We have seen in past chapters that the radiation of sound from typical transducers is basically a fixed quantity. There appears to be little that we can do to affect the sound radiation response. 我们在过去的章节中已经看到，典型换能器的声音辐射基本上是一个固定的量。我们似乎没有什么办法影响声辐射响应。 The size and configuration determines the radiation pattern with the enclosure playing an important role at the lowest frequencies. 尺寸和配置决定了辐射模式，外壳在最低频率时起着重要作用。 If the driver is still radiating above the point where the enclosure is controlling the response then the response is pretty much completely dependent on the driver size. Little else has much of an effect. 如果驱动器的辐射仍高于箱体控制响应的点，那么响应几乎完全取决于驱动器的尺寸。其他方面几乎没有什么影响。
In this chapter, we will study the concept of a waveguide as a directionality controlling device. 在本章中，我们将研究波导作为方向控制装置的概念。
6.1 Historical Notes 6.1 历史说明
It is important to go through the historical development of horn and waveguide theory in order to understand its evolution the current level of our understanding. 我们有必要回顾一下喇叭和波导理论的历史发展，以了解其演变过程和我们目前的理解水平。 The importance of this review stems from long-standing beliefs about waveguides and horns that are not in fact correct. Correcting these beliefs creates extreme limits on their applicability to current issues in their design. 这篇评论的重要性源于人们长期以来对波导和喇叭的看法，而事实上这些看法并不正确。纠正这些观念会极大地限制它们对当前设计问题的适用性。
Horns have been around for centuries and we have no idea when or where they were first used. Horns as musical instruments are certainly centuries old. 号角已经存在了几个世纪，我们不知道它们最早是在何时何地被使用的。作为乐器的号角当然有几百年的历史。 With the advent of the phonograph, the horn was found to play a crucial role in amplifying the sound emitted from the small mechanical motions of the stylus. The horn was responsible for virtually all of the gain in the system. 随着留声机的问世，人们发现号角在放大唱针的微小机械运动所发出的声音方面起着至关重要的作用。号角负责系统中几乎所有的增益。 Its use therefore was principally one of a loading or impedance matching mechanism required to better match the high mechanical impedance of the stylus to the very low mechanical impedance of the medium - air. 因此，它主要是一种负载或阻抗匹配机制，用于更好地将测针的高机械阻抗与介质（空气）的极低机械阻抗相匹配。 The horns role as an acoustic transformer is central to the evolution of horn theory. 号角作为声学变压器的作用是号角理论发展的核心。
When one is interested in the loading properties of a conduit, they need only be concerned with the average distribution of the acoustic variables across the diaphragm and hence across the conduit. 当人们对导管的加载特性感兴趣时，只需关注隔膜上的声学变量的平均分布，从而关注导管上的声学变量的平均分布。 This is, in fact, the assumption that Webster made when he derived what is now known as Webster's Horn Equation 事实上，这就是韦伯斯特在推导出现在被称为韦伯斯特霍恩方程时所作的假设
It is commonly thought that this equation applies only to plane waves since Webster used a plane wave assumption in its derivation. However, this equation is far more broadly applicable than to plane waves alone. 人们通常认为这个方程只适用于平面波，因为韦伯斯特在推导过程中使用了平面波假设。然而，这个方程的适用范围要比平面波广泛得多。 It is actually exact for any geometry where the scale factor of the coordinate of interest is one. The scale factor is a fundamental parameter of all coordinate systems as shown in Morse . (Having a scale factor is a requirement for separability.) The scale factors are known and can only be calculated for separable coordinate systems. The interesting thing to note is that any coordinate system which has a unity scale factor for any of its three dimensions is exact in Webster's formulation. 实际上，对于相关坐标的比例因子为 1 的任何几何体来说，它都是精确的。如莫尔斯 所示，比例因子是所有坐标系的基本参数（比例因子是可分性的要求）。比例因子是已知的，只有可分坐标系才能计算出比例因子。值得注意的是，在韦伯斯特公式中，任何一个坐标系的三个维度的比例因子都是统一的，都是精确的。 Specifically these coordinates are: 具体来说，这些坐标是
all three Cartesian Coordinates 所有三个直角坐标
the axial coordinate in all Cylindrical Coordinate systems (Elliptic, Parabolic and Circular) 所有圆柱坐标系（椭圆坐标系、抛物线坐标系和圆坐标系）中的轴坐标
the radial coordinate in the Cylindrical coordinate system 在圆柱坐标系中的径向坐标
the radial coordinate in Spherical Coordinates 球面坐标中的径向坐标
the radial coordinate in Conical Coordinates 圆锥坐标中的径向坐标
The first two (six coordinates) apply to conduits of constant cross section, which are not very interesting to us. The useful ones are the three radial coordinates for the Cylindrical, Spherical, and Conical coordinate systems. It is extremely important to note two additional items. First, that there are only three useful coordinates in which Webster's equation is exact; and second, that all three of these have wave propagation in the other orthogonal coordinates. 前两个坐标（六个坐标）适用于横截面不变的导管，对我们来说并不十分有趣。有用的坐标是圆柱坐标系、球坐标系和圆锥坐标系的三个径向坐标。还有两点需要特别注意。首先，只有三个有用的坐标系中的韦伯斯特方程是精确的；其次，所有这三个坐标系中的波都在其他正交坐标系中传播。 The importance of this last attribute will become clear later on. 这最后一个属性的重要性稍后会变得很清楚。
If Webster's Equation is only correct in three useful situations then why is there so much literature surrounding its use? That is because the equation is still useful as an approximation to any conduit of varying cross section. 如果韦伯斯特方程只在三种有用的情况下才是正确的，那么为什么会有这么多围绕其使用的文献呢？这是因为对于任何横截面不同的导管来说，该方程仍然是有用的近似值。 In nearly all of the common cases of the application of Webster's equation it is used as an approximation to the actual wave propagation in a flared conduit. 在应用韦伯斯特方程的几乎所有常见情况下，它都被用作扩口导管中实际波传播的近似值。 There has also been a great deal of literature written about the application of Webster's equation to the evaluation of these approximate solutions. 关于应用韦伯斯特方程评估这些近似解，也有大量的文献记载。 Almost nothing has been written about when these approximations are "good" approximations and when we should be suspect of their validity. 关于这些近似值什么时候是 "好 "近似值，什么时候我们应该怀疑它们的有效性，几乎没有任何论述。
The only place where we have seen such a discussion is, once again, in Morse where they state: 我们唯一看到过这种讨论的地方，还是在莫尔斯 中，他们在那里说：
Both pressure and fluid velocity obey this modified Wave Equation, which approximately takes into account the variation of cross sectional size with . The equation is a good approximation as long as the magnitude of the rate of change of with is much smaller than unity (as long as the tube "flares" slowly). 压力和流体速度都服从这个修正的波方程，该方程近似考虑了横截面尺寸随 的变化。只要 随 的变化率远小于统一值（只要管道缓慢 "扩张"），该方程就是一个很好的近似值。
See Morse, Methods of Theoretical Physics 见莫尔斯：《理论物理学方法
See Morse, Methods of Theoretical Physics, pg. 1352 参见莫尔斯：《理论物理学方法》，第 1352 页。
These are very limiting conditions that have been almost universally ignored. Let us look at what they imply about the development of the well know exponential horn. 这些限制性条件几乎被普遍忽视。让我们看看它们对众所周知的指数角的发展有何影响。
Restating Morse, as applied to an exponential horn, let 重述应用于指数角的莫尔斯，让
we can see that, for this example, Webster's equation is "good" only so long as 我们可以看到，在这个例子中，韦伯斯特方程的 "好 "之处在于
the throat area 咽喉部位
the flare rate 耀斑率
The horn contour for this example is shown in Fig. 6-1. This figure shows an 该示例的喇叭轮廓如图 6-1 所示。该图显示了
exponential horn contour of typical shape and length. In Fig. 6-1, we have plotted the value of Eq.(6.1.3) as a function of the axial distance from the throat. 典型形状和长度的指数角轮廓。在图 6-1 中，我们绘制了公式（6.1.3）值与喉管轴向距离的函数关系图。 These figures show that the assumptions for an accurate application of Webster's Equation to an exponential horn are clearly violated for a length of the horn beyond about (Morse says much less than 1.0, but how much less is a matter of choice. We ascribe here to the use of a value of .5 as being the limit of accuracy, the assumptions being completely invalid at the value of 1.0.) 这些数字表明，当喇叭的长度超过 时，韦伯斯特方程对指数喇叭的精确应用显然违反了假定（莫尔斯说远远小于 1.0，但究竟小于多少则是一个选择问题。在此，我们认为 0.5 的值是精确度的极限，假设值为 1.0 时完全无效）。
Figure 6-1 - Plot of Eq.(6.1.3) 图 6-1 - 公式(6.1.3)的曲线图
In Fig. 6-1, we have shown a line drawn tangent to the horn contour and originating at the center of the throat, the acoustic center of the throat's wavefront. 在图 6-1 中，我们绘制了一条与喇叭轮廓相切的直线，该直线的起点是喉管中心，即喉管波面的声学中心。 A simple rule of thumb that we use is that any contour which lies past this point of tangency cannot be accurately described by Webster's equation. This rule of thumb implies a geometrical interpretation of the limitation of Webster's equation. 我们使用的一个简单经验法则是，韦伯斯特方程无法准确描述过切点的任何轮廓。这条经验法则意味着对韦伯斯特方程局限性的几何解释。 The horn equation cannot predict the wavefronts once they are required to diffract around a point along the device that places the receding boundary in the shadow zone of the acoustic center of the originating wavefronts. Fig. 一旦要求波阵面在沿装置的某一点周围衍射，将后退边界置于起源波阵面声学中心的阴影区内，喇叭方程就无法预测波阵面。图 6-1 shows that the only exponential horn which could be accurately represented by the Webster equation is extremely short. A horn of this length is of no practical interest. 6-1 表明，唯一能用韦伯斯特方程准确表示的指数角非常短。这种长度的角没有实际意义。
