Applying the Finite Element Method for
Modelling Loudspeaker Waveguide Directivity
应用有限元方法建模扬声器波导指向性
Master's Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Technology.
硕士论文，作为获得科学与技术硕士学位要求的一部分。
硕士论文，作为获得科学与技术硕士学位要求的一部分。
Isalmi, May 6, 2010
艾萨米，2010 年 5 月 6 日
艾萨米，2010 年 5 月 6 日
Supervisor: 指导老师：
Prof. Vesa Välimäki 教授韦萨·瓦利马基
Instructor: 教师：
Dr. Aki Mäkivirta, D.Sc 马克维塔博士，D.Sc
| Author: 作者： | Juha Holm | ||
| Name of the thesis: 论文名称： Translated Text: |
应用有限元方法建模扬声器波导指向性
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| Date: 日期： 翻译文本： | May 6, 2010 2010 年 5 月 6 日 | ||
| Number of pages: 页数： |
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| Faculty: 教师： | Electronics, Communication and Automation 电子、通信与自动化 |
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| Professorship: 教授职位： | S-89 | ||
| Supervisor: 指导老师： | Prof. Vesa Välimäki 瓦利马基教授 | ||
| Instructors: 教师： | Dr. Aki Mäkivirta, D.Sc 马克维塔博士，D.Sc |
Directivity has a great influence on a loudspeaker's perceived performance. Indoors directivity defines the indirect sound heard which influences the timbre and spatial perception.
直接性对扬声器的感知性能有很大影响。室内直接性定义了听到的间接声音，这影响了音色和空间感知。
Outdoors the directivity defines the sound heard off-axis of the speaker where most of the audience is.
户外，直接性定义了听众大部分所在位置听到的扬声器的非轴向声音。
直接性对扬声器的感知性能有很大影响。室内直接性定义了听到的间接声音，这影响了音色和空间感知。
Outdoors the directivity defines the sound heard off-axis of the speaker where most of the audience is.
户外，直接性定义了听众大部分所在位置听到的扬声器的非轴向声音。
The directivity of a single transducer primarily depends on driver size. Directivity can be modified using an acoustical waveguide. The primary purpose of a waveguide is to control the directivity of the source, but increased efficiency is a favourable side-effect.
单个换能器的方向性主要取决于驱动器的大小。可以通过声学波导来修改方向性。波导的主要目的是控制声源的方向性，但提高效率是其有利的副产品。
单个换能器的方向性主要取决于驱动器的大小。可以通过声学波导来修改方向性。波导的主要目的是控制声源的方向性，但提高效率是其有利的副产品。
This thesis concentrates on applying the Finite Element Method (FEM) to virtually prototype waveguides. The theory of FEM and its usability in acoustics is reviewed. Also theory for horn directivity is discussed.
本论文专注于将有限元方法（FEM）应用于虚拟原型波导。回顾了 FEM 的理论及其在声学中的应用性。还讨论了号角定向理论。
Major emphasis is on reviewing and developing a method for visualizing modelled and measured directivity in a comparable manner.
主要重点是回顾并开发一种方法，以可比的方式可视化模拟和测量的指向性。
本论文专注于将有限元方法（FEM）应用于虚拟原型波导。回顾了 FEM 的理论及其在声学中的应用性。还讨论了号角定向理论。
Major emphasis is on reviewing and developing a method for visualizing modelled and measured directivity in a comparable manner.
主要重点是回顾并开发一种方法，以可比的方式可视化模拟和测量的指向性。
There are three major outputs of the thesis. First, the method of virtual prototyping is validated by comparing and analyzing the measured and FEM modelled prototypes. Also the value of the method as a designing tool is emphasized.
论文有三个主要输出。首先，通过比较和分析测量和有限元模型的原型，验证了虚拟原型方法的有效性。同时，强调了该方法作为设计工具的价值。
Second, a visualization tool is created to enable comparison and analysis of the modelled and measured directivity. Third, a new method is created for combining a FEM model and laser velocimetry of a driver.
第二，创建了一个可视化工具，以实现对模型化和测量的直接性进行比较和分析。第三，创建了一种方法，用于结合 FEM 模型和驾驶员的激光测速仪。
The presented approach increases the accuracy of the model because the driver excitation can be made more realistic.
所提出的这种方法提高了模型的准确性，因为驾驶员的激励可以变得更加真实。
论文有三个主要输出。首先，通过比较和分析测量和有限元模型的原型，验证了虚拟原型方法的有效性。同时，强调了该方法作为设计工具的价值。
Second, a visualization tool is created to enable comparison and analysis of the modelled and measured directivity. Third, a new method is created for combining a FEM model and laser velocimetry of a driver.
第二，创建了一个可视化工具，以实现对模型化和测量的直接性进行比较和分析。第三，创建了一种方法，用于结合 FEM 模型和驾驶员的激光测速仪。
The presented approach increases the accuracy of the model because the driver excitation can be made more realistic.
所提出的这种方法提高了模型的准确性，因为驾驶员的激励可以变得更加真实。
Keywords: Loudspeaker, Directivity, Waveguide, Horn, Acoustics, Finite Element Method, FEM, Modelling
关键词：扬声器，指向性，波导，号角，声学，有限元方法，FEM，建模
关键词：扬声器，指向性，波导，号角，声学，有限元方法，FEM，建模
Acknowledgements 致谢
I want to thank my instructor Dr. Aki Mäkivirta for the numerous and creative discussions we had. It has been unique position for me to get to know a person who has such a combination of experience and jovial passion for learning.
我想感谢我的指导老师阿基·马基维塔博士，感谢我们之间进行的众多富有创意的讨论。能够认识一个既有丰富经验又对学习充满热情的人，对我来说是独一无二的体验。
I also want to thank my supervisor Prof. Vesa Välimäki for the constructive feedback given during the work.
我还想感谢我的导师韦萨·瓦利马基教授，在工作中给予的建设性反馈。
我想感谢我的指导老师阿基·马基维塔博士，感谢我们之间进行的众多富有创意的讨论。能够认识一个既有丰富经验又对学习充满热情的人，对我来说是独一无二的体验。
I also want to thank my supervisor Prof. Vesa Välimäki for the constructive feedback given during the work.
我还想感谢我的导师韦萨·瓦利马基教授，在工作中给予的建设性反馈。
I thank Jussi Väisänen for sharing his experience of designing loudspeakers as a system. I thank Ilkka Rissanen for the creative feedback about the measurement setup. I admire Juha Urhonen's MATLAB skills and I am grateful to him for sharing his knowledge.
我感谢 Jussi Väisänen 分享他作为系统设计扬声器的经验。我感谢 Ilkka Rissanen 对测量设置的创意反馈。我钦佩 Juha Urhonen 的 MATLAB 技能，并对他分享知识表示感激。
I thank Stephen Millar for always questioning the results and sharing his knowledge of CAD skills. I thank Darren Rose for motivating discussions and his work on improving the language of the thesis.
我感谢史蒂芬·米勒总是质疑结果并分享他的 CAD 技能知识。我感谢达伦·罗丝激发讨论并为改进论文语言所做的工作。
我感谢 Jussi Väisänen 分享他作为系统设计扬声器的经验。我感谢 Ilkka Rissanen 对测量设置的创意反馈。我钦佩 Juha Urhonen 的 MATLAB 技能，并对他分享知识表示感激。
I thank Stephen Millar for always questioning the results and sharing his knowledge of CAD skills. I thank Darren Rose for motivating discussions and his work on improving the language of the thesis.
我感谢史蒂芬·米勒总是质疑结果并分享他的 CAD 技能知识。我感谢达伦·罗丝激发讨论并为改进论文语言所做的工作。
Lastly I want to thank Genelec Oy for providing the intellectual environment for the project. I had the privilege to dedicate my work to the subject which the thesis is based.
最后，我想感谢 Genelec Oy 为项目提供了智力环境。我有幸将我的工作专注于论文所基于的主题。
Special thanks for creating this opportunity goes to Dr. Siamäk Naghian, the Director of Research & Development.
特别感谢创建这个机会的 Dr. Siamäk Naghian，研发与发展的主任。
最后，我想感谢 Genelec Oy 为项目提供了智力环境。我有幸将我的工作专注于论文所基于的主题。
Special thanks for creating this opportunity goes to Dr. Siamäk Naghian, the Director of Research & Development.
特别感谢创建这个机会的 Dr. Siamäk Naghian，研发与发展的主任。
Iisalmi, May 6, 2010
伊萨米，2010 年 5 月 6 日
伊萨米，2010 年 5 月 6 日
Contents 目录
Symbols and abbreviations ..... vii
符号和缩写...... vii
1 Introduction 1 引言
1.1 The aim of the work ..... 2
1.1 工作的目标......2
1.2 Outline 1.2 框架
2 Time-harmonic solution of the wave equation.
2 脉冲解的波动方程
3 Using the FEM to solve the wave equation ..... 7
使用 FEM 解决波动方程......7
3.1 Discretization the geometry to elements ..... 8
3.1 将几何形状离散化为元素...... 8
3.2 Selection of the element size ..... 9
3.2 元素尺寸的选择......9
3.3 The concept of degrees of freedom ..... 9
3.3 自由度的概念 ...... 9
3.4 Computational cost example ..... 10
3.4 计算成本示例 ..... 10
3.5 Applications of the finite element method to acoustics ..... 11
3.5 有限元方法在声学中的应用......11
4 The subjective importance of the directivity ..... 15
4 直达性的主观重要性......15
4.1 Human preference for sound ..... 15
4.1 人类对声音的偏好 ... 15
4.2 Loudspeaker and room interaction ..... 16
4.2 扬声器与房间的互动......16
5 Representing directivity ..... 19
5 表示指向性 ..... 19
5.1 Source radiation in space ..... 19
5.1 太空中的源辐射......19
5.2 Directivity factor (Q) ..... 20
5.2 方向性系数（Q）......20
5.3 Directivity index (DI) ..... 21
5.3 方向性指数（DI）...... 21
5.4 Beam width ..... 22
5.4 束宽......22
5.5 Polar diagram ..... 22
5.5 极坐标图......22
5.6 Frequency response ..... 23
5.6 频率响应......23
5.7 Balloon graph ..... 24
5.7 气球图......24
5.8 Directivity plot ..... 24
5.8 直达性图 ..... 24
6 The directivity of a direct radiator ..... 26
6 直接辐射器的指向性......26
7 The directivity of a horn radiator ..... 28
7 号筒扬声器的方向性......28
7.1 The exponential horn. ..... 29
7.1 指数喇叭。......29
7.2 The conical horn ..... 31
7.2 锥形号角 ..... 31
7.3 The exponential and conical horn as a waveguide ..... 31
7.3 指数形和圆锥形喇叭作为波导管......31
8 Using the FEM for modelling waveguides ..... 33
使用 FEM 对波导进行建模......33
8.1 Waveguide geometry ..... 33
8.1 波导几何结构 ..... 33
8.2 Motivation and strategy to simplify geometry ..... 34
8.2 动机和简化几何的策略......34
8.3 Modelling the medium and boundary conditions ..... 34
8.3 模型化介质和边界条件......34
8.3.1 The medium ..... 35
8.3.1 中间......35
8.3.2 Sound hard boundary ..... 35
8.3.2 响应硬边界......35
8.3.3 Axisymmetry ..... 35
8.3.3 轴对称性 ... 35
8.3.4 Normal acceleration ..... 35
8.3.4 正常加速度 ..... 35
8.3.5 Radiation boundary condition ..... 36
8.3.5 辐射边界条件 ..... 36
8.4 Improving the diaphragm movement model ..... 36
8.4 提高膈肌运动模型......36
8.4.1 Motivation for the development of the model ..... 36
8.4.1 模型开发的动机......36
8.4.2 The equivalent analogous circuit for loudspeaker driver ..... 37
8.4.2 响 loudspeaker 驱动 driver 的等效类似电路 ..... 37
8.4.3 Defining location dependent acceleration ..... 38
8.4.3 定义位置相关的加速度......38
8.4.4 Measuring the velocity of the diaphragm ..... 39
8.4.4 测量隔膜的速度......39
8.4.5 Using measured velocity in the FEM model ..... 40
8.4.5 在 FEM 模型中使用测量速度......40
9 Measurement system and visualization ..... 42
测量系统和可视化......42
9.1 The selection of the visualization method ..... 42
9.1 数据可视化方法的选择......42
9.2 Visualizing the directivity of the measurements. ..... 43
9.2 可视化测量的指向性。......43
9.3 Visualizing the directivity of the modelled results ..... 45
9.3 可视化模型结果的方向性......45
10 Analyzing the simulated waveguide ..... 46
10 分析模拟波导......46
11 Verifying modelling accuracy ..... 49
验证模型准确性......49
11.1 Accuracy of the simulated directivity ..... 49
11.1 模拟的方向性准确性......49
11.1.1 Comparison of the measured and modelled directivity ..... 50
11.1.1 测量值与模型预测的方向性比较......50
11.1.2 Analyzing the differences ..... 50
11.1.2 分析差异......50
11.2 Accuracy of the simulated frequency response ..... 51
11.2 模拟频率响应的准确性 ..... 51
11.2.1 Comparing the measured and modelled directivity ..... 52
11.2.1 比较测量值和模型化的方向性......52
11.2.2 Analyzing the differences ..... 53
11.2.2 分析差异......53
12 Conclusions ..... 54
结论......54
12.1 Usability of the simulation ..... 54
12.1 模拟的可用性......54
12.2 Guidelines for a successful waveguide design ..... 55
12.2 成功波导设计的指南......55
12.3 Outputs of the work ..... 55
12.3 工作的输出......55
12.4 Advantages of the virtual prototyping ..... 56
12.4 虚拟原型的优点......56
12.5 Future work ..... 56
12.5 未来工作......56
13 Bibliography ..... 58
参考文献 ..... 58
符号和缩写...... vii
1 Introduction 1 引言
1.1 The aim of the work ..... 2
1.1 工作的目标......2
1.2 Outline 1.2 框架
2 Time-harmonic solution of the wave equation.
2 脉冲解的波动方程
3 Using the FEM to solve the wave equation ..... 7
使用 FEM 解决波动方程......7
3.1 Discretization the geometry to elements ..... 8
3.1 将几何形状离散化为元素...... 8
3.2 Selection of the element size ..... 9
3.2 元素尺寸的选择......9
3.3 The concept of degrees of freedom ..... 9
3.3 自由度的概念 ...... 9
3.4 Computational cost example ..... 10
3.4 计算成本示例 ..... 10
3.5 Applications of the finite element method to acoustics ..... 11
3.5 有限元方法在声学中的应用......11
4 The subjective importance of the directivity ..... 15
4 直达性的主观重要性......15
4.1 Human preference for sound ..... 15
4.1 人类对声音的偏好 ... 15
4.2 Loudspeaker and room interaction ..... 16
4.2 扬声器与房间的互动......16
5 Representing directivity ..... 19
5 表示指向性 ..... 19
5.1 Source radiation in space ..... 19
5.1 太空中的源辐射......19
5.2 Directivity factor (Q) ..... 20
5.2 方向性系数（Q）......20
5.3 Directivity index (DI) ..... 21
5.3 方向性指数（DI）...... 21
5.4 Beam width ..... 22
5.4 束宽......22
5.5 Polar diagram ..... 22
5.5 极坐标图......22
5.6 Frequency response ..... 23
5.6 频率响应......23
5.7 Balloon graph ..... 24
5.7 气球图......24
5.8 Directivity plot ..... 24
5.8 直达性图 ..... 24
6 The directivity of a direct radiator ..... 26
6 直接辐射器的指向性......26
7 The directivity of a horn radiator ..... 28
7 号筒扬声器的方向性......28
7.1 The exponential horn. ..... 29
7.1 指数喇叭。......29
7.2 The conical horn ..... 31
7.2 锥形号角 ..... 31
7.3 The exponential and conical horn as a waveguide ..... 31
7.3 指数形和圆锥形喇叭作为波导管......31
8 Using the FEM for modelling waveguides ..... 33
使用 FEM 对波导进行建模......33
8.1 Waveguide geometry ..... 33
8.1 波导几何结构 ..... 33
8.2 Motivation and strategy to simplify geometry ..... 34
8.2 动机和简化几何的策略......34
8.3 Modelling the medium and boundary conditions ..... 34
8.3 模型化介质和边界条件......34
8.3.1 The medium ..... 35
8.3.1 中间......35
8.3.2 Sound hard boundary ..... 35
8.3.2 响应硬边界......35
8.3.3 Axisymmetry ..... 35
8.3.3 轴对称性 ... 35
8.3.4 Normal acceleration ..... 35
8.3.4 正常加速度 ..... 35
8.3.5 Radiation boundary condition ..... 36
8.3.5 辐射边界条件 ..... 36
8.4 Improving the diaphragm movement model ..... 36
8.4 提高膈肌运动模型......36
8.4.1 Motivation for the development of the model ..... 36
8.4.1 模型开发的动机......36
8.4.2 The equivalent analogous circuit for loudspeaker driver ..... 37
8.4.2 响 loudspeaker 驱动 driver 的等效类似电路 ..... 37
8.4.3 Defining location dependent acceleration ..... 38
8.4.3 定义位置相关的加速度......38
8.4.4 Measuring the velocity of the diaphragm ..... 39
8.4.4 测量隔膜的速度......39
8.4.5 Using measured velocity in the FEM model ..... 40
8.4.5 在 FEM 模型中使用测量速度......40
9 Measurement system and visualization ..... 42
测量系统和可视化......42
9.1 The selection of the visualization method ..... 42
9.1 数据可视化方法的选择......42
9.2 Visualizing the directivity of the measurements. ..... 43
9.2 可视化测量的指向性。......43
9.3 Visualizing the directivity of the modelled results ..... 45
9.3 可视化模型结果的方向性......45
10 Analyzing the simulated waveguide ..... 46
10 分析模拟波导......46
11 Verifying modelling accuracy ..... 49
验证模型准确性......49
11.1 Accuracy of the simulated directivity ..... 49
11.1 模拟的方向性准确性......49
11.1.1 Comparison of the measured and modelled directivity ..... 50
11.1.1 测量值与模型预测的方向性比较......50
11.1.2 Analyzing the differences ..... 50
11.1.2 分析差异......50
11.2 Accuracy of the simulated frequency response ..... 51
11.2 模拟频率响应的准确性 ..... 51
11.2.1 Comparing the measured and modelled directivity ..... 52
11.2.1 比较测量值和模型化的方向性......52
11.2.2 Analyzing the differences ..... 53
11.2.2 分析差异......53
12 Conclusions ..... 54
结论......54
12.1 Usability of the simulation ..... 54
12.1 模拟的可用性......54
12.2 Guidelines for a successful waveguide design ..... 55
12.2 成功波导设计的指南......55
12.3 Outputs of the work ..... 55
12.3 工作的输出......55
12.4 Advantages of the virtual prototyping ..... 56
12.4 虚拟原型的优点......56
12.5 Future work ..... 56
12.5 未来工作......56
13 Bibliography ..... 58
参考文献 ..... 58
Symbols and abbreviations
符号和缩写
| BEM | Boundary Element Method 边界元方法 |
| DOF | Degrees of Freedom 自由度 |
| DUT | Device Under Test 待测设备 |
| EMF | Electromagnetic Force 电磁力 |
| FFT | Fast Fourier Transform 快速傅里叶变换 |
| FEM | Finite Element Method 有限元法 |
| MLS | Maximum Length Sequence 最大长度序列 |
| PA | Public Address 公共广播 |
| PDE | Partial Differential Equation 偏微分方程 |
| SPL | Sound Pressure Level 声压级 |
1 Introduction 1 引言
Producing sound is a simple, but sometimes a complex matter. The first musical instruments date back thousands of years ago. The era of reproduced music began in the late nineteenth century with the introduction of the gramophone and its predecessors.
产生声音是一件简单但有时又复杂的事情。最早的音乐乐器可以追溯到几千年前。复制音乐的时代始于 19 世纪末，随着留声机及其前身的引入。
The era of amplified music began in the 1920s, when Kellogg & Rice patented the moving coil loudspeaker. Virtually all current loudspeakers are based on the same principle introduced back then. Billions of audio devices are built every year.
放大音乐的时代始于 20 世纪 20 年代，当时 Kellogg & Rice 获得了动圈扬声器的专利。几乎所有当前的扬声器都基于那时引入的相同原理。每年都会建造数十亿个音频设备。
Therefore it is fair to say that they are really ubiquitous. Nevertheless, even the most advanced sound systems cannot match the fidelity of live sound at its best. The main driving force behind the thesis is to understand and create tools to improve the listening experience.
因此，可以说它们确实是无处不在的。然而，即使是最先进的音响系统也无法与最佳现场声音的保真度相匹敌。论文的主要驱动力是理解并创造工具来改善听觉体验。
As the story ahead will tell, sometimes it is necessary to go quite far from the original idea of listening experience to be able to find ways to improve it.
正如前方的故事将要讲述的，有时为了找到改进听觉体验的方法，有必要远离原始的听觉体验概念相当远的距离。
产生声音是一件简单但有时又复杂的事情。最早的音乐乐器可以追溯到几千年前。复制音乐的时代始于 19 世纪末，随着留声机及其前身的引入。
The era of amplified music began in the 1920s, when Kellogg & Rice patented the moving coil loudspeaker. Virtually all current loudspeakers are based on the same principle introduced back then. Billions of audio devices are built every year.
放大音乐的时代始于 20 世纪 20 年代，当时 Kellogg & Rice 获得了动圈扬声器的专利。几乎所有当前的扬声器都基于那时引入的相同原理。每年都会建造数十亿个音频设备。
Therefore it is fair to say that they are really ubiquitous. Nevertheless, even the most advanced sound systems cannot match the fidelity of live sound at its best. The main driving force behind the thesis is to understand and create tools to improve the listening experience.
因此，可以说它们确实是无处不在的。然而，即使是最先进的音响系统也无法与最佳现场声音的保真度相匹敌。论文的主要驱动力是理解并创造工具来改善听觉体验。
As the story ahead will tell, sometimes it is necessary to go quite far from the original idea of listening experience to be able to find ways to improve it.
正如前方的故事将要讲述的，有时为了找到改进听觉体验的方法，有必要远离原始的听觉体验概念相当远的距离。
This thesis combines FEM (Finite Element Method) modelling and acoustical understanding of directivity and waveguides. The goal is to use modelling to virtually prototype loudspeakers.
本论文结合了有限元方法（FEM）建模和直接性与波导的声学理解。目标是利用建模来虚拟原型扬声器。
The motivation is to speed up the prototyping process and reduce the amount of costly prototypes. The author also has a stubborn belief that modelling can lead to better performing loudspeakers in the end.
动机是为了加快原型制作过程并减少昂贵原型的数量。作者还坚信，建模最终可以导致性能更好的扬声器。
This can be explained by the low threshold of trying out new ideas by virtual prototyping and also the enhanced understanding given by the advanced visualization methods.
这可以用虚拟原型设计尝试新想法的低门槛以及高级可视化方法提供的增强理解来解释。
本论文结合了有限元方法（FEM）建模和直接性与波导的声学理解。目标是利用建模来虚拟原型扬声器。
The motivation is to speed up the prototyping process and reduce the amount of costly prototypes. The author also has a stubborn belief that modelling can lead to better performing loudspeakers in the end.
动机是为了加快原型制作过程并减少昂贵原型的数量。作者还坚信，建模最终可以导致性能更好的扬声器。
This can be explained by the low threshold of trying out new ideas by virtual prototyping and also the enhanced understanding given by the advanced visualization methods.
这可以用虚拟原型设计尝试新想法的低门槛以及高级可视化方法提供的增强理解来解释。
In general, waveguide and horn are synonyms. The primary purpose of early horn designs was to increase the output of the sound system. Later it became obvious that horns have a beneficial effect on the loudspeaker directivity.
一般来说，波导和号角是同义词。早期号角设计的主要目的是增加音响系统的输出。后来很明显，号角对扬声器的方向性有有益的影响。
For some applications, the directivity characteristics of the horn are the main benefit and increased sensitivity is a side-effect [1]. These horns designed for directivity are called waveguides to emphasize the different design goals compared to horns.
对于某些应用，号角的定向特性是主要优势，而提高灵敏度则是副产品[1]。为了强调与号角设计目标的不同，专门为定向设计的号角被称为波导。
These days the devices used for public address sound reinforcement are called horns. Respectively the devices used in home speakers and studio monitors are called waveguides.
这些天用于公共广播声音增强的设备被称为号角。而用于家庭扬声器和录音室监听的设备则被称为波导。
一般来说，波导和号角是同义词。早期号角设计的主要目的是增加音响系统的输出。后来很明显，号角对扬声器的方向性有有益的影响。
For some applications, the directivity characteristics of the horn are the main benefit and increased sensitivity is a side-effect [1]. These horns designed for directivity are called waveguides to emphasize the different design goals compared to horns.
对于某些应用，号角的定向特性是主要优势，而提高灵敏度则是副产品[1]。为了强调与号角设计目标的不同，专门为定向设计的号角被称为波导。
These days the devices used for public address sound reinforcement are called horns. Respectively the devices used in home speakers and studio monitors are called waveguides.
这些天用于公共广播声音增强的设备被称为号角。而用于家庭扬声器和录音室监听的设备则被称为波导。
Figure 1.1. A loudspeaker with a waveguide - Genelec 8040A.
图 1.1. 带有波导的扬声器 - Genelec 8040A。
图 1.1. 带有波导的扬声器 - Genelec 8040A。
A common goal of a waveguide is to match the directivity of a high frequency transducer to the directivity of a low frequency transducer (Figure 1.1). The waveguide is located on the upper part of the speaker, around the high frequency transducer, called a tweeter.
波导的常见目标是将高频换能器的指向性与低频换能器的指向性相匹配（图 1.1）。波导位于扬声器的上部，围绕高频换能器，称为高音扬声器。
The purpose of the waveguide is to control the dispersion characteristics of the sound source. Above the crossover frequency the directivity of the high frequency driver should be controlled.
波导的目的是控制声源的色散特性。在分频频率以上，高频扬声器的指向性应被控制。
In general, the goal is to achieve constant beam width of the acoustical radiation or slightly decreasing beam width towards high frequencies.
一般来说，目标是实现声辐射的恒定束宽，或者在高频时束宽略有减小。
波导的常见目标是将高频换能器的指向性与低频换能器的指向性相匹配（图 1.1）。波导位于扬声器的上部，围绕高频换能器，称为高音扬声器。
The purpose of the waveguide is to control the dispersion characteristics of the sound source. Above the crossover frequency the directivity of the high frequency driver should be controlled.
