the pressure field on a nearby conformal surface. An alternative approach for this reconstruction on a general structure utilizes the equivalent sources method (ESM). In ESM the acoustic field is represented by a set of point sources located over a surface that is close to the structure surface. This approach is attractive mainly for its simplicity of implementation and speed. In this work ESM as an approximation of BEM based NAH is studied and the necessary conditions for the successful application of this approach in NAH is discussed. A cylindrical fuselage surface excited by a point force as an example to validate the results is used.     

On optimal retreat distance for the equivalent source method-based nearfield acoustical holography

As a basic form of the equivalent source method (ESM) that is used to nearfield acoustical holography (NAH) problems, discrete monopoles are utilized to represent the sound field of interest. When setting up the virtual source distribution, it is vital to maintain a "retreat distance" between the virtual sources and the actual source surface such that reconstruction would not suffer from singularity problems. However, one cannot increase the distance without bound because of the ill-posedness inherent in the reconstruction process with large distance. In prior research, 1-2 times lattice spacing, or the inter-element distance of microphones, is generally recommended as retreat distance in using the ESM-based NAH. While this rule has shown to yield good results in many cases, the optimal choice is a complicated issue that depends on frequency, geometry of the physical source, content of evanescent waves, distribution of sensors and virtual sources, etc. This paper deals about attaining the best compromise between the reconstruction errors induced by the point source singularity; the reconstruction ill-posedness is an interesting problem in its own right. The paper revisits this issue, with the aid of an optimization algorithm based on the golden section search and parabolic interpolation. Numerical simulations were conducted for a baffled planar piston source and a spherically baffled piston source. The results revealed that the retreat distance appropriate for the ESM ranged from 0.4 to 0.5 times the spacing for the planar piston, while from 0.8 to 1.7 times average spacing for the spherical piston. Experiments carried out for a vibrating aluminum plate also revealed that the retreat distance with 0.5 times the spacing yielded better reconstructed velocity than those with 1/20 and 1 times the spacing.     

Planar near-field acoustical holography in a moving medium

Near-field acoustical holography (NAH) is a well-established method to study acoustic radiation near a stationary sound source in a homogeneous, stationary medium. However, the current theory of NAH is not applicable to moving sound sources, such as automobiles and trains. In this paper, the inclusion of a moving medium (i.e., moving source and receiver) is introduced in the wave equation and a new set of equations for plannar NAH is developed. Equations are developed for the acoustic pressure, particle velocity, and intensity when mean flow is either parallel or perpendicular to the hologram plane. If the source and the measurement plane are moving at the same speed, the frequency Doppler effect is absent, but a wave number Doppler effect exists. This leads to errors when reconstructing the acoustic field both towards and away from the source using static NAH. To investigate these errors, a point source is studied analytically using planar NAH with flow in one direction. The effect of the medium moving parallel to the hologram plane is noted by a shift of the radiation circle in wave number space (k-space). A k-space Green's function and a k-space filter are developed that include the effects of the moving medium.     

Approximations of inverse boundary element methods with partial measurements of the pressure field

Boundary element methods (BEMs) based near-field acoustic holography (NAH) requires the measurement of the pressure field over a closed surface in order to recover the normal velocity on a nearby conformal surface. There are practical cases when measurements are available over a patch from the measurement surface in which conventional inverse BEM based NAH (IBEM) cannot be applied directly, but instead as an approximation. In this work two main approximations based on the indirect-implicit methods are considered: Patch IBEM and IBEM with Cauchy data. Patch IBEM can be applied with a continuation procedure, which as its predecessor patch NAH (a well known technique that can be used on separable geometries of the wave equation) continues the pressure field using an iterative procedure, or it can be applied by a direct procedure. On the other hand, IBEM with Cauchy data requires measurements over two conformal patches and it will be shown that this technique will be reliable regardless of the position of the source. The theory behind each method will be justified and validated using a cylindrical surface with numerical data generated by point sources, and using experimental data from a cylindrical fuselage excited by a point force.     

Statistically optimized near field acoustic holography using an array of pressure-velocity probes

Statistically optimized near field acoustic holography (SONAH) differs from conventional near field acoustic holography (NAH) by avoiding spatial Fourier transforms; the processing is done directly in the spatial domain. The main advantage of SONAH compared with NAH is that the usual requirement of a measurement aperture that extends well beyond the source can be relaxed. Both NAH and SONAH are based on the assumption that all sources are on one side of the measurement plane whereas the other side is source free. An extension of the SONAH procedure based on measurement with a double layer array of pressure microphones has been suggested. The double layer technique makes it possible to distinguish between sources on the two sides of the array and thus suppress the influence of extraneous noise coming from the "wrong" side. It has also recently been demonstrated that there are significant advantages in NAH based on an array of acoustic particle velocity transducers (in a single layer) compared with NAH based on an array of pressure microphones. This investigation combines the two ideas and examines SONAH based on an array of pressure-velocity intensity probes through computer simulations as well as experimentally.     