Further support for our "rule of thumb" comes from considering Huygens' principle and the construction of Huygen wavefronts. 考虑惠更斯原理和惠更斯波面的构造，可以进一步支持我们的 "经验法则"。 Beyond the point of tangency, wavefronts, if they are to remain perpendicular to the sides of the conduit (as they must), have to have an apparent acoustic center which is front of the actual throat of the device. 在切点之外，如果波面要保持与导管两侧垂直（这是必须的），就必须有一个位于设备实际喉管前方的表观声学中心。 Huygen's principle allows for un-diffracted wavefronts to be flatter than those from the acoustic center, but is does not allow for a wavefront curvature to be less than the radius to the acoustic center. A little thought will show why this must be true. 惠根原理允许未衍射波面比来自声学中心的波面更平坦，但不允许波面曲率小于到声学中心的半径。只要稍加思考，就会明白为什么这一定是正确的。 So stated another way, our rule of thumb becomes: horn wavefronts with a curvature less than the radius to the throat must have diffracted somewhere along the trip down the device - the diffraction creating a new acoustic center from which wavefronts emerge. 因此，换一种说法，我们的经验法则就变成了：曲率小于喉管半径的喇叭波面，一定是在设备的某个地方发生了衍射--衍射产生了一个新的声学中心，波面就是从这个中心出现的。
It is further interesting to note that in those cases where Webster's Equation is exact, there is never a point on the contour of the horn which is in the shadow 更有趣的是，在韦伯斯特方程精确的情况下，角的轮廓上从来没有一个点处于阴影中
zone, i.e. our "rule of thumb" is never violated. The overwhelming majority of work done in horn theory suffers from a serious question of its validity. 区，也就是说，我们的 "经验法则 "从未被违反过。绝大多数有关角理论的工作都存在严重的有效性问题。
How one gets around this problem leads us into another line of reasoning for which there are two paths. 如何绕过这个问题，会让我们进入另一条推理思路，而这条路有两条。 We could join the exponential section of the above contour to a spherical section continuing out from the point of tangency to the acoustic center line, thus insuring that our rule of thumb was never violated. This does in fact work reasonably well, and is in common usage. 我们可以将上述等值线的指数部分与从声学中心线的切点继续向外延伸的球形部分连接起来，从而确保我们的经验法则不被违反。事实上，这种方法效果相当不错，也很常用。 However, the fact remains that the exponential section is still only an approximation and we really don't know the actual shape of the wavefront at the joining point. As we shall see later, this is a serious limitation to the joining approach. 然而，事实上指数截面仍然只是一个近似值，我们确实不知道连接点处波面的实际形状。正如我们稍后将看到的，这是连接方法的一个严重局限。
Now that we have shown that the horn equation has severe limitations in its applicability to important problems in waveguides, we will discuss how these limitations might affect the expected results of using it. 既然我们已经证明了喇叭方程在波导重要问题的适用性上存在严重的局限性，那么我们将讨论这些局限性会如何影响使用该方程的预期结果。 As we showed in Chap.3, if we know the wave shape of the wavefront as it crosses through a boundary for which we have a radiation solution (flat, spherical or cylindrical), then we can achieve a fairly accurate prediction of the directivity of this source. 正如我们在第 3 章中所展示的，如果我们知道波阵面穿过边界时的波形，而我们又有一个辐射解（平面、球面或圆柱面），那么我们就能相当准确地预测出这个波源的指向性。 However, we must know the précis magnitude and phase of the wavefront at every point in the aperture in order to do this calculation. From the above discussion, we should have serious doubts about the ability of Eq. 但是，我们必须知道孔径中每一点的波面的简要幅度和相位，才能进行计算。从上面的讨论中，我们应该会对公式......的能力产生严重怀疑。 (6.1.1) to give us this information, except, of course in those limited cases where it is exact. (6.1.1) 向我们提供这些信息，当然，在少数情况下提供准确信息除外。 The natural question to ask is: can we develop an equation or an approach which will allow us to know, with some certainty, what the magnitude and phase is at every point within the waveguide? 自然而然要问的问题是：我们能否制定一个方程或一种方法，让我们能够比较肯定地知道波导内每一点的振幅和相位是多少？ The answer is yes, but the price that we must pay for this precious knowledge is a substantial increase in the complexity of the equations and their solution. 答案是肯定的，但我们必须为这些宝贵的知识付出代价，那就是方程及其解法的复杂性大大增加。
6.2 Waveguide Theory 6.2 波导理论
As we discussed above, the early use of a horn was substantially different than what we are attempting to develop here. The early need for horns was as an acoustic loading devices and our interest here is in controlling source directivity. 如上文所述，号角的早期用途与我们在此尝试开发的用途大相径庭。早期对号角的需求是作为一种声学加载装置，而我们在这里感兴趣的是控制声源的指向性。 (Loading essentially became a non-issue with the almost unlimited amplifier power capability available today.) For this reason, we will adapt the terminology that a horn is a device which was developed with Webster's Equation and its approach to calculation (wherein, only the average wavefront shape and the acoustic loading is required) and a waveguide is a device whose principal use is to control the directivity. (如今，放大器的功率几乎不受限制，负载基本上已不再是问题）。因此，我们将采用以下术语：号角是一种利用韦伯斯特方程及其计算方法开发的设备（只需计算平均波面形状和声学负载），而波导则是一种主要用于控制指向性的设备。 A waveguides design is along the lines that we will develop in the following sections, a horns design using Webster's equation. The acoustic loading of a waveguide can usually be calculated without much trouble, but not always. 波导的设计思路与我们将在以下章节中介绍的使用韦伯斯特方程进行喇叭设计的思路一致。波导的声学负载通常可以轻松计算出来，但并非总是如此。 However, since loading is not a central concern for us this limitation is not significant. 不过，由于负载并不是我们关注的核心问题，因此这一限制并不重要。
See Geddes "Waveguide Theory" and "Waveguide Theory Revisited", JAES. 见 Geddes "波导理论 "和 "波导理论重温"，JAES。
We know from earlier chapters that the Wave Equation is always accurate so long as we can apply the proper boundary conditions. This can happen only in one of a limited set of coordinate systems. Take, as an example, the simple case of a conduit in a spherical geometry as shown in Fig. 6-2. The boundary conditions 通过前面几章的学习，我们知道，只要能应用适当的边界条件，波方程总是准确的。这只能在有限坐标系中的一个坐标系中实现。以图 6-2 所示球形几何体中导管的简单情况为例。边界条件
Figure 6-2 - A simple hornwaveguide example 图 6-2 - 一个简单的喇叭波导示例
here are that the conduit is symmetric in , and has a velocity which is zero at some . For now we will not worry about the terminations of this conduit along its axis and simply assume that it is semi-infinite. In other words it has a finite throat, but a mouth at infinity. 在这里，导管在 上是对称的，并且 速度在某个 处为零。现在，我们暂且不考虑导管沿其轴线的终端，只假设它是半无限的。换句话说，它的喉部是有限的，但在无穷远处有一个口。
We can use either the full Wave Equation, or Eq. (6.1.1) in this example because of its simplicity. We have already discussed the horn approach so let's use the full Wave Equation as we developed in Chap.3. We know that the following solution applies in both the Spherical coordinate system and Webster's Equation. 在这个例子中，我们既可以使用完整的波方程，也可以使用公式 (6.1.1)，因为它很简单。我们已经讨论过喇叭口方法，所以还是使用第 3 章中的全波方程吧。 我们知道，以下解法既适用于球面坐标系，也适用于韦伯斯特方程。
We can immediately see from this solution that there is one aspect to the Wave Equation approach that is not present in Webster's approach. That is, we know from the Wave Equation that the wave number is a coupling constant to two other equations in and . We can exclude the coupling to since there is no variation in the boundary conditions, but we cannot simply make the assumption that there is no variation of the waves in . If there is a dependence of the wavefront at the throat (or any point for that matter) then there will be a dependence in the wave as it is propagates down the device. This is significantly different from Webster's approach since Webster's equation does not allow for any dependence. This limitation is actually far more significant than the errors due to the flare rate that we discussed above. 从这一解法中我们可以立即看出，波方程方法有一个方面是韦伯斯特方法所不具备的。也就是说，我们从波浪方程中知道，波数 是 和 中另外两个方程的耦合常数。我们可以排除与 的耦合，因为 的边界条件没有变化，但我们不能简单地假设 中的波没有变化。如果喉部（或任何一点）的波面与 有关，那么波在设备上向下传播时就会与 有关。这与韦伯斯特的方法大相径庭，因为韦伯斯特方程不允许任何 相关性。这一限制实际上比我们上面讨论的耀斑率所造成的误差要大得多。
As an example, consider evaluating a "conical" horn versus a Spherical waveguide (both are as shown in Fig. 6-2) using Webster's equation and the acoustic waveguide approaches respectively. 例如，考虑分别使用韦伯斯特方程和声波导方法评估 "锥形 "喇叭和球形波导（两者如图 6-2 所示）。 Horn theory yields an impedance at the throat, but it yields no information about the amplitude and phase of the wavefront anywhere within the device. The assumption of uniform amplitude across 喇叭理论可以得出喉管处的阻抗，但无法得出设备内任何位置的波面振幅和相位信息。假定整个波面的振幅是均匀的
the device means that Webster's approach predicts the same value at every point on some (unknown) surface which is orthogonal to the horn boundaries. Of course we could argue (correct in some cases) that the wavefront must be a spherical section at every point. 该装置意味着韦伯斯特方法预测了与喇叭边界正交的某个（未知）表面上每一点的相同值。当然，我们也可以认为（在某些情况下是正确的），波阵面在每一点上都必须是一个球形截面。 But what happens if the device is fed at the throat with a plane wave? There is simply no way to answer this question with the tools available to us from Webster. 但是，如果在设备的喉部输入平面波，会发生什么情况呢？韦伯斯特提供的工具根本无法回答这个问题。
Consider now the alternate waveguide approach. From Sec. 3.4 on page 49 we know that the solutions for the radial coordinate are 现在考虑另一种波导法。根据第 49 页第 3.4 节，我们知道径向坐标的解法为
If the driving wavefront is of a spherical shape, then only a single mode will be excited. In that case we get the same answer for a wave propagating down the device for either the Horn Equation or the Wave Equation 如果驱动波面呈球形，则只会激发单模 。在这种情况下，对于沿设备向下传播的波，无论是喇叭方程还是波方程，我们都能得到相同的答案
where , since it is always in the direction in this case. However, we also know that there are an infinite number of other possibilities where . Only the Wave Equation approach offers up this added flexibility in its application. ，因为在这种情况下，它总是朝着 方向。不过，我们也知道，在 的情况下，还有其他无限的可能性。只有波方程方法在应用中提供了这种额外的灵活性。