波导的目的是控制声源的色散特性。在分频频率以上，高频扬声器的指向性应被控制。
In general, the goal is to achieve constant beam width of the acoustical radiation or slightly decreasing beam width towards high frequencies.
一般来说，目标是实现声辐射的恒定束宽，或者在高频时束宽略有减小。
1.1 The aim of the work
1.1 工作的目标
The content of the thesis is based on the work done on modelling acoustical waveguides while working for Genelec Oy. In this thesis are discussed the problems and solutions encountered when developing a method for modelling waveguide directivity.
论文的内容基于在 Genelec Oy 工作期间进行的声波导模拟能力。在本论文中，讨论了在开发波导指向性建模方法时遇到的问题和解决方案。
论文的内容基于在 Genelec Oy 工作期间进行的声波导模拟能力。在本论文中，讨论了在开发波导指向性建模方法时遇到的问题和解决方案。
Analytical solutions are available for only a waveguide geometries and those are not feasible designs for the targeted directivity characteristics. Nevertheless there are some rules of thumb available for waveguide design.
分析解仅适用于波导几何形状，这些设计对于目标定向特性并不可行。然而，对于波导设计有一些经验法则可供参考。
These rules are discussed and also evaluated during the test case of the thesis.
这些规则在论文的测试案例中被讨论和评估。
分析解仅适用于波导几何形状，这些设计对于目标定向特性并不可行。然而，对于波导设计有一些经验法则可供参考。
These rules are discussed and also evaluated during the test case of the thesis.
这些规则在论文的测试案例中被讨论和评估。
By definition, the accuracy of a model is always second to the real world. The selected method for validating the accuracy of the model is to compare it against the real world. There is no ready made tool for visualizing measurements and modelled results in comparable form.
定义上，模型的准确性始终次于现实世界。验证模型准确性的选定方法是将其与现实世界进行比较。没有现成的工具可以以可比较的形式可视化测量和模型结果。
Therefore a major emphasis is needed on developing and creating tools for visualizing directivity.
因此，需要重点开发和创建用于可视化指向性的工具。
定义上，模型的准确性始终次于现实世界。验证模型准确性的选定方法是将其与现实世界进行比较。没有现成的工具可以以可比较的形式可视化测量和模型结果。
Therefore a major emphasis is needed on developing and creating tools for visualizing directivity.
因此，需要重点开发和创建用于可视化指向性的工具。
One aim of the work is to verify the accuracy of the method. Therefore reasoning is needed to evaluate the sources of the error of the model. Inaccuracy in the modeling result does not necessarily make the tool useless.
工作的一个目标是验证方法的准确性。因此，需要推理来评估模型误差的来源。建模结果的不准确性并不一定意味着工具无用。
However it is essential to understand the limits of the tool.
然而，理解该工具的局限性是至关重要的。
工作的一个目标是验证方法的准确性。因此，需要推理来评估模型误差的来源。建模结果的不准确性并不一定意味着工具无用。
However it is essential to understand the limits of the tool.
然而，理解该工具的局限性是至关重要的。
A test case is created for comparing the measured and modelled results. The shape of the waveguide is chosen to be far from the optimal design. The reasoning for the selection is that there would be more acoustical phenomena present.
创建一个测试案例用于比较测量结果和模型结果。波导的形状被选择得远离最优设计。选择的原因是会有更多的声学现象出现。
Modelling should be able to expose these undesired effects. After all, the motivation for the use of modelling is to reveal and minimize these unwanted effects.
建模应该能够揭示这些不期望的效果。毕竟，使用建模的动机就是为了揭示并最小化这些不希望出现的效果。
创建一个测试案例用于比较测量结果和模型结果。波导的形状被选择得远离最优设计。选择的原因是会有更多的声学现象出现。
Modelling should be able to expose these undesired effects. After all, the motivation for the use of modelling is to reveal and minimize these unwanted effects.
建模应该能够揭示这些不期望的效果。毕竟，使用建模的动机就是为了揭示并最小化这些不希望出现的效果。
1.2 Outline 1.2 框架
The outline of the thesis is divided into twelve chapters. The first chapter contains the introduction, aim and outline of the thesis. The second chapter discusses the acoustic wave equation, which is the basis of modelling acoustic fields.
论文大纲分为十二章。第一章包含论文的引言、目标和大纲。第二章讨论声波方程，这是建模声场的基础。
The third chapter discusses the use of FEM for solving the wave equation, splitting complex geometry to small elements and previous work done in the field of transducer and waveguide modelling.
第三章讨论了使用 FEM 解决波动方程的方法，将复杂几何结构分解为小元素，并回顾了在换能器和波导建模领域的先前工作。
The fourth chapter concentrates on the sound source and room interaction, its subjective importance and how it is related to the directivity of the source. Also other aspects of sound quality are reviewed in order to widen the perspective.
第四章集中探讨了声源与房间交互、其主观重要性以及与声源的方向性之间的关系。同时，还回顾了其他声音质量方面的内容，以拓宽视野。
The fifth chapter introduces the directivity visualization methods used now and in the past. The sixth chapter discusses the directivity of a circular piston source. This reveals the directivity characteristics typical for the most commonly used sound sources.
第五章介绍了当前和过去使用的方向性可视化方法。第六章讨论了圆形活塞声源的方向性。这揭示了最常用声源的典型方向性特征。
The seventh chapter expands the directivity discussion to horns. The directivity of several well known horn profiles is discussed.
第七章扩展了对号角的指向性讨论。讨论了几种广为人知的号角轮廓的指向性。
论文大纲分为十二章。第一章包含论文的引言、目标和大纲。第二章讨论声波方程，这是建模声场的基础。
The third chapter discusses the use of FEM for solving the wave equation, splitting complex geometry to small elements and previous work done in the field of transducer and waveguide modelling.
第三章讨论了使用 FEM 解决波动方程的方法，将复杂几何结构分解为小元素，并回顾了在换能器和波导建模领域的先前工作。
The fourth chapter concentrates on the sound source and room interaction, its subjective importance and how it is related to the directivity of the source. Also other aspects of sound quality are reviewed in order to widen the perspective.
第四章集中探讨了声源与房间交互、其主观重要性以及与声源的方向性之间的关系。同时，还回顾了其他声音质量方面的内容，以拓宽视野。
The fifth chapter introduces the directivity visualization methods used now and in the past. The sixth chapter discusses the directivity of a circular piston source. This reveals the directivity characteristics typical for the most commonly used sound sources.
第五章介绍了当前和过去使用的方向性可视化方法。第六章讨论了圆形活塞声源的方向性。这揭示了最常用声源的典型方向性特征。
The seventh chapter expands the directivity discussion to horns. The directivity of several well known horn profiles is discussed.
第七章扩展了对号角的指向性讨论。讨论了几种广为人知的号角轮廓的指向性。
The eighth chapter covers the methods used in the FEM modelling. The ninth chapter discusses measurement methods for the physical prototype and visualization methods to enable the comparison of the measured and modelled waveguide.
第八章涵盖了在 FEM 建模中使用的方法。第九章讨论了物理原型的测量方法和可视化方法，以实现测量波导和模型波导之间的比较。
第八章涵盖了在 FEM 建模中使用的方法。第九章讨论了物理原型的测量方法和可视化方法，以实现测量波导和模型波导之间的比较。
The tenth chapter analyzes the phenomena found in the FEM modelled waveguide. The emphasis is on introducing the modelling as an engineering tool.
第十章分析了 FEM 建模波导中发现的现象。重点在于介绍建模作为工程工具的作用。
第十章分析了 FEM 建模波导中发现的现象。重点在于介绍建模作为工程工具的作用。
In the eleventh chapter the results of the model and measurement are shown and analyzed. The comparison is done for the directivity of the waveguide and also for the frequency response of the waveguide. The twelfth chapter contains the conclusion of the work.
第十一章展示了模型和测量的结果，并进行了分析。比较了波导的方向性和波导的频率响应。第十二章包含了工作的结论。
The purpose of the chapter is to summarize the results and discuss future of the FEM modelling.
本章的目的是总结结果并讨论 FEM 建模的未来。
第十一章展示了模型和测量的结果，并进行了分析。比较了波导的方向性和波导的频率响应。第十二章包含了工作的结论。
The purpose of the chapter is to summarize the results and discuss future of the FEM modelling.
本章的目的是总结结果并讨论 FEM 建模的未来。
2 Time-harmonic solution of the wave equation
2. 频域解的波动方程
The basis of solving acoustic fields is the three-dimensional wave equation (Equation (2.1). The wave equation is the basis when deriving the physics necessary for solving FEM. The medium is assumed to be lossless. The wave equation is based on the conservation of mass.
解决声场的基础是三维波动方程（式（2.1）。波动方程是推导用于求解 FEM 所需物理原理的基础。假设介质为无耗散的。波动方程基于质量守恒。
Therefore no fluid flow should be present when it is used [2]. On the right hand side of the wave equation there is a possible monopole source
因此，在使用时不应存在流体流动[2]。波动方程的右侧可能存在一个单极源
解决声场的基础是三维波动方程（式（2.1）。波动方程是推导用于求解 FEM 所需物理原理的基础。假设介质为无耗散的。波动方程基于质量守恒。
Therefore no fluid flow should be present when it is used [2]. On the right hand side of the wave equation there is a possible monopole source
因此，在使用时不应存在流体流动[2]。波动方程的右侧可能存在一个单极源
The wave equation can be significantly simplified if the pressure is time-harmonic. In other words, the modelling is done in the frequency domain and transient phenomena are left out.
波动方程在压力为时谐的情况下可以显著简化。换句话说，建模是在频率域中进行的，瞬态现象被排除在外。
波动方程在压力为时谐的情况下可以显著简化。换句话说，建模是在频率域中进行的，瞬态现象被排除在外。
For a time-harmonic wave, the pressure in three-dimensional space must be time harmonic (Equation (2.2). The variable
对于时谐波，三维空间中的压力必须是时谐波（式（2.2）。变量
对于时谐波，三维空间中的压力必须是时谐波（式（2.2）。变量
If the source and the problem are time-harmonic, the wave-equation can be written in time-harmonic form (Equation (2.3) [3].
如果源和问题都是时谐的，波动方程可以写成时谐形式（式（2.3）[3]）。
如果源和问题都是时谐的，波动方程可以写成时谐形式（式（2.3）[3]）。
Where
在
在
The frequency domain solution of the wave equation presented above is also known as the Helmholtz equation.
上述波动方程的频域解也被称为亥姆霍兹方程。
上述波动方程的频域解也被称为亥姆霍兹方程。
Reducing the analysis to 2D axisymmetric geometry significantly reduces the computational cost of solving the model. The computational cost for various
将分析简化为二维轴对称几何显著降低了求解模型的计算成本。各种不同情况的计算成本如下：
geometries is discussed in the Chapter 3.3. Fortunately many acoustic problems are 2D axisymmetrical, for example loudspeaker driver cone and waveguide.
第 3.3 章讨论了几何学。幸运的是，许多声学问题具有 2D 轴对称性，例如扬声器驱动器锥体和波导。
3D geometry is axisymmetric if it can be represented by a 2D profile and defined axis of rotation which extrudes it to a 3D geometry.
三维几何学如果可以通过二维轮廓和定义的旋转轴将其延伸为三维几何学，则称为轴对称。
将分析简化为二维轴对称几何显著降低了求解模型的计算成本。各种不同情况的计算成本如下：
geometries is discussed in the Chapter 3.3. Fortunately many acoustic problems are 2D axisymmetrical, for example loudspeaker driver cone and waveguide.
第 3.3 章讨论了几何学。幸运的是，许多声学问题具有 2D 轴对称性，例如扬声器驱动器锥体和波导。
3D geometry is axisymmetric if it can be represented by a 2D profile and defined axis of rotation which extrudes it to a 3D geometry.
三维几何学如果可以通过二维轮廓和定义的旋转轴将其延伸为三维几何学，则称为轴对称。
For a time-harmonic wave, the pressure in the 2 D axisymmetric space (Equation (2.4) is dependent on the radial coordinate
对于时谐波，二维轴对称空间（式（2.4）依赖于径向坐标
对于时谐波，二维轴对称空间（式（2.4）依赖于径向坐标
It is noteworthy that azimuthal angle
值得注意的是，方位角
值得注意的是，方位角
3 Using the FEM to solve the wave equation
使用 FEM 解决波动方程
The use of the FEM for engineering problems originated in the mechanical engineering and the aeronautical industry as early the late 1950s [4]. FEM was first used for applications where failure of the end product would be costly and even fatal.
有限元法在工程问题中的应用起源于 20 世纪 50 年代末的机械工程和航空航天行业 [4]。有限元法最初用于那些产品失效会带来高昂代价甚至致命后果的应用场景。
Examples of such applications are bridges and aircraft.
此类应用的例子包括桥梁和飞机。
有限元法在工程问题中的应用起源于 20 世纪 50 年代末的机械工程和航空航天行业 [4]。有限元法最初用于那些产品失效会带来高昂代价甚至致命后果的应用场景。
Examples of such applications are bridges and aircraft.
此类应用的例子包括桥梁和飞机。
An analytical solution of an acoustic field is only available to a few simple geometries. For more complex geometries, a method called the FEM can be used.
声场的解析解仅对少数简单的几何形状可用。对于更复杂的几何形状，可以使用一种称为有限元法（FEM）的方法。
The basic idea of the method is to divide a geometrically complex system into smaller individual components, which are called elements. The solution for these small and geometrically simple elements is straightforward to obtain.
方法的基本思想是将几何上复杂的系统分解为更小的独立组件，这些组件被称为元素。这些小且几何上简单的元素的解决方案很容易获得。
声场的解析解仅对少数简单的几何形状可用。对于更复杂的几何形状，可以使用一种称为有限元法（FEM）的方法。
The basic idea of the method is to divide a geometrically complex system into smaller individual components, which are called elements. The solution for these small and geometrically simple elements is straightforward to obtain.
方法的基本思想是将几何上复杂的系统分解为更小的独立组件，这些组件被称为元素。这些小且几何上简单的元素的解决方案很容易获得。
Error in the FEM model is caused by three reasons. First is the inaccuracy in the geometry. Second is the inaccuracy of the physics. Third is the finite element size, which causes computational error.
有限元模型中的误差由三个原因引起。首先是几何形状的不准确性。其次是物理学的不准确性。第三是有限元的大小，这会导致计算误差。
有限元模型中的误差由三个原因引起。首先是几何形状的不准确性。其次是物理学的不准确性。第三是有限元的大小，这会导致计算误差。
Usually the inaccuracies of geometry and physics dominate the overall error. Considering the computational cost, it is not reasonable to push the error caused by finite element size much below the overall error [5].
通常，几何和物理的不准确性主导了整体误差。考虑到计算成本，将有限元尺寸引起的误差推到整体误差以下太多是不合理的[5]。
通常，几何和物理的不准确性主导了整体误差。考虑到计算成本，将有限元尺寸引起的误差推到整体误差以下太多是不合理的[5]。
最高频率进行建模
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Sampling rate 采样率 | 时间域计算
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频率分辨率
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时间域解决方案
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20 kHz | 40 kHz | 256 samples 256 个样本 | 156 Hz 156 赫兹 | |||||||||
最高频率进行建模
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- | 频域计算（1 kHz 到 20 kHz）
|
频率分辨率
|
||||||||||
频域解
|
20 kHz | - | 95 | 200 Hz |
Table 3.1. Computational cost of the time-domain modelling versus frequency domain modelling.
表 3.1. 时域建模与频域建模的计算成本。
表 3.1. 时域建模与频域建模的计算成本。
With the acoustic simulation, usually the frequency response is the most interesting result.
通过声学模拟，通常频率响应是最有趣的结果。
One approach to obtain the frequency response would be the same as with most modern acoustic measurement systems: somehow achieve the impulse response of the system and calculate frequency response with a fast fourier transform (FFT) of
获取频率响应的一种方法与大多数现代声学测量系统相同：以某种方式获得系统的冲激响应，并使用快速傅立叶变换（FFT）计算频率响应
the impulse response. Similarly it is also possible to model the impulse response of the system and obtain frequency response. However the computational cost of this approach is very high.
响应。同样，也可以建模系统的冲激响应并获得频率响应。然而，这种方法的计算成本非常高。
The computational cost of calculating time-domain solutions for the FEM is much higher than calculating frequency domain solutions.
计算 FEM 在时域解决方案的计算成本远高于在频域解决方案的计算成本。
For a time domain solution 256 calculations are needed in time domain compared to the 95 calculations of the frequency domain solution (Table 3.1). Therefore it is more feasible to directly calculate the amplitudes at the frequencies of interest.
对于时间域解决方案，与频率域解决方案（表 3.1）所需的 95 次计算相比，时间域中需要进行 256 次计算。因此，直接计算感兴趣的频率上的振幅更为可行。
通过声学模拟，通常频率响应是最有趣的结果。
One approach to obtain the frequency response would be the same as with most modern acoustic measurement systems: somehow achieve the impulse response of the system and calculate frequency response with a fast fourier transform (FFT) of
获取频率响应的一种方法与大多数现代声学测量系统相同：以某种方式获得系统的冲激响应，并使用快速傅立叶变换（FFT）计算频率响应
the impulse response. Similarly it is also possible to model the impulse response of the system and obtain frequency response. However the computational cost of this approach is very high.
响应。同样，也可以建模系统的冲激响应并获得频率响应。然而，这种方法的计算成本非常高。
The computational cost of calculating time-domain solutions for the FEM is much higher than calculating frequency domain solutions.
计算 FEM 在时域解决方案的计算成本远高于在频域解决方案的计算成本。
For a time domain solution 256 calculations are needed in time domain compared to the 95 calculations of the frequency domain solution (Table 3.1). Therefore it is more feasible to directly calculate the amplitudes at the frequencies of interest.
对于时间域解决方案，与频率域解决方案（表 3.1）所需的 95 次计算相比，时间域中需要进行 256 次计算。因此，直接计算感兴趣的频率上的振幅更为可行。
3.1 Discretization the geometry to elements
3.1 将几何形状离散化为元素
The fundamental idea of the FEM is to divide the geometrically complicated partial differential equation problem down to a coupled group of smaller problems. Each small problem is called an element. The process of dividing the geometry is called meshing.
有限元法的基本思想是将几何上复杂的偏微分方程问题分解为一组较小的耦合问题。每个小问题被称为一个元素。将几何进行分解的过程称为网格划分。
There are several locations to specify the physics on each mesh element.
在每个网格元素上指定物理的地点有几个。
The basic approach is to specify the wave equation state parameters to the corners of the element (Figure 3.1). It is also possible to specify the states to the edges of the element or even to the centre of the element. These state parameters are called degrees of freedom (DOF).
基本方法是将波方程状态参数指定到元素的角点（图 3.1）。也可以将状态指定到元素的边缘，甚至指定到元素的中心。这些状态参数被称为自由度（DOF）。
Using higher order elements improves the accuracy of the simulation, because there are more state parameters calculated per element. Of course the computational cost is also increased. In the models in this thesis a second order Lagrange element is used.
使用高阶元素可以提高模拟的准确性，因为每个元素计算的状态参数更多。当然，计算成本也会增加。在本论文的模型中，使用了二次拉格朗日元素。
Therefore there is a degree of freedom in each corner of the element and also at the sides of the elements.
因此，每个元素的每个角落以及元素的两侧都存在一定的自由度。
有限元法的基本思想是将几何上复杂的偏微分方程问题分解为一组较小的耦合问题。每个小问题被称为一个元素。将几何进行分解的过程称为网格划分。
There are several locations to specify the physics on each mesh element.
在每个网格元素上指定物理的地点有几个。
The basic approach is to specify the wave equation state parameters to the corners of the element (Figure 3.1). It is also possible to specify the states to the edges of the element or even to the centre of the element. These state parameters are called degrees of freedom (DOF).
基本方法是将波方程状态参数指定到元素的角点（图 3.1）。也可以将状态指定到元素的边缘，甚至指定到元素的中心。这些状态参数被称为自由度（DOF）。
Using higher order elements improves the accuracy of the simulation, because there are more state parameters calculated per element. Of course the computational cost is also increased. In the models in this thesis a second order Lagrange element is used.
使用高阶元素可以提高模拟的准确性，因为每个元素计算的状态参数更多。当然，计算成本也会增加。在本论文的模型中，使用了二次拉格朗日元素。
Therefore there is a degree of freedom in each corner of the element and also at the sides of the elements.
因此，每个元素的每个角落以及元素的两侧都存在一定的自由度。
Figure 3.1. Degrees of freedom of each mesh element. Adopted from [3].
图 3.1. 每个网格元素的自由度。引自[3]。
图 3.1. 每个网格元素的自由度。引自[3]。
3.2 Selection of the element size
3.2 元素尺寸的选择
The distribution of the geometry to elements is called meshing. The maximum size of the element is limited by two factors. The first limiting factor is the shortest wavelength to be calculated. The second limitation is the size of the geometry details to be modelled.
网格的几何分布称为网格划分。元素的最大尺寸受到两个因素的限制。第一个限制因素是需要计算的最短波长。第二个限制是需要建模的几何细节的大小。
网格的几何分布称为网格划分。元素的最大尺寸受到两个因素的限制。第一个限制因素是需要计算的最短波长。第二个限制是需要建模的几何细节的大小。
Usually these constraints are related to each other. Small details become interesting only when they are comparable to the wavelength of interest. Of course this depends on the phenomenon.
通常这些限制是相互关联的。小细节只有在与感兴趣的波长可比时才会变得有趣。当然，这取决于现象。
For example, Helmholtz resonance might a have significant influence on the modelled result, even if the port opening is small related to the wavelength.
例如，赫尔姆霍兹共振可能对模型结果产生显著影响，即使端口开口相对于波长很小。
通常这些限制是相互关联的。小细节只有在与感兴趣的波长可比时才会变得有趣。当然，这取决于现象。
For example, Helmholtz resonance might a have significant influence on the modelled result, even if the port opening is small related to the wavelength.
例如，赫尔姆霍兹共振可能对模型结果产生显著影响，即使端口开口相对于波长很小。
Maximum element size should be smaller than one-sixth of the wavelength of the acoustical wave (Equation (3.1) [5]. According to the Nyquist theorem, the element size should be smaller than half of the wavelength so that solution would have any meaning [3].
最大元素尺寸应小于声波（方程（3.1）[5]）波长的六分之一。根据奈奎斯特定理，元素尺寸应小于波长的一半，以便解决方案才有意义[3]。
最大元素尺寸应小于声波（方程（3.1）[5]）波长的六分之一。根据奈奎斯特定理，元素尺寸应小于波长的一半，以便解决方案才有意义[3]。
3.3 The concept of degrees of freedom
3.3 自由度的概念
The number of degrees of freedom determines the computational cost of solving the model. It is dependent on the mesh element count of the model. This is contradictory to the need for a detailed model and large air space to approximate far field conditions.
自由度的数量决定了求解模型的计算成本。它依赖于模型的网格元素数量。这与需要详细模型和大量空间来近似远场条件的需求相矛盾。
The number of degrees of freedom is also highly dependent on whether the problem is 1D, 2D or 3D (Table 3.2). An approximation of the degrees of freedom of the model can be calculated, if constant
自由度的数量在很大程度上取决于问题是一维、二维还是三维（表 3.2）。如果已知常数
自由度的数量决定了求解模型的计算成本。它依赖于模型的网格元素数量。这与需要详细模型和大量空间来近似远场条件的需求相矛盾。
The number of degrees of freedom is also highly dependent on whether the problem is 1D, 2D or 3D (Table 3.2). An approximation of the degrees of freedom of the model can be calculated, if constant
自由度的数量在很大程度上取决于问题是一维、二维还是三维（表 3.2）。如果已知常数
| Geometry 几何学 | 乘以常数 y
|
模型域大小 A
|
波长
|
||||||
一维或一维轴对称
|
12 | Length 长度 | 1 | ||||||
二维或二维轴对称
|
144 | Area 区域 | Squared 平方 | ||||||
| 3D | 1828 | Volume 体积 | Cubed 立方 |
Table 3.2. Factors affecting the degrees of freedom with 1D, 2D and 3D geometries [5].
表 3.2. 影响一维、二维和三维几何结构自由度的因素 [5]
表 3.2. 影响一维、二维和三维几何结构自由度的因素 [5]
According to the table and equation presented, the degrees of freedom can be significantly reduced if a 3D geometry can be reduced to a 2 D geometry.
根据表格和给出的方程，如果可以将 3D 几何简化为 2D 几何，自由度可以显著降低。
It is possible to solve an axisymmetric 3D problem in a 2D domain without extra computational cost. This feature is very fortunate in the field of acoustics. Often geometries related to acoustics are axisymmetric or at least a reasonable axisymmetric approximation can be made.
有可能在二维域中解决轴对称 3D 问题而无需额外的计算成本。这一特性在声学领域非常幸运。通常与声学相关的几何形状是轴对称的，或者至少可以做出合理的轴对称近似。
This is true also in the case of waveguide design.
这在波导设计的情况下也是正确的。
根据表格和给出的方程，如果可以将 3D 几何简化为 2D 几何，自由度可以显著降低。
It is possible to solve an axisymmetric 3D problem in a 2D domain without extra computational cost. This feature is very fortunate in the field of acoustics. Often geometries related to acoustics are axisymmetric or at least a reasonable axisymmetric approximation can be made.
有可能在二维域中解决轴对称 3D 问题而无需额外的计算成本。这一特性在声学领域非常幸运。通常与声学相关的几何形状是轴对称的，或者至少可以做出合理的轴对称近似。
This is true also in the case of waveguide design.
这在波导设计的情况下也是正确的。
3.4 Computational cost example
3.4 计算成本示例
An approximation of the degrees of freedom for 1D, 2D and 3D problems were introduced in the previous chapter. However it is not straightforward to understand the difference in computational cost between a 2D axisymmetric and a full 3D model.
前一章中介绍了 1D、2D 和 3D 问题的自由度近似值。然而，理解 2D 轴对称模型与完整 3D 模型在计算成本之间的差异并不直观。
Therefore the following example is shown to justify the use axisymmetry whenever possible.
因此，以下示例用于证明在可能的情况下使用轴对称性。
前一章中介绍了 1D、2D 和 3D 问题的自由度近似值。然而，理解 2D 轴对称模型与完整 3D 模型在计算成本之间的差异并不直观。
Therefore the following example is shown to justify the use axisymmetry whenever possible.
因此，以下示例用于证明在可能的情况下使用轴对称性。
| Geometry 几何学 | 模型几何体的边长
|
模型域大小
|
20 kHz 波长
|
波长指数
|
自由度
|
|||||||||||||
二维轴对称
|
0.4 |
|
3.4 | 2 | 78000 | |||||||||||||
| 3D | 0.4 |
|
3.4 | 3 | 23000000 |
Table 3.3. Example calculation of degrees of freedom with 2D-axisymmetric and 3D model.
