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J Panzer J 潘泽尔R&D TEAM, Salgen, Germany
P Macey P 梅西PACSYS, Nottingham, UK 英国诺丁汉 PACSYS 公司

This paper reports on the investigation of the sound pressure response and radiation impedance of a loudspeaker horn driven by an electro-dynamic transducer.

The investigation has three parts, the measurement and further the simulation using two independent programs, using finite and boundary element methods. Issues associated with meshing for the BEM and FEM and the experimental setup are discussed.
调查包括三个部分:测量和使用有限元和边界元方法的两个独立程序进行的模拟。讨论了与 BEM 和 FEM 的网格划分以及实验装置有关的问题。


In this paper we present the modeling of the frequency response of a given horn. Our curiosity is focused on the results of the simulation software applications used and how the results would compare.

Also presented is the measured directivity, which provides a further point of comparison. The simulation tools used solve the wave-equation in the frequency domain for given boundary conditions.

As a result we can obtain observations of the sound pressure response and the radiation impedance.
To understand the performance of the horn, independent of its environment, and without any contribution from diffraction of baffle edges and room acoustic effects, it is informative to study its performance when it is mounted in an infinite rigid baffle and radiating into a half space.
For comparison of the calculation results we selected two quantities. Firstly, the lumped acoustic radiation impedance at the throat of the waveguide. Secondly, the sound pressure level at a certain fixed distance at various angles in front of the baffle.
The lumped acoustic radiation impedance is notoriously difficult to measure. However, it is straightforward to extract from a simulation model.

The radiation impedance is an important parameter for the design of a waveguide because the radiated power is proportional the radiation resistance. Further, its curves are ideal candidates when it comes to compare the results of simulation software.

This is so because the calculation of the lumped radiation impedance involves the whole acoustic field as an integral value.
The reason, why we selected the sound pressure is that it is easily reproduced by calculation, and also because we have had available the measurement device for taking directivity sound pressure measurements.


The device under test is a little horn or waveguide of over-all dimensions:
Mouth width height
Throat diameter 喉咙直径
Length 长度
Figure 1: Sketch of waveguide
图 1:波导草图
Figure 1 shows a sketch of the view into the mouth, a cut of the view of the curved sides and a cut of the view of the linear sides.
图 1 显示了口腔内视角草图、弧形侧面视角剪切图和直线侧面视角剪切图。
The flair of the horn does not follow any particular mathematical function. It has been designed with the help of a CAD-tool. Basically the waveguide opens linearly with a slight curvature in the first quarter along the horn.
喇叭的喇叭口并不遵循任何特定的数学函数。它是在 CAD 工具的帮助下设计的。基本上,波导管是线性打开的,在喇叭的前四分之一处有轻微的弯曲。

Additionally there is a bump on two opposite sides yielding a constriction along the center of the horn.

3 整块声辐射阻抗

During development of a waveguide for a loudspeaker the designer typically has a close look at the curve of the lumped acoustic radiation impedance. This is so because the acoustic output power of the horn is proportional to the real part of this impedance.

Hence, the variations of the impedance curve will inevitably have an effect on the sound quality of the speaker. When we speak here of the radiation impedance we mean the lumped acoustic impedance of the piston mode at the throat:
with volume velocity and sound pressure . We assume here that the acoustic velocity is constant (pistonic) over the cross-sectional area of the throat and points into the same direction. In formula 1 the sound pressure is thought to be a mean value over . A plausible meanvalue can be obtained with the help of the acoustic power, which is in general:
声速 和声压 。在此,我们假定声速 在喉管 横截面上是恒定的(活塞式),并指向同一方向。在公式 1 中,声压被认为是 上的平均值。一个可信的平均值可以借助声功率来获得,一般来说,声功率为:......:
here, the factor " " comes from the crest-factor of a sinusoidal wave where we assume and to be peak values of a sinuidal signal. The acoustic velocity is a vector and we take the conjugate complex value. can be any area. Then is the acoustic power through that area.
在这里,系数 " "来自正弦波的波峰因数,我们假定 为正弦信号的峰值。声速是一个矢量,我们取其共轭复数值。 可以是任何面积。那么 就是通过该区域的声功率。
If we combine formula 1 and formula 2 under the assumptions of uniform velocity (pistonic, ) and being the cross-section at the throat then this would yield for the radiation impedance:
如果我们在匀速(活塞式, )和 为喉部横截面的假设条件下将公式 1 和公式 2 结合起来,就可以得出辐射阻抗:
hence, the lumped radiation impedance at the throat would be proportional to the mean-value of the pressure, if the horn is excited by a uniform acoustic velocity. This condition can be easily satisfied inside a simulation algorithm.