When the throat is driven by a wavefront which does not coincide with a spherical section of the same radius as the throat, then the wavefront can be expanded into a series of admissible wavefronts prior to propagation down the device. 当喉管受到与喉管半径相同的球面截面不重合的波阵面驱动时，波阵面就会在沿设备向下传播之前扩展成一系列可接受的波阵面。 Since we have solutions for all of these waves, we can develop the final solution as a sum over these various wave orders. 由于我们有所有这些波的解，因此可以将最终解作为这些不同波阶的总和。
Consider a plane wave excitation at the throat. It is well know that a plane wave can be expanded into an infinite series of spherical waves. This series is 考虑在喉管处激发平面波。众所周知，平面波可以扩展为无限的球面波系列。这个系列是
Basically, the Legendre Polynomials form an expansion with the weighting factors given by the terms to the left of them. Eq. (6.2.7) is applicable to a plane pressure wave - a scalar function. 基本上，Legendre 多项式形成了一个展开式，其左侧项给出了加权因子。公式 (6.2.7) 适用于平面压力波--一个标量函数。 If we had a planar velocity source at the throat (i.e., a flat piston) then we would have to match this to the radial and angular velocities at the throat. 如果喉管处有一个平面速度源（即平面活塞），那么我们就必须将其与喉管处的径向速度和角速度相匹配。 This would not actually be too difficult, except that there is yet another problem with this approach, so we will leave this discussion for later in order to address the solution to our current problem. 这实际上并不难，只是这种方法还有另一个问题，所以我们将留待以后再讨论，以便解决目前的问题。
Unfortunately, the Legendre Polynomials, as shown in Eq. (6.2.7), do not fit the boundary conditions of our waveguide, that is, they do not have a zero slope 不幸的是，如公式 (6.2.7) 所示，Legendre 多项式不符合我们波导的边界条件，即它们的斜率不是零
See Morse, Any text 见莫尔斯，任何文本
at the walls of the waveguide, for all . The normal Legendre Polynomials must have a separation constant which is an integer because of periodicity. The new functions can no longer require this constant to be an integer. (Why?) Compare the two plots in Fig. 6-3. The right side of the plot shows the normal Legendre Polynomials over a arc. The left side shows the Modified Legendre Polynomials (modified because of the new separation constant ) that meet the boundary conditions at the walls of the waveguide. The new polynomials can be used to expand any axi-symmetric source at the throat - they form a complete orthogonal set. It is worth noting the similarity of Fig. 6-3 with Fig.4-3 on page 73. 在波导壁上， ，所有 。由于周期性的原因，普通 Legendre 多项式必须有一个整数的分离常数 。新函数不再要求该常数为整数。(为什么？）比较图 6-3 中的两幅图。图的右侧显示的是 弧线上的普通 Legendre 多项式。左侧显示的是符合波导壁边界条件的修正 Legendre 多项式（由于使用了新的分离常数 ）。新的多项式可用于扩展喉部的任何轴对称声源--它们构成了一个完整的正交集合。值得注意的是，图 6-3 与第 73 页的图 4-3 非常相似。
Thus far we have seen that by utilizing the full machinery of the Wave Equation we can match any velocity distribution placed at the throat of a waveguide so long as this waveguide lies along a coordinate surface of one of the separable coordinate systems (although we have as yet only looked at a very simple one). 到目前为止，我们已经看到，通过利用波方程的全部机制，我们可以匹配波导喉部的任何速度分布，只要该波导位于一个可分离坐标系的坐标面上（尽管我们还只是研究了一个非常简单的坐标系）。 We have also seen that this wavefront matching cannot be accomplished by using Webster's horn equation; the machinery to do so just does not exist in that formulation. 我们还看到，这种波面匹配无法通过韦伯斯特的角方程来实现，因为在该方程中不存在实现波面匹配的机制。 Of course we could force the throat wavefront to match the lowest order mode of the horn and then the horn equation would be accurate, exact in fact. 当然，我们可以迫使喉管波面与喇叭的低阶模式相匹配，这样喇叭方程就会准确无误。
Figure 6-3 - Normal Legendre Polynomials (right) for waveguide (left) 图 6-3 - 波导（左）的法线 Legendre 多项式（右）。
The problem is that there are no sources which have a velocity profile that matches any of the three geometries which have an exact horn solution! 问题是，没有任何信号源的速度曲线与三种几何图形中的任何一种相匹配，而这三种几何图形都有精确的喇叭解！
In order to continue we have a choice of three alternatives: 为了继续前进，我们有三种选择：
accurately use the horn equation with unrealistic sources 用不现实的声源准确地使用喇叭方程
use approximate solutions for real sources, or 使用真实源的近似解，或
obtain exact solutions for realistic sources but restricting ourselves to the use of a few prescribed geometries (separable coordinate systems) for which an exact analysis is possible 为现实源获取精确解，但仅限于使用少数几个规定的几何图形（可分离坐标系），对这些几何图形可以进行精确分析
The first choice is not of interest to us. The second choice may be workable and we will investigate that alternative later, but for now we will choose the third option in order to get exact answers, which we can use later as a comparison to the approximate solutions. 我们对第一种选择不感兴趣。第二种选择可能可行，我们稍后会研究这种选择，但现在我们会选择第三种选择，以便得到精确的答案，稍后我们可以用它与近似解进行比较。 We will also get a better understanding of the nature of the exact solution. 我们还将更好地了解精确解的性质。
Returning to the above example, we can see that a planar source at the throat of a Spherical waveguide will have more than a single mode of propagation due to the required fitting of this source to the waveguide at the throat. 回到上面的例子，我们可以看到，球形波导喉部的平面声源将不止一种传播模式，这是因为该声源需要与喉部的波导相匹配。 By expanding the source velocity in a series of modified Legendre Polynomials, we can determine the contribution of the various modes of the waveguide. We saw an example of this in Sec. 通过在一系列修正的 Legendre 多项式中展开源速度，我们可以确定波导中各种模式的贡献。我们在第二章中看到了一个例子。 4.5 on page 83 where we expanded a spherical wavefront in terms of a set of plane aperture modes. We are now doing the reverse, namely, expanding a plane wave in an aperture in terms of a set of finite angular spherical modes. 在第 83 页的 4.5 节中，我们用一组平面孔径模来扩展球面波。现在我们反过来，用一组有限角球面模来扩展孔径中的平面波。 The two processes are completely analogous albeit reversed. 这两个过程虽然相反，但完全类似。
Each of the waveguide modes will propagate with a different phase and amplitude which can readily be calculated as 每种波导模式都会以不同的相位和振幅传播，其计算公式为
The eigenvalues need to be determined specifically for each waveguide since they vary with the angle of the walls. The function is the same Spherical Hankel Function (of the second kind - outgoing) that we have seen before except that now these functions have a non-integer order. 由于特征值 会随波导壁角度的变化而变化，因此需要根据每个波导的具体情况来确定。函数 与我们之前看到的球形汉克尔函数（第二类--传出）相同，只是现在这些函数具有非整数阶。 The calculation of these functions is not difficult although the details are beyond the scope of this text and covered elsewhere 5 . 这些函数的计算并不困难，尽管细节超出了本文的范围，在其他地方也有涉及5。
As we saw in Sec.4.2, the modes radiate (propagate in the current case) with efficiencies which vary with frequency. Fig. 6-4 shows the modal impedances for the first three modes in a Spherical waveguide. From this figure, we can see precisely the differences in horn theory and waveguide theory. 正如我们在第 4.2 节中所看到的，这些模式的辐射（在当前情况下是传播）效率随频率而变化。图 6-4 显示了 球形波导中前三个模态的模态阻抗。从图中我们可以清楚地看到喇叭理论和波导理论的不同之处。
Horn theory yields only the solid line shown in this figure, which is exactly the same as the waveguide calculation for this lowest order mode. Above (where is the radius of the waveguide's throat), the first mode begins to cut-in. 喇叭理论只得到了图中所示的实线，这与波导对这一最低阶模式的计算结果完全相同。在 以上（其中 为波导喉部半径），第一个模式开始切入。
5. See Zhang, Computation of Special Functions 5.见 Zhang，《特殊函数的计算》。
Figure 6-4 - Real and imaginary parts of the modal impedances for a 图 6-4 - a 的模态阻抗的实部和虚部
Below this frequency both the horn theory of Webster and waveguide theory will yield nearly identical results (a small difference is due to the finite imaginary part of the higher order modes). 在此频率以下，韦伯斯特的号角理论和波导理论将得出几乎相同的结果（由于高阶模式的有限虚部，两者之间存在微小差异）。 Above this frequency the first mode (which would be quite significant for a piston source driving a Spherical waveguide) has an even greater proportional effect on the wavefront than the zero order mode, and could hardly be ignored for accurate results. 在此频率以上，第一种模式（对于驱动球形波导的活塞源来说非常重要）对波面的比例影响甚至大于零阶模式，因此很难忽略它以获得准确的结果。 It is in this region (above cut-in of the first mode) that horn theory has serious shortcomings. Its validity becomes progressively worse as the frequency goes up and even more modes cut-in. Waveguide theory remains accurate to as high a value as one cares to calculate its modes. 正是在这一区域（第一模切入以上），喇叭理论存在严重缺陷。随着频率的升高和更多模式的切入，其有效性会逐渐降低。波导理论在计算其模式时，只要愿意，其精确度仍可达到很高的数值。 This is a significant difference in accuracy for a large directivity controlling device. 对于大型指向性控制设备来说，这在精度上是一个很大的差别。
Now that we have seen why waveguide theory is preferable to horn theory for high frequency directivity controlling devices, we will investigate the various separable coordinate systems for which waveguide theory is directly applicable in order to determine which ones have useful geometries. 既然我们已经了解了在高频指向性控制设备方面波导理论优于喇叭理论的原因，那么我们将研究波导理论可直接适用的各种可分离坐标系，以确定哪些坐标系具有有用的几何形状。
6.3 Waveguide Geometries 6.3 波导几何形状
We have already discussed several of the separable coordinate systems, but below is a table of the complete set of 11 along with the type of source that is required at the throat for a pure zero order mode. 我们已经讨论过几种可分离坐标系，下面是完整的 11 种坐标系以及纯零阶模式喉部所需的声源类型表。
Name
Coordinate
Source
Aperture
Source
Curvature
Mouth
Curvature
Rectangular
any
rectangle
flat
flat
Circular Cylinder 圆柱形
radial