表 3.3. 二维轴对称和三维模型的自由度计算示例。
表 3.3. 二维轴对称和三维模型的自由度计算示例。
Assume an axisymmetric system that can be modelled either as a full 3D model or a 2D axisymmetric model. The length of the model edges is 0.4 meters and the highest frequency of interest is 20 kHz .
假设一个轴对称系统，可以被建模为完整的 3D 模型或 2D 轴对称模型。模型边长为 0.4 米，感兴趣的最高频率为 20 kHz。
Degrees of freedom of the 2D-axisymmetric and full 3D geometry can be approximated with the known theory (Equation (3.2 and Table 3.2). The number of the DOF for 2D axisymmetric model is 78000 (Table 3.3). Solving 78 thousand degrees of freedom takes about 10 seconds per frequency with a modern desktop computer.
二维轴对称和完整三维几何的自由度可以通过已知理论（公式（3.2 和表 3.2）来近似计算。二维轴对称模型的自由度数量为 78000（表 3.3）。解决 78 千个自由度，每频率大约需要现代台式电脑 10 秒。
The memory requirement is less than 1 GB .
内存需求小于 1GB。
假设一个轴对称系统，可以被建模为完整的 3D 模型或 2D 轴对称模型。模型边长为 0.4 米，感兴趣的最高频率为 20 kHz。
Degrees of freedom of the 2D-axisymmetric and full 3D geometry can be approximated with the known theory (Equation (3.2 and Table 3.2). The number of the DOF for 2D axisymmetric model is 78000 (Table 3.3). Solving 78 thousand degrees of freedom takes about 10 seconds per frequency with a modern desktop computer.
二维轴对称和完整三维几何的自由度可以通过已知理论（公式（3.2 和表 3.2）来近似计算。二维轴对称模型的自由度数量为 78000（表 3.3）。解决 78 千个自由度，每频率大约需要现代台式电脑 10 秒。
The memory requirement is less than 1 GB .
内存需求小于 1GB。
For a full 3D model the degrees of freedom is 23 million. The problem is too large to be solved with a desktop computer because of the memory requirement. According
对于完整的 3D 模型，自由度为 2300 万。问题太大，无法用台式计算机解决，因为需要内存。
to this example it can be concluded that 3D modelling is limited either to small frequencies or small geometries (or supercomputers).
对于这个例子，可以得出结论，3D 模型制作要么局限于较低的频率，要么局限于较小的几何尺寸（或者需要超级计算机）。
对于完整的 3D 模型，自由度为 2300 万。问题太大，无法用台式计算机解决，因为需要内存。
to this example it can be concluded that 3D modelling is limited either to small frequencies or small geometries (or supercomputers).
对于这个例子，可以得出结论，3D 模型制作要么局限于较低的频率，要么局限于较小的几何尺寸（或者需要超级计算机）。
The limit of the degrees of freedom to be calculated with a desktop computer is from one to two million. However, with long calculation times the FEM modelling is rather a verification tool than an interactive design tool.
计算用台式计算机可计算的自由度极限是从一百万到二百万。然而，由于计算时间较长，FEM 模型更适合作为验证工具，而不是交互式设计工具。
计算用台式计算机可计算的自由度极限是从一百万到二百万。然而，由于计算时间较长，FEM 模型更适合作为验证工具，而不是交互式设计工具。
Figure 3.2. Mesh around the waveguide.
图 3.2. 波导周围的网格。
图 3.2. 波导周围的网格。
In this thesis prototype the highest frequency of interest is 20 kHz . According to the one sixth of the wavelength rule (Equation (3.1) the maximum element size is 0.5 mm . Figure 3.2 shows the mesh around the waveguide.
在本论文原型中，最高的频率兴趣点为 20 kHz。根据波长的六分之一规则（公式（3.1），最大元件尺寸为 0.5 mm。图 3.2 显示了波导周围的网格。
This mesh is used in all the calculations of the FEM based models. It easy to see that the element size is quite constant over the domain. However, around the tweeter surround and faceplate the mesh is denser. This is caused by the small details in geometry around that area.
这种网格用于基于 FEM 的所有计算。很容易看出，网格尺寸在整个领域内相当一致。然而，在扬声器周围和面板附近，网格更密集。这是由于该区域周围几何细节较小所导致的。
The small details have an influence on the directivity at high frequencies. Therefore the extra computational cost caused by the small details is justified.
小细节对高频段的方向性有影响，因此，小细节导致的额外计算成本是合理的。
在本论文原型中，最高的频率兴趣点为 20 kHz。根据波长的六分之一规则（公式（3.1），最大元件尺寸为 0.5 mm。图 3.2 显示了波导周围的网格。
This mesh is used in all the calculations of the FEM based models. It easy to see that the element size is quite constant over the domain. However, around the tweeter surround and faceplate the mesh is denser. This is caused by the small details in geometry around that area.
这种网格用于基于 FEM 的所有计算。很容易看出，网格尺寸在整个领域内相当一致。然而，在扬声器周围和面板附近，网格更密集。这是由于该区域周围几何细节较小所导致的。
The small details have an influence on the directivity at high frequencies. Therefore the extra computational cost caused by the small details is justified.
小细节对高频段的方向性有影响，因此，小细节导致的额外计算成本是合理的。
3.5 Applications of the finite element method to acoustics
3.5 有限元方法在声学中的应用
The use of FEM for acoustical modelling is not a new idea. This chapter discusses the most important papers and applications. Emphasis is on the topics of waveguide and transducer modelling.
有限元法在声学建模中的应用并非新概念。本章讨论了最重要的论文和应用。重点在于波导和换能器建模的话题。
有限元法在声学建模中的应用并非新概念。本章讨论了最重要的论文和应用。重点在于波导和换能器建模的话题。
In the 1980s a Japanese research group published papers of applying FEM for solving loudspeaker related acoustical problems. In 1982, Kyouno used vibroacoustic coupling for compression driver and horn [6].
20 世纪 80 年代，日本的一个研究小组发表了关于使用有限元方法解决扬声器相关声学问题的论文。1982 年，Kyouno 使用振动声学耦合技术对压缩驱动器和号角进行了研究 [6]。
He explained the theoretical background of coupling the mechanical and acoustical domain with FEM. An elastical diaphragm of a compression driver was modelled in the mechanical domain with two-way coupling to the acoustical domain of a horn.
他解释了将机械和声学领域通过有限元法耦合的理论背景。压缩驱动器的弹性膜在机械领域进行了建模，并与号角的声学领域通过双向耦合进行了连接。
Both cone vibrations and the near field acoustic field in the horn was investigated. Far field sound pressure was approximated with an analytical equation. Enlightening conclusions were made based on the model and measurements.
圆锥振动和号角附近的声场进行了研究。远场声压通过分析方程近似计算。基于模型和测量，得出了一些启发性的结论。
First, the diaphragm vibration has little effect on the directivity characteristics of a system with a compression driver. Second, the acoustic load coupling to the diaphragm has great effect on its vibration.
首先，隔膜振动对压缩驱动器系统的指向性特性影响较小。其次，声学负载耦合到隔膜对其振动影响巨大。
Thirdly, the assumption of plane wave or spherical wave shape of radiation is not valid at high frequencies - therefore the analytical solutions are not valid either.
第三，辐射在高频时的平面波或球面波假设不成立——因此，分析解也不成立。
20 世纪 80 年代，日本的一个研究小组发表了关于使用有限元方法解决扬声器相关声学问题的论文。1982 年，Kyouno 使用振动声学耦合技术对压缩驱动器和号角进行了研究 [6]。
He explained the theoretical background of coupling the mechanical and acoustical domain with FEM. An elastical diaphragm of a compression driver was modelled in the mechanical domain with two-way coupling to the acoustical domain of a horn.
他解释了将机械和声学领域通过有限元法耦合的理论背景。压缩驱动器的弹性膜在机械领域进行了建模，并与号角的声学领域通过双向耦合进行了连接。
Both cone vibrations and the near field acoustic field in the horn was investigated. Far field sound pressure was approximated with an analytical equation. Enlightening conclusions were made based on the model and measurements.
圆锥振动和号角附近的声场进行了研究。远场声压通过分析方程近似计算。基于模型和测量，得出了一些启发性的结论。
First, the diaphragm vibration has little effect on the directivity characteristics of a system with a compression driver. Second, the acoustic load coupling to the diaphragm has great effect on its vibration.
首先，隔膜振动对压缩驱动器系统的指向性特性影响较小。其次，声学负载耦合到隔膜对其振动影响巨大。
Thirdly, the assumption of plane wave or spherical wave shape of radiation is not valid at high frequencies - therefore the analytical solutions are not valid either.
第三，辐射在高频时的平面波或球面波假设不成立——因此，分析解也不成立。
In 2001, Martin Opitz described three important tools for optimizing miniature loudspeakers for mobile applications [7]. FEM was used to optimize the force factor Bl and linearity and suspension compliance linearity. The structure of the magnet circuit was optimized with FEM.
2001 年，马丁·奥皮茨描述了优化微型扬声器用于移动应用的三个重要工具 [7]。有限元法（FEM）被用于优化磁通量因数 Bl 和线性度以及悬挂合规线性度。磁路结构通过有限元法进行了优化。
Mobile transducers should be flat as possible, but the efficiency of the magnet system sacrifices if the iron parts saturate because of small material thickness. Therefore these two contradictory requirements should be optimized.
移动换能器应尽可能平坦，但磁体系统的效率会因材料厚度较小而导致铁部件饱和而牺牲。因此，这两个相互矛盾的要求应该得到优化。
Also the membrane material thickness and geometry is optimized with a mechanical FEM model as is the linearity of the suspension. Mechanical vibrations were coupled to the acoustical world by using BEM (Boundary Element Method).
膜材料的厚度和几何形状也通过机械有限元模型进行了优化，悬架的线性度也进行了优化。通过使用边界元法（BEM），机械振动被耦合到了声学世界中。
Results were compared to physical prototype measured with laser velocitymetry.
结果与使用激光速度计测量的物理原型进行了比较。
2001 年，马丁·奥皮茨描述了优化微型扬声器用于移动应用的三个重要工具 [7]。有限元法（FEM）被用于优化磁通量因数 Bl 和线性度以及悬挂合规线性度。磁路结构通过有限元法进行了优化。
Mobile transducers should be flat as possible, but the efficiency of the magnet system sacrifices if the iron parts saturate because of small material thickness. Therefore these two contradictory requirements should be optimized.
移动换能器应尽可能平坦，但磁体系统的效率会因材料厚度较小而导致铁部件饱和而牺牲。因此，这两个相互矛盾的要求应该得到优化。
Also the membrane material thickness and geometry is optimized with a mechanical FEM model as is the linearity of the suspension. Mechanical vibrations were coupled to the acoustical world by using BEM (Boundary Element Method).
膜材料的厚度和几何形状也通过机械有限元模型进行了优化，悬架的线性度也进行了优化。通过使用边界元法（BEM），机械振动被耦合到了声学世界中。
Results were compared to physical prototype measured with laser velocitymetry.
结果与使用激光速度计测量的物理原型进行了比较。
Mark Dodd has done extensive research on research on modelling loudspeakers with FEM. In 2002, Dodd modelled loudspeaker motor thermal behaviour with the FEM. He modelled the heat spreading from voice coil with axisymmetric geometry of the complete driver.
马克·多德对使用有限元法模拟扬声器的研究进行了广泛的研究。2002 年，多德使用有限元法模拟了扬声器电机的热行为。他使用了完整驱动器的轴对称几何模型来模拟音圈的热量传播。
Both static and transient cases were studied. He explained the four heat paths: radiation, conduction, natural convection and forced convection.
两种情况都进行了研究，静态和动态情况。他解释了四种热传递路径：辐射、传导、自然对流和强制对流。
Convection is the most difficult phenomenon to model, because the physics model changes from low velocity laminar flow to high velocity turbulent flow. Dodd found agreement between modelled and measured results.
对流是最难建模的现象，因为其物理模型从低速层流流动转变为高速湍流流动。Dodd 发现了模型预测结果与测量结果之间的吻合。
In 2003 Dodd published his first paper with electro-magneto-mechanical-acoustical interaction [8]. Because full twoway interaction between domains would have been too complicated to calculate, the
2003 年，多德发表了他关于电磁机械声学相互作用的第一篇论文[8]。由于在各个领域之间进行完整的双向相互作用计算过于复杂，因此进行了简化处理。
problem was divided into parts and parameterized one-way results were used for coupling. The design case was developing a phase-plug for compression driver. Blocked coil impedance was calculated with transient magnetic FEM, force factor with magneto-static FEM.
问题被分解为部分，并通过参数化单向结果进行耦合。设计案例是开发压缩驱动器的相位塞。使用瞬态磁 FEM 计算了阻尼线圈的阻抗，使用磁静力学 FEM 计算了磁力因子。
Input voltage and voice coil length could be defined. These parameters were enough to define the force affecting the fully coupled vibroacoustic FEM model and thus obtain the full model of the driver.
输入电压和音圈长度可以定义。这些参数足以定义完全耦合振动声学有限元模型所影响的力，从而获得完整的扬声器模型。
Compression drivers are always combined with a horn, which defines the acoustical impedance seen by the compression driver throat.
压缩驱动器总是与号角结合使用，这定义了压缩驱动器喉部所看到的声学阻抗。
This variable was eliminated by modelling and measuring the drivers in an impedance tube, which provides a purely resistive termination to the horn mouth. Then phase-plug cavity geometry was optimized with the model described above.
这个变量通过在阻抗管中建模和测量驱动器来消除，这为号嘴口提供了一个纯电阻终止。然后，使用上述模型优化了相位塞腔体几何结构。
The accuracy of the final design was also compared with measurements. A simplified compression driver model was coupled with a horn. There was basic agreement with the model and measurements, but the causes for differences were not analyzed.
最终设计的准确性也与测量结果进行了比较。使用了一个简化压缩驱动器模型与号角耦合。模型与测量结果基本一致，但差异的原因没有进行分析。
In 2006, Dodd expanded the FEM based optimization to diaphragm and waveguide geometry [9]. He analyzed the theoretical solution of A planar piston in an infinite baffle. Unfortunately he did not compare the result with the well known analytical solution.
2006 年，Dodd 将基于 FEM 的优化扩展到了隔板和波导几何结构[9]。他分析了无限遮挡中平面活塞的理论解。不幸的是，他没有将结果与广为人知的解析解进行比较。
Also the radiation of a hemispherical diaphragm was studied. With analysis of a finite length conical waveguide, he pointed out the problem caused by the mouth reflection. The last example was about a realistic dome and waveguide shape.
半球形隔板的辐射也被研究了。通过分析有限长度的圆锥波导，他指出了口反射引起的问题。最后一个例子是关于一个实际的穹顶和波导形状。
The contours in the directivity sonogram behaved very well. He also presented a directivity sonogram of the impulse response of the system. This approach gave insight to the shape of wavefronts in a waveguide.
直接性声谱图中的轮廓表现得非常好。他还展示了一个系统冲激响应的直接性声谱图。这种方法提供了波导中波前形状的见解。
马克·多德对使用有限元法模拟扬声器的研究进行了广泛的研究。2002 年，多德使用有限元法模拟了扬声器电机的热行为。他使用了完整驱动器的轴对称几何模型来模拟音圈的热量传播。
Both static and transient cases were studied. He explained the four heat paths: radiation, conduction, natural convection and forced convection.
两种情况都进行了研究，静态和动态情况。他解释了四种热传递路径：辐射、传导、自然对流和强制对流。
Convection is the most difficult phenomenon to model, because the physics model changes from low velocity laminar flow to high velocity turbulent flow. Dodd found agreement between modelled and measured results.
对流是最难建模的现象，因为其物理模型从低速层流流动转变为高速湍流流动。Dodd 发现了模型预测结果与测量结果之间的吻合。
In 2003 Dodd published his first paper with electro-magneto-mechanical-acoustical interaction [8]. Because full twoway interaction between domains would have been too complicated to calculate, the
2003 年，多德发表了他关于电磁机械声学相互作用的第一篇论文[8]。由于在各个领域之间进行完整的双向相互作用计算过于复杂，因此进行了简化处理。
problem was divided into parts and parameterized one-way results were used for coupling. The design case was developing a phase-plug for compression driver. Blocked coil impedance was calculated with transient magnetic FEM, force factor with magneto-static FEM.
问题被分解为部分，并通过参数化单向结果进行耦合。设计案例是开发压缩驱动器的相位塞。使用瞬态磁 FEM 计算了阻尼线圈的阻抗，使用磁静力学 FEM 计算了磁力因子。
Input voltage and voice coil length could be defined. These parameters were enough to define the force affecting the fully coupled vibroacoustic FEM model and thus obtain the full model of the driver.
输入电压和音圈长度可以定义。这些参数足以定义完全耦合振动声学有限元模型所影响的力，从而获得完整的扬声器模型。
Compression drivers are always combined with a horn, which defines the acoustical impedance seen by the compression driver throat.
压缩驱动器总是与号角结合使用，这定义了压缩驱动器喉部所看到的声学阻抗。
This variable was eliminated by modelling and measuring the drivers in an impedance tube, which provides a purely resistive termination to the horn mouth. Then phase-plug cavity geometry was optimized with the model described above.
这个变量通过在阻抗管中建模和测量驱动器来消除，这为号嘴口提供了一个纯电阻终止。然后，使用上述模型优化了相位塞腔体几何结构。
The accuracy of the final design was also compared with measurements. A simplified compression driver model was coupled with a horn. There was basic agreement with the model and measurements, but the causes for differences were not analyzed.
最终设计的准确性也与测量结果进行了比较。使用了一个简化压缩驱动器模型与号角耦合。模型与测量结果基本一致，但差异的原因没有进行分析。
In 2006, Dodd expanded the FEM based optimization to diaphragm and waveguide geometry [9]. He analyzed the theoretical solution of A planar piston in an infinite baffle. Unfortunately he did not compare the result with the well known analytical solution.
2006 年，Dodd 将基于 FEM 的优化扩展到了隔板和波导几何结构[9]。他分析了无限遮挡中平面活塞的理论解。不幸的是，他没有将结果与广为人知的解析解进行比较。
Also the radiation of a hemispherical diaphragm was studied. With analysis of a finite length conical waveguide, he pointed out the problem caused by the mouth reflection. The last example was about a realistic dome and waveguide shape.
半球形隔板的辐射也被研究了。通过分析有限长度的圆锥波导，他指出了口反射引起的问题。最后一个例子是关于一个实际的穹顶和波导形状。
The contours in the directivity sonogram behaved very well. He also presented a directivity sonogram of the impulse response of the system. This approach gave insight to the shape of wavefronts in a waveguide.
直接性声谱图中的轮廓表现得非常好。他还展示了一个系统冲激响应的直接性声谱图。这种方法提供了波导中波前形状的见解。
In 2007, Biba et al used FEM for developing the moving parts of a headphone transducer [10]. The mechanical vibrations of the transducer membrane were modelled with FEM and vibro-acoustic interaction with the BEM.
2007 年，Biba 等人使用有限元法（FEM）开发了耳机换能器的移动部分[10]。换能器膜的机械振动通过 FEM 进行建模，与 BEM 结合考虑了振动声学交互作用。
Visco-thermal effects were modelled in the narrow regions of the model. Cushion and similar damping elements were modelled with frequency dependent transfer impedances.
粘温效应在模型的狭窄区域进行了建模。缓冲器和类似的减震元件通过频率相关的传输阻抗进行了建模。
The model was done in real 3D because the surround had corrugations and therefore axisymmetric modelling could not be used. The publication was divided to three phases. First, the moving parts of the transducer were modelled with FEM.
模型是在真实的 3D 中完成的，因为周围有波纹，因此无法使用轴对称建模。出版物被分为三个阶段。首先，使用 FEM 对换能器的移动部分进行了建模。
Modelling results were compared to the average acceleration derived from measurements in a vacuum. As usual, there was agreement with results although the high frequency modelling is not very exact. Second, the air load was included in the
建模结果与在真空环境中测量得出的平均加速度进行了比较。如常，结果是一致的，尽管高频建模的准确性不是非常高。其次，空气负载被包含在了
model. The influence of the slit elements and magnet circuit behind the transducer was investigated. Also a phase plug was added in front of the transducer. Third, the cushioning surround of the transducer was added. Adding details to the model also adds sources of error.
模型。研究了转换器背后的狭缝元件和磁路的影响。此外，在转换器前方添加了一个相位塞。第三，添加了转换器的缓冲环绕。在模型中添加细节也会引入误差源。
However, the basic phenomena were visible both in measured and modelled curves. After all, this paper was a rare insight to headphone transducer design.
然而，在测量和建模的曲线中，基本现象都清晰可见。毕竟，这篇论文是对耳机换能器设计难得的深入探讨。
2007 年，Biba 等人使用有限元法（FEM）开发了耳机换能器的移动部分[10]。换能器膜的机械振动通过 FEM 进行建模，与 BEM 结合考虑了振动声学交互作用。
Visco-thermal effects were modelled in the narrow regions of the model. Cushion and similar damping elements were modelled with frequency dependent transfer impedances.
粘温效应在模型的狭窄区域进行了建模。缓冲器和类似的减震元件通过频率相关的传输阻抗进行了建模。
The model was done in real 3D because the surround had corrugations and therefore axisymmetric modelling could not be used. The publication was divided to three phases. First, the moving parts of the transducer were modelled with FEM.
模型是在真实的 3D 中完成的，因为周围有波纹，因此无法使用轴对称建模。出版物被分为三个阶段。首先，使用 FEM 对换能器的移动部分进行了建模。
Modelling results were compared to the average acceleration derived from measurements in a vacuum. As usual, there was agreement with results although the high frequency modelling is not very exact. Second, the air load was included in the
建模结果与在真空环境中测量得出的平均加速度进行了比较。如常，结果是一致的，尽管高频建模的准确性不是非常高。其次，空气负载被包含在了
model. The influence of the slit elements and magnet circuit behind the transducer was investigated. Also a phase plug was added in front of the transducer. Third, the cushioning surround of the transducer was added. Adding details to the model also adds sources of error.
模型。研究了转换器背后的狭缝元件和磁路的影响。此外，在转换器前方添加了一个相位塞。第三，添加了转换器的缓冲环绕。在模型中添加细节也会引入误差源。
However, the basic phenomena were visible both in measured and modelled curves. After all, this paper was a rare insight to headphone transducer design.
然而，在测量和建模的曲线中，基本现象都清晰可见。毕竟，这篇论文是对耳机换能器设计难得的深入探讨。
In 2007, Backman presented a paper where he compared analytical solutions of various acoustical phenomena to more realistic FEM models [11]. He studied the impedances of transmission lines. The main point was to excite the transmission line with a non-planar wave.
2007 年，Backman 发表了一篇论文，他在论文中将各种声学现象的分析解与更现实的 FEM 模型进行了比较[11]。他研究了传输线的阻抗。主要观点是用非平面波激发传输线。
The conclusion was that one-dimensional solutions successfully predict the few lowest nodes of the impedance tube. Second, he modelled the acoustical length of the port with several flare radii.
结论是，一维解决方案成功预测了阻抗管的几个最低节点。其次，他用几个喇叭半径建模了出口的声学长度。
The conclusion was, that the lumped parameter model is accurate for predicting the fundamental resonance frequency of a box with port, but not accurate enough to predict the open pipe resonances of the port if a back wall or flared edges are present.
结论是，集中参数模型在预测带通口的盒子的基本共振频率方面是准确的，但不足以预测通口的开放管共振，如果有后墙或喇叭形边缘存在。
One detail to criticize is that Backman used acoustic simulation software, which does not take in account the turbulent airflow inside the port [2].
需要批评的一点是，Backman 使用了声学模拟软件，该软件并未考虑港口内部湍流气流的影响 [2]。
2007 年，Backman 发表了一篇论文，他在论文中将各种声学现象的分析解与更现实的 FEM 模型进行了比较[11]。他研究了传输线的阻抗。主要观点是用非平面波激发传输线。
The conclusion was that one-dimensional solutions successfully predict the few lowest nodes of the impedance tube. Second, he modelled the acoustical length of the port with several flare radii.
结论是，一维解决方案成功预测了阻抗管的几个最低节点。其次，他用几个喇叭半径建模了出口的声学长度。
The conclusion was, that the lumped parameter model is accurate for predicting the fundamental resonance frequency of a box with port, but not accurate enough to predict the open pipe resonances of the port if a back wall or flared edges are present.
结论是，集中参数模型在预测带通口的盒子的基本共振频率方面是准确的，但不足以预测通口的开放管共振，如果有后墙或喇叭形边缘存在。
One detail to criticize is that Backman used acoustic simulation software, which does not take in account the turbulent airflow inside the port [2].
需要批评的一点是，Backman 使用了声学模拟软件，该软件并未考虑港口内部湍流气流的影响 [2]。
4 The subjective importance of the directivity
4 直达性的主观重要性
The purpose of this chapter is to motivate the importance of the source directivity to the sound perceived by the listener.
本章的目的是强调声源的方向性对于听众感知声音的重要性。
First is a literature review on the factors affecting to the human preference of the sound and what is the contribution of the directivity Second is discussed the loudspeaker and room interaction.
第一部分是对影响人类对声音偏好的因素的文献综述，以及直接性对贡献的讨论。第二部分是讨论扬声器与房间的相互作用。
本章的目的是强调声源的方向性对于听众感知声音的重要性。
First is a literature review on the factors affecting to the human preference of the sound and what is the contribution of the directivity Second is discussed the loudspeaker and room interaction.
第一部分是对影响人类对声音偏好的因素的文献综述，以及直接性对贡献的讨论。第二部分是讨论扬声器与房间的相互作用。
4.1 Human preference for sound
4.1 人类对声音的偏好
Comparison of loudspeakers' performance has been under discussion as long as they have been built. It would be desirable that, the subjective performance of a speaker could be evaluated with objective measurements.
扬声器性能的比较自它们被构建以来一直是讨论的话题。理想情况下，扬声器的主观性能可以用客观测量来评估。
Toole has done extensive work on evaluating loudspeaker subjective performance [12][13][14]. His findings were unambiguous. In general, flat on-axis frequency response is preferred over inconsistent response [13].
Toole 在评估扬声器的主观性能方面进行了大量工作 [12][13][14]。他的发现是明确的。一般来说，轴向频率响应平坦的扬声器更受青睐，而不是响应不一致的扬声器 [13]。
Likewise a low level of nonlinear distortion is preferred over high a level of distortion [14]. Toole also paid great attention to the loudspeaker and room interaction, which is dependent on the loudspeaker directivity.