However, for a measurement the uniform velocity condition would be

Proceedings of the Institute of Acoustics

more difficult to achieve, especially at frequencies where the wavelength is small compared to the dimension of the throat.
Having available the impedance one can calculate the acoustic power of the piston mode of the horn. Its real part is proportional to the acoustic power radiated into the far-field:
有了阻抗 ,我们就可以计算出喇叭活塞模式的声功率。其实际部分与辐射到远场的声功率成正比:
One can show that the lumped acoustic radiation impedance approaches always a certain value asymptotically at high frequencies, which is
Formula 5 can be used to normalize the radiation impedance, so that the real part of any curve would approach the value of one at high frequencies. The imaginary goes to zero.
公式 5 可用来对辐射阻抗进行归一化处理,这样任何曲线的实部在高频时都会接近 1 的值。虚部为零。
Figure 2: Calculated normalized lumped radiation resistance at the throat of example horn as shown in figure 1.
图 2:图 1 所示喇叭喉部的计算归一化叠加辐射阻抗。
PAFEC red and AKABAK blue.
Figure 2 shows the result of the calculation of formula 3 . Displayed is the real part of the normalized lumped radiation impedance at the throat of the horn if excited by uniform driving velocity (pistonmode).
图 2 显示了公式 3 的计算结果。图中显示的是在均匀驱动速度(活塞模式)激励下,喇叭喉部归一化叠加辐射阻抗的实部。

The normalization is done by applying formula 5 , hence the curves should approach the value of one at high frequencies. The curves of Figure 2 can be regarded as a transmission characteristic and, hence, have a high-pass character.
归一化是通过应用公式 5 来实现的,因此曲线在高频时应接近 1 的值。图 2 的曲线可视为传输特性,因此具有高通特性。

The stop-band is at low frequencies below approximately . At high frequencies the radiated power will follow the spectrum curve of the velocity at the throat. Between and the horn radiates selectively. For example at the radiated sound power would be more than twice than at high frequencies. At the transmission is strongly attenuated. These fluctuations could be altered by changing the horn geometry.
阻带位于低频处,大约低于 。在高频率下,辐射功率将遵循喉部速度的频谱曲线。在 之间,喇叭会选择性地辐射。例如,在 ,辐射声功率是高频的两倍多。在 处,声波的传播会强烈衰减。这些波动可以通过改变喇叭的几何形状来改变。
In this paper we want to draw the attention to the fact that there are two curves which are almost identical. The red curve is the result of the PAFEC-simulator whereas the blue curve results from the AKABAK-simulator.
在本文中,我们希望提请注意有两条曲线几乎完全相同。红色曲线是 PAFEC 模拟器的结果,而蓝色曲线则是 AKABAK 模拟器的结果。

These two software tools calculate the same response of the device under test. Their results are almost identical although the internal working of these software tools is different.


For the measurement of the directivity the horn is driven by a compression driver. The exit of its phase plug has a diameter of which fits perfectly to the throat diameter of the waveguide.
为了测量指向性,号角由压缩驱动器驱动。其相位插头的出口直径为 ,与波导管的喉部直径完全吻合。

Proceedings of the Institute of Acoustics

In this paper we focus on the normalized directivity. The normalization hides the influence of the properties of the driver. Hence, there is no need to describe the compression driver in detail.

The response is linear because adjusting the drive voltage did not affect the directivity pattern.
Figure 3: Directivity measurement set-up
图 3:指向性测量装置
The horn is fitted into a rotatable baffle of size as shown in figure 3 . The directivity is the sound pressure in the far-field at various position of equal distance between the microphone and the center of the mouth. The distance of the microphone is . The baffle is rotated in the horizontal plane in 5 deg steps. The other dimension of rotation comes from a rotation about the on-axis. In this way a whole so-called balloon directivity measurement could be performed.
如图 3 所示,喇叭安装在一个尺寸为 的可旋转障板上。指向性是指在传声器与口腔中心距离相等的不同位置上的远场声压。传声器的距离为 。障板以 5 度为单位在水平面内旋转。另一个旋转维度来自于围绕轴线的旋转。这样就可以进行整个所谓的气球指向性测量。