基于声压-振速测量的平面近场声全息实验研究

常规的近场声全息均是采用全息面声压或质点振速作为输入量求解,由于采用单一输入量无法分离来自全息面背向声波的干扰,因此要求所有声源均位于全息面的同一侧,即测量声场为自由声场,这种要求大大限制了近场声全息的实际应用.基于声压-速度测量的近场声全息以全息面上声压和质点振速同时作为输入量,通过建立和求解两侧声源在全息面上的声压和质点振速耦合关系,可以实现全息面两侧声波的分离,从而解决上述问题.文中在前期对声场分离技术研究的基础上,基于欧拉公式和有限差分近似,推导了新的基于声压-速度测量的平面近场声全息理论公式.随后通过实验检验了该方法在有背景源干扰情况下实现声场分离和重建的有效性.     

Real-time near-field acoustic holography for continuously visualizing nonstationary acoustic fields

Near-field acoustic holography (NAH) is an effective tool for visualizing acoustic sources from pressure measurements made in the near-field of sources using a microphone array. The method involving the Fourier transform and some processing in the frequency-wavenumber domain is suitable for the study of stationary acoustic sources, providing an image of the spatial acoustic field for one frequency. When the behavior of acoustic sources fluctuates in time, NAH may not be used. Unlike time domain holography or transient method, the method proposed in the paper needs no transformation in the frequency domain or any assumption about local stationary properties. It is based on a time formulation of forward sound prediction or backward sound radiation in the time-wavenumber domain. The propagation is described by an analytic impulse response used to define a digital filter. The implementation of one filter in forward propagation and its inverse to recover the acoustic field on the source plane implies by simulations that real-time NAH is viable. Since a numerical filter is used rather than a Fourier transform of the time-signal, the emission on a point of the source may be rebuilt continuously and used for other post-processing applications.     

Underwater hybrid near-field acoustical holography based on the measurement of vector hydrophone array

Hybrid near-field acoustical holography(NAH) is developed for reconstructing acoustic radiation from a cylindrical source in a complex underwater environment. In hybrid NAH,we combine statistically optimized near-field acoustical holography(SONAH) and broadband acoustical holography from intensity measurements(BAHIM) to reconstruct the underwater cylindrical source field. First,the BAHIM is utilized to regenerate as much acoustic pressures on the hologram surface as necessary,and then the acoustic pressures are taken as input to the formulation implemented numerically by SONAH. The main advantages of this technology are that the complex pressure on the hologram surface can be reconstructed without reference signal,and the measurement array can be smaller than the source,thus the practicability and efficiency of this technology are greatly enhanced. Numerical examples of a cylindrical source are demonstrated. Test results show that hybrid NAH can yield a more accurate reconstruction than conventional NAH. Then,an experiment has been carried out with a vector hydrophone array. The experimental results show the advantage of hybrid NAH in the reconstruction of an acoustic field and the feasibility of using a vector hydrophone array in an underwater NAH measurement,as well as the identification and localization of noise sources.     

Use of nonsingular boundary integral formulation for reducing errors due to near-field measurements in the boundary element method based near-field acoustic holography

In the conformal near-field acoustic holography (NAH) using the boundary element method (BEM), the transfer matrix relating the vibro-acoustic properties of source and field depends solely on the geometrical condition of the problem. This kind of NAH is known to be very powerful in dealing with the sources having irregular shaped boundaries. When the vibro-acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components. The conventional acoustic BEM based on the Kirchhoff-Helmholtz integral equation has the singularity problem in the close near field of the source surface. This problem stems from the singular kernel of the Green function of the boundary integral equation (BIE) and the singularity can influence the reconstruction accuracy greatly. In this paper, the nonsingular BIE is introduced to the NAH calculation and the holographic BIE is reformulated. The effectiveness of nonsingular BEM has been investigated for the reduction of reconstruction error. Through interior and exterior examples, it is shown that the resolution of predicted field pressure could be improved in the close near field by employing the nonsingular BIE. Because the BEM-based NAH inevitably requires the field pressure measured in the close proximity to the source surface, the present approach is recommended for improving the resolution of the reconstructed source field.     