同样，较低的非线性失真水平优于较高的失真水平 [14]。Toole 也非常重视扬声器和房间的相互作用，这取决于扬声器的直接性。
These finding are discussed in detail in the following paragraphs.
这些发现将在以下段落中详细讨论。
扬声器性能的比较自它们被构建以来一直是讨论的话题。理想情况下，扬声器的主观性能可以用客观测量来评估。
Toole has done extensive work on evaluating loudspeaker subjective performance [12][13][14]. His findings were unambiguous. In general, flat on-axis frequency response is preferred over inconsistent response [13].
Toole 在评估扬声器的主观性能方面进行了大量工作 [12][13][14]。他的发现是明确的。一般来说，轴向频率响应平坦的扬声器更受青睐，而不是响应不一致的扬声器 [13]。
Likewise a low level of nonlinear distortion is preferred over high a level of distortion [14]. Toole also paid great attention to the loudspeaker and room interaction, which is dependent on the loudspeaker directivity.
同样，较低的非线性失真水平优于较高的失真水平 [14]。Toole 也非常重视扬声器和房间的相互作用，这取决于扬声器的直接性。
These finding are discussed in detail in the following paragraphs.
这些发现将在以下段落中详细讨论。
According to Toole, a controlled change of frequency response towards an off-axis direction is preferred over abrupt changes [13]. This concept is called controlled directivity. The concept is very loosely specified. Defining the directivity is further discussed in Chapter 5.
根据 Toole 的观点，频率响应在偏离轴向的方向上的受控变化比突然变化更可取[13]。这个概念被称为受控定向性。这个概念的定义非常宽松。在第 5 章中进一步讨论了定向性的定义。
根据 Toole 的观点，频率响应在偏离轴向的方向上的受控变化比突然变化更可取[13]。这个概念被称为受控定向性。这个概念的定义非常宽松。在第 5 章中进一步讨论了定向性的定义。
The sound heard in a room is dependent on the room, the speaker and the signal transmitted to the speaker. In this work the emphasis is on the speaker performance and how it can be evaluated.
房间中听到的声音依赖于房间、演讲者和传送给演讲者的信号。在这项工作中，重点是演讲者的性能以及如何对其进行评估。
However room acoustics is briefly covered to highlight the importance of the loudspeaker directivity characteristics. The effect of the loudspeaker signal source is out of the scope of the thesis.
然而，本论文简要涉及了室内声学，以强调扬声器直接性特征的重要性。扬声器信号源的影响超出了论文的范围。
房间中听到的声音依赖于房间、演讲者和传送给演讲者的信号。在这项工作中，重点是演讲者的性能以及如何对其进行评估。
However room acoustics is briefly covered to highlight the importance of the loudspeaker directivity characteristics. The effect of the loudspeaker signal source is out of the scope of the thesis.
然而，本论文简要涉及了室内声学，以强调扬声器直接性特征的重要性。扬声器信号源的影响超出了论文的范围。
4.2 Loudspeaker and room interaction
4.2 扬声器与房间的互动
The following is an analysis of the factors affecting the measured impulse response of a speaker in a room.
以下是关于房间中扬声器测量瞬态响应受到的影响因素的分析。
The assumptions are that the speaker is in a room, its acoustical axis is towards the listening position and there are not obstacles in the line of sight between the speaker and the listening position (Figure 4.1). The impulse response specifies change between the input signal of the system and the pressure at the listening position.
假设是：说话者位于一个房间内，其声学轴线指向听音位置，且在扬声器和听音位置之间没有障碍物（见图 4.1）。脉冲响应规定了系统输入信号与听音位置处的压力之间的变化。
The only missing variable is the listener. Therefore impulse response should closely correlate to the sound perceived by the listener.
唯一缺失的变量是听众。因此，冲激响应应与听众感知的声音高度相关。
以下是关于房间中扬声器测量瞬态响应受到的影响因素的分析。
The assumptions are that the speaker is in a room, its acoustical axis is towards the listening position and there are not obstacles in the line of sight between the speaker and the listening position (Figure 4.1). The impulse response specifies change between the input signal of the system and the pressure at the listening position.
假设是：说话者位于一个房间内，其声学轴线指向听音位置，且在扬声器和听音位置之间没有障碍物（见图 4.1）。脉冲响应规定了系统输入信号与听音位置处的压力之间的变化。
The only missing variable is the listener. Therefore impulse response should closely correlate to the sound perceived by the listener.
唯一缺失的变量是听众。因此，冲激响应应与听众感知的声音高度相关。
Figure 4.1. Direct sound (green), boundary reflections (magenta, red, orange) at a listening position of a music studio.
图 4.1. 直达声（绿色），边界反射（品红色，红色，橙色）在音乐工作室的聆听位置。
图 4.1. 直达声（绿色），边界反射（品红色，红色，橙色）在音乐工作室的聆听位置。
One approach to analyze the loudspeaker and room interaction is to look at the energy decay at the listening position (Figure 4.2). The energy decay curve is a logarithmic presentation of a squared impulse response.
分析扬声器和房间相互作用的一种方法是观察听音位置的能量衰减（图 4.2）。能量衰减曲线是以平方脉冲响应的对数形式呈现的。
As the figure shows, the sound heard at the listening position can be divided to three parts.
如图所示，听音位置听到的声音可以分为三部分。
分析扬声器和房间相互作用的一种方法是观察听音位置的能量衰减（图 4.2）。能量衰减曲线是以平方脉冲响应的对数形式呈现的。
As the figure shows, the sound heard at the listening position can be divided to three parts.
如图所示，听音位置听到的声音可以分为三部分。
Figure 4.2. The energy decay curve of a loudspeaker in a room. Measured at the listening position of a music studio.
图 4.2. 房间中扬声器的能量衰减曲线。在音乐工作室的聆听位置测量。
图 4.2. 房间中扬声器的能量衰减曲线。在音乐工作室的聆听位置测量。
First, the direct sound arrives.
首先，直接声音到达。
With a reasonable listening arrangement, there are no obstacles in the line of sight between the listening position and the speaker (Figure 4.1). Likewise the speaker acoustical axis is turned towards the listening position, which is now on referred to on-axis response.
通过合理的听音布局，听音位置与演讲者之间视线无障碍（图 4.1）。同样，演讲者的声学轴线转向听音位置，现在将其称为轴向响应。
Therefore the direct sound depends only on the loudspeaker on-axis frequency response. The direct sound is marked in Figure 4.2, which usually is the highest peak of the impulse response.
因此，直接声音仅取决于扬声器的轴向频率响应。直接声音在图 4.2 中标记，通常是最高的瞬态响应峰值。
首先，直接声音到达。
With a reasonable listening arrangement, there are no obstacles in the line of sight between the listening position and the speaker (Figure 4.1). Likewise the speaker acoustical axis is turned towards the listening position, which is now on referred to on-axis response.
通过合理的听音布局，听音位置与演讲者之间视线无障碍（图 4.1）。同样，演讲者的声学轴线转向听音位置，现在将其称为轴向响应。
Therefore the direct sound depends only on the loudspeaker on-axis frequency response. The direct sound is marked in Figure 4.2, which usually is the highest peak of the impulse response.
因此，直接声音仅取决于扬声器的轴向频率响应。直接声音在图 4.2 中标记，通常是最高的瞬态响应峰值。
Second, the early reflections arrive (Figure 4.2). These are usually first order reflections from the side walls, floor and ceiling (Figure 4.1). The time difference between the direct sound and early reflections is dependent on the wave travel time difference.
第二，早期的回声到达（图 4.2）。这些通常是来自侧面墙壁、地板和天花板的第一级反射（图 4.1）。直接声音和早期回声之间的时间差取决于波的传播时间差。
Early reflections contribute to the spatial and tonal perception of the sound. The spatial effect is more easily understood by considering the time domain signal presented.
早期的反思有助于对声音的空间和音调感知。空间效果通过考虑时间域信号来更容易理解。
The time delay and amplitude of the reflection gives a clue to the auditory system about the spatial space. Early reflections also affect the tonal balance. The sound absorption of the reflecting surfaces is frequency dependent.
反射的时间延迟和幅度为听觉系统提供了关于空间位置的线索。早期的反射也影响音调平衡。反射表面的声吸收依赖于频率。
With a steady state signal, the direct and delayed reflected sound interferes at the listening position, which causes a comb filtering effect in the frequency domain.
稳态信号下，在听音位置直接和延迟反射的声音相互干扰，导致频率域中的梳状滤波效应。
第二，早期的回声到达（图 4.2）。这些通常是来自侧面墙壁、地板和天花板的第一级反射（图 4.1）。直接声音和早期回声之间的时间差取决于波的传播时间差。
Early reflections contribute to the spatial and tonal perception of the sound. The spatial effect is more easily understood by considering the time domain signal presented.
早期的反思有助于对声音的空间和音调感知。空间效果通过考虑时间域信号来更容易理解。
The time delay and amplitude of the reflection gives a clue to the auditory system about the spatial space. Early reflections also affect the tonal balance. The sound absorption of the reflecting surfaces is frequency dependent.
反射的时间延迟和幅度为听觉系统提供了关于空间位置的线索。早期的反射也影响音调平衡。反射表面的声吸收依赖于频率。
With a steady state signal, the direct and delayed reflected sound interferes at the listening position, which causes a comb filtering effect in the frequency domain.
稳态信号下，在听音位置直接和延迟反射的声音相互干扰，导致频率域中的梳状滤波效应。
Third, the diffuse reverberation is left (Figure 4.2). Reverberation consists of a countless number of reflections from the room boundaries e.g. it is diffuse. The diffuse reverberation contributes to the perceived spatial experience.
第三，留下扩散反射（图 4.2）。反射包括无数次从房间边界反射，例如它是扩散的。扩散反射有助于形成感知的空间体验。
It takes a certain time for diffuse reverberation to build up. Therefore there is a silent part in the impulse response between the early reflections and constantly attenuating diffuse
散射回波的积累需要一定的时间。因此，在早期反射和不断衰减的散射之间，冲激响应中存在一段静默部分。
reverberation. Diffuse reverberation also affects the tonal balance of the sound, because absorption coefficients of the room material are frequency dependent. It is common that the reverberation time is longer at the low frequencies.
回声。漫反射回声也会影响声音的音调平衡，因为房间材料的吸收系数是频率相关的。通常情况下，低频时的回声时间更长。
第三，留下扩散反射（图 4.2）。反射包括无数次从房间边界反射，例如它是扩散的。扩散反射有助于形成感知的空间体验。
It takes a certain time for diffuse reverberation to build up. Therefore there is a silent part in the impulse response between the early reflections and constantly attenuating diffuse
散射回波的积累需要一定的时间。因此，在早期反射和不断衰减的散射之间，冲激响应中存在一段静默部分。
reverberation. Diffuse reverberation also affects the tonal balance of the sound, because absorption coefficients of the room material are frequency dependent. It is common that the reverberation time is longer at the low frequencies.
回声。漫反射回声也会影响声音的音调平衡，因为房间材料的吸收系数是频率相关的。通常情况下，低频时的回声时间更长。
When analyzing the three phenomena seen in the impulse response, it is noteworthy that only the direct sound is defined by the on-axis frequency response. Despite of the fact, the on-axis response is one of the most used measures to evaluate loudspeaker performance.
分析冲击响应中出现的三种现象时，值得注意的是，只有直达声由轴向频率响应定义。尽管如此，轴向响应仍然是评估扬声器性能最常用的指标之一。
Early reflections are dependent on the room acoustics and off-axis response of the speaker at the relevant angle. Diffuse reverberation is dependent on room acoustics and total power emitted by the speaker i.e. power response.
早期的反思依赖于房间的声学特性和与相关角度相匹配的扬声器的离轴响应。扩散回声依赖于房间的声学特性和扬声器发出的总功率，即功率响应。
分析冲击响应中出现的三种现象时，值得注意的是，只有直达声由轴向频率响应定义。尽管如此，轴向响应仍然是评估扬声器性能最常用的指标之一。
Early reflections are dependent on the room acoustics and off-axis response of the speaker at the relevant angle. Diffuse reverberation is dependent on room acoustics and total power emitted by the speaker i.e. power response.
早期的反思依赖于房间的声学特性和与相关角度相匹配的扬声器的离轴响应。扩散回声依赖于房间的声学特性和扬声器发出的总功率，即功率响应。
To conclude, it is generally agreed that subjective loudspeaker performance is largely specified by following characteristics: on-axis frequency response, directivity and distortion [1] [14]. This thesis concentrates on the directivity.
总结而言，普遍认为主观扬声器性能主要由以下特性规定：轴上频率响应、方向性和失真 [1] [14]。本论文着重探讨方向性。
总结而言，普遍认为主观扬声器性能主要由以下特性规定：轴上频率响应、方向性和失真 [1] [14]。本论文着重探讨方向性。
5 Representing directivity
5 表示指向性
This chapter presents the known ways to visualize directivity. In the simplest form the directivity is presented as a single scalar number that is a function of frequency (Directivity factor (Q), Directivity index (DI) and Beam width).
本章介绍了已知的直接性可视化方法。在最简单的形式中，直接性被表示为一个单一的标量数字，它是频率的函数（直接性因子（Q）、直接性指数（DI）和束宽）。
It is noteworthy that the mathematics behind these presentation methods is most complicated, because of the need to compress a four-dimensional information to one scalar number.
值得注意的是，这些展示方法背后的数学原理最为复杂，因为需要将四维信息压缩为一个标量数字。
本章介绍了已知的直接性可视化方法。在最简单的形式中，直接性被表示为一个单一的标量数字，它是频率的函数（直接性因子（Q）、直接性指数（DI）和束宽）。
It is noteworthy that the mathematics behind these presentation methods is most complicated, because of the need to compress a four-dimensional information to one scalar number.
值得注意的是，这些展示方法背后的数学原理最为复杂，因为需要将四维信息压缩为一个标量数字。
The second step is to show the frequency response at the specific points of the space (Frequency response) or one polar arc of the directivity at a specific frequency (Polar diagram).
第二步是展示空间特定点的频率响应（频率响应）或特定频率下直接性的一条极坐标图（极图）。
The problem with these plots is that many lines are needed present comprehensive information about the directivity.
这些图表的问题在于，需要许多线条来呈现指向性全面的信息。
第二步是展示空间特定点的频率响应（频率响应）或特定频率下直接性的一条极坐标图（极图）。
The problem with these plots is that many lines are needed present comprehensive information about the directivity.
这些图表的问题在于，需要许多线条来呈现指向性全面的信息。
The third group are the 3D graphs. Either a full 3D pressure response balloon of the radiated sound at a specific frequency can be presented (Balloon graph) or equal amplitude contours of one polar plane as function of frequency (Directivity plot).
第三组是 3D 图。可以呈现特定频率下辐射声波的完整 3D 压力响应气球（气球图）或频率函数下一个极平面的等幅轮廓（方向性图）。
The limitation of the balloon graph is that only one frequency can be shown at a time. The limitation of the directivity plot is that only one plane can be shown at once.
气球图的限制在于一次只能显示一个频率。直接性图的限制在于一次只能显示一个平面。
第三组是 3D 图。可以呈现特定频率下辐射声波的完整 3D 压力响应气球（气球图）或频率函数下一个极平面的等幅轮廓（方向性图）。
The limitation of the balloon graph is that only one frequency can be shown at a time. The limitation of the directivity plot is that only one plane can be shown at once.
气球图的限制在于一次只能显示一个频率。直接性图的限制在于一次只能显示一个平面。
5.1 Source radiation in space
5.1 太空中的源辐射
Consider a sound source in a space that is located at the origin of the axes (Figure 5.1). Around the source are the polar arcs of the horizontal plane and vertical plane. The sound source is radiating to space and thus its radiation is a three-dimensional problem.
考虑一个位于坐标轴原点的声源（图 5.1）。在声源周围是水平面和垂直面的极坐标弧。声源向空间辐射，因此其辐射是一个三维问题。
Its radiation is also a function of frequency and therefore a full expression of a directivity is four-dimensional problem. Unfortunately four-dimensional problems cannot be visualized in an understandable form.
其辐射也与频率有关，因此直接性表达是一个四维问题。不幸的是，四维问题无法以可理解的形式可视化。
Therefore there have been several methods compressing the directivity information to one constant, onedimensional, two-dimensional or three-dimensional graph. But it is noteworthy that no graph can express the source directivity completely.
因此，已经提出了几种方法将方向性信息压缩为一个常数、一维、二维或三维的图表。值得注意的是，没有任何图表能够完全表达源的方向性。
The fewer dimensions, the more compromises have to be made.
维度越少，需要做出的妥协就越多。
考虑一个位于坐标轴原点的声源（图 5.1）。在声源周围是水平面和垂直面的极坐标弧。声源向空间辐射，因此其辐射是一个三维问题。
Its radiation is also a function of frequency and therefore a full expression of a directivity is four-dimensional problem. Unfortunately four-dimensional problems cannot be visualized in an understandable form.
其辐射也与频率有关，因此直接性表达是一个四维问题。不幸的是，四维问题无法以可理解的形式可视化。
Therefore there have been several methods compressing the directivity information to one constant, onedimensional, two-dimensional or three-dimensional graph. But it is noteworthy that no graph can express the source directivity completely.
因此，已经提出了几种方法将方向性信息压缩为一个常数、一维、二维或三维的图表。值得注意的是，没有任何图表能够完全表达源的方向性。
The fewer dimensions, the more compromises have to be made.
维度越少，需要做出的妥协就越多。
Figure 5.1. Loudspeaker and the planes. Loudspeaker in the origin, vertical plane (blue), horizontal plane (green).
图 5.1. 扬声器和平面。扬声器在原点，垂直平面（蓝色），水平平面（绿色）。
图 5.1. 扬声器和平面。扬声器在原点，垂直平面（蓝色），水平平面（绿色）。
The target of expressing directivity with numbers or a graph has been to compress the four-dimensional information to form that would be informative and describe the directivity characteristics enough and at the same time being understandable and easy to compare with other sources
使用数字或图表表达直接性的目标是将四维信息压缩为足够描述直接性特征、同时易于理解并与其它来源进行比较的信息
使用数字或图表表达直接性的目标是将四维信息压缩为足够描述直接性特征、同时易于理解并与其它来源进行比较的信息
5.2 Directivity factor (Q)
5.2 方向性系数（Q）
Directivity factor can be seen as a ratio between on-axis pressure and the total sound power radiated by the speaker. In exact form, it is ratio between the sound power
直接性系数可以被视为扬声器辐射的总声功率与轴上压力的比率。确切地说，它是
直接性系数可以被视为扬声器辐射的总声功率与轴上压力的比率。确切地说，它是
Where
在方程（5.2）中定义了
在方程（5.2）中定义了
Figure 5.2. Coordinate system. Adopted from [16].
图 5.2. 坐标系。从[16]采用。
图 5.2. 坐标系。从[16]采用。
Some very useful room acoustic parameters can be calculated when Directivity factor Q is known. The sound pressure level
已知直接性系数 Q 时，可以计算出一些非常有用的房间声学参数。当已知声源的直接性系数
已知直接性系数 Q 时，可以计算出一些非常有用的房间声学参数。当已知声源的直接性系数
The reverberation radius
反射半径
反射半径
5.3 Directivity index (DI)
5.3 方向性指数（DI）
Directivity index (Equation (5.6) is the directivity factor Q in a logarithmic form. It is more commonly used than the directivity factor. It is defined as [15].
直接性指数（式（5.6）中的直接性因子 Q 以对数形式表示。它比直接性因子更常用。它被定义为[15]。）
直接性指数（式（5.6）中的直接性因子 Q 以对数形式表示。它比直接性因子更常用。它被定义为[15]。）
The DI index is widely used in the loudspeaker industry. Its benefit is that it is rather easy to calculate from the commonly measured on-axis frequency response and horizontal and vertical polar measurements. It can be presented as a single curve that is a function of frequency.
DI 指数在扬声器行业被广泛使用。其优点是可以从通常测量的轴向频率响应和水平、垂直极化测量中相对容易地计算出来。它可以表示为一个频率的函数的单曲线。
DI 指数在扬声器行业被广泛使用。其优点是可以从通常测量的轴向频率响应和水平、垂直极化测量中相对容易地计算出来。它可以表示为一个频率的函数的单曲线。
The drawback is that DI simplifies the off-axis radiation. The root of this problem is that sound power is integrated over all angles.
缺点是 DI 简化了偏离轴线的辐射。这个问题的根源在于，声音功率是在所有角度上进行积分的。
Therefore sources with very different radiation characteristic may still have equal on-axis pressure response and power response, and therefore have equal directivity indeces.
因此，具有非常不同辐射特性的源仍然可能具有相同的轴向压力响应和功率响应，因此具有相同的直接性指数。
缺点是 DI 简化了偏离轴线的辐射。这个问题的根源在于，声音功率是在所有角度上进行积分的。
Therefore sources with very different radiation characteristic may still have equal on-axis pressure response and power response, and therefore have equal directivity indeces.
因此，具有非常不同辐射特性的源仍然可能具有相同的轴向压力响应和功率响应，因此具有相同的直接性指数。
5.4 Beam width 5.4 光束宽度
Beam width presents the width of the main lobe of the sound source. The main lobe is defined as an angle of -6 dB attenuation compared to the on-axis response that is a function of frequency. This information is already available in polar plots.
束宽表示声源主瓣的宽度。主瓣定义为与频率相关的轴向响应相比衰减-6 dB 的角度。这些信息已经在极坐标图中已经可用。
Therefore the beam width can be seen as post-processing of polar plots to show the directivity information in a single graph. The benefit is good frequency resolution.
因此，波束宽度可以被视为对极坐标图的后处理，以在单个图表中显示直接性信息。好处是具有良好的频率分辨率。
The downside is that only the -6 dB curve is available, which does not comprehensively describe directivity characteristic of the source.
缺点是仅提供了-6 dB 曲线，无法全面描述声源的指向性特性。
束宽表示声源主瓣的宽度。主瓣定义为与频率相关的轴向响应相比衰减-6 dB 的角度。这些信息已经在极坐标图中已经可用。
Therefore the beam width can be seen as post-processing of polar plots to show the directivity information in a single graph. The benefit is good frequency resolution.
因此，波束宽度可以被视为对极坐标图的后处理，以在单个图表中显示直接性信息。好处是具有良好的频率分辨率。
The downside is that only the -6 dB curve is available, which does not comprehensively describe directivity characteristic of the source.
缺点是仅提供了-6 dB 曲线，无法全面描述声源的指向性特性。
The beam width graph usability is at best at public address systems for outdoor events. Then the beam width provides indication of how large area can be relatively uniformly covered with a single source.
阵列宽度图的可用性在户外活动的公共广播系统中最好。然后，阵列宽度提供了单个来源可以相对均匀覆盖多大面积的指示。
阵列宽度图的可用性在户外活动的公共广播系统中最好。然后，阵列宽度提供了单个来源可以相对均匀覆盖多大面积的指示。
5.5 Polar diagram 5.5 极坐标图
The polar diagram shows the amplitude of the source on a horizontal or a vertical plane (Figure 5.1). The polar plot depicts amplitude of a source at a certain frequency as a function of angle (Figure 5.3). Usually the polar response is normalized to the on-axis response.
极坐标图显示在水平或垂直平面上源的振幅（图 5.1）。极坐标图表示在特定频率下源的振幅随角度的变化（图 5.3）。通常，极坐标响应会归一化到轴向响应。
Therefore on-axis response has 0 dB and amplitudes at the polar angles are relative and usually lower than the on-axis amplitude.
因此，轴向响应为 0 分贝，极角处的振幅相对且通常低于轴向振幅。
极坐标图显示在水平或垂直平面上源的振幅（图 5.1）。极坐标图表示在特定频率下源的振幅随角度的变化（图 5.3）。通常，极坐标响应会归一化到轴向响应。
Therefore on-axis response has 0 dB and amplitudes at the polar angles are relative and usually lower than the on-axis amplitude.
因此，轴向响应为 0 分贝，极角处的振幅相对且通常低于轴向振幅。
Figure 5.3. Polar diagram of a 2-way speaker at 3 kHz . Horizontal (red) and vertical (blue) planes.
图 5.3. 3 kHz 时的 2 路扬声器的极坐标图。水平（红色）和平面（蓝色）。
图 5.3. 3 kHz 时的 2 路扬声器的极坐标图。水平（红色）和平面（蓝色）。
The advantage of the polar plot is the accurate angle resolution. Single graphs are also easy to compare. The disadvantage is that only few frequencies can be shown in one graph with good readability.
极坐标图的优点是准确的角度分辨率。单个图表也易于比较。缺点是一张图表中只能显示少量可读性良好的频率。
Therefore several graphs are needed to see a large enough frequency range, which compromises the readability of the result.
因此需要几个图表来观察足够大的频率范围，这会牺牲结果的可读性。
极坐标图的优点是准确的角度分辨率。单个图表也易于比较。缺点是一张图表中只能显示少量可读性良好的频率。
Therefore several graphs are needed to see a large enough frequency range, which compromises the readability of the result.
因此需要几个图表来观察足够大的频率范围，这会牺牲结果的可读性。
5.6 Frequency response 5.6 频率响应
A simple way to visualize directivity is to plot several frequency responses at different locations on the polar planes (Figure 5.4). The advantage of this visualization is a good frequency resolution.
可视化直接性的一种简单方法是在极坐标平面上的不同位置绘制多个频率响应（图 5.4）。这种可视化的优势是具有良好的频率分辨率。
The source has to be measured only at the angles of interest. Therefore the measurement is fast and can be made without a turntable. The disadvantage is the limited angle resolution, because only a few responses can be shown with good readability.
源必须仅在感兴趣的角 度进行测量。因此，测量速度快，无需使用旋转台。缺点是角度分辨率有限，因为只能显示有限的响应，且具有良好的可读性。
可视化直接性的一种简单方法是在极坐标平面上的不同位置绘制多个频率响应（图 5.4）。这种可视化的优势是具有良好的频率分辨率。
The source has to be measured only at the angles of interest. Therefore the measurement is fast and can be made without a turntable. The disadvantage is the limited angle resolution, because only a few responses can be shown with good readability.
源必须仅在感兴趣的角 度进行测量。因此，测量速度快，无需使用旋转台。缺点是角度分辨率有限，因为只能显示有限的响应，且具有良好的可读性。
Figure 5.4. Frequency responses of horizontal plane.
图 5.4. 水平面的频率响应。
图 5.4. 水平面的频率响应。
5.7 Balloon graph 5.7 气球图
Imagine that the every point on the surface of the balloon (Figure 5.5) has its own frequency response. If only one frequency is viewed at the time, a graph may be made where the colour and position of the points describe the amplitude in that direction.