However, in this paper we regard only the polar measurement of the horizontal plane (azimuth deg) and of the vertical plane (azimuth deg).
不过,在本文中,我们只考虑水平面(方位 度)和垂直面(方位 度)的极坐标测量。
Figure 4: Frequency-directivity contour plot of example horn in the horizontal plane (linear walls)
图 4:水平面(线性墙)上示例喇叭的频率-指向性等值线图
Figure 4 displays the measurement of normalized sound pressure level taken at 19 microphone positions in a regular angular range between 0 deg to 90 deg. The map is mirrored to display the whole range from -90 deg to 90 deg. The colored contours range from to . For normalization each spectrum of the sound pressure is divided by the spectrum in on-axis direction ( 0 deg). The fine outline displays the contour at which is also called the beamwidth curve. At low frequencies the horn should radiated almost omni-directional. The slight deviations are caused by diffraction of the finite baffle and other insufficiency of the measurement situation.
图 4 显示了在 0 度到 90 度之间的规则角度范围内 19 个传声器位置测量的归一化声压级。该图经过镜像处理,显示了从 -90 度到 90 度的整个范围。彩色等值线范围从 。为了归一化,每个声压频谱都除以同轴方向(0 度)的频谱。细轮廓显示的是 处的等值线,也称为波束宽度曲线。在低频情况下,喇叭的辐射几乎是全向的。轻微的偏差是由于有限障板的衍射和其他测量条件的不足造成的。

Proceedings of the Institute of Acoustics

Figure 5: Frequency-directivity contour plot of example horn in the vertical plane (curved walls)
图 5:垂直面(弧形壁)上示例喇叭的频率-指向性等值线图
Figure 5 shows the directivity in the vertical plane. In comparing the plot to the one of figure 4 it is obvious that the beamwidth is broader. This is likely caused by the smaller aperture due to the bumps inside the horn.
图 5 显示了垂直面的指向性。与图 4 相比,波束宽度明显变宽。这可能是由于喇叭内部的凹凸导致孔径变小所致。

4.1 Comparison of Directivity Calculations
4.1 指向性计算的比较

Both simulation applications are able to calculate the sound pressure at the same locations as used for the measurement. The virtual microphone is placed at constant distance from the center of the horn mouth. Starting at the on-axis direction calculations are done at various angles in the horizontal and vertical planes. The angular distance between the positions is 5 deg.
这两个模拟应用程序都能计算与测量相同位置的声压。虚拟传声器放置在距离喇叭口中心一定距离的位置 。从轴上方向开始,在水平面和垂直面的不同角度进行计算。各位置之间的角度距离为 5 度。
Figure 6: Directivity in horizontal plane (linear walls) PAFEC red and AKABAK blue at frequencies
图 6: PAFEC 红色和 AKABAK 蓝色在水平面(线性墙)上的指向性频率
Figure 5 shows the angular distribution of the sound pressure level in the horizontal plane. The polar-plot shows curves at and . At the horn radiates almost omnidirectional. At beam-forming starts. At radiation to the side is attenuated and most energy is radiated on-axis. The red curve is the result of the PAFEC-simulator and the blue curve is the result of the AKABAK-simulator. The plot demonstrates the similarity of the calculation results.
图 5 显示声压级在水平面上的角度分布。极坐标图显示了 的曲线。在 处,喇叭几乎是全向辐射。在 处开始形成波束。在 处,向两侧的辐射被衰减,大部分能量沿轴向辐射。红色曲线是 PAFEC 模拟器的结果,蓝色曲线是 AKABAK 模拟器的结果。该图显示了计算结果的相似性。
Figure 7: Directivity in horizontal plane (curved walls)
图 7:水平面内的指向性(弧形墙壁)
red and blue
红色和 蓝色
at frequencies  频率
Figure 5 shows the overlay of the simulated curves in the vertical plane. The broader pattern is caused by the bump which yields a constricted aperture.
图 5 显示了垂直面上模拟曲线的叠加。较宽的图案是由产生收缩孔径的凸起造成的。

4.2 Comparison of Measurement and Simulation
4.2 测量与模拟的比较

Figure 8: Directivity in horizontal plane (linear walls)
图 8:水平面的指向性(线性墙壁)
measured versus simulated
at frequencies  频率
Figure 8 demonstrates an overlay of simulated and measured directivity curves in the horizontal plane which curves are predominantly caused of the linear walls of the horn.
图 8 显示了水平面内模拟和测量的指向性曲线,这些曲线主要是由喇叭的线性壁造成的。