rectangle
cylindrical
cylindrical
Elliptic Cylinder 椭圆柱
radial
rectangle
Flat
cylindrical
Parabolic Cylinder 抛物线圆柱体
none
Spherical
radial
circular
spherical
spherical
Conical
radial
elliptical
spherical
spherical
Parabolic
none
Prolate Spheroidal 扁球形
radial
rectangle
cylindrical
spherical
Oblate Spheroidal 扁球形
radial
circular
flat
spherical
Ellipsoidal
radial
elliptical
flat
spherical
Paraboloidal
none
Table 6.1: Useful waveguides for Separable Coordinates 表 6.1：可分离坐标的有用波导
All of the useful coordinates are radial and all of the mouth apertures are the same as the throat apertures (not shown). The source apertures are either rectangular or elliptical (circular being a special case of elliptical). 所有有用坐标都是径向坐标，所有口孔都与喉孔相同（未显示）。声源孔径为矩形或椭圆形（圆形是椭圆形的特例）。 The mouth curvatures (radiation wavefronts) can only be spherical or cylindrical, flat being of little interest. This last feature is the main reason why we studied the geometries that we did in Chap. 4. If the apertures are circular then the physical device must be axi-symmetric. 孔口曲率（辐射波面）只能是球形或圆柱形，扁平的孔口曲率并不重要。如果孔径是圆形的，那么物理装置必须是轴对称的。 The wave propagation need not be axi-symmetric, however. We will not look into this possibility since it is rather unusual in practice. 不过，波的传播不一定是轴对称的。我们将不研究这种可能性，因为它在实践中并不常见。
It is also possible to combine waveguides to create new devices. For example, the Prolate Spheroidal (PS) waveguide takes a square cylindrically curved wavefront at its throat, which is exactly what an Elliptic Cylinder waveguide produces. 还可以将波导组合起来，创造出新的设备。例如，Prolate Spheroidal（PS）波导在其喉部产生方形圆柱弯曲波面，这正是椭圆圆柱波导所产生的波面。 A waveguide created as a combination of two waveguides in these two coordinate systems would take a square flat wavefront as input and produce a square spherical one. 由这两个坐标系中的两个波导组合而成的波导，将以方形平面波面作为输入，并产生一个方形球面波面。
It is interesting to note that the horn equation is only exact when the input and output wavefront curvatures remain unchanged. By this we mean that the location of the center of radius of the wavefront for both the throat and the 值得注意的是，只有当输入和输出波面曲率保持不变时，喇叭方程才是精确的。这意味着波面半径中心的位置对于喉部和
mouth does not move in space. This is exactly what it means to have a unity scale factor. Unfortunately, geometries that do not have unity scale factors are significantly more difficult to analyze - the price that we must pay for the higher accuracy of the waveguide approach. 嘴巴在空间中不会移动。这正是比例系数为一的含义。遗憾的是，不具有统一比例系数的几何图形分析起来要困难得多，这也是我们必须为波导法的高精度付出的代价。 We have already looked at a Spherical waveguide in some detail and now we will investigate a waveguide that is based on the Oblate Spheroidal (OS) coordinate system in order to compare and contrast its characteristics with those that we have already studied. 我们已经对球面波导进行了比较详细的研究，现在我们将研究一种基于扁球面 (OS) 坐标系的波导，以便将其特性与我们已经研究过的波导进行比较和对比。
6.4 The Oblate Spheroidal Waveguide 6.4 橢圓球面波導
Proceeding as in the previous sections, the first calculations that we need to do for an OS waveguide are to determine the wave functions (or Eigenfunctions) in this coordinate system. 与前几节一样，我们需要对 OS 波导进行的第一项计算是确定该坐标系中的波函数（或特征函数）。 Unlike the previous case of the Spherical Wave Equation, the wave functions for the OS coordinate system are not as readily available. 与之前的球面波方程不同，操作系统坐标系的波函数并不容易获得。 The unique thing about those coordinate systems that do not have unity scale factors is; even though the equations separate in the spatial coordinates they remain coupled through the separation constants (the wavenumber or time coordinate). 这些坐标系没有统一的比例因子，其独特之处在于：即使方程在空间坐标中分离，它们仍然通过分离常数（波长或时间坐标）耦合在一起。 We have not encountered this complication in any of the problems that we have studied thus far. The wave functions in both the radial coordinate and the angular coordinate will be found to depend on a common parameter , where is the inter-focal separation distance (See Fig.1-2 on pg. 6). (We must be careful not to confuse this (non-italic-bold) with the wave velocity , of the same letter. The use of is historical and the authors do not feel privileged enough to change it.) 在我们迄今研究的问题中，还没有遇到过这种复杂情况。径向坐标和角坐标的波函数都将取决于一个共同的参数 ，其中 是焦距间距（见第 6 页图 1-2）。(我们必须注意不要将 （非大写加粗）与同一字母的波速 混淆。 的使用是有历史渊源的，作者认为没有足够的特权去改变它）。
The separated Wave Equation in OS Coordinates is OS 坐标中的分离波方程为
and
For brevity, we have already simplified these equations by assuming axi-symmetric wave propagation around the waveguide and set the value of ( is the traditional constant for the coordinate and found in most texts on OS and PS wave functions). All of the published information on the OS wave functions assumes periodicity in , which is a different boundary condition than what we require here. We must apply a zero velocity (zero gradient) boundary condition at the walls just as we did in the previous section. Unfortunately, none of the published tables and subroutines which are readily available can be applied to our problem. 为了简洁起见，我们已经简化了这些方程，假定波导周围的波传播是轴对称的，并设置了 的值 ( 是 坐标的传统常数，可在大多数有关 OS 和 PS 波函数的文章中找到)。所有出版的 OS 波函数资料都假定 具有周期性，这与我们这里所要求的边界条件不同。我们必须在壁面上应用 速度为零（梯度为零）的边界条件，就像上一节所做的那样。遗憾的是，已出版的表格和子程序都无法应用于我们的问题。
We will need to revise the techniques used in the published literature and apply them to the specific boundary conditions for our particular problem. To do this, we will use a differential equation solution technique known as "shooting." 我们需要修改已发表文献中使用的技术，并将其应用于我们特定问题的具体边界条件。为此，我们将使用一种称为 "射击 "的微分方程求解技术。
We must first calculate the Eigenvalues for the boundary conditions of our problem. These boundary conditions are 我们必须首先计算问题边界条件的特征值 。这些边界条件是
and
The first condition allows us to only consider functions which are symmetric about . This means that only even values of will be considered. By starting at and we "shoot" to the point , where is the design angle of the waveguide, and enforce a boundary condition on the slope of these functions to be zero at that point. A point of clarification here: the design angle is not necessarily the "coverage" angle of the device. 第一个条件允许我们只考虑关于 对称的函数。这意味着只考虑 的偶数值。从 和 开始，我们 "射击 "到 点，其中 是波导的设计角，并在该点强制执行这些函数斜率为零的边界条件。这里有一点需要说明：设计角并不一定是设备的 "覆盖 "角。 This issue will be investigated in more detail later on, but for now, it is important to note that here refers to the physical angle of the walls of the waveguide. 稍后我们将对这一问题进行更详细的研究，但现在需要注意的是，这里 指的是波导壁的物理角度。
The Eigenvalues are known for , from the spherical case, and they are . It is also known that the Eigenvalues will decrease as increases at a constant rate. Using these approximations to the Eigenvalues as starting values, we calculate the exact Eigenvalues for any given value of by "shooting" from one boundary to the other. The Eigenvalue is adjusted until a satisfactory match has been achieved between the boundary conditions at the two end points. Once we have the Eigenvalue we simply use standard numerical integration to compile the angular functions (where ). 根据球面情况， 的特征值是已知的，它们是 。此外，我们还知道，随着 以恒定速率增加，特征值也会减小。利用这些特征值的近似值作为起始值，我们通过从一个边界到另一个边界的 "射击"，计算出 的任何给定值的精确特征值。对特征值进行调整，直到两个端点的边界条件达到令人满意的匹配为止。得到特征值后，我们只需使用标准数值积分来编制角度函数 （其中 ）。
Using the Eigenvalues calculated from the above calculations we can also numerically integrate the radial functions starting at with a slope of zero and an arbitrary value. It can be shown that this will give the correct form for the radial functions, but not the correct scaling. The radial functions must be scaled to match the Spherical Hankel Functions for large , because the magnitudes of both functions must asymptotically approach each other at large distances from the source. This process is not too difficult for us to contemplate, but it is a heck of a lot of work for the computer! 利用上述计算得出的特征值，我们还可以对径向函数进行数值积分，从 开始，斜率为零，取任意值。可以看出，这将给出径向函数的正确形式，但不是正确的缩放。必须对径向函数进行缩放，以便在 时与球面汉克尔函数相匹配，因为在距离源很远的地方，这两个函数的大小必须渐近地相互接近。这个过程对我们来说并不难，但对计算机来说却是一项艰巨的工作！
Fig. 6-5 shows the Oblate Spheroidal angular wave functions at for a waveguide. These functions are similar to the corresponding wave functions for a Spherical waveguide due to the relatively low frequency (c value). 图 6-5 显示了 波导在 处的 Oblate Spheroidal 角波函数。由于频率（c 值）相对较低，这些函数与球面波导的相应波函数相似。 This similarity between the functions supports our contention that at low frequencies nearly all shapes of waveguides with similar flare rates act just about the same, the flare rate being the only aspect of importance - i.e. it sets the location of the low- 函数之间的这种相似性支持了我们的论点，即在低频下，几乎所有具有相似耀斑率的波导形状的作用都差不多，耀斑率是唯一重要的方面，即它决定了低频的位置。
Figure 6-5- Angular wave functions for case, 图 6-5- 情况下的角波函数、
est frequency of significant transmission usually called cutoff. Of note in this figure is the fact that the lowest order mode, which is always independent of angle for a Spherical waveguide, is beginning to become curved with respect to in the OS waveguide. This means that even if we drive an OS waveguide with a flat piston, one which perfectly matches the aperture requirements for this waveguide, it will still generate higher order modes. This effect becomes greater as the frequency increases. 在球面波导中，最低阶模式始终与角度无关，而在 OS 波导中，最低阶模式开始变得弯曲。图中值得注意的是，对于球形波导来说，最低阶模式总是与角度无关，而在 OS 波导中，最低阶模式开始变得相对于 而弯曲。这意味着，即使我们用完全符合该波导孔径要求的扁平活塞驱动 OS 波导，它仍会产生高阶模式。频率越高，这种影响越大。