想象一下，气球（图 5.5）表面的每一个点都有自己的频率响应。如果一次只观察一个频率，可以制作一个图表，其中点的颜色和位置描述了该方向的振幅。
This can be also seen as a 3D equivalent for traditional polar plots introduced before. The advantage of this visualization is that full 3D radiation can be shown at once. The disadvantage is that only one frequency can be presented per graph.
这也可以被视为传统极坐标图的 3D 等价物。这种可视化的优势在于可以一次性展示完整的 3D 辐射。缺点是每张图只能展示一个频率。
想象一下，气球（图 5.5）表面的每一个点都有自己的频率响应。如果一次只观察一个频率，可以制作一个图表，其中点的颜色和位置描述了该方向的振幅。
This can be also seen as a 3D equivalent for traditional polar plots introduced before. The advantage of this visualization is that full 3D radiation can be shown at once. The disadvantage is that only one frequency can be presented per graph.
这也可以被视为传统极坐标图的 3D 等价物。这种可视化的优势在于可以一次性展示完整的 3D 辐射。缺点是每张图只能展示一个频率。
Figure 5.5. Balloon graph of two-way speaker at 3 kHz
图 5.5. 3 kHz 时双向扬声器的气球图
图 5.5. 3 kHz 时双向扬声器的气球图
5.8 Directivity plot 5.8 直达性图
The directivity contour plot can be seen as an extension of the beam width curve. Information about the beam width is enhanced by adding the contours of other dB limits. Usually the resolution is from 0 dB to -21 dB in 3 dB increments.
直接性轮廓图可以被视为波束宽度曲线的扩展。通过添加其他分贝限制的轮廓，波束宽度的信息得到了增强。通常，分辨率从 0 分贝到-21 分贝，以 3 分贝的增量进行。
The readability of the plot has been improved by adding colours between the amplitude contours. Adding the colours can also been seen as converting the graph to a three dimensional representation.
通过在幅度轮廓之间添加颜色，故事情节的可读性得到了提高。添加颜色也可以被视为将图表转换为三维表示。
In a way, the directivity contour diagram combines the advantages of polar plots and beam width curves. The downside is the trade off between the frequency resolution and the readability of the graph.
以某种方式，直接性轮廓图结合了极坐标图和波束宽度曲线的优点。缺点是频率分辨率和图表可读性之间的权衡。
The amplitude resolution of the contours has to be large enough to maintain the readability.
轮廓的振幅分辨率必须足够大，以保持可读性。
直接性轮廓图可以被视为波束宽度曲线的扩展。通过添加其他分贝限制的轮廓，波束宽度的信息得到了增强。通常，分辨率从 0 分贝到-21 分贝，以 3 分贝的增量进行。
The readability of the plot has been improved by adding colours between the amplitude contours. Adding the colours can also been seen as converting the graph to a three dimensional representation.
通过在幅度轮廓之间添加颜色，故事情节的可读性得到了提高。添加颜色也可以被视为将图表转换为三维表示。
In a way, the directivity contour diagram combines the advantages of polar plots and beam width curves. The downside is the trade off between the frequency resolution and the readability of the graph.
以某种方式，直接性轮廓图结合了极坐标图和波束宽度曲线的优点。缺点是频率分辨率和图表可读性之间的权衡。
The amplitude resolution of the contours has to be large enough to maintain the readability.
轮廓的振幅分辨率必须足够大，以保持可读性。
Figure 5.6. Directivity of direct radiating two-way speaker.
图 5.6. 直接辐射式双向扬声器的方向性。
图 5.6. 直接辐射式双向扬声器的方向性。
An example of a directivity plot (Figure 5.6) is from two-way speaker horizontal plane measurements. The plot is for a horizontal angle from
直接性图（图 5.6）的一个例子是从双路扬声器水平平面测量得到的。图示的是从
直接性图（图 5.6）的一个例子是从双路扬声器水平平面测量得到的。图示的是从
6 The directivity of a direct radiator
6 直接辐射器的指向性
The directivity of the circular piston source is relevant for several reasons, although the topic of the thesis is waveguide directivity. It is common combine direct radiating sources and sources with a waveguide in a loudspeaker.
圆柱形活塞源的方向性对于几个原因很重要，尽管论文的主题是波导的方向性。通常会将直接辐射源与波导结合在扬声器中使用。
Usually the goal of the design is to match the directivity of these sources. Therefore it is essential to understand the elements affecting direct radiating source directivity.
通常设计的目标是匹配这些源的指向性。因此，理解影响直接辐射源指向性的元素是至关重要的。
Also, in some cases the directivity of a waveguide can be described as the directivity of a plane circular piston.
此外，在某些情况下，波导的直接性可以描述为平面圆形活塞的直接性。
圆柱形活塞源的方向性对于几个原因很重要，尽管论文的主题是波导的方向性。通常会将直接辐射源与波导结合在扬声器中使用。
Usually the goal of the design is to match the directivity of these sources. Therefore it is essential to understand the elements affecting direct radiating source directivity.
通常设计的目标是匹配这些源的指向性。因此，理解影响直接辐射源指向性的元素是至关重要的。
Also, in some cases the directivity of a waveguide can be described as the directivity of a plane circular piston.
此外，在某些情况下，波导的直接性可以描述为平面圆形活塞的直接性。
The directivity of a loudspeaker is highly frequency dependent. The analytical solution for directivity of circular piston source in an infinite baffle does exist (Equation (6.2) [15]).
扬声器的方向性高度依赖于频率。无限障板中圆形活塞源的方向性分析解确实存在（式（6.2）[15]）。
The circular piston source radiation directivity is dependent on the variable called wave number
圆盘活塞源辐射的方向性依赖于称为波数的变量
扬声器的方向性高度依赖于频率。无限障板中圆形活塞源的方向性分析解确实存在（式（6.2）[15]）。
The circular piston source radiation directivity is dependent on the variable called wave number
圆盘活塞源辐射的方向性依赖于称为波数的变量
With the wave number known, the directivity at an angle
已知波数，通过一阶贝塞尔函数（公式（6.2）），可以在角度
已知波数，通过一阶贝塞尔函数（公式（6.2）），可以在角度
One way to express the directivity of a circular piston source is to show the polar plots at certain ka-numbers (Figure 6.1). The following three features can be found when analyzing the directivity of a circular piston source.
表达环形活塞源的方向性的一种方式是在特定的 ka 数下展示极坐标图（图 6.1）。在分析环形活塞源的方向性时，可以发现以下三个特征。
First, a circular piston source is omnidirectional when the wave number
首先，当波数
Thus sound radiated from different parts of the piston is in and out of phase which causes constructive and destructive interference in off-axis directions.
因此，活塞的不同部分发出的声音在轴外方向上相位不同，导致了相长和相消干涉。
表达环形活塞源的方向性的一种方式是在特定的 ka 数下展示极坐标图（图 6.1）。在分析环形活塞源的方向性时，可以发现以下三个特征。
First, a circular piston source is omnidirectional when the wave number
首先，当波数
Thus sound radiated from different parts of the piston is in and out of phase which causes constructive and destructive interference in off-axis directions.
因此，活塞的不同部分发出的声音在轴外方向上相位不同，导致了相长和相消干涉。
Figure 6.1. Directivity of a circular piston source. Adopted from [16].
图 6.1. 圆形活塞源的方向性。引自[16]。
图 6.1. 圆形活塞源的方向性。引自[16]。
Direct radiator directivity performance has a major influence on loudspeaker design. These acoustical phenomena limit the usability of direct radiating sources for loudspeakers.
直接辐射器的直接性性能对扬声器设计有重大影响。这些声学现象限制了直接辐射源在扬声器中的应用。
直接辐射器的直接性性能对扬声器设计有重大影响。这些声学现象限制了直接辐射源在扬声器中的应用。
Based on the analysis of the circular piston directivity, three conclusions can be made considering a loudspeaker design. First, the beam width of a large cone might be too narrow at high frequencies to cover the desired area or then the side lobes limit the usability.
基于对循环活塞指向性的分析，在扬声器设计中可以得出以下三个结论。首先，大锥体在高频时的束宽可能过于狭窄，无法覆盖所需区域，或者侧瓣限制了其可用性。
Second, the beam width of the piston might be too broad for the application, thus causing high SPL levels at an undesired location. This is problem arises especially with low frequency sources.
第二，活塞的光束宽度可能对于应用来说太宽，因此在不期望的位置产生过高的声压级（SPL）。这个问题尤其出现在低频源中。
Third, ratio between direct sound and power response of the direct radiator is a function of frequency. Therefore the ratio between direct and reflected sound is not constant as a function of frequency.
第三，直接辐射器的直接声音与功率响应之间的比率是频率的函数。因此，直接声音与反射声音之间的比率不是频率的常数。
基于对循环活塞指向性的分析，在扬声器设计中可以得出以下三个结论。首先，大锥体在高频时的束宽可能过于狭窄，无法覆盖所需区域，或者侧瓣限制了其可用性。
Second, the beam width of the piston might be too broad for the application, thus causing high SPL levels at an undesired location. This is problem arises especially with low frequency sources.
第二，活塞的光束宽度可能对于应用来说太宽，因此在不期望的位置产生过高的声压级（SPL）。这个问题尤其出现在低频源中。
Third, ratio between direct sound and power response of the direct radiator is a function of frequency. Therefore the ratio between direct and reflected sound is not constant as a function of frequency.
第三，直接辐射器的直接声音与功率响应之间的比率是频率的函数。因此，直接声音与反射声音之间的比率不是频率的常数。
7 The directivity of a horn radiator
7 号角扬声器的方向性
The directivity of a horn has been of interest as long horn loaded sources have been made. Therefore it is surprising how little information about the directivity of horns can be found in the literature. Again, the probable reason for this is the lack analytical solutions.
号角的方向性自号角负载源被制造以来就一直是一个研究焦点。因此，令人惊讶的是，在文献中能找到关于号角方向性的信息竟然如此之少。再次，这种可能性的原因可能是缺乏分析解决方案。
Therefore horn directivity has been more of an engineering problem solved with prototypes and intuition. The academic world has shown relatively little interesting for this topic. Most papers published concentrate on the final shape achieved for a commercial application.
因此，角直接性更多地是一个工程问题，通过原型和直觉来解决。学术界对此话题表现出相对较少的兴趣。大多数发表的论文主要集中在商业应用最终实现的形状。
Very little is discussed about the reasoning that led to the solution, probably because of trade secrets.
很少讨论到导致解决方案的推理过程，可能是因为涉及商业机密。
号角的方向性自号角负载源被制造以来就一直是一个研究焦点。因此，令人惊讶的是，在文献中能找到关于号角方向性的信息竟然如此之少。再次，这种可能性的原因可能是缺乏分析解决方案。
Therefore horn directivity has been more of an engineering problem solved with prototypes and intuition. The academic world has shown relatively little interesting for this topic. Most papers published concentrate on the final shape achieved for a commercial application.
因此，角直接性更多地是一个工程问题，通过原型和直觉来解决。学术界对此话题表现出相对较少的兴趣。大多数发表的论文主要集中在商业应用最终实现的形状。
Very little is discussed about the reasoning that led to the solution, probably because of trade secrets.
很少讨论到导致解决方案的推理过程，可能是因为涉及商业机密。
Despite the efforts made for finding sources for horn directivity, the success of finding solid text was thin. The most comprehensive source found for this thesis was Olson's work on the subject published back in 1957 [15].
尽管在寻找号角直接性来源方面做出了努力，但找到坚实文本的成功率很低。为本论文找到的最全面的来源是奥尔森于 1957 年发表的关于该主题的工作 [15]。
The theory presented in this chapter is largely based on Olson's research.
本章所阐述的理论主要基于奥尔森的研究。
尽管在寻找号角直接性来源方面做出了努力，但找到坚实文本的成功率很低。为本论文找到的最全面的来源是奥尔森于 1957 年发表的关于该主题的工作 [15]。
The theory presented in this chapter is largely based on Olson's research.
本章所阐述的理论主要基于奥尔森的研究。
When discussing the directivity of a horn, it is essential to keep in mind the directivity characteristics of a direct radiator presented in Chapter 6. It is common to combine direct radiator and horn loaded radiator in loudspeaker systems.
讨论号角的指向性时，需要记住第 6 章中直接辐射器的指向性特性。在扬声器系统中，通常会将直接辐射器和号角加载辐射器结合使用。
讨论号角的指向性时，需要记住第 6 章中直接辐射器的指向性特性。在扬声器系统中，通常会将直接辐射器和号角加载辐射器结合使用。
The theory part of the horn directivity chapter is divided into two parts. First the directivity of an exponential horn is discussed. Exponential horn has been historically of great interest because of two reasons.
号角直接性章节的理论部分分为两部分。首先讨论指数号角的直接性。指数号角历史上因其两个原因而备受关注。
First, there is an analytical solution for designing an exponential horn. Second, exponential horn has good impedance match between source mechanical impedance and the impedance of air.
首先，存在设计指数号角的解析解。其次，指数号角在源机械阻抗和空气阻抗之间具有良好的阻抗匹配。
号角直接性章节的理论部分分为两部分。首先讨论指数号角的直接性。指数号角历史上因其两个原因而备受关注。
First, there is an analytical solution for designing an exponential horn. Second, exponential horn has good impedance match between source mechanical impedance and the impedance of air.
首先，存在设计指数号角的解析解。其次，指数号角在源机械阻抗和空气阻抗之间具有良好的阻抗匹配。
The second topic of discussion is the directivity of a conical horn. The conical horn has been used a lot because it is the most straightforward geometry of the horn imaginable. It also shows an interesting phenomenon toward high frequencies.
讨论的第二个主题是圆锥形号筒的方向性。圆锥形号筒被大量使用，因为它是最直观的号筒几何形状。它在高频段表现出一种有趣的现象。
讨论的第二个主题是圆锥形号筒的方向性。圆锥形号筒被大量使用，因为它是最直观的号筒几何形状。它在高频段表现出一种有趣的现象。
7.1 The exponential horn
7.1 指数喇叭
The cross-sectional area
截面积
截面积
Figure 7.î. Exponential horn with constant mouth size and varying length. Adopted from [15].
图 7.î. 指数喇叭，口部尺寸恒定，长度变化。引自[15]。
图 7.î. 指数喇叭，口部尺寸恒定，长度变化。引自[15]。
One approach to research the directivity properties of a horn is to compare polar plots of several horn geometries (Figure 7.1). The throat and mouth circumferences are constant, but the length and flare rate are varying.
研究号角的指向性特性的方法之一是对比几个号角几何形状的极坐标图（图 7.1）。喉部和口部的周长是恒定的，但长度和扩张率是变化的。
The beam width decreases towards high frequencies, which is a universal feature for exponential horn. The following problems arise because of this feature. First, the narrow beam at high frequencies limits the area that can be covered with a single horn unit.
波束宽度在高频时减小，这是指数号角的普遍特征。由于这个特征，出现了一些问题。首先，高频时的窄波束限制了单个号角单元可以覆盖的区域。
Second, the power response of the source decreases towards high frequencies, which affects the room response of the speaker.
第二，源的功率响应在高频时减小，这影响了扬声器的房间响应。
研究号角的指向性特性的方法之一是对比几个号角几何形状的极坐标图（图 7.1）。喉部和口部的周长是恒定的，但长度和扩张率是变化的。
The beam width decreases towards high frequencies, which is a universal feature for exponential horn. The following problems arise because of this feature. First, the narrow beam at high frequencies limits the area that can be covered with a single horn unit.
波束宽度在高频时减小，这是指数号角的普遍特征。由于这个特征，出现了一些问题。首先，高频时的窄波束限制了单个号角单元可以覆盖的区域。
Second, the power response of the source decreases towards high frequencies, which affects the room response of the speaker.
第二，源的功率响应在高频时减小，这影响了扬声器的房间响应。
The number at the right side of each polar diagram indicates the size of a circular piston, which would have equal directivity to this particular horn. At 1000 Hz , the length of the exponential horn does not affect the directivity of the horn.
每个极坐标图右侧的数字表示一个圆形活塞的大小，该活塞具有与特定号角相同的指向性。在 1000 Hz 时，指数号角的长度不会影响号角的指向性。
The directivity of the horn is equal to the directivity of a circular piston with diameter of 12 inches, which is exactly the diameter of the horn mouth. For frequencies from 2
号角的指向性等于直径为 12 英寸的圆形活塞的指向性，而这正是号角口的直径。对于从 2 Hz 的频率
每个极坐标图右侧的数字表示一个圆形活塞的大小，该活塞具有与特定号角相同的指向性。在 1000 Hz 时，指数号角的长度不会影响号角的指向性。
The directivity of the horn is equal to the directivity of a circular piston with diameter of 12 inches, which is exactly the diameter of the horn mouth. For frequencies from 2
号角的指向性等于直径为 12 英寸的圆形活塞的指向性，而这正是号角口的直径。对于从 2 Hz 的频率
kHz to 10 kHz the horn length does matter. With a longer horn, the beam width is slightly decreased at mid and high frequency range.
从 1kHz 到 10kHz，号角长度很重要。使用更长的号角，在中频和高频范围内，光束宽度会略微减小。
从 1kHz 到 10kHz，号角长度很重要。使用更长的号角，在中频和高频范围内，光束宽度会略微减小。
Figure 7.2. Exponential horn with varying mouth size. Adopted from [15].
图 7.2. 变口部大小的指数喇叭。从[15]改编。
图 7.2. 变口部大小的指数喇叭。从[15]改编。
Another approach is to keep the flare rate m constant (Figure 7.2). Mouth size and length are varying. Again mouth the size is defining the directivity at low frequencies.
另一种方法是保持发散率 m 不变（图 7.2）。口的大小和长度在变化。再次，口的大小定义了低频时的方向性。
With 6 inch and 12 inch mouth diameter horns the directivity at 1 kHz is comparable to the directivity of a direct radiator with an equal diameter. With a long horn and a large mouth, the directivity is more constant as a function of frequency.
六英寸和十二英寸口直径的号角在 1 kHz 时的指向性与具有相同直径的直接辐射器的指向性相当。使用长号角和大口部，指向性在频率函数中更为稳定。
This is because the horn length and mouth are large compared to the wavelength and therefore the directivity is defined by the horn geometry rather than size of the mouth.
这是因为号嘴长度和口部尺寸相对于波长较大，因此直接性由号嘴的几何形状而非口部尺寸定义。
另一种方法是保持发散率 m 不变（图 7.2）。口的大小和长度在变化。再次，口的大小定义了低频时的方向性。
With 6 inch and 12 inch mouth diameter horns the directivity at 1 kHz is comparable to the directivity of a direct radiator with an equal diameter. With a long horn and a large mouth, the directivity is more constant as a function of frequency.
六英寸和十二英寸口直径的号角在 1 kHz 时的指向性与具有相同直径的直接辐射器的指向性相当。使用长号角和大口部，指向性在频率函数中更为稳定。
This is because the horn length and mouth are large compared to the wavelength and therefore the directivity is defined by the horn geometry rather than size of the mouth.
这是因为号嘴长度和口部尺寸相对于波长较大，因此直接性由号嘴的几何形状而非口部尺寸定义。
7.2 The conical horn
7.2 锥形角
The conical horn has different directivity characteristics (Figure 7.3). In the figure is presented polar plots of two conical horns with varying length, but constant mouth and throat size. The effect of the different profiles is not very intuitive.
圆锥形号角具有不同的定向特性（图 7.3）。图中展示了长度不同但口部和喉部尺寸恒定的两个圆锥形号角的极坐标图。不同轮廓的效果不太直观。
The following observations can be made of the directivity.
以下是对直接性的一些观察。
圆锥形号角具有不同的定向特性（图 7.3）。图中展示了长度不同但口部和喉部尺寸恒定的两个圆锥形号角的极坐标图。不同轮廓的效果不太直观。
The following observations can be made of the directivity.
以下是对直接性的一些观察。
Figure 7.3. Directivity of a conical horn with constant mouth size and varying length. Adopted from [15].
图 7.3. 长度可变、口部尺寸恒定的圆锥号角的方向性。参考自[15]。
图 7.3. 长度可变、口部尺寸恒定的圆锥号角的方向性。参考自[15]。
The low frequency directivities are equal. It is dependent only on the mouth size. The short horn has a wider beam width at high frequencies than long horn has.
低频直接性相等，仅依赖于口部大小。短号角在高频下的束宽比长号角宽。
低频直接性相等，仅依赖于口部大小。短号角在高频下的束宽比长号角宽。
The beam width is narrowest at the mid frequencies. This is also called midrange beaming [15]. The longer horn has a narrower beam in the 4 kHz and 7 kHz diagrams.
波束宽度在中频段最窄。这也被称为中频束射 [15]。较长的号角在 4 kHz 和 7 kHz 图中的波束更窄。
波束宽度在中频段最窄。这也被称为中频束射 [15]。较长的号角在 4 kHz 和 7 kHz 图中的波束更窄。
It is safe to assume, that the low frequency directivity of a conical horn is defined by its mouth size as it is with exponential horn geometry.
可以安全地假设，圆锥号角的低频定向性由其口部尺寸定义，就像指数号角几何学那样。
可以安全地假设，圆锥号角的低频定向性由其口部尺寸定义，就像指数号角几何学那样。
The directivity of a conical horn is a strongly varying function of frequency and therefore it is not suitable for waveguide as such.
圆锥号角的直接性是频率的一个强烈变化函数，因此它不适合用作波导。
圆锥号角的直接性是频率的一个强烈变化函数，因此它不适合用作波导。
7.3 The exponential and conical horn as a waveguide
7.3 指数和圆锥形喇叭作为波导
The beamwidth of an exponential horn and a conical horn strongly vary as a function frequency. Therefore they are not suitable for a loudspeaker with controlled directivity characteristics. Nevertheless some useful conclusion can be made based on the theory presented before.
指数喇叭和圆锥喇叭的波束宽度随频率函数强烈变化。因此，它们不适合具有受控直接性特性的扬声器。然而，基于之前呈现的理论，可以得出一些有用的结论。
指数喇叭和圆锥喇叭的波束宽度随频率函数强烈变化。因此，它们不适合具有受控直接性特性的扬声器。然而，基于之前呈现的理论，可以得出一些有用的结论。
Low frequency directivity is dependent only on the mouth size. The cutoff limit is
低频定向性仅依赖于口部尺寸。截止限制为
低频定向性仅依赖于口部尺寸。截止限制为
Because of this, an interesting conclusion can be made about minimum waveguide size for multi-way systems if controllable directivity is desired. Consider a three-way system, with a direct radiating woofer, a midrange driver in a waveguide and a tweeter in a waveguide.
由于这个原因，对于希望实现可控方向性的多路系统，可以得出关于最小波导尺寸的有趣结论。考虑一个三路系统，其中包含一个直接辐射的低音炮，一个在波导中的中频驱动器，以及一个在波导中的高音扬声器。
由于这个原因，对于希望实现可控方向性的多路系统，可以得出关于最小波导尺寸的有趣结论。考虑一个三路系统，其中包含一个直接辐射的低音炮，一个在波导中的中频驱动器，以及一个在波导中的高音扬声器。
The midrange waveguide size is dependent on the crossover frequency and the woofer directivity, which is related to its size. If ka
中频波导尺寸依赖于分频频率和低音炮的方向性，后者与其尺寸有关。如果在分频频率下，低音炮的 ka
If ka > 1 for the woofer at the crossover frequency, the midrange waveguide circumference has to be equal to or larger than the wavelength. Then the directivity of the midrange can be adjusted by the waveguide geometry.
如果在分频频率下，低音扬声器的 ka 大于 1，中频波导的周长必须等于或大于波长。然后，可以通过波导的几何形状来调整中频的指向性。
中频波导尺寸依赖于分频频率和低音炮的方向性，后者与其尺寸有关。如果在分频频率下，低音炮的 ka
If ka > 1 for the woofer at the crossover frequency, the midrange waveguide circumference has to be equal to or larger than the wavelength. Then the directivity of the midrange can be adjusted by the waveguide geometry.
如果在分频频率下，低音扬声器的 ka 大于 1，中频波导的周长必须等于或大于波长。然后，可以通过波导的几何形状来调整中频的指向性。
The tweeter waveguide directivity has to match the midrange waveguide directivity at the crossover frequency. With practical crossover frequencies, ka
推特波导的方向性必须在分频频率处与中频波导的方向性相匹配。考虑到实际的分频频率，ka
推特波导的方向性必须在分频频率处与中频波导的方向性相匹配。考虑到实际的分频频率，ka
Neither exponential nor conical horn shapes are suitable for a constant directivity waveguide as such. The exponential horn beam width decreases towards high frequencies and the beam at high frequencies is too narrow for most applications.
指数形或圆锥形喇叭形状都不适合用于恒定定向波导，因为这样的设计不适合。指数形喇叭的束宽在高频时会减小，高频时的束太窄，不适合大多数应用。
The conical horn beam width is narrow at mid frequencies, which is also referred to as midrange beaming. Still these two profiles are a good starting point for horn design and understanding the influence of horn geometry.
圆锥形号角宽度在中频段较窄，这通常被称为中频束射。尽管如此，这两个轮廓仍然是号角设计和理解号角几何结构影响的良好起点。
There are also analytical solutions available for other shapes of horns, but they are even less fit for achieving constant directivity.
对于其他形状的号角，也存在分析解决方案，但它们甚至不太适合实现恒定的方向性。
指数形或圆锥形喇叭形状都不适合用于恒定定向波导，因为这样的设计不适合。指数形喇叭的束宽在高频时会减小，高频时的束太窄，不适合大多数应用。
The conical horn beam width is narrow at mid frequencies, which is also referred to as midrange beaming. Still these two profiles are a good starting point for horn design and understanding the influence of horn geometry.
圆锥形号角宽度在中频段较窄，这通常被称为中频束射。尽管如此，这两个轮廓仍然是号角设计和理解号角几何结构影响的良好起点。
There are also analytical solutions available for other shapes of horns, but they are even less fit for achieving constant directivity.
对于其他形状的号角，也存在分析解决方案，但它们甚至不太适合实现恒定的方向性。
8 Using the FEM for modelling waveguides
使用 FEM 对波导进行建模
8.1 Waveguide geometry 8.1 波导几何
The simulation method and measurement were tested with a physical waveguide. The shape of the waveguide was chosen to be simple.
仿真方法和测量方法使用物理波导进行了测试。波导的形状被选择为简单。
The reasoning for this is that the geometry would be easy to manufacture and there would be no measurable geometric error between the modelled and physical prototypes.