Figure 9: Directivity in horizontal plane (curved walls) measured versus simulated
图 9:水平面(弧形墙)的指向性测量值与模拟值对比
at frequencies  频率
Figure 9 shows an overlay of simulated and measured curves in the vertical plane.
图 9 显示了垂直面上模拟和测量曲线的叠加。
There is a good agreement between the simulated and the measured results. The deviations close to 90 deg are due to the fact that for the measurement the infinite baffle is finite after all.
模拟结果与测量结果非常吻合。接近 90 度的偏差是由于在测量中,无限障板毕竟是有限的。

5 模拟建模

The horn is mounted with its mouth flush in an infinite baffle. The infinite baffle is reflecting and its boundary condition means that the component of the acoustic velocity which is normal to its plane should be zero everywhere.

The Sommerfeld radiation condition must be satisfied, to ensure that the pressure field in the half space consists of outgoing waves.

5.1 Simulation Methods and Subdomain Modelling
5.1 仿真方法和子域建模

Many simulation methods are used in acoustics including finite difference, finite volume, finite element (FEM) and boundary element (BEM). In the current work FEM and BEM simulations are used.

Each of these methods has strengths and weaknesses and it is consequently sometimes beneficial to split the acoustic domain into subdomains.

Proceedings of the Institute of Acoustics

Figure 10: Subdomain modelling.
图 10:子域建模。
Left: Splitting at mouth of horn.
Right: Splitting at hemispherical surface
Two such decompositions are shown in figure 10. Finite difference and finite volume approaches are obtained directly from the Helmholtz equation, using differencing operations to approximate derivatives.
图 10 显示了两种这样的分解。有限差分法和有限体积法直接从亥姆霍兹方程中获得,使用差分运算来近似导数。

The finite element method is also closely related to the underlying differential equation, which is multiplied by a weighting function and integrated by parts.

The domain is decomposed into small elements such that the pressure can be assumed to vary as a linear combination of some suitable basis functions, e. g. low order polynomials, within each element. Applying the Galerkin method produces the FEM linear equations.
将域分解为小元素,这样就可以假定压力在每个元素内以一些合适的基函数(如低阶多项式)的线性组合形式变化。应用 Galerkin 方法可得出有限元线性方程。

The boundary element method attempts to solve a derived integral equation, using a set of local basis functions on the bounding surface of the acoustic volume; within the domain itself the solution of the Helmholtz equation is ensured by the properties of the Green's function.

Only FEM and BEM are considered further, as they are used for the results in this study, see also for example [4] and [5]. Both methods produce a set of linear equations which are solved to determine the pressure at the nodes in the model.

For FEM a large set of sparse equations are produced, as nodes are required on elements throughout the volume. For BEM a smaller dense set of equations is produced from the nodes on the surface elements.
对于 FEM,由于需要在整个体积的元素上设置节点,因此会产生大量稀疏方程组。对于 BEM,曲面元素上的节点会产生较小的密集方程组。

In both cases the element size has to be small enough to adequately represent the pressure variation, which in turn is usually related to the acoustic wavelength.

Analysis at higher frequencies requires smaller elements, more nodes and hence has greater computational requirements: CPU time, memory and disk space.
更高频率下的分析需要更小的元素、更多的节点,因此计算要求更高:CPU 时间、内存和磁盘空间。
The surface of the horn has a sharp fold, particularly near the throat, such that the radius of curvature is smaller than the element side-length which is needed from frequency-based considerations. There is hence a concern that the geometry may not be adequately represented.

This can be checked using mesh convergence.
The geometry and boundary conditions of the idealized problem have two planes of symmetry. It is thus possible to reduce the problem size by using a quarter model. All the simulations in this paper were using quarter models.

5.2 AKABAK 5.2 阿卡巴科

The AKABAK simulator [1] calculates the acoustic field inside the horn and in front of the baffle with the help of the boundary element method. The calculation of the Helmholtz Integral is usually a two stage process.
AKABAK 模拟器[1]借助边界元法计算喇叭内部和障板前的声场。亥姆霍兹积分的计算通常分为两个阶段。

Firstly, one has to solve for certain unknown parameters of the integral. If available, one can commence with the calculation of the observation points. In order to solve for the unknown parameters one divides the surface of the acoustic boundary into small elements.

Typically, the mesh-density needs to be found experimentally, one only knows that the result becomes exact with infinitely small elements. Otherwise we get an approximation.

Because only surface values of pressure and velocity are to be integrated, the mesh need to be of two dimensions only [3].