Fig. 6-6 shows the OS coordinate waveguide radial functions for both the real and imaginary parts for the case at . (The Spherical radial wave function - Spherical Hankel Function of the second kind - is also shown in these plots for reference.) 图 6-6 显示了 情况下 的 OS 坐标波导径向函数的实部和虚部（球面径向波函数--第二类球面汉克尔函数--也显示在这些图中，以供参考）。
The imaginary parts of the radial wave functions can be very difficult to develop owing to the near singularity at the origin. 由于原点处的奇异性，径向波函数的虚部可能很难展开。 The slope of these functions is known (from the Wronskian as shown below) but the values at the origin, which yields the proper asymptotic scaling, must be determined. 这些函数的斜率是已知的（如下图所示，来自弗伦斯基函数），但必须确定原点处的值，这样才能得到适当的渐近比例。 Convergence of these functions requires a very high degree of accuracy in finding this value at the origin - about 15 significant digits for the mode at . This makes it almost impossible to calculate these functions on a computer using standard iterative techniques for small values of at the higher modal orders. The imaginary parts of the wave functions are usually required in order to calculate the modal radiation impedances. We will see a way around this situation shortly. 这些函数的收敛需要非常高的精度，才能找到原点处的这个值--对于 的模态 ，需要约 15 个有效位数。因此，对于模态阶数较高的小值 ，几乎不可能在计算机上使用标准迭代技术计算这些函数。为了计算模态辐射阻抗，通常需要波函数的虚部。我们很快就会看到解决这种情况的方法。
Figure 6-6 - The OS radial wavefunctions for a waveguide at 图 6-6 - 波导在以下位置的 OS 径向波函数
Fig.6-7 shows the angular wave functions for the same waveguide as in the previous figure but at a value of . Here we see that lowest order wave function is becoming even more curved relative to the flat aperture at the throat. This means that there will be a significant amount of the mode present when this aperture is driven by a flat wavefront. 图 6-7 显示了与上图相同波导的角波函数，但波导值为 。在这里我们可以看到，相对于喉部的平面孔径，最低阶波函数变得更加弯曲。这意味着，当该孔径由平坦波面驱动时，将存在大量的 模式。
Figure 6-7 - Angular wave functions for OS waveguide at 图 6-7 - OS 波导的角波函数，位于
Fig. 6-8 shows the radial wave functions for the case at . 图 6-8 显示了 情况下的径向波函数， 。
We can now compute the modal impedances for the radiation modes as follows. 现在我们可以计算出辐射模式的模态阻抗如下。
Calculate the radial wave functions finding the value required at the origin to yield the correct asymptotic values at high . 计算径向波函数，找出原点所需的值，以便在高 时得出正确的渐近值。
Note that the Wronskian (a characteristic of all PDE's) for the OS Coordinates is 请注意，OS 坐标的 Wronskian（所有 PDE 的特征）为
which when evaluated at yields 对 进行求值，得到
Using this know slope for the radial wave function of the second kind at the origin, we can use ordinary integration to calculate the function to some large value of . Once again we compare the magnitude of this function to that of the Spherical Bessel Functions and iterate the 利用原点处第二类径向波函数的已知斜率，我们可以使用普通积分来计算函数到 的某个大值。我们再次将该函数的大小与球面贝塞尔函数的大小进行比较，并迭代出
Figure 6-8 - The radial wave functions for a waveguide at 图 6-8 - 波导在以下位置的径向波函数
starting value until the two functions match amplitudes at these higher radial values. 直到两个函数在这些较高径向值上的振幅相匹配。
This is a tremendous amount of work, and the results are known . We will not elaborate on the details of these calculations, but we will show the results. Fig. 6-9 shows the modal impedances of the waveguide for the first two modes. The third mode would not appear on this graph's scale. Thus only the first two modes are of significance for this waveguide over the bandwidth of interest. The second mode, , enters the picture at a value of , corresponding to a frequency of, (for a one-inch radius throat) 这是一项艰巨的工作，其结果已在 上公布。我们将不再详述这些计算的细节，但会展示计算结果。图 6-9 显示了 波导前两种模式的模态阻抗。第三个模式 不会出现在该图的比例尺上。因此，在所关注的带宽范围内，只有前两种模式对该波导具有重要意义。第二种模式 的值为 ，对应的频率为（对于半径为一英寸的喉管）。
Above this value, the second order mode will propagate with an equal or greater amount of influence than the "plane" wave mode, . In the vicinity of , this waveguide will becoming heavily dependent on the specific configuration of the components that control the wavefront at the throat (phase plug, etc.) Below about the wavefront geometry at the throat aperture is of little importance since only the lowest order mode - basically the average of the veloc- 在此值以上，二阶模式的传播影响将等于或大于 "平面 "波模式， 。在 附近，这种波导将在很大程度上取决于控制喉部波面的部件（相位塞等）的具体配置。在 以下，喉部孔径处的波面几何形状并不重要，因为只有最低阶的模式--基本上是速度--的平均值--才是最重要的。
See Geddes, "Acoustic Waveguide Theory Revisited", JAES 见 Geddes，"声波导理论再探"，JAES
ity of wavefront across the aperture - will propagate. At low frequencies, the details of the throat wavefront are irrelevant. 在低频情况下，喉管波面的细节并不重要。在低频情况下，喉管波面的细节无关紧要。
The mode in the OS waveguide exhibits an impedance characteristic which is similar to that for a simple Spherical waveguide or conical horn. 模式在 OS 波导中的阻抗特性与简单球形波导或锥形喇叭的阻抗特性相似。 We must be careful in this comparison however, because even though the impedance and transfer characteristics for the OS waveguide are similar to those for a Spherical waveguide there are still significant differences. 不过，我们在进行比较时必须谨慎，因为尽管 OS 波导的阻抗和传输特性与球形波导相似，但两者之间仍存在显著差异。
Fig. 6-10 shows the velocity distribution at the mouth for the waveguide. (These are the velocity amplitudes normal to the spherical surface defined by the mouth.) These velocities are dependent on both the frequency and the angle . Note that the velocity gets greater at the center, and that this effect increases rapidly with frequency after about where the second mode is becoming significant. This velocity distribution calculation at the mouth is one of the most important distinctions between waveguide theory and horn theory. 图 6-10 显示了 波导口处的速度分布。(这些是波导口球面法线方向的速度振幅）。这些速度取决于频率和角度 。请注意，中心位置的速度会变大，而且在 之后，这种影响会随着频率的增加而迅速增大，此时第二种模式会变得非常重要。口部的速度分布计算是波导理论与喇叭理论之间最重要的区别之一。 Waveguide theory predicts a significant variation of the wavefront amplitudes across the mouth of the device even when driven by a uniform velocity distribution at the throat. Horn theory can only predict amplitudes which are independent of angle, which is clearly incorrect. 根据波导理论的预测，即使在喉管处由均匀的速度分布驱动，整个装置口的波面振幅也会有明显的变化。喇叭口理论只能预测与角度无关的振幅，这显然是不正确的。
Consider now a waveguide. Several things happen when we increase the waveguide coverage angle. First, the modes cut in at a lower value of as shown in Fig. 6-11. The second mode is now significant, above about and we can see that the third mode will be a factor in the passband of the device. The 现在考虑一个 波导。当我们增大波导覆盖角时，会发生几种情况。首先，如图 6-11 所示，模式会在 的较低值处截止。现在，第二种模式在 以上非常重要，我们可以看到，第三种模式 将成为器件通带的一个因素。图 6-11
Figure 6-11 - The impedances for the waveguide 图 6-11 - 波导的阻抗
next aspect of the angle increase is that the wave functions vary in to a greater extent with the larger angle. All of these effects add up to cause an even greater focusing of the wavefront velocities towards the center of the mouth. We have not yet shown whether this is good or bad, but it is important to note the effect. 角度增大的另一个方面是，随着角度的增大，波函数在 上的变化程度也会增大。所有这些影响加在一起，会使波面速度更加集中于口腔中心。我们还没有证明这是好是坏，但注意到这种影响是很重要的。
If instead of driving the throat with a wavefront of constant amplitude we taper this amplitude as shown in Fig. 如果我们不使用恒定振幅的波阵面来驱动喉管，而是将其振幅变细，如图所示。 6-12, then the net effect will be to create a distribution of the velocity at the mouth which has a far more uniform distribution than one fed with a flat throat velocity distribution . This is an important result, for it means that better control of the sound radiation coverage of a waveguide can be achieved by manipulating the velocity distribution at the throat. Horn theory could never have predicted this result. 6-12，那么净效果将是在口部产生一个速度分布，其分布比一个平坦的喉部速度分布 要均匀得多。这是一个重要的结果，因为它意味着通过操纵喉部的速度分布，可以更好地控制波导的声辐射覆盖范围。喇叭理论从未预测到这一结果。 Phasing plugs in compression drivers in common use today are principally designed to create a flat velocity distribution, because horn theory did not have the sophistication to consider anything else. 目前常用的压缩驱动器中的相位塞主要是为了产生平坦的速度分布，因为当时的号角理论还不成熟，无法考虑其他因素。 The implications of this result to the phasing plug design is that, in essence, it must be part of the waveguide design and not part of the compression driver design. 这一结果对相位塞设计的影响是，从本质上讲，它必须是波导设计的一部分，而不是压缩驱动器设计的一部分。 In the future phasing plugs will certainly be made to better adapt the device driving the waveguide to the requirements of the waveguide itself. The phasing plug is a variable in the design problem, not a fixed component. 未来，相位调节插头的制造必将使波导驱动装置更好地适应波导本身的要求。相位调节插头是设计问题中的一个变量，而不是一个固定部件。
At this point, it would be a good idea to review the key aspects of the waveguide theory developed in this chapter: 在此，我们不妨回顾一下本章所阐述的波导理论的主要方面：
See Geddes, "Acoustic Waveguide Theory - Revisited", JAES. 见 Geddes，"Acoustic Waveguide Theory - Revisited"，JAES。
Figure 6-12 - Proposed throat velocity distribution for a flatter velocity distribution at the mouth 图 6-12 - 建议的喉部速度分布，使河口处的速度分布更平缓