这是因为几何形状易于制造，且模型与物理原型之间的几何误差可测。
The properties of the waveguide were chosen to be such that it would have clear directivity characteristics, but its performance would be far from optimal. In other words, the waveguide shape was chosen to be such that it has problems to look for and to solve.
波导的特性被选择为具有明确的方向性特征，但其性能远非最优。换句话说，波导的形状被选择为存在需要寻找和解决的问题。
仿真方法和测量方法使用物理波导进行了测试。波导的形状被选择为简单。
The reasoning for this is that the geometry would be easy to manufacture and there would be no measurable geometric error between the modelled and physical prototypes.
这是因为几何形状易于制造，且模型与物理原型之间的几何误差可测。
The properties of the waveguide were chosen to be such that it would have clear directivity characteristics, but its performance would be far from optimal. In other words, the waveguide shape was chosen to be such that it has problems to look for and to solve.
波导的特性被选择为具有明确的方向性特征，但其性能远非最优。换句话说，波导的形状被选择为存在需要寻找和解决的问题。
The sound source consists of two parts, which are the waveguide and the tweeter (Figure 8.1, right). The waveguide is a conical shape with a
声源由两部分组成，即波导管和高音扬声器（图 8.1，右侧）。波导管呈圆锥形，与对称轴相切的角度为
声源由两部分组成，即波导管和高音扬声器（图 8.1，右侧）。波导管呈圆锥形，与对称轴相切的角度为
Figure 8.1. Model boundaries (left) and waveguide boundaries (right). Dimensions are in meters.
图 8.1. 模型边界（左）和波导边界（右）。尺寸单位为米。
图 8.1. 模型边界（左）和波导边界（右）。尺寸单位为米。
The waveguide is surrounded with air (Figure 8.1, left). The radius of the air medium is 0.4 meters. The radius of the air space to be modelled is related to the computational cost of the model. The computational cost of solving a model is further discussed in Chapter 3.3.
波导被空气包围（图 8.1，左侧）。空气介质的半径为 0.4 米。所建模的空气空间的半径与模型的计算成本有关。解决模型的计算成本在第 3.3 章中进一步讨论。
波导被空气包围（图 8.1，左侧）。空气介质的半径为 0.4 米。所建模的空气空间的半径与模型的计算成本有关。解决模型的计算成本在第 3.3 章中进一步讨论。
8.2 Motivation and strategy to simplify geometry
8.2 激励和简化几何学的策略
The computational cost of the model is highly dependent on the size and complexity of the geometry. The computational cost of the model can be reduced in the following ways.
模型的计算成本高度依赖于几何体的大小和复杂性。可以通过以下方式降低模型的计算成本。
模型的计算成本高度依赖于几何体的大小和复杂性。可以通过以下方式降低模型的计算成本。
Use a 2D axisymmetrical model whenever possible. This is the most significant factor in minimizing model size.
尽可能使用二维轴对称模型。这是最小化模型尺寸最重要的因素。
尽可能使用二维轴对称模型。这是最小化模型尺寸最重要的因素。
Reduce the area of simulation. For acoustics, this means usually the radius of the modelled air space around the simulated device.
减少模拟区域的面积。对于声学而言，这意味着通常模拟设备周围所建模的空气空间的半径。
减少模拟区域的面积。对于声学而言，这意味着通常模拟设备周围所建模的空气空间的半径。
Reduce the details of the geometry. Small details increase the element count of the model. Optimization of the geometry is a matter of knowing what details really affect the result of the simulation.
减少几何学的细节。小细节会增加模型的元素数量。几何优化的问题在于知道哪些细节真正影响模拟的结果。
In general, the closer the detail is to the sound source, the greater the effect. Also larger details have a larger effect than small details.
一般来说，细节越接近声源，其效果越大。同样，较大的细节比较小的细节产生更大的效果。
减少几何学的细节。小细节会增加模型的元素数量。几何优化的问题在于知道哪些细节真正影响模拟的结果。
In general, the closer the detail is to the sound source, the greater the effect. Also larger details have a larger effect than small details.
一般来说，细节越接近声源，其效果越大。同样，较大的细节比较小的细节产生更大的效果。
8.3 Modelling the medium and boundary conditions
8.3 模型化介质和边界条件
Figure 8.2. The boundaries of an axisymmetric waveguide in 4 sterians airspace.
图 8.2。在四维球形空间中，轴对称波导的边界。
图 8.2。在四维球形空间中，轴对称波导的边界。
Boundaries are the geometry edges surrounding the air medium. Boundary condition has to be specified for each boundary.
边界是围绕空气介质的几何边缘。每个边界都需要指定边界条件。
边界是围绕空气介质的几何边缘。每个边界都需要指定边界条件。
8.3.1 The medium 8.3.1 中介体
The medium is the area or volume in the model where the waves propagate. The properties of the medium have to be correct to achieve accurate results. The medium for acoustical modelling is air. Its density is
介质是模型中波传播的区域或体积。介质的性质必须正确以获得准确的结果。声学建模的介质是空气。其密度为
介质是模型中波传播的区域或体积。介质的性质必须正确以获得准确的结果。声学建模的介质是空气。其密度为
8.3.2 Sound hard boundary
8.3.2 坚硬边界
The sound hard boundary is a boundary with infinite acoustical impedance. Particle velocity at the boundary is zero. Therefore also particle acceleration is zero at the boundary. The sound hard boundary condition is specified by subtracting the dipole source
声硬边界是一个声学阻抗无限大的边界。边界处的粒子速度为零。因此，边界处的粒子加速度也为零。声硬边界条件通过从压力的梯度
声硬边界是一个声学阻抗无限大的边界。边界处的粒子速度为零。因此，边界处的粒子加速度也为零。声硬边界条件通过从压力的梯度
This boundary condition is used to define all the rigid surfaces of the model
此边界条件用于定义模型中所有刚性表面
此边界条件用于定义模型中所有刚性表面
8.3.3 Axisymmetry 8.3.3 轴对称
This is quite self-explanatory. In other words it is the line which is tangential to the axis of revolution. Of course this boundary needs only to be defined for axisymmetric models.
这相当一目了然。换句话说，它是与旋转轴相切的线。当然，这个边界只需要在轴对称模型中定义。
这相当一目了然。换句话说，它是与旋转轴相切的线。当然，这个边界只需要在轴对称模型中定义。
8.3.4 Normal acceleration
8.3.4 正常加速度
This is the boundary condition that defines the excitation inside the model. The equation for constant acceleration (Equation (8.2) is same as for the sound hard boundary (Equation (8.1) except the acceleration at the boundary is now specified to
这是模型内部激发的边界条件。恒定加速度的方程（方程（8.2））与声硬边界（方程（8.1））相同，除了现在在边界处的加速度被指定为
这是模型内部激发的边界条件。恒定加速度的方程（方程（8.2））与声硬边界（方程（8.1））相同，除了现在在边界处的加速度被指定为
The most straightforward value would be constant acceleration
最直接的价值是恒定加速度
acceleration source. First, the diaphragm moves as rigid surface. No breakups of the cone, dome or surround are present. Second, the effect of the voice coil inductance is excluded.
加速源。首先，隔膜作为刚性表面移动。没有锥形、顶点或环绕的断裂。其次，排除了音圈电感的影响。
Third, electromagnetic field (EMF) from the movement of the voice coil in magnet gap and other dynamic effects are excluded. Fourth, there is no coupling between the acoustical impedance of the air and the diaphragm motion.
第三，排除了音圈在磁隙中的移动和其他动态效应产生的电磁场（EMF）的影响。第四，空气的声阻抗与膜片运动之间没有耦合。
Pressure at the diaphragm surface does not damp movement of the cone. Therefore the resonances in the acoustic domain tend to be over-exaggerated compared to the reality.
扬声器膜片表面的压力不会削弱锥体的运动。因此，在声学领域中，共振往往会过度夸大现实情况。
最直接的价值是恒定加速度
acceleration source. First, the diaphragm moves as rigid surface. No breakups of the cone, dome or surround are present. Second, the effect of the voice coil inductance is excluded.
加速源。首先，隔膜作为刚性表面移动。没有锥形、顶点或环绕的断裂。其次，排除了音圈电感的影响。
Third, electromagnetic field (EMF) from the movement of the voice coil in magnet gap and other dynamic effects are excluded. Fourth, there is no coupling between the acoustical impedance of the air and the diaphragm motion.
第三，排除了音圈在磁隙中的移动和其他动态效应产生的电磁场（EMF）的影响。第四，空气的声阻抗与膜片运动之间没有耦合。
Pressure at the diaphragm surface does not damp movement of the cone. Therefore the resonances in the acoustic domain tend to be over-exaggerated compared to the reality.
扬声器膜片表面的压力不会削弱锥体的运动。因此，在声学领域中，共振往往会过度夸大现实情况。
With so many simplifications made, it is fair to question if constant acceleration is realistic enough to model the transducer part of the simulation. However, of the mentioned simplifications, only cone breakups affect the directivity of the system.
在进行了如此多的简化之后，确实可以质疑恒定加速度是否足够现实，用于模拟仿真中的换能器部分。然而，在提到的简化中，只有圆锥分裂会影响系统的指向性。
在进行了如此多的简化之后，确实可以质疑恒定加速度是否足够现实，用于模拟仿真中的换能器部分。然而，在提到的简化中，只有圆锥分裂会影响系统的指向性。
On the other hand, it may be a desirable feature to the designer to be able to exclude the nonidealities of the transducer from the contribution of the waveguide.
另一方面，设计师可能希望能够从波导的贡献中排除转换器的非理想特性，这是一个可欲的特性。
另一方面，设计师可能希望能够从波导的贡献中排除转换器的非理想特性，这是一个可欲的特性。
An external source can be coupled to the model. It is also possible to define
外部源可以与模型耦合。同时，可以将
外部源可以与模型耦合。同时，可以将
8.3.5 Radiation boundary condition
8.3.5 辐射边界条件
Radiation at the boundary can be specified as zero. Therefore there can be no reflections back to the model air domain. This boundary condition is used at the edges of the air domain to represent infinite space outside the modelled space.
边界处的辐射可以规定为零。因此，不可能有反射回到模型空气域。这种边界条件在空气域的边缘使用，以代表模型化空间之外的无限空间。
边界处的辐射可以规定为零。因此，不可能有反射回到模型空气域。这种边界条件在空气域的边缘使用，以代表模型化空间之外的无限空间。
8.4 Improving the diaphragm movement model
8.4 提高膈肌运动模型
8.4.1 Motivation for the development of the model
8.4.1 模型开发的动机
Accuracy of the tweeter radiation used in the FEM model limits the accuracy of the frequency responses of the model. At high frequencies the tweeter radiation starts to dominate the directivity characteristics over the waveguide.
FEM 模型中使用的扬声器辐射的准确性限制了模型频率响应的准确性。在高频时，扬声器辐射开始在波导上主导直接性特性。
Therefore the accuracy of the tweeter model is vital for achieving realistic directivity results at high frequencies. The most difficult part of the tweeter model is to approximate the different radiation characteristics of the several radiating parts.
因此，扬声器模型的准确性对于实现高频段的现实指向性结果至关重要。扬声器模型中最困难的部分是近似多个辐射部分的不同的辐射特性。
The source of difficulty is that the radiation is dependent on the geometry, location in the geometry and frequency.
困难的根源在于辐射依赖于几何形状、几何中的位置以及频率。
FEM 模型中使用的扬声器辐射的准确性限制了模型频率响应的准确性。在高频时，扬声器辐射开始在波导上主导直接性特性。
Therefore the accuracy of the tweeter model is vital for achieving realistic directivity results at high frequencies. The most difficult part of the tweeter model is to approximate the different radiation characteristics of the several radiating parts.
因此，扬声器模型的准确性对于实现高频段的现实指向性结果至关重要。扬声器模型中最困难的部分是近似多个辐射部分的不同的辐射特性。
The source of difficulty is that the radiation is dependent on the geometry, location in the geometry and frequency.
困难的根源在于辐射依赖于几何形状、几何中的位置以及频率。
This chapter is shows several levels of accuracy improvement for of the tweeter radiation model. First the analogous circuits related to the tweeter and its radiation is introduced.
本章展示了扬声器辐射模型准确度提升的几个层次。首先，介绍了与扬声器及其辐射相关的类比电路。
Second, the use of a simple normal acceleration to model the tweeter excitation in the model is discussed. Third, the use of velocity measurements of a known driver as diaphragm excitation is shown.
第二，讨论了在模型中使用简单的线性加速度来模拟扬声器激发的问题。第三，展示了使用已知驱动器的速度测量作为膜激发的应用。
Fourth, is a discussion on how to couple the acoustical and mechanical domain to include the effect of the acoustic pressure on the diaphragm motion.
第四，讨论如何将声学和机械领域结合，以包括声压对膜片运动的影响。
本章展示了扬声器辐射模型准确度提升的几个层次。首先，介绍了与扬声器及其辐射相关的类比电路。
Second, the use of a simple normal acceleration to model the tweeter excitation in the model is discussed. Third, the use of velocity measurements of a known driver as diaphragm excitation is shown.
第二，讨论了在模型中使用简单的线性加速度来模拟扬声器激发的问题。第三，展示了使用已知驱动器的速度测量作为膜激发的应用。
Fourth, is a discussion on how to couple the acoustical and mechanical domain to include the effect of the acoustic pressure on the diaphragm motion.
第四，讨论如何将声学和机械领域结合，以包括声压对膜片运动的影响。
8.4.2 The equivalent analogous circuit for loudspeaker driver
8.4.2 响 loudspeaker 驱动 driver 的等效类比电路
(b)
(a)
(c)
Figure 8.3. Analogous circuit of a transducer. Divided to (a) electrical, (b) mechanical, and (c) acoustical domain. Adopted from [18].
图 8.3. 转换器的类似电路。分为（a）电气，（b）机械，和（c）声学领域。从[18]采用。
图 8.3. 转换器的类似电路。分为（a）电气，（b）机械，和（c）声学领域。从[18]采用。
The operation of a transducer can be expressed with an equivalent analogous circuit. There are several ways of presenting the analogous circuit.
转换器的操作可以用等效的类比电路来表达。呈现类比电路的方法有几种。
In the version used, the electrical, mechanical and acoustical domains are separate and then coupled with controlled voltage or current sources (Figure 8.3). The electrical domain consists of the voltage generator
在所使用的版本中，电气、机械和声学领域是分开的，然后通过受控电压或电流源耦合（图 8.3）。电气领域包括电压发生器
calculated by multiplying the pressure
计算方法是通过将膜表面的压力
The approach presented crystallizes the idea of modelling the acoustical domain with FEM and how it is connected to the mechanical and electrical domains of the transducer.
所呈现的方法将声学领域用 FEM 建模的概念以及它与换能器的机械和电气领域之间的连接具体化。
转换器的操作可以用等效的类比电路来表达。呈现类比电路的方法有几种。
In the version used, the electrical, mechanical and acoustical domains are separate and then coupled with controlled voltage or current sources (Figure 8.3). The electrical domain consists of the voltage generator
在所使用的版本中，电气、机械和声学领域是分开的，然后通过受控电压或电流源耦合（图 8.3）。电气领域包括电压发生器
calculated by multiplying the pressure
计算方法是通过将膜表面的压力
The approach presented crystallizes the idea of modelling the acoustical domain with FEM and how it is connected to the mechanical and electrical domains of the transducer.
所呈现的方法将声学领域用 FEM 建模的概念以及它与换能器的机械和电气领域之间的连接具体化。
8.4.3 Defining location dependent acceleration
8.4.3 定义位置相关的加速度
The simplest method to excite the FEM model is to specify the diaphragm as constant acceleration boundary condition. The tweeter geometry model (Figure 8.4) has an aluminium dome part, inner half of the suspension, outer half of the suspension, and rigid parts.
激发 FEM 模型最简单的方法是将隔板指定为恒定加速度边界条件。扬声器高音单元几何模型（图 8.4）包含一个铝制球顶部分、悬挂的内半部分、悬挂的外半部分以及刚性部分。
The simplest approach to specify the dome motion is to specify a constant acceleration for all the moving parts. In that case the modelled directivity becomes quite realistic for wavelengths much longer than the dimensions of the moving parts.
指定穹顶运动的最简单方法是为所有移动部件指定一个恒定加速度。在这种情况下，所建模的方向性对于波长远大于移动部件尺寸的情况变得相当真实。
激发 FEM 模型最简单的方法是将隔板指定为恒定加速度边界条件。扬声器高音单元几何模型（图 8.4）包含一个铝制球顶部分、悬挂的内半部分、悬挂的外半部分以及刚性部分。
The simplest approach to specify the dome motion is to specify a constant acceleration for all the moving parts. In that case the modelled directivity becomes quite realistic for wavelengths much longer than the dimensions of the moving parts.
指定穹顶运动的最简单方法是为所有移动部件指定一个恒定加速度。在这种情况下，所建模的方向性对于波长远大于移动部件尺寸的情况变得相当真实。
Figure 8.4. The geometry of the tweeter in a model. Aluminium dome (green), inner half of the suspension (cyan), outer half of the suspension (magenta) and rigid parts (blue).
图 8.4. 模型中扬声器的几何结构。铝质球顶（绿色），悬挂的内半部分（青色），悬挂的外半部分（品红色）和刚性部分（蓝色）。
图 8.4. 模型中扬声器的几何结构。铝质球顶（绿色），悬挂的内半部分（青色），悬挂的外半部分（品红色）和刚性部分（蓝色）。
The next improvement is to specify different accelerations for the dome and the surround. Acceleration of the surround has its maximum at the point where it is attached to the dome and the minimum close to the fixing point.
下一步的改进是为穹顶和环绕结构指定不同的加速度。环绕结构的加速度在其与穹顶连接的点达到最大，在接近固定点的区域达到最小。
For example, the acceleration can be specified to be 0.75 times for the inner part of the suspension and 0.25 for the outer part of the suspensions of the acceleration of the dome.
例如，悬架系统的内部部分的加速度可以指定为 0.75 倍，而悬架系统的外部部分的加速度可以指定为 0.25 倍。
下一步的改进是为穹顶和环绕结构指定不同的加速度。环绕结构的加速度在其与穹顶连接的点达到最大，在接近固定点的区域达到最小。
For example, the acceleration can be specified to be 0.75 times for the inner part of the suspension and 0.25 for the outer part of the suspensions of the acceleration of the dome.
例如，悬架系统的内部部分的加速度可以指定为 0.75 倍，而悬架系统的外部部分的加速度可以指定为 0.25 倍。
The root of the problem for defining a realistic diaphragm movement is that the material properties of the dome and surround differ. The aluminium dome moves as a rigid piston in its passband, which is usually about 1 kHz to 20 kHz .
问题的根本在于定义实际的隔膜运动，因为隔膜和周围材料的物理属性不同。铝制隔膜在其频带内作为一个刚性活塞移动，通常这个频带大约在 1 kHz 到 20 kHz 之间。
The first mechanical eigenmode of the dome is called the dome break-up frequency. Then, the dome no longer moves as a rigid piston. For an aluminium dome this frequency is typically 22 kHz to 30 kHz [19].
球顶的第一机械固有模态称为球顶破裂频率。然后，球顶不再作为一个刚性活塞移动。对于铝制球顶，这个频率通常在 22 kHz 到 30 kHz 之间 [19]。
However already much below the break-up frequency there is a phase difference between the moving parts of the dome. Usually this phase difference becomes significant about one octave below the break up frequency. For an aluminium dome the range is from 11 kHz to 15 kHz .
然而，在崩溃频率以下的很多情况下，圆顶的移动部分之间就已经存在相位差。通常，这个相位差在崩溃频率大约低一个八度音时变得显著。对于铝制圆顶，这个范围是从 11 kHz 到 15 kHz。
The sound radiated from the different parts of the dome does not sum up perfectly because of the destructive interference caused by the phase difference. A typical tweeter surround is made of fabric mixed with adhesives.
球顶不同部分发出的声音由于相位差引起的破坏性干涉并不完全相加。典型的高音喇叭包覆层由粘合剂混合的织物制成。
The combination is not very rigid, but the damping properties are excellent. Therefore the surround has resonances in the tweeter frequency range but they are well damped.
组合不够刚性，但阻尼性能极佳。因此，环绕扬声器在高音频率范围内有共振，但这些共振被很好地阻尼。
问题的根本在于定义实际的隔膜运动，因为隔膜和周围材料的物理属性不同。铝制隔膜在其频带内作为一个刚性活塞移动，通常这个频带大约在 1 kHz 到 20 kHz 之间。
The first mechanical eigenmode of the dome is called the dome break-up frequency. Then, the dome no longer moves as a rigid piston. For an aluminium dome this frequency is typically 22 kHz to 30 kHz [19].
球顶的第一机械固有模态称为球顶破裂频率。然后，球顶不再作为一个刚性活塞移动。对于铝制球顶，这个频率通常在 22 kHz 到 30 kHz 之间 [19]。
However already much below the break-up frequency there is a phase difference between the moving parts of the dome. Usually this phase difference becomes significant about one octave below the break up frequency. For an aluminium dome the range is from 11 kHz to 15 kHz .
然而，在崩溃频率以下的很多情况下，圆顶的移动部分之间就已经存在相位差。通常，这个相位差在崩溃频率大约低一个八度音时变得显著。对于铝制圆顶，这个范围是从 11 kHz 到 15 kHz。
The sound radiated from the different parts of the dome does not sum up perfectly because of the destructive interference caused by the phase difference. A typical tweeter surround is made of fabric mixed with adhesives.
球顶不同部分发出的声音由于相位差引起的破坏性干涉并不完全相加。典型的高音喇叭包覆层由粘合剂混合的织物制成。
The combination is not very rigid, but the damping properties are excellent. Therefore the surround has resonances in the tweeter frequency range but they are well damped.
组合不够刚性，但阻尼性能极佳。因此，环绕扬声器在高音频率范围内有共振，但这些共振被很好地阻尼。
8.4.4 Measuring the velocity of the diaphragm
8.4.4 测量隔膜的速度
Velocity responses of an aluminium dome and a fabric surround of a tweeter are measured with a laser velocity meter (Figure 8.5). It is clearly seen that the aluminium dome has a higher velocity throughout the frequency range.
铝合金球顶和扬声器的织物环绕的加速度响应使用激光加速度计（图 8.5）进行测量。很明显，铝合金球顶在整个频率范围内具有更高的加速度。
However, the difference is decreases towards high frequencies. Therefore the radiation caused by the surround starts to dominate at high frequencies.
然而，这种差异在高频时会减小。因此，在高频时，周围的辐射开始占据主导地位。
Then the combined radiation approaches the radiation of a ring radiation source, which is much more directive than a piston source. This phenomenon does have an influence on the tweeter high frequency response and directivity.
然后，组合辐射接近环辐射源的辐射，这比活塞源的辐射更具有定向性。这种现象确实会影响高音扬声器的高频响应和指向性。
Directivity of the transducer increases toward the high frequencies because of the ring radiator phenomenon.
扬声器的指向性随着高频的增加而增加，这是由于环形辐射器现象所致。
铝合金球顶和扬声器的织物环绕的加速度响应使用激光加速度计（图 8.5）进行测量。很明显，铝合金球顶在整个频率范围内具有更高的加速度。
However, the difference is decreases towards high frequencies. Therefore the radiation caused by the surround starts to dominate at high frequencies.
然而，这种差异在高频时会减小。因此，在高频时，周围的辐射开始占据主导地位。
Then the combined radiation approaches the radiation of a ring radiation source, which is much more directive than a piston source. This phenomenon does have an influence on the tweeter high frequency response and directivity.
然后，组合辐射接近环辐射源的辐射，这比活塞源的辐射更具有定向性。这种现象确实会影响高音扬声器的高频响应和指向性。
Directivity of the transducer increases toward the high frequencies because of the ring radiator phenomenon.
扬声器的指向性随着高频的增加而增加，这是由于环形辐射器现象所致。
Figure 8.5. Measured velocity
图 8.5. 铝合金球顶扬声器的顶盖（黑色）和环绕（蓝色）的测量速度
图 8.5. 铝合金球顶扬声器的顶盖（黑色）和环绕（蓝色）的测量速度
The velocities of the dome and surround were measured with a laser velocity meter. The transducer was excited with an maximum length sequence (MLS) signal as done in acoustic measurements. The MLS is a pseudo random noise, which consists of a series of impulses.
穹顶和周围环境的速度使用激光速度计测量。传感器通过最大长度序列（MLS）信号激发，类似于声学测量中的做法。MLS 是一种伪随机噪声，由一系列脉冲组成。
One impulse contains all frequencies and in the frequency domain it is white noise. MLS contains a series of these impulses to improve the energy of the excitation signal and improve signal to noise ratio [20].
一个脉冲包含所有频率，在频域中它是白噪声。MLS 包含一系列这样的脉冲，以提高激励信号的能量并改善信噪比 [20]。
穹顶和周围环境的速度使用激光速度计测量。传感器通过最大长度序列（MLS）信号激发，类似于声学测量中的做法。MLS 是一种伪随机噪声，由一系列脉冲组成。
One impulse contains all frequencies and in the frequency domain it is white noise. MLS contains a series of these impulses to improve the energy of the excitation signal and improve signal to noise ratio [20].
一个脉冲包含所有频率，在频域中它是白噪声。MLS 包含一系列这样的脉冲，以提高激励信号的能量并改善信噪比 [20]。
The velocity signal of the transducer was sent back to the measurement system, instead of a pressure microphone signal. The boundary condition for the moving part is the acceleration
换能器的速度信号被发送回测量系统，而不是压力麦克风信号。移动部分的边界条件是加速度
换能器的速度信号被发送回测量系统，而不是压力麦克风信号。移动部分的边界条件是加速度
8.4.5 Using measured velocity in the FEM model
8.4.5 使用有限元模型中的测量速度
The intention was that by combining measurements and simulations a very accurate result could be achieved. Considering again Figure 8.3, now
意图是通过结合测量和模拟，可以达到非常准确的结果。再次考虑图 8.3，现在
calculated (Equation (8.4) by multiplying the pressure at the diaphragm
计算（公式 8.4），通过将膜片处的压力
意图是通过结合测量和模拟，可以达到非常准确的结果。再次考虑图 8.3，现在
calculated (Equation (8.4) by multiplying the pressure at the diaphragm
计算（公式 8.4），通过将膜片处的压力
Of course the pressure at the diaphragm is not constant, but is location dependent. Fortunately the FEM is able to integrate the complex pressure over the diaphragm surface.
当然，膈膜处的压力不是恒定的，而是依赖于位置。幸运的是，FEM 能够对膈膜表面的复杂压力进行积分。
The pressure at the diaphragm boundary can be integrated, thereby obtaining the exact force caused by the acoustic pressure.