Proceedings of the Institute of Acoustics

5.3 Subdomain Modelling 5.3 子域建模

There exists a special Green-function which satisfies automatically the condition of zero normal velocity on the infinite baffle plane. Hence, there is no need to mesh this boundary.

However, this function would be valid only for boundary elements which are in or in front of the baffle.
Figure 11: Subdomain modeling
图 11:子域建模
If there are acoustic boundaries behind the infinite baffle the calculation of the acoustic field becomes more complicated. As our horn ends in the plane of the baffle all walls of the horn are behind the infinite baffle.

The trick to be applied consists of dividing the radiation domain into two subdomains as sketched in figure 11. In-between one creates an interface where at any point there is guarantied a continuity of parameters. The interface is acoustically transparent.
要使用的技巧包括将辐射域划分为两个子域,如图 11 所示。在这两个子域之间创建一个界面,在该界面上的任何一点都能保证参数的连续性。该界面在声学上是透明的。

On its surface the pressure of subdomain 1 is equal to the surface pressure of subdomain 2. Likewise, should the acoustic velocity be equal. Here we consider the velocity-component normal to the plane of the interface.

The negative sign comes in because one regards the vectors pointing into the subdomain.

5.4 Driving 5.4 驾驶

The specified boundary condition is concerned only with the surface velocity. The reflecting boundaries feature a zero normal velocity. However, the cross-section at the throat of the horn has an imprinted velocity.

This motion is specified here to be uniformly distributed and simply of value one, as labeled in figure 11 by velocity . The surface pressure is the parameter of solving.
这种运动在这里被指定为均匀分布且简单取值为 1,如图 11 中速度 所示。表面压力是求解参数。

5.5 Meshing 5.5 网格划分

The geometry of the acoustic boundary is meshed with the help of GMSH [3], which is an external meshing tool specialized for producing elements for the boundary and finite element analysis. Only a 2D-mesh needs to be produced as we consider only the surface.
声学边界的几何形状是在 GMSH [3] 的帮助下进行网格划分的,GMSH 是一种外部网格划分工具,专门用于为边界和有限元分析生成元素。由于我们只考虑了表面,因此只需要生成一个二维网格。

The imported mesh is then refined by AKABAK to make sure all elements are smaller than a specified edge length. For the shown simulation a mesh of 1906 triangles was used for the boundaries of the horn and 1040 for the interface. We assume a constant pressure over each element.
然后通过 AKABAK 对导入的网格进行细化,以确保所有元素都小于指定的边长。在所示模拟中,喇叭边界使用了 1906 个三角形网格,界面使用了 1040 个三角形网格。我们假设每个元素上的压力都是恒定的。

Experimentation of varying the mesh-density showed that results could be regarded sufficiently accurate up to a frequency of .
改变网格密度的实验结果表明,在 的频率范围内,可以认为结果足够精确。

Proceedings of the Institute of Acoustics

5.6 PAFEC 5.6 泛非教育论坛

Two modeling strategies were employed to analyze the baffled horn in PAFEC VibroAcoustics [2].
PAFEC VibroAcoustics [2] 中采用了两种建模策略来分析障板喇叭。

In the first approach the subdomain splitting of figure 10 -left was used with 10 -noded quadratic tetrahedral acoustic finite elements used in subdomain 1 and a Rayleigh integral boundary element composed of 6-noded quadratic triangles.
在第一种方法中,采用了图 10 左侧的子域分割方法,在子域 1 中使用了 10 个编码的二次四面体声学有限元,以及一个由 6 个编码的二次三角形组成的瑞利积分边界元。

The model BEM1 had 55096 acoustic finite element degrees of freedom and 2453 acoustic boundary element degrees of freedom. Based on a criterion of 3 quadratic elements/wavelength should be valid up to , however because of the small radius of curvature on the "fold", the size of elements may perhaps not have adequately represented the geometry.
BEM1 模型有 55096 个声学有限元自由度和 2453 个声学边界元自由度。根据 3 个二次元/波长的标准, ,但由于 "褶皱 "的曲率半径较小,元件的大小可能无法充分反映几何形状。

In many situations, where there is not constricted geometry, deviations by distances small compared with the acoustic wavelength should not significantly affect the results. To confirm this, a finer mesh density model BEM2 was run.
在许多情况下,如果几何形状不受限制,与声波波长相比距离较小的偏差应该不会对结果产生重大影响。为了证实这一点,我们运行了更精细的网格密度模型 BEM2。

