Combined Helmholtz equation-least squares method for reconstructing acoustic radiation from arbitrarily shaped objects

A combined Helmholtz equation-least squares (CHELS) method is developed for reconstructing acoustic radiation from an arbitrary object. This method combines the advantages of both the HELS method and the Helmholtz integral theory based near-field acoustic holography (NAH). As such it allows for reconstruction of the acoustic field radiated from an arbitrary object with relatively few measurements, thus significantly enhancing the reconstruction efficiency. The first step in the CHELS method is to establish the HELS formulations based on a finite number of acoustic pressure measurements taken on or beyond a hypothetical spherical surface that encloses the object under consideration. Next enough field acoustic pressures are generated using the HELS formulations and taken as the input to the Helmholtz integral formulations implemented through the boundary element method (BEM). The acoustic pressure and normal component of the velocity at the discretized nodes on the surface are then determined by solving two matrix equations using singular value decomposition (SVD) and regularization techniques. Also presented are in-depth analyses of the advantages and limitations of the CHELS method. Examples of reconstructing acoustic radiation from separable and nonseparable surfaces are demonstrated.     

Radiation force on absorbing targets and power measurements of a high intensity focused ultrasound (HIFU) source

Based on the analytic expressions for the radiated field of a circular concave piston given by Hasegawa et al.,an integral for calculation of the radiation force on a plane absorbing target in a spherically focused field is derived.A general relation between acoustic power P and normal radiation force Fn is obtained under the condition of kr 1.Numerical computation is carried out by using the symbolic computation program for practically focused sources and absorbing circular targets.The results show that,for a given source,there is a range of target positions where the radiation force is independent of the target's position under the assumption that the contribution of the acoustic field behind the target to the radiation force can be neglected.The experiments are carried out and confirm that there is a range of target positions where the measured radiation force is basically independent of the target's position even at high acoustic power (up to 700 W).It is believed that when the radiation force method is used to measure the acoustic power radiated from a focused source,the size of the target must be selected in such a way that no observable sound can be found in the region behind the target.     

Reduction of sound radiation by using force radiation modes

The location of a vibration source within a machine is sometimes found to have a significant effect upon its radiated acoustic power. It is known that a simple reduction of vibration cannot always reduce the radiated acoustic power, so that treatments based on analysis of a structure’s vibration modes are not always effective. At the same time, radiation mode analysis is known to be a powerful tool for interpreting sound radiation since those modes are independent of a structure’s surface vibration. However, knowledge of the radiation modes alone cannot be used directly to understand the relationship between vibration source location and acoustic power radiation. In this paper, it is shown that the radiation mode concept can be extended to understand the relationship between acoustic power and driving force distribution by considering the product of the structure’s mobility matrix and the radiation modes: the resulting functions are here defined to be force radiation modes (f_rad-modes). An example is presented in which the acoustic power radiated by a simply-supported, baffled beam is reduced by using guidance provided by the structure’s force radiation modes. The results demonstrate that the force radiation modes can be used to guide the reduction of radiated acoustic power by changing the driving force location without the need to perform additional calculations or experiments.     

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.     