All waveguides (as well as horns) have higher order modes. The fact that horn theory neither predicts nor is able to deal with this situation is a serious failing of the theory. 所有波导（以及喇叭）都有高阶模式。喇叭理论既不能预测也不能处理这种情况，这是理论的严重缺陷。
The wavefront geometry (magnitude and phase distribution) at the throat of the waveguide is critical to its performance at higher frequencies. 波导喉部的波面几何形状（幅度和相位分布）对波导在较高频率下的性能至关重要。
The loading aspects of nearly all waveguide/horn devices is, for all practical purposes, the same. The total encompassed solid angle of radiation is really the only factor influencing the loading. 实际上，几乎所有波导/喇叭装置的负载都是一样的。总的辐射实体角是影响负载的唯一因素。
Horn theory is adequate only for the low frequency aspects of waveguides - well below the first mode cut-in and even then it gives no indication as to what the wavefront shape is at the mouth. 喇叭口理论仅适用于波导的低频方面--远低于第一模切入点，即便如此，它也无法说明波导口处的波面形状。
We will now return to the discussion that we had in Sec. 6.2 about an approximate method for the evaluation of a waveguide which does not conform to a separable coordinate system and yet retains those features from waveguide theory that are necessary for acceptable results. 现在，我们将回到第 6.2 节中关于波导评估近似方法的讨论，该方法不符合可分离坐标系，但保留了波导理论中可接受结果所必需的特征。
The question is: can we find a way to do an approximate numerical calculation while still retaining the main features of waveguide theory? 问题是：我们能否找到一种既能进行近似数值计算，又能保留波导理论主要特征的方法？ Clearly, any new technique must include the possibility for higher order modes, and it must be able to predict the actual wavefront distributions at the output (mouth) for any given distribution at the input (throat). 显然，任何新技术都必须包括高阶模式的可能性，而且必须能够预测输入端（喉管）的任何给定分布在输出端（口）的实际波面分布。
The way to do this is a modification of the obvious technique of breaking a waveguide down into a series of finite spherical sections. 这种方法是对将波导分解成一系列有限球形截面的显而易见的技术进行修改。 This is an old technique but we will add one new feature - we will track all of the modes, including the higher order ones, as they progress through the elements. 这是一项老技术，但我们将增加一项新功能--我们将跟踪所有模式，包括高阶模式，因为它们在元素中不断变化。 We will only develop and outline this technique because time and space constraints will not allow us to show an example of its application. To do a thorough study of this technique and its implications would require far too much space. 由于时间和篇幅所限，我们无法举例说明这一技术的应用，因此我们将只对其进行阐 述和概述。如果要对这一技术及其影响进行深入研究，将需要太多的篇幅。 We will disclose the techniques and leave it to the reader to develop the applications. 我们将公开相关技术，并让读者自行开发应用。
Consider the geometry shown in Fig. 6-13. The waveguide is broken into four sections where each section is a section of a cone in Spherical Coordinates. Calculation of the first order mode down this waveguide is trivial. 请看图 6-13 所示的几何图形。波导被分成四段，每段都是球面坐标中锥体的一段。计算该波导的一阶模式非常简单。 It is done by simply multiplying together the T-matrices for each section to yield the composite matrix which represents the whole waveguide. 只需将每个部分的 T 矩阵相乘，即可得到代表整个波导的复合矩阵。 The problem with this approach is that it ignores the presence and propagation of higher order modes within the waveguide, just as horn theory does. 这种方法的问题在于，它忽略了波导内高阶模式的存在和传播，就像喇叭理论一样。
Figure 6-13 - A simple waveguide broken into sections and a detail of a single junction 图 6-13 - 分成几个部分的简单波导和单个结点的细节
It can be seen in the right hand side of this figure that at each junction between the sections, the wavefronts are not contiguous - the radius of these waves must changes between the two sections at each junction. 从图中右侧可以看到，在每个部分之间的交界处，波面并不连续--在每个交界处的两个部分之间，这些波的半径必须发生变化。 In order for this wavefront to propagate from one section into the next section we must match the wavefronts by creating higher order mode modes in the second section. These higher order modes are required for the wavefronts to match at the junction of the two sections. 为了让波面从一个部分传播到下一个部分，我们必须在第二个部分产生高阶模式，从而匹配波面。这些高阶模式是波面在两段交界处匹配的必要条件。 We will then have two (or more) modes which must be propagated through each of the following sections. This same situation will occur at each and every junction, thus continually increasing the higher order mode content of the wavefront. 这样，我们就会有两个（或更多）模式必须通过下面的每个部分传播。同样的情况会出现在每一个交界处，从而不断增加波面的高阶模式含量。
The only way for this higher order mode creation to not occur would be for there to be a wavefront radius source point which did not move in space, as opposed to one that is changing in each section. 不产生这种高阶模式的唯一方法是，波面半径源点在空间中不移动，而不是在每个截面中不断变化。 It is now readily apparent that any waveguide which has a changing location of the origin for the wavefront radius will require the presence of higher order modes to account for this changing origin. 现在很明显，任何波导，如果波面半径的原点位置不断变化，都需要高阶模式的存在来解释这种原点的变化。 In separable coordinates this changing radius location is exactly what the coordinate scale factors account for and exactly why the equations have become so much more complicated. This is another way of looking at the results that we elaborated on in the previous sections. 在可分离坐标中，这种半径位置的变化正是坐标比例因子所要考虑的，也正是方程变得如此复杂的原因。这也是我们在前几节中阐述的结果的另一种理解方式。
The entire concept of one-parameter (1P) waves is thus shattered by the realization that there can only ever be three waveguides, none of which are of interest to us, in which there can be true behavior and then only if we feed them with non-existent sources. All other geometries and source configurations will have higher order modes no matter how we attempt to minimize them. 因此，单参数（1P）波的整个概念都被打破了，因为我们发现只有三种波导（我们对其中的任何一种都不感兴趣）可能存在真正的 行为，而且只有当我们向它们馈入不存在的源时才会出现这种行为。所有其他几何形状和声源配置都会产生高阶模式，无论我们如何试图将其最小化。
Furthermore, we cannot circumvent this problem by making more sections and thus a smaller change between sections, resulting in less higher order mode creation at each junction. 此外，我们无法通过增加分段数来规避这一问题，因为分段数越多，分段之间的变化就越小，从而导致每个交界处产生的高阶模式越少。 The smaller values of the higher order mode components are multiplied many more times by the increased number of junctions, resulting in the exact same result that we have described above. 高阶模式成分的较小值与增加的连接点数量相乘，结果与上述结果完全相同。 There is simply no way around the conclusion that, in order to be accurate all waveguide calculations, we must include the presence of higher order modes or they are seriously flawed. 我们无法回避这样一个结论：为了保证所有波导计算的准确性，我们必须将高阶模式的存在包括在内，否则计算就会存在严重缺陷。
With this realization in mind we can be thankful that we have developed the machinery to deal with this complication, namely the T-matrix. By adding two more dimensions to the -matrices - for each higher order mode that we want to calculate - we can accommodate this new complication. We will have larger matrices to deal with ( for two modes, for three modes and so on), but we'll let the computer deal with that problem. 有鉴于此，我们可以庆幸我们已经开发出了处理这一复杂问题的机制，即 T 矩阵。通过为 - 矩阵增加两个维度--针对我们想要计算的每一个高阶模式--我们就可以适应这一新的复杂性。我们将需要处理更大的矩阵（ 用于两种模式， 用于三种模式，依此类推），但我们会让计算机来处理这个问题。
The T-matrix for the two mode spherical element is easily derived from the Spherical Wave Equations by using techniques identical to those we used in Sec.5.2 on page 95, only now we are using two modes in Spherical Coordinates. 通过使用与第 95 页第 5.2 节中相同的技术，可以很容易地从球面波方程中推导出二模球面元素的 T 矩阵，只是现在我们使用的是球面坐标中的两种模式。
Without belaboring the details in the derivation of the T-matrix for this problem, the results are 无需赘述该问题 T 矩阵的推导细节，结果如下
the modal pressure at junction 交界处的模态压力
the modal velocity at junction 交界处的模态速度
mode number 模式编号
section-junction number, section has junctions and 节交点编号，节 的交点 和
and where 以及
the Spherical Hankel Function of the first kind of order the Spherical Hankel Function of the second kind of order 第一类阶的球形汉克尔函数 第二类阶的球形汉克尔函数
the Eigenvalue for mode with the specific angle of section 模式的特征值与特定截面角度的关系
the derivative of the function with respect to its argument 函数相对于其参数的导数
the radius to the entry aperture for section 段入口孔径的半径
the radius to the exit aperture for section 段出口孔径的半径
This is a complex result - one best left to a computer to sort out. In the case of the lowest order mode it can be simplified significantly and turns out to be 这是一个复杂的结果，最好由计算机来解决。在最低阶模式的情况下，可以大大简化，结果是
From the numerical standpoint it is best to leave the matrix in the form shown in Eq. (6.5.14) above. The wave functions can be made as a single call to a subrou- 从数值的角度来看，最好将矩阵保留为上式 (6.5.14) 所示的形式。波函数的计算只需调用一次子程序即可。
tine with the correct arguments. In fact only two calls need to be made for each mode since the subroutines return four values, the values of the functions of the first and second types along with their derivatives, all of which are required. 使用正确的参数调用子程序。事实上，每种模式只需调用两次，因为子程序会返回四个值，即第一和第二类函数的值及其导数，所有这些值都是必需的。