在隔膜边界处的压力可以进行积分，从而获得由声压引起的精确力。
The last step for including diaphragm pressure in the model is to calculate the difference in the acceleration caused by the force caused by the pressure at the diaphragm
在模型中包含膈肌压力的最后一步是计算由膈肌
当然，膈膜处的压力不是恒定的，而是依赖于位置。幸运的是，FEM 能够对膈膜表面的复杂压力进行积分。
The pressure at the diaphragm boundary can be integrated, thereby obtaining the exact force caused by the acoustic pressure.
在隔膜边界处的压力可以进行积分，从而获得由声压引起的精确力。
The last step for including diaphragm pressure in the model is to calculate the difference in the acceleration caused by the force caused by the pressure at the diaphragm
在模型中包含膈肌压力的最后一步是计算由膈肌
The goal is to combine the measured velocity
目标是将测量的位移速度
目标是将测量的位移速度
9 Measurement system and visualization
9 测量系统和可视化
9.1 The selection of the visualization method
9.1 数据可视化方法的选择
Post-processing of the result is significant part of modelling work. First, a meaningful graph should be made so that results can be analyzed and understood. Secondly, the post-processing method should be such that it can be used to visualize physical measurements for comparison.
结果的后处理是建模工作的重要部分。首先，应该制作一个有意义的图表，以便分析和理解结果。其次，后处理方法应该能够用于可视化物理测量以进行比较。
结果的后处理是建模工作的重要部分。首先，应该制作一个有意义的图表，以便分析和理解结果。其次，后处理方法应该能够用于可视化物理测量以进行比较。
Various methods for visualizing directivity were introduced in the Chapter 5. For this thesis, the directivity plot was chosen for visualizing the results.
第 5 章介绍了多种直接性可视化的方法。对于本论文，选择了直接性图来可视化结果。
The directivity plot is introduced in Chapter 5.8. There are several reasons for selecting the directivity plot as the visualization method.
第 5.8 章介绍了方向图。选择方向图作为可视化方法的原因有几个。
第 5 章介绍了多种直接性可视化的方法。对于本论文，选择了直接性图来可视化结果。
The directivity plot is introduced in Chapter 5.8. There are several reasons for selecting the directivity plot as the visualization method.
第 5.8 章介绍了方向图。选择方向图作为可视化方法的原因有几个。
First, there were no readily available comparison methods for viewing the results from modelling software and physical measurement setups. Writing a postprocessing program was inevitable.
首先，没有现成的比较方法可以用来查看建模软件和物理测量设置的结果。编写后处理程序是不可避免的。
首先，没有现成的比较方法可以用来查看建模软件和物理测量设置的结果。编写后处理程序是不可避免的。
Second, the numerical data of the modelled results can be exported with a very good angular resolution. The frequency resolution depends on how many solutions were calculated in the solving process.
第二，模型结果的数值数据可以以非常好的角度分辨率导出。频率分辨率取决于在求解过程中计算了多少个解。
第二，模型结果的数值数据可以以非常好的角度分辨率导出。频率分辨率取决于在求解过程中计算了多少个解。
Third, the physical model can be measured in the anechoic chamber. The measurement system consists of a turntable and a PC-based measurement program. The measurement program controlled the turntable and excitation signal automatically.
第三，物理模型可以在屏蔽室中测量。测量系统由转台和基于 PC 的测量程序组成。测量程序自动控制转台和激励信号。
第三，物理模型可以在屏蔽室中测量。测量系统由转台和基于 PC 的测量程序组成。测量程序自动控制转台和激励信号。
The code for data processing and graph plotting were written in a numerical data processing environment as part of the thesis project (Figure 9.1). Data processing codes had to be individually written for modelling and measurement processing, since the input data formats were different.
数据处理和图形绘制的代码作为论文项目的一部分，在数值数据处理环境中编写（图 9.1）。数据处理代码需要分别编写用于建模和测量处理，因为输入数据格式不同。
The code for plotting the results is shared.
绘图结果的代码是共享的。
数据处理和图形绘制的代码作为论文项目的一部分，在数值数据处理环境中编写（图 9.1）。数据处理代码需要分别编写用于建模和测量处理，因为输入数据格式不同。
The code for plotting the results is shared.
绘图结果的代码是共享的。
Figure 9.1. The block diagram of data flow to create directivity plots from measured on modelled data.
图 9.1。从测量数据和模型数据创建方向性图的数据流程的块图。
图 9.1。从测量数据和模型数据创建方向性图的数据流程的块图。
9.2 Visualizing the directivity of the measurements
9.2 可视化测量的定向性
The creation of the directivity plot is divided to three parts which are data acquisition, processing the impulses to frequency responses and plotting of the frequency responses (Figure 9.1).
直接性图的创建分为三部分：数据采集、将脉冲处理为频率响应以及绘制频率响应（图 9.1）。
直接性图的创建分为三部分：数据采集、将脉冲处理为频率响应以及绘制频率响应（图 9.1）。
Figure 9.2. Prototype waveguide with a tweeter.
图 9.2. 原型波导与高音扬声器。
图 9.2. 原型波导与高音扬声器。
First, the acoustic response of the transducer in the waveguide is measured. The measurements are performed in an anechoic chamber (Figure 9.2). The low frequency limit of the anechoic room is approximately 100 Hz .
首先，测量波导中换能器的声学响应。测量在消音室中进行（图 9.2）。消音室的低频极限约为 100 Hz。
Loudspeakers are usually omnidirectional at frequencies (below 100 Hz ). Therefore the limit is not critical with directivity measurements. In this case the waveguide is used for tweeter (Figure 9.3), whose frequency range is above 1 kHz .
扬声器在频率（低于 100 Hz）下通常是全向的，因此直接性测量的限制并不关键。在这种情况下，使用波导管来处理高音扬声器（如图 9.3 所示），其频率范围在 1 kHz 以上。
Rotation of the waveguide to different angles during the measurement is executed with turn table. The turn table is automatically controlled by the measurement program. Measurements are made at
测量过程中，波导在不同角度的旋转通过转盘执行。转盘由测量程序自动控制。从 1#到 2#，以
polar arc is enough, because the waveguide is axisymmetric. Impulse responses of the measurements are stored and named according to the measurement angle.
极化弧足够，因为波导是对称的。测量的脉冲响应被存储并根据测量角度命名。
首先，测量波导中换能器的声学响应。测量在消音室中进行（图 9.2）。消音室的低频极限约为 100 Hz。
Loudspeakers are usually omnidirectional at frequencies (below 100 Hz ). Therefore the limit is not critical with directivity measurements. In this case the waveguide is used for tweeter (Figure 9.3), whose frequency range is above 1 kHz .
扬声器在频率（低于 100 Hz）下通常是全向的，因此直接性测量的限制并不关键。在这种情况下，使用波导管来处理高音扬声器（如图 9.3 所示），其频率范围在 1 kHz 以上。
Rotation of the waveguide to different angles during the measurement is executed with turn table. The turn table is automatically controlled by the measurement program. Measurements are made at
测量过程中，波导在不同角度的旋转通过转盘执行。转盘由测量程序自动控制。从 1#到 2#，以
polar arc is enough, because the waveguide is axisymmetric. Impulse responses of the measurements are stored and named according to the measurement angle.
极化弧足够，因为波导是对称的。测量的脉冲响应被存储并根据测量角度命名。
Figure 9.3. The measurement setup of the waveguide. Waveguide is on top of a microphone stand. Stand is on a turning table. Microphone is located in the top right of the figure.
图 9.3. 波导的测量设置。波导位于麦克风支架上方。支架位于旋转台上。麦克风位于图的右上角。
图 9.3. 波导的测量设置。波导位于麦克风支架上方。支架位于旋转台上。麦克风位于图的右上角。
Second, the impulse responses are imported to software to perform a numerical data processing. The frequency responses are calculated with an FFT. All off-axis frequency responses are normalized to the on-axis response. Thus only the directivity can be examined.
第二，响应脉冲被导入软件进行数值数据处理。使用 FFT 计算频率响应。所有偏离轴的频率响应都归一化到轴向响应。因此，只能检查指向性。
Also data of the measurement angles, frequency range and title are created.
测量角度、频率范围和标题的数据也被创建。
第二，响应脉冲被导入软件进行数值数据处理。使用 FFT 计算频率响应。所有偏离轴的频率响应都归一化到轴向响应。因此，只能检查指向性。
Also data of the measurement angles, frequency range and title are created.
测量角度、频率范围和标题的数据也被创建。
Third, the directivity plot is created. The inputs are the frequency responses, their respective frequencies and angles. The amplitude contour interval is 3 dB . The directivity is presented from
第三，绘制指向性图。输入是频率响应、各自的频率和角度。幅度轮廓间隔为 3 dB。指向性从
第三，绘制指向性图。输入是频率响应、各自的频率和角度。幅度轮廓间隔为 3 dB。指向性从
The directivity plot is at its best for examining the directivity of the source. The 3 dB amplitude resolution does limit its usability for amplitude comparison. Therefore it is convenient to exclude the on-axis frequency response information from the graph.
直接性图对于检查源的直接性非常有效。3 dB 振幅分辨率确实限制了其在振幅比较方面的使用性。因此，从图表中排除轴向频率响应信息是方便的。
This is done by normalizing all frequency responses to the on-axis frequency
这通过将所有频率响应标准化到轴向频率来完成
response. The result of this normalization is equivalent to the directivity of a source with flat on-axis response.
响应。这种归一化的结果等同于轴向响应平坦的声源的方向性。
直接性图对于检查源的直接性非常有效。3 dB 振幅分辨率确实限制了其在振幅比较方面的使用性。因此，从图表中排除轴向频率响应信息是方便的。
This is done by normalizing all frequency responses to the on-axis frequency
这通过将所有频率响应标准化到轴向频率来完成
response. The result of this normalization is equivalent to the directivity of a source with flat on-axis response.
响应。这种归一化的结果等同于轴向响应平坦的声源的方向性。
9.3 Visualizing the directivity of the modelled results
9.3. 可视化模型结果的方向性
The pressure at the outer boundary of the modelled domain (Figure 8.2) is exported from the modelling software to a numerical format. The exported data contains information of the complex amplitude, frequency and angle.
模型域外边界（图 8.2）的压力由建模软件导出为数值格式。导出的数据包含复振幅、频率和角度的信息。
模型域外边界（图 8.2）的压力由建模软件导出为数值格式。导出的数据包含复振幅、频率和角度的信息。
Directivity data in numerical format is imported to a software to perform numerical analysis. Directivity plot is created with same tool as for measured results (Figure 9.1).
直接性数据以数字格式导入软件进行数值分析。使用与测量结果相同的工具创建直接性图表（图 9.1）。
直接性数据以数字格式导入软件进行数值分析。使用与测量结果相同的工具创建直接性图表（图 9.1）。
10 Analyzing the simulated waveguide
10 分析模拟波导
There are two purposes for this chapter. First is to represent the results found in the directivity plot of the modelled waveguide. Second purpose is to compare the results to known theory of the horns (Chapter 7) and analyze the reasons for the phenomena found in the graphs.
本章有两大目的。首先，展示模型波导直接性图中找到的结果。其次，将结果与号角已知理论（第 7 章）进行比较，并分析图表中发现的现象的原因。
The idea is to demonstrate that the modelling tool can be used to understand the reasons behind the phenomena found in the measurement results.
要展示的是，该建模工具可以用来理解测量结果中发现的现象背后的原因。
本章有两大目的。首先，展示模型波导直接性图中找到的结果。其次，将结果与号角已知理论（第 7 章）进行比较，并分析图表中发现的现象的原因。
The idea is to demonstrate that the modelling tool can be used to understand the reasons behind the phenomena found in the measurement results.
要展示的是，该建模工具可以用来理解测量结果中发现的现象背后的原因。
Figure 10.1. Directivity plot of the modelled waveguide.
图 10.1. 模型波导的直接性图。
图 10.1. 模型波导的直接性图。
The directivity plot (Figure 10.1) of the modelled waveguide can be used to analyze the phenomena related to waveguide design. The directivity characteristics of this particular waveguide can be divided to four frequency ranges.
模型模拟的波导的直接性图（图 10.1）可用于分析与波导设计相关的现象。此特定波导的直接性特性可以分为四个频率范围。
模型模拟的波导的直接性图（图 10.1）可用于分析与波导设计相关的现象。此特定波导的直接性特性可以分为四个频率范围。
At low frequencies (from 1 to 2 kHz ) the waveguide directivity is close to the directivity of a circular piston. The wavelength (from
在低频段（从 1 kHz 到 2 kHz）时，波导的指向性接近圆形活塞的指向性。波长（从
在低频段（从 1 kHz 到 2 kHz）时，波导的指向性接近圆形活塞的指向性。波长（从
At medium frequencies (from 2 to 5 kHz ) the beamwidth decreases towards high frequencies. The wavelength (from 17 cm to 7 cm ) is shorter than the circumference of the waveguide mouth. Now the shape of the waveguide profile is dominating the directivity.
在中频段（从 2 kHz 到 5 kHz）范围内，波束宽度向高频段减小。波长（从 17 cm 到 7 cm）短于波导口的周长。现在，波导剖面的形状对方向性起主导作用。
Again this is congruent with the horn theory.
这再次与角理论相吻合。
在中频段（从 2 kHz 到 5 kHz）范围内，波束宽度向高频段减小。波长（从 17 cm 到 7 cm）短于波导口的周长。现在，波导剖面的形状对方向性起主导作用。
Again this is congruent with the horn theory.
这再次与角理论相吻合。
At high frequencies (from 5 to 12 kHz ) the diffraction dominates the directivity. The conical waveguide used has a sharp edge at the mouth. The sharp edge can be seen as a impedance discontinuity, which causes diffraction. It is known from the theory that
在高频率（从 5 到 12 kHz）下，衍射主导了方向性。所使用的锥形波导在口部有一个锐边。锐边可以被视为阻抗不连续性，导致衍射。根据理论，我们知道
the diffraction can be seen as a new sound source [15]. In the far field, this new source is out of phase with the direct sound at certain frequency.
衍射可以被视为一个新的声源[15]。在远场，这个新声源在某些频率下与直接声波相位相反。
This phenomenon is almost solely an on-axis frequency response problem, because the off-axis responses are fairly unaffected except the
这种现象几乎完全是一个轴向频率响应问题，因为离轴响应除了
There are two explanations for this. First explanation is related to the model geometry. The dome is not a point source. Therefore the distance from different parts of the dome to the mouth is not constant.
对于这一点有两种解释。第一种解释与模型几何有关。穹顶不是一个点源，因此穹顶不同部分到嘴巴的距离不是恒定的。
Second explanation is that in theory a plane wave pressure field is assumed. In reality the pressure field is more complex (Figure 10.2) and plane wave approximation is not adequate. The shape of the pressure wave is also frequency dependent.
第二种解释是在理论上假设了平面波压力场。实际上，压力场更为复杂（图 10.2），平面波近似不够准确。压力波的形状也依赖于频率。
在高频率（从 5 到 12 kHz）下，衍射主导了方向性。所使用的锥形波导在口部有一个锐边。锐边可以被视为阻抗不连续性，导致衍射。根据理论，我们知道
the diffraction can be seen as a new sound source [15]. In the far field, this new source is out of phase with the direct sound at certain frequency.
衍射可以被视为一个新的声源[15]。在远场，这个新声源在某些频率下与直接声波相位相反。
This phenomenon is almost solely an on-axis frequency response problem, because the off-axis responses are fairly unaffected except the
这种现象几乎完全是一个轴向频率响应问题，因为离轴响应除了
There are two explanations for this. First explanation is related to the model geometry. The dome is not a point source. Therefore the distance from different parts of the dome to the mouth is not constant.
对于这一点有两种解释。第一种解释与模型几何有关。穹顶不是一个点源，因此穹顶不同部分到嘴巴的距离不是恒定的。
Second explanation is that in theory a plane wave pressure field is assumed. In reality the pressure field is more complex (Figure 10.2) and plane wave approximation is not adequate. The shape of the pressure wave is also frequency dependent.
第二种解释是在理论上假设了平面波压力场。实际上，压力场更为复杂（图 10.2），平面波近似不够准确。压力波的形状也依赖于频率。
Figure 10.2. Sound pressure around the waveguide at 10 kHz .
图 10.2. 10 kHz 时波导周围的声压。
图 10.2. 10 kHz 时波导周围的声压。
The severity of the diffraction problem is emphasized by the axisymmetry (Figure 10.3) because the distance from the tweeter dome to the edge is equal in all azimuth angles. An asymmetrical waveguide would smear the diffraction problem to a broader frequency range.
衍射问题的严重性通过轴对称性（图 10.3）得到强调，因为扬声器球顶到边缘的距离在所有方位角下都相等。不对称的波导会将衍射问题扩散到更宽的频率范围内。
衍射问题的严重性通过轴对称性（图 10.3）得到强调，因为扬声器球顶到边缘的距离在所有方位角下都相等。不对称的波导会将衍射问题扩散到更宽的频率范围内。
Figure 10.3. Sound pressure level around the waveguide at 8200 Hz .
图 10.3. 在 8200 Hz 时波导周围的声压级。
图 10.3. 在 8200 Hz 时波导周围的声压级。
At very high frequencies (from 12 kHz to 20 kHz ) the dome shape starts to affect the directivity, but also the influence of the edge diffraction is present. At 13 kHz , the pressure diffracted at the edge of the waveguide is in phase with the on-axis response.
在非常高频率（从 12 kHz 到 20 kHz）下，球面形状开始影响指向性，但边缘衍射的影响也存在。在 13 kHz 时，波导边缘衍射的压力与轴向响应相位一致。
The sum of these sources causes a bump in the on-axis response. Therefore the directivity of the source is increased. Again at
这些来源的总和导致轴向响应出现峰值。因此，声源的方向性增强。在
The aluminium dome approximately moves as a rigid piston within the tweeter frequency range. However the surround made of fabric is not rigid at high frequencies.
铝合金穹顶在扬声器频率范围内大约像一个刚性活塞移动。然而，高频率时，织物制成的边带并不是刚性的。
在非常高频率（从 12 kHz 到 20 kHz）下，球面形状开始影响指向性，但边缘衍射的影响也存在。在 13 kHz 时，波导边缘衍射的压力与轴向响应相位一致。
The sum of these sources causes a bump in the on-axis response. Therefore the directivity of the source is increased. Again at
这些来源的总和导致轴向响应出现峰值。因此，声源的方向性增强。在
The aluminium dome approximately moves as a rigid piston within the tweeter frequency range. However the surround made of fabric is not rigid at high frequencies.
铝合金穹顶在扬声器频率范围内大约像一个刚性活塞移动。然而，高频率时，织物制成的边带并不是刚性的。
11 Verifying modelling accuracy
验证模型准确性
In this chapter are compared the simulated and measured results. First step is to analyze the differences between simulation and measurement and discuss the possible sources of dissimilarities. The second step is to analyze the simulated and measured frequency responses.
本章比较了模拟和测量结果。第一步是分析模拟和测量之间的差异，并讨论差异可能的来源。第二步是分析模拟和测量的频率响应。
If the modelled directivity is accurate, then the frequency response should reveal the accuracy of the tweeter driver model used.
如果模型预测的方向性准确，那么频率响应应该揭示所使用的高音扬声器驱动模型的准确性。
本章比较了模拟和测量结果。第一步是分析模拟和测量之间的差异，并讨论差异可能的来源。第二步是分析模拟和测量的频率响应。
If the modelled directivity is accurate, then the frequency response should reveal the accuracy of the tweeter driver model used.
如果模型预测的方向性准确，那么频率响应应该揭示所使用的高音扬声器驱动模型的准确性。
11.1 Accuracy of the simulated directivity
11.1 模拟方向性精度
The first purpose of this chapter is to present the differences between the measured (Figure 11.1) and simulated directivity (Figure 11.2) of the prototype waveguide. The second purpose is to analyze the difference between these two.
本章的第一目的是展示原型波导的测量（图 11.1）和模拟方向性（图 11.2）之间的差异。第二目的是分析这两个差异。
本章的第一目的是展示原型波导的测量（图 11.1）和模拟方向性（图 11.2）之间的差异。第二目的是分析这两个差异。
Figure 11.1. Measured directivity plot of the waveguide.
图 11.1. 波导的直接性测量图谱。
图 11.1. 波导的直接性测量图谱。
Figure 11.2. Modelled directivity plot of the waveguide
图 11.2. 波导的模型方向性图
图 11.2. 波导的模型方向性图
The first impression is that the measured and modelled directivities are similar. Next the differences are presented in four frequency ranges.
第一印象是，测量和建模的指向性相似。接下来，在四个频率范围内呈现差异。
第一印象是，测量和建模的指向性相似。接下来，在四个频率范围内呈现差异。
11.1.1 Comparison of the measured and modelled directivity
11.1.1 测量值与模型预测的方向性比较
At low frequencies (from 1 to 2 kHz ) the beamwidth of the measured directivity is first increasing and then decreasing. The beamwidth of the modelled results is constantly decreasing.
在低频段（从 1 kHz 到 2 kHz）中，测量的直接性带宽首先增加然后减少。模型结果的带宽始终在减少。
在低频段（从 1 kHz 到 2 kHz）中，测量的直接性带宽首先增加然后减少。模型结果的带宽始终在减少。
At mid frequencies (from 2 to 5 kHz ) the beamwidth of the measured results is constantly decreasing with little ripple. The -12 dB and -15 dB contours shows decreasing directivity at 5 kHz .
在中频段（从 2 kHz 到 5 kHz），测量结果的束宽始终在减少，波动很小。-12 dB 和-15 dB 的轮廓线显示在 5 kHz 时直接性减少。
Modelled directivity is also decreasing and showing the decrease in directivity at 5 kHz .
模型直接性也在减少，并在 5 kHz 处显示出直接性的减少。
The distance between equal amplitude contours is smaller in the measurement - in other words the directivity of the source is increasing more rapidly when moving towards the off-axis direction.
测量中等幅值轮廓之间的距离较小，换句话说，当向偏离轴方向移动时，源的指向性增加得更快。
在中频段（从 2 kHz 到 5 kHz），测量结果的束宽始终在减少，波动很小。-12 dB 和-15 dB 的轮廓线显示在 5 kHz 时直接性减少。
Modelled directivity is also decreasing and showing the decrease in directivity at 5 kHz .
模型直接性也在减少，并在 5 kHz 处显示出直接性的减少。
The distance between equal amplitude contours is smaller in the measurement - in other words the directivity of the source is increasing more rapidly when moving towards the off-axis direction.
测量中等幅值轮廓之间的距离较小，换句话说，当向偏离轴方向移动时，源的指向性增加得更快。
At high frequencies (from 5 to 12 kHz ) the beamwidth of the measured directivity is rapidly increasing. The centre of the bump is at 10 kHz . The modelled directivity show similar phenomena. The increased beamwidth consist of two merging peaks. Peaks are at 8 kHz and 9.5 kHz .
在高频率（从 5 kHz 到 12 kHz）下，测量的直接性束宽迅速增加。凸起的中心位于 10 kHz。模型预测的直接性显示出类似的现象。增加的束宽由两个合并的峰值组成。峰值位于 8 kHz 和 9.5 kHz。
在高频率（从 5 kHz 到 12 kHz）下，测量的直接性束宽迅速增加。凸起的中心位于 10 kHz。模型预测的直接性显示出类似的现象。增加的束宽由两个合并的峰值组成。峰值位于 8 kHz 和 9.5 kHz。
At very high frequencies (from 12 to 20 kHz ) the measured directivity is first increasing and then decreasing. The modelled directivity shows the same phenomena but with more ripple.
在非常高的频率（从 12 到 20 kHz）下，测量的指向性首先增加然后减少。模型预测的指向性显示出相同的现象，但波动更大。
在非常高的频率（从 12 到 20 kHz）下，测量的指向性首先增加然后减少。模型预测的指向性显示出相同的现象，但波动更大。
11.1.2 Analyzing the differences
11.1.2 分析差异
At low frequencies (from 1 to 2 kHz ) the measured directivity has more ripple. This is probably caused by the measurement jig used under the waveguide (which shown in Figure 9.3).
在低频率（从 1 到 2 kHz）下，测量的直接性有更多波动。这可能由在波导下使用的测量夹具引起（如图 9.3 所示）。
在低频率（从 1 到 2 kHz）下，测量的直接性有更多波动。这可能由在波导下使用的测量夹具引起（如图 9.3 所示）。
At mid frequencies (from 2 to 5 kHz ) the space between equal amplitude contours is narrower in the modelled directivity. The only reasonable explanation is the 40 cm distance used in model. Thus the far field approximation of the analysis is not valid.
在中频段（从 2 到 5 kHz）中，模型直接性中的等幅轮廓之间的空间更窄。唯一的合理解释是模型中使用的 40 cm 距离。因此，分析的远场近似无效。
在中频段（从 2 到 5 kHz）中，模型直接性中的等幅轮廓之间的空间更窄。唯一的合理解释是模型中使用的 40 cm 距离。因此，分析的远场近似无效。
At high frequencies (from 5 to 12 kHz ) the width of the bump is very similar, although the exact amplitude contours have differences. The difference between the amplitude contours is largest at large angles. The diffraction causes a resonance
在高频率（从 5 到 12 kHz）下，凸起的宽度非常相似，尽管幅度轮廓的具体差异存在。幅度轮廓之间的差异在大角度时最大。衍射导致了共振。
between the low acoustical impedance at the edge of the waveguide mouth and the high acoustical impedance at the throat of the waveguide. There are no losses included in the model but the outer boundary of the modelled air space.
在波导口边缘的低声学阻抗与波导喉部的高声学阻抗之间。模型中不包含任何损耗，但考虑了所模拟的空气空间的外部边界。
Therefore resonant phenomena tend to be exaggerated in the simulation.
因此，在模拟中共振现象往往会有所夸大。
在高频率（从 5 到 12 kHz）下，凸起的宽度非常相似，尽管幅度轮廓的具体差异存在。幅度轮廓之间的差异在大角度时最大。衍射导致了共振。
between the low acoustical impedance at the edge of the waveguide mouth and the high acoustical impedance at the throat of the waveguide. There are no losses included in the model but the outer boundary of the modelled air space.
在波导口边缘的低声学阻抗与波导喉部的高声学阻抗之间。模型中不包含任何损耗，但考虑了所模拟的空气空间的外部边界。
Therefore resonant phenomena tend to be exaggerated in the simulation.