This had 339045 acoustic finite element degrees of freedom and 9212 boundary element degrees of freedom, should be valid to and was much closer the geometry in the problematic area of the sharp fold near to the throat.
它有 339045 个声学有限元自由度和 9212 个边界元自由度,有效期应为 ,而且更接近喉部附近尖锐褶皱问题区域的几何形状。
Figure 12: Throat impedance computed by method BEM1 & BEM2
图 12:用 BEM1 和 BEM2 方法计算的喉咙阻抗
Figure 12 compares the throat impedance for the two models. The agreement is good. BEM2 is used as a reference standard below in figure 14
图 12 比较了两种模型的喉部阻抗。两者的一致性很好。下图 14 以 BEM2 为参考标准

Proceedings of the Institute of Acoustics

Figure 13: Throat impedance computed by WEE1 & WEE2
图 13:WEE1 和 WEE2 计算的喉咙阻抗
In the second approach the subdomain splitting was as in figure 10-right. Acoustic finite elements were used in subdomain 1 extending from the throat to a hemispherical surface and wave envelope elements were used in subdomain 2 for the remainder of the half space.
第二种方法的子域划分如图 10 右所示。从喉部延伸到半球形表面的子域 1 中使用了声学有限元,而半空间的其余部分则在子域 2 中使用了波包络有限元。

Wave envelope elements are similar to finite elements but extend out to infinity.

The basis functions are outward traveling waves, monopole, dipole, etc. A wave envelope element based approach can be faster than a BEM-based approach, but the accuracy is affected by the radius of the spherical surface at the interface between the subdomains and the number of terms used for outward traveling waves (= radial order).
基函数包括向外行波、单极子、偶极子等。基于波包络元的方法可能比基于 BEM 的方法更快,但精度会受到子域之间界面的球面半径和用于向外行波的项数(= 径向阶)的影响。

Model WEE1 had the hemispherical surface extending out to and had radial order 2 and a total of 173915 acoustic degrees of freedom. Model WEE2 had a hemispherical radius of , radial order 6 and a total of 229368 acoustic degrees of freedom. Both meshes were valid to , based on the 3 quadratic elements/wavelength criterion. Figure 13 compares throat impedances for WEE1 & WEE2.
模型 WEE1 的半球面延伸至 ,径向阶数为 2,共有 173915 个声学自由度。模型 WEE2 的半球半径为 ,径向阶数为 6,共有 229368 个声学自由度。根据 3 个二次元/波长准则,两个网格的有效范围均为 。图 13 比较了 WEE1 和 WEE2 的喉部阻抗。
Figure 14: Throat impedance computed by BEM2 & WEE2
图 14:BEM2 和 WEE2 计算的喉咙阻抗
Figure 14 compares the models BEM2 & WEE2. The model WEE1 becomes less accurate at higher frequencies.
图 14 比较了 BEM2 和 WEE2 模型。频率越高,WEE1 模型的精确度越低。

It is concluded that a wave envelope element approach can produce accurate results up to high frequency, but it is necessary to take care with the hemispherical radius and radial order.


We would like to extend the work to model the electro-dynamic compression driver so we can compare absolute sound pressure values of measurement and simulation.

In that scenario AKABAK would model the compression driver with the help of the lumped element method whereas PAFEC would use the finite element method as a modeling algorithm.
在这种情况下,AKABAK 将借助集合元素法对压缩驱动器进行建模,而 PAFEC 将使用有限元法作为建模算法。


The second author gratefully acknowledges the assistance of his colleague John King by doing some of the model preparation work.
第二作者感谢他的同事约翰-金(John King)的协助,为他做了一些模型准备工作。


  1. AKABAK, software simulation tool, www.randteam.de.
  2. PAFEC level 8.8, Acoustics manual, Strelley Hall, Nottingham, NG8 6PE, UK, on request.
    PAFEC 8.8 级,声学手册,Strelley Hall, Nottingham, NG8 6PE, UK,如有需要,请联系。
  3. GMSH, software for 2D and 3D meshing, www.gmsh.info.
  4. Atalla N, Sgard F: Finite Element and Boundary Methods in Structural Acoustics and Vibration; CRC Press, 2017.
    Atalla N, Sgard F:结构声学和振动中的有限元和边界方法》;CRC Press,2017 年。
  5. Marburg S, and Nolte B: Computational Acoustics of Noise Propagation in Fluids: Finite and Boundary Element Methods, Springer 2008.
    Marburg S 和 Nolte B:流体中噪声传播的计算声学:有限元和边界元方法》,施普林格出版社,2008 年。