Shape design sensitivity analysis for the radiated noise from the thin-body

Many industrial applications generally use thin-body structures in their design. To calculate the radiated noise from vibrated structure including thin bodies, the conventional boundary element method (BEM) using the Helmholtz integral equation is not an effective resolution. Thus, many researchers have studied to resolve the thin-body problem in various physical fields. No major study in the design sensitivity analysis (DSA) fields for thin-body acoustics, however, has been reported.A continuum-based shape DSA method is presented for the radiated noise from the thin-body. The normal derivative integral equation is employed as an analysis formulation. And, for the acoustic shape design sensitivity formulation, the equation is differentiated directly by using material derivative concept. To solve the normal derivative integral equation, the normal velocities on the surface should be calculated. In the acoustic shape sensitivity formulation, not only the normal velocities on the surface are required but also derivative coefficients of the normal velocities (structural shape design sensitivity) are also required as the input. Hence, the shape design sensitivity of structural velocities on the surface, with respect to the shape change, should be calculated. In this research, the structural shape design sensitivities are also obtained by using a continuum approach. And both a modified interpolation function and the Cauchy principle value are used to regularize the singularities generated from the acoustic shape design sensitivity formulation.A simple annular disk is considered as a numerical example to validate the accuracy and efficiency of the shape design sensitivity equations derived in this research. The commercial BEM code, SYSNOISE, is utilized to confirm the results of the developed in-house code based on a normal derivative integral equation. To validate the calculated design sensitivity results, central finite difference method (FDM) is employed. The error between FDM and the analytical result are less than 3%. This comparison demonstrates that the proposed design sensitivities of the radiated pressure are very accurate.     

Research on sound radiation from body in free field

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Numerical analysis of immersed finite cylindrical shells using a coupled BEM/FEM and spatial spectrum approach

This study numerically analyzes submerged cylindrical shells using a coupled boundary element method (BEM) with finite element method (FEM) in conjunction with the wave number theory, in which the spatial Fourier transform of surface velocity for cylinders is directly related to pressure in a far field. The acoustic loading is formulated using a symmetric complex matrix derived from a boundary integral equation where the symmetry is based on an acoustic reciprocal principle for surface acoustics. In this formulation the acoustic loading matrix is a large acoustic element whose degree of freedom is connected to the normal displacement of the vibrating structures. The coupled BEM/FEM equation is a banded, symmetric matrix, and thus its bandwidth can be minimized using a proper algorithm. This formulation significantly increases numerical efficiency. The computed normal velocity is thus transformed to wave number representation to examine acoustic radiation. A finite plane cylindrical shell, without attached stiffeners, and a shell with internal ring stiffeners are chosen to demonstrate the present analysis procedure. The far field pressure computed directly from the integral equation and predicted by wave number theory correlates closely with increasing vibrating frequency. Meanwhile, the influences of the internal ring structures on acoustic radiation are examined using the wave number theory, which helps in understanding how internal structures influence radiated noise.     

Experimental validations of the HELS method for reconstructing acoustic radiation from a complex vibrating structure

This paper presents experimental validations of the Helmholtz Equation Least Squares (HELS) method [Wang and Wu, J. Acoust. Soc. Am. 102, 2020-2032 (1997); Wu and Wang, U.S. Patent Number 5712805 (1998); Wu, J. Acoust. Soc. Am. 107, 2511-2522 (2000)] on reconstruction of the radiated acoustic pressures from a complex vibrating structure. The structure under consideration has geometry and dimensions similar to those of a real passenger vehicle front end. To simulate noise radiation from a vehicle, a high fidelity loudspeaker installed inside the structure at the location of the engine is employed to generate both random and harmonic acoustic excitations. The radiated acoustic pressures are measured over a finite planar surface above the structure by a microphone. The measured data are taken as input to the HELS formulation to reconstruct the acoustic pressures on the top surface of the structure as well as in the field. The reconstructed acoustic pressures are then compared with measured ones at the same locations. Also shown are comparisons of the reconstructed and measured acoustic pressure spectra at various locations on the surface. Results show that satisfactory reconstruction can be obtained on the top surface of the structure subject to both random and harmonic excitations. Moreover, the more measurements and the closer their distances to the source surface, the more accurate the reconstruction. The efficiency of the HELS method may decrease with increasing of the excitation frequency. This high frequency difficulty is inherent in all expansion theories.     

Predict acoustic field of vibrating beams by Helmholtz integral equation and BEM

New method of predicting the acoustic field radiation of vibratingbodies is presented.It is based on the Helmholtz integral equation and bounda-ry element method and applicable to the use of computers.The acoustic field ofthe cantilever beams has been calculated and measured.Test results haveproved the effectiveness of the predicting method.     