Optimal two-layer directive microphone array with application in near-field acoustical holography

Conventional near-field acoustical holography (NAH) is generally based on the free-field assumption, which can cause errors when interfering sources are present in practical environments. To cope with this problem, previous research suggested a combined pressure-velocity approach for NAH that provides certain advantages to rejection of interferences. This paper revisits this idea in a broader context of optimal array design and examines the feasibility of using unidirectional microphones in each channel of the array such that the robustness of inverse reconstruction is enhanced against interfering sources. As indicated in the simulation, the numerical noise in finite difference estimation of particle velocity can nullify the advantage of the well-conditioned velocity-based reconstruction. In the proposed approach, the characteristics of each array channel consisting of two microphones are tailored by taking into account not only the directivity, but also the robustness against self-noise. An objective function based on directivity index and white noise gain is exploited in a linear quadratic optimization of a two-element end-fire array. The proposed optimal array is validated in conjunction with the equivalent source model (ESM) -based NAH through numerical simulations, with an interfering source positioned behind the array. The results have shown the directive optimal array has yielded improved quality of images in comparison with conventional approaches in the presence of an interfering source and sensor noise.     

联合波叠加法的全息理论与实验研究

当空间声场中同时存在多个相干声源时，运用常规近场声全息方法无法重建每个相干声源表面的声学信息，当然也无法预测每个声源单独产生的空间声场，相干声场的全息重建与预测已成为全息技术推广应用过程中亟待解决的问题.在提出联合波叠加法并将其应用于空间声场变换的基础上，对其进行了实验研究.通过对实际相干声场的全息重建与预测，验证了常规波叠加法在相干声场重建中的局限性、联合波叠加法在相干声场全息重建与预测过程的可行性和准确性，还研究了Tikhonov正则化方法在抑制声学逆问题的非适定性中的有效性和滤波系数的选择原则的可行性，以提高全息重建与预测的精度. 关键词： 近场声全息 联合波叠加 相干声场 Tikhonov正则化     

Effect of sensor proximity over the non-conformal hologram plane in the near-field acoustical holography based on the inverse boundary element method

Near-field acoustical holography (NAH) is a useful tool for the identification and visualization of vibro-acoustic sources. In particular, NAH can be applied to many practical sources having irregular shape if the inverse boundary element method (BEM) is employed. Once the relation between the source and the radiated field is defined in the transfer matrix modeled by the BEM, the reconstruction of acoustic parameters on the source surface can be conducted by the multiplication of the inverse transfer matrix and the field data measured over the hologram surface. The usual practical way to measure the field data radiated from an irregular shaped source is to adopt a regular hologram surface, which can be a flat, cylindrical, or spherical shape, for the measurement ease. Then, the hologram surface is not conformal to the source surface and the resulting transfer matrix becomes further ill-posed than the conformal case. To investigate the effect of sensor proximity and distance variability on the reconstruction error, simulation and measurement were conducted for an interior problem comprised of a parallelepiped rigid box with a vibrating end plate. Flat, tilted, and randomized hologram surfaces were adopted in the test. It was shown that the reconstruction error is greatly affected by the conditioning of transfer matrix which is related to the positioning of sensor. To improve the reconstruction accuracy, an investigation was conducted to find proper field points among a large number of overdetermined field points. The number of field data was reduced gradually under various reduction schemes using condition number, effective independence value, and sensor distance. It was demonstrated that the quality of reconstruction result given by the non-conformal measurement can be improved by removing some field points that contribute to the ill-conditioning of the inverse problem. A small improvement of the reconstruction accuracy was observed by reducing the field points in the overdetermined situation. However, further reduction of the field points, becoming an underdetermined situation, yielded a drastic improvement of the reconstruction accuracy.     

Equivalent source method for identification of panel acoustic contribution in irregular enclosed sound field

For the interior sound field formed by the complex vibrating structure,an identification approach of panel acoustic contribution based on equivalent source method(ESM)was presented.The normal velocity on the surface of vibrating structure was first reconstructed by using interior nearfield acoustic holography based on ESM and the prediction of whole interior enclosed sound field was realized.Then the sound pressure produced by each panel at the interested field point was respectively replaced by the radiated pressure of the enclosed interior sound field which is formed by the equivalent virtual sources located near the surface of the cavity.Combining with the reconstructed normal surface velocity,the acoustic contribution of each panel to any position in the cavity was obtained by transforming the complex enclosed non-free field into the simple interior free field.Numerical simulations and experiments are conducted,and the influences of the number of the equivalent sources and the distance between them and the reconstructed surface have been investigated.The results show that the proposed method is easier to be implemented with the same accuracy than the traditional analysis method.     