The procedure for the application of this technique is to start at the throat, calculating the modal contributions of the wavefront found at that location in terms of the angular modes appropriate for that section. 应用该技术的程序是从喉部开始，根据适合该截面的角模计算在该位置发现的波面的模态贡献。 These modes are then propagated to the next junction using the matrices shown above, at which point the wavefront (the sum of the modal wavefronts) is again expanded in terms of the angular modes appropriate for the next section. 然后，这些模态通过上述矩阵传播到下一个交界处，此时波面（模态波面之和）将再次按照适合下一部分的角模态展开。 In order to do these calculations we will need to know the Eigenvalues as a function of the sections angle. 为了进行这些计算，我们需要知道特征值与截面角度的函数关系。
These Eigenvalues can be calculated using the same shooting technique from the previous section. Then a simple equation is fit to the data. The exact values (circles) and the fitted equation are both shown in Fig. 6-14. The equation for this fit is 这些特征值可以用上一节中的相同拍摄技术计算出来。然后对数据进行简单的方程拟合。精确值（圆圈）和拟合方程均如图 6-14 所示。拟合方程为
Figure 6-14 - Eigenvalues for a Spherical waveguide versus angle 图 6-14 - 球形波导的特征值与角度的关系
With the eigenvalues known it is an easy matter to generate the "allowed" angular functions in each section. In order to find the contribution of each mode at each junction, we must pick a surface on which we can expand both sets of functions - those in the incoming section into those in the outgoing section. 有了已知的特征值，就很容易在每个截面上生成 "允许的 "角函数 。为了找到每个模式在每个交界处的贡献，我们必须选择一个曲面，在这个曲面上我们可以将两组函数（入射部分的函数和出射部分的函数）展开。 The simplest surface on which to do this is a planar one through the junction. 要做到这一点，最简单的表面就是通过交界处的平面。
Expanding the velocity function , as shown in Fig. 6-13, at junction 展开速度函数 ，如图 6-13 所示，在交界处
where is obtained from Eq.(6.5.13). We can calculate the complex velocity distribution of a spherical wave onto the flat disk at junction . We need to slightly modify the values that we found in Sec. 4.2 on page 75 to account for the presence of the higher order modes as 其中 由公式(6.5.13)得出。我们可以计算球面波在交界处平盘上的复速度分布 。我们需要对第 75 页第 4.2 节中的数值稍作修改，以考虑到高阶模式的存在，即
the normal velocity function on the planar disk at junction 交界处平面圆盘上的法向速度函数
the length from the virtual apex of element to junction 从元素 的虚拟顶点到交界处的长度
This equation represents the complex normal velocity distribution across the waveguide at junction . 该方程表示交界处 波导上的复合法向速度分布。
We now need to expand this velocity function into the angular modes of section . The modal velocity contributions at the input to section , will then be 现在，我们需要将这一速度函数扩展到 部分的角模态中。输入 的模态速度贡献为
the angle of the nth section 第 n 节的角度
angle of section 断面角
The calculation procedure then becomes: 这样，计算程序就变成了
calculate the modal contributions at the throat 计算喉管处的模态贡献
propagate these modes to junction 1 using Eq. (6.5.13) 利用公式 (6.5.13) 将这些模式传播到交界处 1
using Eq.(6.5.17), find the angular velocity 利用公式（6.5.17），求角速度
find the normal velocity on the matching disk from Eq. (6.5.18) 根据公式 (6.5.18) 求出匹配圆盘上的法向速度
expand in terms of the new angular functions in section 1 from Eq.(6.5.19) 根据公式（6.5.19），用第 1 节中的新角函数展开
loop through all sections to end junction - the mouth 穿过所有部分，到达终点交界处--口部
It is interesting to note that the above problem can also be worked in reverse. 有趣的是，上述问题也可以反过来解决。 That is, we can start with a desired mouth wavefront and work back to the throat in order to determine what the wavefront should be at that surface in order to achieve the wavefront that we want at the mouth. It is simply a matter of taking 也就是说，我们可以从所需的口腔波面开始，回溯到喉咙，以确定该表面的波面，从而获得我们想要的口腔波面。只需将
the inverse of the matrix from the throat to the mouth. The problem is that we have not said anything about what the mouth wavefront should be in a particular situation. We have looked at the issue of analyzing a given device, but not, finding the optimum device. 的倒数。问题在于，我们并没有提到在特定情况下，口腔波阵面应该是怎样的。我们研究了分析给定设备的问题，但没有找到最佳设备。 We will find in later sections that the entire problem can be worked in reverse, namely, we can specify a desired polar response field, calculate the required mouth velocity distribution to achieve that field, and finally work backward to find a throat distribution needed to achieve the desired polar response. 我们将在后面的章节中发现，整个问题可以反过来处理，即我们可以指定一个所需的极地响应场，计算出实现该场所需的口部速度分布，最后逆向找出实现所需的极地响应所需的喉部分布。 With any luck we could design a phasing/amplitude plug for the driver to give us the throat velocity that we want. 如果运气好的话，我们可以为驱动器设计一个相位/振幅插头，以获得我们想要的喉管速度。
6.6 Treatment of Mouth Diffraction 6.6 口衍射处理
Up to this point we have not talked about the diffraction of the waveguides wavefront at the mouth of the device. There will always be a termination of the waveguide since no waveguide can be of infinite length. 到此为止，我们还没有讨论波导波面在设备口的衍射问题。由于波导的长度不可能是无限的，因此波导总会有一个终端。 We need to understand how the mouth diffraction will affect the polar response. First, however, we should understand what our target polar response needs to be in order to know what mouth velocity we will want. 我们需要了解口腔衍射将如何影响极性响应。不过，首先我们应该了解我们的目标极响应需要达到什么程度，才能知道我们需要什么样的口腔速度。 The obvious question "what polar response do we want?" will not be resolved here; there probably is not a single answer. However, we will attempt to shed some light on this topic in the chapter on room acoustics. 对于 "我们需要什么样的极性响应？"这个显而易见的问题，我们在此不做讨论；因为答案可能并不唯一。不过，我们将在 "房间声学 "一章中尝试阐明这一问题。 For now, let's just simply assume that what we want is a controllable radiation pattern of some nominal angle, say , since this is the angle that we have been using in our examples thus far. 现在，让我们简单地假设，我们想要的是某个标称角度的可控辐射模式，例如 ，因为这是我们迄今为止在示例中一直使用的角度。
It may not be obvious that we can work the polar radiation problem backwards. 我们可以倒推极地辐射问题，这一点可能并不明显。 Namely, that given a desired polar response we can use our modal radiation tools in reverse to calculate the velocity distribution required on a flat or more appropriate to a waveguide, a spherical surface. 也就是说，给定所需的极性响应后，我们可以反向使用模态辐射工具来计算平面或更适合波导的球面上所需的速度分布。 The process is to expand the desired field into its fundamental radiation modes and then to calculate (in reverse) what the amplitude and phase of these modes would have to be at the source to yield the desired radiation pattern. 这一过程是将所需的场扩展为基本辐射模式，然后（反向）计算这些模式在声源处的振幅和相位，以获得所需的辐射模式。
Knowing that this reverse problem is a transform, let's consider some characteristics of the problem that we should expect. The most important is the k-space equivalent of the well known energy-time trade-off in sound measurements. 既然这个反向问题是一个变换问题，那么我们就来考虑一下这个问题应该具有的一些特征。最重要的是，k 空间相当于声音测量中众所周知的能量-时间权衡。 The more confined, narrower, the polar response is the broader we should expect the source velocity to have to be to achieve this result. This implies that since our source is finite in size, we could easily ask for a polar pattern which cannot be achieved by the given source. 极性响应越局限、越窄，我们就应该期望声源速度越宽才能实现这一结果。这意味着，由于声源的大小是有限的，我们可以很容易地要求一个特定声源无法实现的极性模式。 We should expect a gradual rate of change of the polar response with angle if we want to achieve a smooth velocity distribution across the mouth of the waveguide and hence across the throat. 如果我们想在波导口处实现平滑的速度分布，进而在整个喉管处实现平滑的速度分布，我们就应该期望极性响应随角度的变化而逐渐变化。 This last aspect is analogous to a filter, where a sharper cutoff requires more coefficients in the filter - in essence, a "high order" filter. 最后一个方面类似于滤波器，更清晰的截止点需要更多的滤波器系数，本质上就是 "高阶 "滤波器。 High order velocity distributions mean a lot of radiation modes, but we already know that we will have three or at most four modes within the waveguide to work with and then only above the frequency of 高阶速度分布意味着大量的辐射模式，但我们已经知道，在波导内我们只能使用三种或最多四种模式，而且只能在"...... "频率以上使用。
the particular mode's cut-in/off. So clearly the best that we could hope to achieve is a fairly low order polar response, i.e. a gradual change in response with angle. 特定模式的切入/关闭。因此，我们所能达到的最佳效果显然是相当低阶的极性响应，即响应随角度的变化而逐渐变化。
As an example of the characteristics that we have been talking about, consider the unrealistic polar response where the pressure is 举例说明我们一直在讨论的特性，考虑一下不切实际的极性响应，在这种响应中，压力是
We can calculate the required radiation modes as follows. From Chap. 3 we know 我们可以计算出所需的辐射模式如下。根据第 3 章我们知道
the (unknown) modal velocity components at the spherical surface 球面上的（未知）模态速度分量
the spherical radius, basically the length of the waveguide 球面半径，基本上是波导的长度
the distance to the measurement surface 到测量面的距离
from which, using orthogonality for the Legendre Polynomials and a know function , we will find 由此，利用 Legendre 多项式的正交性和已知函数 ，我们可以求出
where 其中
If we are interested only in a far field directivity function, then the denominator terms in Eq. (6.6.21) constitute a complex constant that is the same for every mode except for a factor , which comes from the large argument approximation to the Hankel Functions. Any term which is independent of simply scales the coefficients. Since we will be looking at normalized directivity functions, these constants will therefore be unimportant. (If we want to look at the frequency response at some point then, of course, we will need to retain these terms.) 如果我们只对远场指向性函数感兴趣，那么公式 (6.6.21) 中的分母项就构成了一个复常数，该复常数对每种模式都是相同的，除了来自汉克尔函数大参数近似的系数 。任何与 无关的项都只是简单地缩放系数。由于我们要研究的是归一化指向性函数，因此这些常数并不重要。(当然，如果我们想在某个时刻查看频率响应，则需要保留这些项）。