因此，在模拟中共振现象往往会有所夸大。
At very high frequencies (from 12 to 20 kHz ) the dome shape and edge diffraction are combined. Increasing directivity caused by in-phase diffraction can be seen at 13 kHz both in the simulation and the measurement.
在非常高的频率（从 12 到 20 kHz）下，穹顶形状和边缘衍射结合在一起。在 13 kHz 时，由同相衍射引起的直接性增加在模拟和测量中都能看到。
Also the shapes of decreasing directivity above 16 kHz are similar.
以上 16 kHz 以上的直接性减小的形状也相似。
The error between the model and the measurement should be highest at high frequencies (e.g. short wavelengths), because the limited detail in the model geometry and finite element size are comparable to the wavelength.
模型与测量之间的误差应在高频（例如短波长）时最高，因为模型几何形状的细节有限，以及有限元大小与波长相当。
在非常高的频率（从 12 到 20 kHz）下，穹顶形状和边缘衍射结合在一起。在 13 kHz 时，由同相衍射引起的直接性增加在模拟和测量中都能看到。
Also the shapes of decreasing directivity above 16 kHz are similar.
以上 16 kHz 以上的直接性减小的形状也相似。
The error between the model and the measurement should be highest at high frequencies (e.g. short wavelengths), because the limited detail in the model geometry and finite element size are comparable to the wavelength.
模型与测量之间的误差应在高频（例如短波长）时最高，因为模型几何形状的细节有限，以及有限元大小与波长相当。
11.2 Accuracy of the simulated frequency response
11.2 模拟频率响应的准确性
Main emphasize was on to model the driver excitation in the model. The selected method to validate the accuracy of the driver model is to compare the frequency response of the model against the reality.
主要强调的是在模型中模拟驾驶员激励。验证驾驶员模型准确性的选定方法是将模型的频率响应与实际情况进行比较。
主要强调的是在模型中模拟驾驶员激励。验证驾驶员模型准确性的选定方法是将模型的频率响应与实际情况进行比较。
The frequency responses of the modelled (Figure 11.3) and measured (Figure 11.4) prototypes are presented at several angles. Frequency responses are presented with a 2 dB amplitude scale to enable critical evaluation of the results.
模型（图 11.3）和测量（图 11.4）原型的频率响应在多个角度下呈现。频率响应使用 2 分贝幅度尺度进行展示，以方便对结果进行关键性评估。
Often a 10 dB scale is used with frequency responses, which would give the illusion of more congruent results. The directivity of the system was discussed in the previous chapter.
通常使用 10 分贝的尺度来表示频率响应，这会给人一种更一致的结果的错觉。系统的指向性在上一章中进行了讨论。
Therefore phenomena in the frequency responses are emphasized in this chapter, although the directivity of the system can be also analyzed by comparing the off-axis frequency responses.
因此，在本章中强调了频率响应中的现象，尽管可以通过比较轴外频率响应来分析系统的指向性。
模型（图 11.3）和测量（图 11.4）原型的频率响应在多个角度下呈现。频率响应使用 2 分贝幅度尺度进行展示，以方便对结果进行关键性评估。
Often a 10 dB scale is used with frequency responses, which would give the illusion of more congruent results. The directivity of the system was discussed in the previous chapter.
通常使用 10 分贝的尺度来表示频率响应，这会给人一种更一致的结果的错觉。系统的指向性在上一章中进行了讨论。
Therefore phenomena in the frequency responses are emphasized in this chapter, although the directivity of the system can be also analyzed by comparing the off-axis frequency responses.
因此，在本章中强调了频率响应中的现象，尽管可以通过比较轴外频率响应来分析系统的指向性。
Figure 11.3. The modelled frequency responses of waveguide. From
图 11.3。波导的模型频率响应。从
图 11.3。波导的模型频率响应。从
Figure 11.4. The measured frequency responses of the waveguide. From
图 11.4. 波导的测量频率响应。从
图 11.4. 波导的测量频率响应。从
11.2.1 Comparing the measured and modelled directivity
11.2.1 比较测量值和模型化的方向性
Modelled frequency response show a peak of 102 dB at 2 kHz . With measured response the peak is one dB less. The peak is sharper with the modelled result.
模型频率响应显示在 2 kHz 处有一个 102 dB 的峰值。通过测量得到的响应峰值低 1 dB。模型结果的峰值更尖锐。
模型频率响应显示在 2 kHz 处有一个 102 dB 的峰值。通过测量得到的响应峰值低 1 dB。模型结果的峰值更尖锐。
Above 3 kHz , the slope of the response is similar in the measured and modelled results. The difference from 3 kHz to 7 kHz is 7 dB both for modelled and measured response. The peak at 4 kHz is present in both graphs, but the peak is 1 dB sharper for the modelled response.
在 3 kHz 以上，响应的斜率在测量结果和模型结果中相似。从 3 kHz 到 7 kHz，模型和测量响应之间的差异均为 7 dB。4 kHz 处的峰值在两个图表中都存在，但模型响应的峰值比测量响应更尖锐 1 dB。
The diffraction problem around 9 kHz is also present in the both graphs. For modelled results it consists of two merging notches whereas for the measured response it is one notch. The on-axis response decreases above 12 kHz .
9 kHz 周围的衍射问题在两幅图中都存在。对于模拟结果，它由两个合并的凹槽组成，而对于测量响应，它只是一个凹槽。轴向响应在 12 kHz 以上降低。
The measured result shows more ripple at high frequencies. The modelled angle contours are further apart from each other which equals less directivity.
测量结果显示高频段的纹波更大。模拟的角度轮廓彼此之间的距离更大，这相当于直接性更小。
在 3 kHz 以上，响应的斜率在测量结果和模型结果中相似。从 3 kHz 到 7 kHz，模型和测量响应之间的差异均为 7 dB。4 kHz 处的峰值在两个图表中都存在，但模型响应的峰值比测量响应更尖锐 1 dB。
The diffraction problem around 9 kHz is also present in the both graphs. For modelled results it consists of two merging notches whereas for the measured response it is one notch. The on-axis response decreases above 12 kHz .
9 kHz 周围的衍射问题在两幅图中都存在。对于模拟结果，它由两个合并的凹槽组成，而对于测量响应，它只是一个凹槽。轴向响应在 12 kHz 以上降低。
The measured result shows more ripple at high frequencies. The modelled angle contours are further apart from each other which equals less directivity.
测量结果显示高频段的纹波更大。模拟的角度轮廓彼此之间的距离更大，这相当于直接性更小。
11.2.2 Analyzing the differences
11.2.2 分析差异
The peak at 2 kHz of the modelled response is caused by the driver mechanical resonance frequency. In the tweeter model it is expected that the driver is operating above its resonance frequency, which is called the mass controlled region.
模型响应的 2 kHz 峰值是由扬声器机械共振频率引起的。在高音扬声器模型中，预期扬声器在高于其共振频率下运行，这被称为质量控制区域。
The acoustical pressure at the dome surface is coupled to the mass of the dome (Equation (8.6)). Therefore the tweeter driver model is not valid close to or below the resonance frequency of the driver.
球顶表面的声压与球顶的质量相关（方程（8.6））。因此，扬声器驱动模型在接近或低于驱动器的共振频率时无效。
Also the fast decreasing slope of the on-axis response above 2 kHz is also caused by this invalid assumption.
轴向响应在 2 kHz 以上的快速下降斜率也是由这个无效假设引起的。
模型响应的 2 kHz 峰值是由扬声器机械共振频率引起的。在高音扬声器模型中，预期扬声器在高于其共振频率下运行，这被称为质量控制区域。
The acoustical pressure at the dome surface is coupled to the mass of the dome (Equation (8.6)). Therefore the tweeter driver model is not valid close to or below the resonance frequency of the driver.
球顶表面的声压与球顶的质量相关（方程（8.6））。因此，扬声器驱动模型在接近或低于驱动器的共振频率时无效。
Also the fast decreasing slope of the on-axis response above 2 kHz is also caused by this invalid assumption.
轴向响应在 2 kHz 以上的快速下降斜率也是由这个无效假设引起的。
The sharpness of the 4 kHz peak at the modelled result can only be explained by the lack of damping in the resonant phenomena in the model.
模型结果中 4 kHz 峰值的尖锐度只能用模型中共振现象缺乏阻尼来解释。
模型结果中 4 kHz 峰值的尖锐度只能用模型中共振现象缺乏阻尼来解释。
The diffraction problem is also resonance related. The possible explanation for the second notch of the modelled result is the lack of internal damping. At very high frequencies (from 12 kHz to 20 kHz ) the modelled results show more ripple.
衍射问题也与共振有关。模型结果中第二个凹陷的可能解释是内部阻尼的缺乏。在非常高的频率（从 12 kHz 到 20 kHz）下，模型结果显示出更多的波动。
Peak of the bump at 12 kHz is 92 dB for both results, but the modelled bump does have a peak in the middle.
对于两个结果而言，12 kHz 处的突起峰值均为 92 dB，但模型预测的突起确实有一个中间峰值。
衍射问题也与共振有关。模型结果中第二个凹陷的可能解释是内部阻尼的缺乏。在非常高的频率（从 12 kHz 到 20 kHz）下，模型结果显示出更多的波动。
Peak of the bump at 12 kHz is 92 dB for both results, but the modelled bump does have a peak in the middle.
对于两个结果而言，12 kHz 处的突起峰值均为 92 dB，但模型预测的突起确实有一个中间峰值。
12 Conclusions 结论：12
The purpose of this chapter is to summarize the accuracy and reliability of the FEM method for waveguide modelling. It also outlines the achievements of the work. Lastly the advantages of the tool with engineering problems and the possible future developments are discussed.
本章的目的是总结有限元方法在波导建模中的准确性和可靠性。同时概述了工作的成就。最后，讨论了该工具在工程问题中的优势以及可能的未来发展方向。
本章的目的是总结有限元方法在波导建模中的准确性和可靠性。同时概述了工作的成就。最后，讨论了该工具在工程问题中的优势以及可能的未来发展方向。
12.1 Usability of the simulation
12.1 模拟的易用性
The goal of the thesis is set in the first chapter. The goal is to use, further develop and verify a modelling method for designing waveguide directivity. The finite element method was used to model a conical waveguide.
论文的目标在第一章中设定。目标是使用、进一步发展并验证一种用于设计波导方向性的建模方法。有限元方法被用于模拟一个锥形波导。
The results were compared to a physical prototype with equal dimensions. The main phenomena of interest were: change of directivity as a function of frequency, diffraction caused by the waveguide mouth edge and the resonant phenomena inside the waveguide.
结果与具有相同尺寸的物理原型进行了比较。主要感兴趣的现象包括：频率函数中的直接性变化，由波导口边缘引起的衍射以及波导内部的共振现象。
论文的目标在第一章中设定。目标是使用、进一步发展并验证一种用于设计波导方向性的建模方法。有限元方法被用于模拟一个锥形波导。
The results were compared to a physical prototype with equal dimensions. The main phenomena of interest were: change of directivity as a function of frequency, diffraction caused by the waveguide mouth edge and the resonant phenomena inside the waveguide.
结果与具有相同尺寸的物理原型进行了比较。主要感兴趣的现象包括：频率函数中的直接性变化，由波导口边缘引起的衍射以及波导内部的共振现象。
All the main significant phenomena found in the measured directivity could be found in the modelled directivity, although the accuracy of the frequencies and the amplitudes were varying.
在测量的指向性中发现的所有主要显著现象，都可以在模型化的指向性中找到，尽管频率和幅度的准确性有所变化。
Differences between the amplitudes were less than one equal amplitude contour, which means 3 dB accuracy. Model tends to exaggerate the resonant phenomena. This is not necessarily a problem.
幅度之间的差异小于一个等幅度轮廓，这意味着 3 dB 的准确性。模型倾向于夸大共振现象。这不一定是一个问题。
If the design is optimized in the model to achieve minimum resonance problems, then the real world prototype should be excellent in terms of resonance problems.
如果在模型中对设计进行优化以实现最小的共振问题，那么实际世界的原型在共振问题方面应该表现得非常优秀。
在测量的指向性中发现的所有主要显著现象，都可以在模型化的指向性中找到，尽管频率和幅度的准确性有所变化。
Differences between the amplitudes were less than one equal amplitude contour, which means 3 dB accuracy. Model tends to exaggerate the resonant phenomena. This is not necessarily a problem.
幅度之间的差异小于一个等幅度轮廓，这意味着 3 dB 的准确性。模型倾向于夸大共振现象。这不一定是一个问题。
If the design is optimized in the model to achieve minimum resonance problems, then the real world prototype should be excellent in terms of resonance problems.
如果在模型中对设计进行优化以实现最小的共振问题，那么实际世界的原型在共振问题方面应该表现得非常优秀。
The frequency responses of the measured and modelled results show the same phenomena. Differences were less than 2 dB .
测量结果和模型结果的频率响应显示相同的现象。差异小于 2 分贝。
测量结果和模型结果的频率响应显示相同的现象。差异小于 2 分贝。
Is the model accurate enough to be used in the daily world of engineering problems? It depends on the purpose. For optimizing and visualizing purposes it has shown its value.
模型在工程问题的日常世界中是否足够准确以供使用？这取决于目的。对于优化和可视化目的，它已经证明了其价值。
Often it is enough to be able to visualize the problem by seeing the acoustic field at a certain frequency. Then fast changes to the geometry will show how the phenomenon changes.
通常，只需通过在特定频率下可视化声场就能解决问题。然后，对几何形状的快速变化就能显示出现象如何改变。
However prototypes are needed for the final design if the shown accuracy of the model is not enough.
然而，如果模型的显示精度不足以满足最终设计的要求，那么需要原型。
模型在工程问题的日常世界中是否足够准确以供使用？这取决于目的。对于优化和可视化目的，它已经证明了其价值。
Often it is enough to be able to visualize the problem by seeing the acoustic field at a certain frequency. Then fast changes to the geometry will show how the phenomenon changes.
通常，只需通过在特定频率下可视化声场就能解决问题。然后，对几何形状的快速变化就能显示出现象如何改变。
However prototypes are needed for the final design if the shown accuracy of the model is not enough.
然而，如果模型的显示精度不足以满足最终设计的要求，那么需要原型。
12.2 Guidelines for a successful waveguide design
12.2 波导设计成功的指南
The design of the prototype waveguide was intentionally far from optimal. Still there is room for conclusions to be made about designing a waveguide with good performance.
原型波导的设计故意远离最优。尽管如此，仍然可以得出关于设计具有良好性能波导的结论。
According to the analysis of the results, the following conclusions can be made for achieving successful waveguide design.
根据结果的分析，可以得出以下结论以实现成功的波导设计。
原型波导的设计故意远离最优。尽管如此，仍然可以得出关于设计具有良好性能波导的结论。
According to the analysis of the results, the following conclusions can be made for achieving successful waveguide design.
根据结果的分析，可以得出以下结论以实现成功的波导设计。
Sharp transitions in geometry should be avoided so as to not excite diffraction. It is shown that smooth a transition to the enclosure baffle is necessary or a severe diffraction problem will occur.
几何中的急剧变化应避免，以免激发衍射。研究表明，向围栏的平滑过渡是必要的，否则将出现严重的衍射问题。
几何中的急剧变化应避免，以免激发衍射。研究表明，向围栏的平滑过渡是必要的，否则将出现严重的衍射问题。
At low frequencies the directivity of the waveguide is comparable to the directivity of a circular piston with a size equal to the mouth of the waveguide. This is consistent with the horn theory.
在低频段，波导的指向性类似于波导口尺寸等于波导口的圆形活塞的指向性。这与号筒理论一致。
在低频段，波导的指向性类似于波导口尺寸等于波导口的圆形活塞的指向性。这与号筒理论一致。
When the waveguide mouth circumference is comparable to the wavelength, the geometry of the waveguide dominates the directivity. In this frequency range the diffraction problem is worst.
当波导口的周长与波长相当，波导的几何形状主导了方向性。在这个频率范围内，衍射问题最严重。
当波导口的周长与波长相当，波导的几何形状主导了方向性。在这个频率范围内，衍射问题最严重。
When the circumference of the driver is comparable to the wavelength, the geometry of the driver starts to dominate the directivity characteristics.
当扬声器的直径与波长相当时，扬声器的几何形状开始主导其指向性特性。
当扬声器的直径与波长相当时，扬声器的几何形状开始主导其指向性特性。
An asymmetrical geometry would reduce the diffraction problem by smearing it to a broader frequency range.
非对称几何会通过将其扩散到更宽的频率范围来减少衍射问题。
非对称几何会通过将其扩散到更宽的频率范围来减少衍射问题。
12.3 Outputs of the work
12.3 工作的输出
There are two unique outputs of the work for which I did not find references in the bibliography. One of them is related to the directivity visualization tool created. It enables an intelligible way to compare the measured and modelled results.
在工作中，我未在参考文献中找到关于两项独特输出的引用。其中一项与我创建的直接性可视化工具相关。它提供了一种清晰的方法来比较测量和建模结果。
One favourable feature is that directivity plots are published by many loudspeaker manufacturers. Therefore there is already plenty of material with which to compare the modelled results without making measurements.
有利的一点是，许多扬声器制造商都发布了指向性图，因此已经有大量的资料可以用来比较模拟能果，而无需进行测量。
The real value of this work is to have a tool for virtually prototyping waveguides. Directivity plots give instant feedback about how the change in the design affected the performance.
这项工作的真正价值在于拥有一种工具，可以对波导进行几乎原型化。直接性图表提供了即时反馈，说明设计变化如何影响性能。
With compact graphs it is possible to compare the directivity of several prototypes at a glance, whether they are virtual or real.
通过紧凑的图表，可以一目了然地比较多个原型的指向性，无论是虚拟的还是实际的。
在工作中，我未在参考文献中找到关于两项独特输出的引用。其中一项与我创建的直接性可视化工具相关。它提供了一种清晰的方法来比较测量和建模结果。
One favourable feature is that directivity plots are published by many loudspeaker manufacturers. Therefore there is already plenty of material with which to compare the modelled results without making measurements.
有利的一点是，许多扬声器制造商都发布了指向性图，因此已经有大量的资料可以用来比较模拟能果，而无需进行测量。
The real value of this work is to have a tool for virtually prototyping waveguides. Directivity plots give instant feedback about how the change in the design affected the performance.
这项工作的真正价值在于拥有一种工具，可以对波导进行几乎原型化。直接性图表提供了即时反馈，说明设计变化如何影响性能。
With compact graphs it is possible to compare the directivity of several prototypes at a glance, whether they are virtual or real.
通过紧凑的图表，可以一目了然地比较多个原型的指向性，无论是虚拟的还是实际的。
The second unique output is the improved transducer model. The idea is to combine measurements and modelling. The tweeter output can be also modelled, but as stated before: a modelled result is always second to the real world.
第二项独特的输出是改进的换能器模型。想法是将测量和建模结合起来。扬声器输出也可以建模，但正如之前所陈述的：建模的结果始终次于现实世界。
The accuracy of the modelled result was improved by combining a laser velocity measurement with the model.
模型结果的准确性通过将激光速度测量与模型结合而得到提高。
第二项独特的输出是改进的换能器模型。想法是将测量和建模结合起来。扬声器输出也可以建模，但正如之前所陈述的：建模的结果始终次于现实世界。
The accuracy of the modelled result was improved by combining a laser velocity measurement with the model.
模型结果的准确性通过将激光速度测量与模型结合而得到提高。
12.4 Advantages of the virtual prototyping
12.4 虚拟原型的优点
There are several advantages when using FEM modelling. First is the speed of testing the new ideas. It is convenient to try several approaches to find a solution to a problem. Also there is not the problem of storing the tested ideas as with physical prototypes.
使用 FEM 建模有几大优势。首先是测试新想法的速度。可以尝试多种方法来找到解决问题的方案，非常方便。此外，与物理原型相比，不需要存储测试过的想法的问题。
使用 FEM 建模有几大优势。首先是测试新想法的速度。可以尝试多种方法来找到解决问题的方案，非常方便。此外，与物理原型相比，不需要存储测试过的想法的问题。
The second advantage is the improved visualization of the problem. In a model the full information of the pressure and particle velocity are always known in the modelled area. The information can be processed to achieve the most informative figure of the problem.
第二项优势是问题的可视化改进。在模型中，所模拟区域内的压力和粒子速度的全部信息总是已知的。这些信息可以被处理以获得问题最具有信息量的图示。
This improves the insight and intuition related to the acoustical problem. The insight given by the colourful graphs is a great extension to have when intuition is not enough to understand the problem.
这提高了对声学问题的理解和直觉。彩色图表提供的洞察力是当直觉不足以理解问题时的一个很好的补充。
第二项优势是问题的可视化改进。在模型中，所模拟区域内的压力和粒子速度的全部信息总是已知的。这些信息可以被处理以获得问题最具有信息量的图示。
This improves the insight and intuition related to the acoustical problem. The insight given by the colourful graphs is a great extension to have when intuition is not enough to understand the problem.
这提高了对声学问题的理解和直觉。彩色图表提供的洞察力是当直觉不足以理解问题时的一个很好的补充。
The third advantage is related to time and money. With virtual prototypes it is possible to reduce the amount of physical prototypes. This may reduce the cost of the prototypes and also the development time of the product.
第三大优势与时间和金钱有关。使用虚拟原型可以减少物理原型的数量。这可能降低原型的成本，同时缩短产品的开发时间。
第三大优势与时间和金钱有关。使用虚拟原型可以减少物理原型的数量。这可能降低原型的成本，同时缩短产品的开发时间。
12.5 Future work 12.5 未来工作
In general, there seems to be a certain trend for which areas are most suitable for FEM modelling. One common factor is that there is no analytical solution to the problem. The second factor is that prototypes are hard to manufacture or hard to measure.
一般来说，对于哪些领域最适合使用 FEM 建模，似乎存在一定的趋势。一个常见的因素是，问题没有解析解。第二个因素是，原型难以制造或难以测量。
In the field of acoustics, usually the measurement is the troublesome phase. Most problems are 3D. Therefore a huge amount of individual points need to be measured for accurate information of the prototype.
在声学领域，通常测量的是麻烦的相位。大多数问题都是三维的。因此，为了获得原型的准确信息，需要测量大量的单个点。
一般来说，对于哪些领域最适合使用 FEM 建模，似乎存在一定的趋势。一个常见的因素是，问题没有解析解。第二个因素是，原型难以制造或难以测量。
In the field of acoustics, usually the measurement is the troublesome phase. Most problems are 3D. Therefore a huge amount of individual points need to be measured for accurate information of the prototype.
在声学领域，通常测量的是麻烦的相位。大多数问题都是三维的。因此，为了获得原型的准确信息，需要测量大量的单个点。
One common problem area is the Helmholtz resonance of a vented enclosure. Current equations for calculating proper port geometry make assumptions that lead to inaccurate results. By using an FEM model the port geometry can be defined right
一个常见的问题领域是排气室的赫尔姆霍兹共振。当前用于计算适当端口几何形状的方程式做出的假设导致了不准确的结果。通过使用 FEM 模型，可以准确定义端口几何形状。
from the first prototype. It is also possible to use FEM for fluid flow simulation to optimize the port geometry for minimal air turbulence, which causes port noises.
从第一个原型开始。也可以使用 FEM 进行流体流动模拟，以优化港口几何形状，使其产生最少的空气湍流，从而减少港口噪音。
一个常见的问题领域是排气室的赫尔姆霍兹共振。当前用于计算适当端口几何形状的方程式做出的假设导致了不准确的结果。通过使用 FEM 模型，可以准确定义端口几何形状。
from the first prototype. It is also possible to use FEM for fluid flow simulation to optimize the port geometry for minimal air turbulence, which causes port noises.
从第一个原型开始。也可以使用 FEM 进行流体流动模拟，以优化港口几何形状，使其产生最少的空气湍流，从而减少港口噪音。
Another possible area of simulation is the enclosure geometry. Diffraction from the enclosure edges is a known problem but few manufacturers pay attention to it.
另一个可能的模拟领域是围护结构几何形状。边缘衍射是一个已知的问题，但很少有制造商关注这个问题。
另一个可能的模拟领域是围护结构几何形状。边缘衍射是一个已知的问题，但很少有制造商关注这个问题。
There is one future improvement for the transducer model presented in Chapter 8.4. The model used in the experiment assumes that the driver is operating above its mechanical resonance frequency. Therefore the movement of the diaphragm is mass controlled.
在第 8.4 章提出的换能器模型中，有一个未来的改进。实验中使用的模型假设驱动器在其机械共振频率之上运行。因此，膜片的运动是质量控制的。
The model does not give correct results if the model is used near the resonance frequency or below it. With a tweeter this is not usually a problem, because the intention is that the resonance frequency of the driver is well below the used frequency range.
模型在接近共振频率或在其下方使用时不会给出正确的结果。对于高音扬声器，这通常不是问题，因为设计的目的是让驱动器的共振频率远低于使用的频率范围。
This assumption is not valid when considering woofers. Therefore a different approach is needed if this method is going to be used for a woofer model.
这一假设在考虑低音扬声器时并不成立。因此，如果要使用这种方法来建立低音扬声器模型，需要采取不同的方法。
At least two different solutions exist. The first solution would use the impedance response of the woofer to specify the motional impedance. In the second solution, Thiele-Small parameters could be used to replace the simple mass with a damped spring-mass resonator.
至少有两种不同的解决方案。第一种解决方案会利用低音炮的阻抗响应来指定动量阻抗。在第二种解决方案中，可以使用 Thiele-Small 参数来替换简单的质量，用阻尼弹簧-质量共振器来代替。
在第 8.4 章提出的换能器模型中，有一个未来的改进。实验中使用的模型假设驱动器在其机械共振频率之上运行。因此，膜片的运动是质量控制的。
The model does not give correct results if the model is used near the resonance frequency or below it. With a tweeter this is not usually a problem, because the intention is that the resonance frequency of the driver is well below the used frequency range.
模型在接近共振频率或在其下方使用时不会给出正确的结果。对于高音扬声器，这通常不是问题，因为设计的目的是让驱动器的共振频率远低于使用的频率范围。
This assumption is not valid when considering woofers. Therefore a different approach is needed if this method is going to be used for a woofer model.
这一假设在考虑低音扬声器时并不成立。因此，如果要使用这种方法来建立低音扬声器模型，需要采取不同的方法。
At least two different solutions exist. The first solution would use the impedance response of the woofer to specify the motional impedance. In the second solution, Thiele-Small parameters could be used to replace the simple mass with a damped spring-mass resonator.
至少有两种不同的解决方案。第一种解决方案会利用低音炮的阻抗响应来指定动量阻抗。在第二种解决方案中，可以使用 Thiele-Small 参数来替换简单的质量，用阻尼弹簧-质量共振器来代替。
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