Dynamic acoustic radiation force acting on cylindrical shells: theory and simulations

An object placed in an acoustic field is known to experience a force due to the transfer of momentum from the wave to the object itself. This force is known to be steady when the incident field is considered to be continuous with constant amplitude. One may define the dynamic (oscillatory) radiation force for a continuous wave-field whose intensity varies slowly with time. This paper extends the theory of the dynamic acoustic radiation force resulting from an amplitude-modulated progressive plane wave-field incident on solid cylinders to the case of solid cylindrical shells with particular emphasis on their thickness and contents of their hollow regions. A new factor corresponding to the dynamic radiation force is defined as Y(d) and stands for the dynamic radiation force per unit energy density and unit cross sectional surface. The results of numerical calculations are presented, indicating the ways in which the form of the dynamic radiation force function curves are affected by variations in the material mechanical parameters and by changes in the interior fluid inside the shell's hollow region. It was shown that the dynamic radiation force function Y(d) deviates from the static radiation force function for progressive waves Y(p) when the modulation frequency increases. These results indicate that the theory presented here is broader than the existing theory on cylinders.     

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.]     

Comparison of laser interferometry and radiation force method of measuring ultrasonic power

The aim of this paper is to compare two different methods for the calculation of the ultrasonic output power of underwater transducers: the radiation force balance, which is the standard method, and the laser heterodyne interferometry, which is rather used to depict displacement or velocity distributions of the acoustic field. Here it is shown that the latter can also be used to calculate the acoustic time-average power with an uncertainty of about 22%, the radiation force balance giving an uncertainty of 12% (with 95% confidence). The interferometry experiments performed with two transducers working at 2.25 MHz and 8.25 MHz showed that they produce different acoustic fields (respectively Gaussian and Lorentz-sigmoidal distributions). Taking into account the acoustic field profiles, the acoustic time-average power from interferometry was calculated. It was found very similar to the time-average power measured with the radiation force balance in the plane-wave assumption.     

Excitation of the acoustic and leaky waves in the atmosphere by a sub-surface seismic source

We study excitation of acoustic, leaky, and surface waves by a time-harmonic force source located in a homogeneous isotropic elastic half-space contacting a homogeneous gas. The force acts in the normal direction to the interface between the media. We consider the case where the sound velocity in the gas is less than the velocity of the Rayleigh wave propagating along the surface of the solid. An expression is derived for the period-averaged radiation power of the surface Stoneley wave. The total radiation power is calculated for the acoustic wave in the gas and for the leaky pseudo-Rayleigh wave. Variations in the radiation powers of the surface and leaky waves are analyzed as functions of the source depth. If the velocities of compressional and shear waves in the elastic medium significantly exceed the sound velocity in the gas, then the radiation power of the Stoneley wave turns out to be a factor of 10⁶–10⁸ smaller than the radiation powers of other waves. The radiation power of the Stoneley wave decreases monotonically with increasing source depth, and the decrease becomes more pronounced with the increase in the difference between the acoustic impedances of the contacting media. If the shear-wave velocity in the solid is close to the sound velocity in the gas, then the radiation power of the Stoneley wave is comparable with the radiation powers of other waves and exhibits maximum at a certain source depth. For some parameters of the gas and the solid, and for certain source depths, the Stoneley wave carries away more than a half of the total radiation power. It is shown that, for certain relations between the parameters of the media, the radiation power of the Stoneley wave increases due to redistribution of the radiated power from the pseudo-Rayleigh leaky wave. The total power of these waves remains approximatly constant and, with accuracy of the order of 10⁻³, is equal to the radiation power of the Rayleigh wave at the vacuum-solid interface. It is shown that the acoustic-wave power which can be transmitted to the upper layers of the atmosphere during an earthquake does not exceed 0.01% of the total power radiated at a given frequency. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 7, pp. 577–592, July 2006.     