复杂封闭声场面板声学贡献度识别的等效源法

针对由复杂结构振动形成的封闭空间声场,提出了一种基于等效源法的面板声学贡献度分析方法。该方法首先利用基于等效源法的内部声全息技术,重构出振动结构表面的法向振速并实现对整个内部封闭声场的预测。再将振动结构的每个面板在腔体内部场点产生的声压分别用位于空腔表面附近的等效源在该点产生的辐射声压代替,将复杂的封闭非自由声场问题转化为简单的内部自由场问题,结合重建出的结构表面法向振速进而识别出封闭振动结构各面板对腔体内任意位置的声学贡献度。通过对复杂结构内声场的数值仿真和验证实验,分析了等效源的数量及与重建面距离等参数对重建精度的影响,结果表明所提方法不仅能够达到传统数值分析方法的计算精度,而且具有更简单的求解过程。      

Data completion method for the characterization of sound sources

This paper presents a different approach to solve the inverse acoustic problem. This problem is an "ill-posed" problem since the solution is very sensitive to measurement precision. A classical way to solve this problem consists in inversing a propagation operator which relates structure quantities (acoustic pressures or gradients) to near-field quantities (acoustic pressures or gradients). This can be achieved by using near-field acoustical holography (NAH) in separable coordinate systems. In order to overcome this limitation, the inverse boundary element method (IBEM) can be implemented to recover all acoustic quantities in a three-dimensional space and on an arbitrary three-dimensional source surface. In this paper, the data completion method (DCM) is developed: the acoustic gradients and pressures are known on a surface surrounding the source, but are unknown on its structure. The solution is given by the resolution of the Helmholtz formulation applied on the empty domain between the two boundaries made by the measurements quantities and the structure of the source. The conventional method applies directly the integral formulation for the empty domain. Another way of solving this Helmholtz formulation can be achieved by splitting it in two well-posed subproblems in a Steklov-Poincare?'s formulation. The data completion method allows one to solve the problem with acoustic perturbations due to sources on the exterior domain, or due to a confined domain, without altering the results.     

Implementation issues of the nearfield equivalent source imaging microphone array

This paper revisits a nearfield microphone array technique termed nearfield equivalent source imaging (NESI) proposed previously. In particular, various issues concerning the implementation of the NESI algorithm are examined. The NESI can be implemented in both the time domain and the frequency domain. Acoustical variables including sound pressure, particle velocity, active intensity and sound power are calculated by using multichannel inverse filters. Issues concerning sensor deployment are also investigated for the nearfield array. The uniform array outperformed a random array previously optimized for far-field imaging, which contradicts the conventional wisdom in far-field arrays. For applications in which only a patch array with scarce sensors is available, a virtual microphone approach is employed to ameliorate edge effects using extrapolation and to improve imaging resolution using interpolation. To enhance the processing efficiency of the time-domain NESI, an eigensystem realization algorithm (ERA) is developed. Several filtering methods are compared in terms of computational complexity. Significant saving on computations can be achieved using ERA and the frequency-domain NESI, as compared to the traditional method. The NESI technique was also experimentally validated using practical sources including a 125 cc scooter and a wooden box model with a loudspeaker fitted inside. The NESI technique proved effective in identifying broadband and non-stationary sources produced by the sources.     

Patch nearfield acoustic holography combined with sound field separation technique applied to a non-free field

The conventional nearfield acoustic holography(NAH) is usually based on the assumption of free-field conditions, and it also requires that the measurement aperture should be larger than the actual source. This paper is to focus on the problem that neither of the above-mentioned requirements can be met, and to examine the feasibility of reconstructing the sound field radiated by partial source, based on double-layer pressure measurements made in a non-free field by using patch NAH combined with sound field separation technique. And also, the sensitivity of the reconstructed result to the measurement error is analyzed in detail. Two experiments involving two speakers in an exterior space and one speaker inside a car cabin are presented. The experimental results demonstrate that the patch NAH based on single-layer pressure measurement cannot obtain a satisfied result due to the influences of disturbing sources and reflections, while the patch NAH based on double-layer pressure measurements can successfully remove these influences and reconstruct the patch sound field effectively.     

Hybrid near-field acoustic holography

Hybrid near-field acoustical holography (NAH) is developed for reconstructing acoustic radiation from an arbitrary object in a cost-effective manner. This hybrid NAH is derived from a modified Helmholtz equation least squares (HELS) formula that expands the acoustic pressure in terms of outgoing and incoming waves. The expansion coefficients are determined by solving an overdetermined linear system of equations obtained by matching the assumed-form solution to measured acoustic pressures through the least squares. Measurements are taken over a conformal surface around a source at close range so that the evanescent waves can be captured. Next, the modified HELS is utilized to regenerate as much acoustic pressures on the conformal surface as necessary and take them as input to the Helmholtz integral formulation implemented numerically by boundary element method (BEM). The acoustic pressures and normal velocities on the source surface are reconstructed by using a modified Tikhnov regularization (TR) with its regularization parameter determined by generalized cross validation (GCV) method. Results demonstrate that this hybrid NAH combines the advantages of HELS and inverse BEM. This is because a majority of the input data are regenerated but not measured, thus the efficiency of reconstruction is greatly enhanced. Meanwhile, the accuracy of reconstruction is ensured by the Helmholtz integral theory and modified TR together with GCV method, provided that HELS converges fast enough on the measurement surface. Numerical examples of reconstructing acoustic quantities on the surface of a simplified engine block are demonstrated. [Work supported by NSF.]