船尾伴流场-导管-螺旋桨互作用噪声预报研究

研究了导管螺旋桨低频离散谱噪声辐射机理和预报方法。依据线性声学原理,导管螺旋桨噪声场为螺旋桨直接辐射噪声与导管散射噪声之和,并利用速度势面元法分析流场,得到导管螺旋桨非定常力,将其作为FW-H方程的源项,求解得到螺旋桨直接辐射声。导管散射声通过Kirchoff积分方程求解获得。由于导管桨的导管是短导管,其算例分析计算表明,低频情况下导管散射声级远小于螺旋桨直接辐射声级。并将导管螺旋桨离散谱噪声级与测量所得的实桨离散谱噪声级进行了比较,证实导管螺旋桨离散谱噪声理论预报结果能够较合理的反映实桨离散谱噪声的量值。      

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.     

利用源强密度声辐射模态重建声场

为了利用声场中少量测点声压数据精确重建复杂结构的辐射声场,提出了源强密度声辐射模态分析理论和声场重建公式.在结构表面定义的空间上,利用以源强密度分布函数为参量的结构辐射声功率泛函表达式定义了一个线性自伴正辐射算子,该算子的特征函数为结构的源强密度声辐射模态.然后通过对矩形平板和带有半球帽的圆柱体的源强密度声辐射模态的分析,证明了源强密度声辐射模态具有空间滤波特性,并利用该性质建立了声场重建公式.球体仿真和平板实验验证了所提出的声场重建方法的可行性和稳健性.基于源强密度声辐射模态的声场重建方法简单,利用较少测点数据就可以获得较高的声场重建精度,特别适合于复杂结构的低频声场重建.     

SiO2波导布拉格器件中声表面波及声光相互作用分析

针对SiO2光波导声光布拉格器件,计算了SiO2非对称平板波导TE模式的横向场分布;给出了SiO2/ZnO/Air层状介质结构的性能方程、运动方程和麦克斯韦方程,推导出这种层状结构的特征方程,并结合所满足的边界条件,得到了各层介质的位移及电磁场分布;计算了声表面波所引起的光学相对介质隔离率张量的变化,最后讨论了声光衍射效率和光场与声场的重叠积分、声功率、声频率、声孔径和光波导参数之间的关系。结果表明,在低频范围内光场与声表面波场重叠良好;低阶模的重叠积分始终大于高阶模重叠积分,最低阶模与声表面波相互作用最强,所需声功率最小;当声功率一定时,增加声孔径可以提高衍射效率。     

Estimation of broadband acoustic power due to rib forces on a reinforced panel under turbulent boundary layer-like pressure excitation. II. Applicability and validation

The previous paper showed that, when the attachment forces on a rib-reinforced panel subjected to turbulent boundary layer excitation can be considered to radiate independently, the rib-related acoustic power in a broad (e.g., one-third octave) frequency band can be estimated as the product of the average mean squared force, the real part of the radiation admittance of an attachment force, and the number of ribs. This paper shows that the radiation condition is always approximated when the acoustic wavelength is less than twice the rib spacing of a periodically reinforced panel, and generally applies at lower frequencies where the acoustic wavelength is less than four times the rib spacing. The procedure is used to estimate the broadband acoustic power radiated per rib of an infinite periodically reinforced membrane and plate in water, and the results are shown to agree with those of "exact" calculations.     

Structural-acoustic coupling in complex shaped cavities

Structural-acoustic interaction is a topic of increasing interest in the study of vehicle noise control. During recent years particular attention has been devoted towards analytical and numerical methods for the analysis of acoustic problems of complex shaped cavities. Among them non-local methods are very appropriate for the low and medium frequency range. The integral formulation of the Helmholtz equation allows the sound pressure level, radiated by a vibrating panel, to be determined. The method may be used to account for the acoustic structural coupling and can be straightforwardly extended to consider interconnected cavities, internal vibrating panels, internal barriers and absorbing materials. Comparisons with experimental results as well as with modal methods are presented which show satisfactory agreement.