Finally, we can simplify Eq. (6.6.22) to get 最后，我们可以简化公式 (6.6.22) 得到
Once we have these velocity modes the velocity distribution in the mouth will be 一旦我们获得了这些速度模式，口腔中的速度分布将是
Fig. 6-15 shows the calculated mouth velocity profile required to achieve the desired polar pattern. 图 6-15 显示了为实现理想的极化模式所需的计算口速度曲线。
These results need some explanation. First, they are not stable with different values of , the number of modes in the calculations. The velocity profiles become unstable - wide oscillations, as one adds more and more terms in the 这些结果需要一些解释。首先，随着 （计算中的模态数）值的不同，这些结果并不稳定。随着计算中的项数越来越多，速度曲线会变得不稳定--出现宽幅振荡。
expansion. This is understandable since the higher order terms are attempting to fit the sharp discontinuity in the polar response. Then another problem occurs for the specific case shown here in that these velocity patterns are frequency dependent. 扩展。这是可以理解的，因为高阶项是为了适应极性响应中的尖锐不连续性。然而，在本文所示的特定情况下，会出现另一个问题，即这些速度模式与频率有关。 If we are going to "sculpt" a mouth velocity we must pick one of these curves. We could attempt to match different profiles at different frequencies, but we would quickly find this unworkable. 如果我们要 "雕刻 "口腔速度，就必须选择其中的一条曲线。我们可以尝试在不同频率下匹配不同的曲线，但很快就会发现这行不通。 It is reasonable to expect to be able to create an approximately frequency independent mouth wavefront by the proper design of a waveguide and its phasing plug (or perhaps a slight frequency dependence if it follows the natural modal changes that we saw in section 6.4), but it is unreasonable to assume that we could have a specific frequency dependence in a prescribed manner, such as in the figure above. 通过波导及其相位插头的合理设计，我们有理由期望能够产生一个大致与频率无关的口波面（如果按照我们在第 6.4 节中看到的自然模态变化，或许会有轻微的频率依赖性），但假设我们能够以规定的方式（如上图所示）获得特定的频率依赖性是不合理的。
Another issue is the fact that the velocity profile goes all the way around the sphere, which is unrealizable in practice. Even if it were realizable it is still undesirable. The limits of our velocity control, namely, the mouth size for a device of a given length, will force us to terminate the velocity profile at this same angle. The method for terminating the velocity profile is critical to achieving our desired results. 另一个问题是，速度曲线会一直绕着球体旋转，这在实践中是不可能实现的。即使可以实现，也是不可取的。我们对速度控制的限制，即 特定长度装置的口部尺寸，将迫使我们在同一角度终止速度曲线。终止速度曲线的方法对于实现我们的预期结果至关重要。
We can pick a profile from Fig. 6-15, terminate it at , and recalculate a new set of modified 's. We can then use this new set of coefficients to calculate an expected polar pattern. A polar map of the response for an abruptly terminated mouth velocity is shown in Fig. 6-16. The waveguide here is a device with a length of . The mouth would be about across, in an enclosure (a sphere) in diameter, which is a fairly large device. 我们可以从图 6-15 中选取一个剖面，在 终止，然后重新计算一组新的修正 's。然后，我们就可以使用这组新系数计算出预期的极性模式。图 6-16 显示了突然终止的口速度响应的极坐标图。这里的波导是 ，长度为 。在直径为 的外壳（球体）中，口的直径约为 ，这是一个相当大的装置。
Once again we need to elaborate on some characteristics of this plot. There are 50 modes used in the summation for this pressure response. Remember that these are the modes for the sphere which is much larger than the waveguide or its mouth. 我们需要再次详细说明该图的一些特征。在压力响应的求和中使用了 50 种模式。请记住，这些是球体的模式，而球体要比波导或波导口大很多。 This means that many more modes are required for convergence of the solution. By changing the number of modes used these plots and noting where the plots change (i.e. at what frequency) we can estimate the number of modes required for accurate results at any frequency. 这意味着需要更多的模式才能使解法收敛。通过改变这些曲线图所使用的模式数，并注意曲线图的变化位置（即频率），我们可以估算出在任何频率下获得准确结果所需的模式数。 The results shown in Fig. 6-16 are estimated to be good to about . We can see a distinct change in the response at about this frequency. The peculiar response at about . is real. 图 6-16 中显示的结果估计在 左右良好。我们可以看到，在这个频率附近，响应发生了明显的变化。在 左右的特殊响应是真实的。
Figure 6-16 - The polar response map for a waveguide with an abruptly terminated mouth 图 6-16 - 带有突然终止口的波导的极响应图
Finally, these polar maps are all normalized to a level of on axis. This brings up an important point that perhaps we should wait to make until we talk about room acoustics, but we will discuss it here. 最后，这些极坐标图都以 为轴进行了归一化处理。这就引出了一个重要的问题，也许我们应该等到讨论房间声学时再讨论这个问题，但我们还是要在这里讨论一下。 It is always possible to electronically equalize the response of any loudspeaker system along any horizontal line in one of these polar maps. We almost always do this along the axial line. 任何扬声器系统都有可能沿着这些极坐标图中的任何一条水平线实现电子均衡响应。我们几乎总是沿着轴向线进行电子均衡。 It is important to realize, however, that it is impossible to electronically correct the entire polar map. Its simply cannot be done. The implication here is that electronics can only go so far in the correction of a loudspeaker's problems. The rest is up to the designer. 不过，重要的是要认识到，不可能对整个极地图进行电子校正。这根本不可能做到。这里的意思是说，电子设备在纠正扬声器问题方面只能做到这一步。其余的则取决于设计者。
Remembering the basics of the transform relationships for the velocity-polar response, perhaps it would be more logical to design to a polar pattern that does not change as abruptly with angle as the "step response" polar pattern used 考虑到速度-极性响应变换关系的基本原理，或许更合理的做法是设计一种极性模式，使其不会像 "阶跃响应 "极性模式那样随角度的变化而突然改变。
above. We will find it convenient to deal with polar response functions of the following form 以上。我们会发现，处理以下形式的极性响应函数比较方便
the rate at which the polar response falls off with angle 极性响应随角度下降的速率
A value of gives a point at about as shown in Fig. 6-17. The polar response is independent of frequency so a polar map (level versus angle and frequency) is not required. 如图 6-17 所示， ，则 点约在 。极性响应与频率无关，因此不需要极性图（电平与角度和频率的关系）。
The calculated velocity distribution for this exponential polar response is shown in Fig. 6-18.This curve is created using 20 modes in the summation. The instability of the inverse calculation can clearly be seen in the curve. At some number of modes all of the curves become this complex. Here we can see that if the curve is not oscillatory then it will be identical to all the over curves. This is attractive since there is now no ambiguity about which curve to choose as our target distribution. 这种指数极性响应的计算速度分布如图 6-18 所示。该曲线在求和时使用了 20 个模式。从 曲线中可以明显看出反演计算的不稳定性。当模式数达到一定数量时，所有曲线都会变得如此复杂。在这里我们可以看到，如果曲线不是振荡的，那么它将与所有的过度曲线相同。这一点很有吸引力，因为现在我们在选择哪条曲线作为我们的目标分布时不会再有歧义了。 The fact that the velocity wraps completely around the source is still a problem. Otherwise this velocity distribution appears to be realizable at all a frequencies. 速度完全环绕源的事实仍然是个问题。否则，这种速度分布在所有频率下似乎都是可以实现的。
It is interesting to note that the velocity distribution in Fig. 6-18 goes well beyond the point on the sphere, which is in stark contrast to the established principle of "line of sight" directivity of a spherical source. The wide angular velocities required to yield a coverage pattern suggests the possibility of using 值得注意的是，图 6-18 中的速度分布远远超出了球面上的 点，这与既定的球面源 "视线 "指向性原则形成了鲜明对比。产生 覆盖模式所需的宽角速度表明有可能使用
Figure 6-18 - The source velocity magnitude for a smooth polar pattern 图 6-18 - 平滑极化模式的声源速度大小
a much wider waveguide angle to yield an improved narrower angle response, an option that we will be developing later. 我们稍后还将开发一种更宽的波导角，以获得更好的窄角响应。
If we terminate the waveguide at the baffle in an abrupt manner we would find that we would not get a polar response any better than what we saw in Fig. 6-16. This implies that abruptly terminating the velocity profile in the spherical surface of the source is not something that we would ever want to do. 如果我们在障板处突然终止波导，我们会发现极性响应不会比图 6-16 中看到的更好。这意味着，在声源的球面上突然终止速度曲线并不是我们想要做的事情。 There may be a better way. 也许有更好的办法。
If we flare the waveguide into the baffle by applying a large radius ("large" being something that we will define in a moment) from the body of the waveguide into the baffle, instead of an abrupt termination, then we will find that we can get a polar response much more to our liking. 如果我们从波导体到障板之间采用一个大半径（"大 "我们稍后再定义），而不是突然终止，将波导扩展到障板，那么我们就会发现极性响应会更符合我们的要求。 Unfortunately, flaring a waveguide into the baffle is not a precise thing to describe mathematically. For our purposes here, we will assume that the flare has the function of tapering the velocity found at the mouth in a gradual fashion, as shown in Fig.6-19. This plot has an abscissa that is an angle, but it can also be thought of as the radius out from the center of the mouth to the outside edge in the hypothetical spherical surface of the enclosure, i.e. 遗憾的是，将波导扩口插入障板并不是一件可以用数学方法精确描述的事情。在此，我们假定扩口的作用是使波导口处的速度逐渐变小，如图 6-19 所示。该图的横座标是一个角度，但也可以认为是从口部中心到外壳假定球面外缘的半径，即 the waveguides mouth. 波导口。
The predicted polar response map for this new velocity distribution is shown in Fig. 6-20. This response is nearly ideal with nearly constant coverage at ( points) from about and up. 这种新速度分布的预测极地响应图见图 6-20。这种响应近乎理想，在 ( 点) 上的覆盖范围几乎恒定，从约 及以上。
There are several things to note from the results to this point. First, it is possible, and reasonable, to do a waveguide design backwards by specifying the desired 从目前的结果来看，有几点值得注意。首先，反向进行波导设计是可能的，也是合理的。
Figure 6-19-Velocity distribution for flared waveguide 图 6-19-扩口波导的速度分布
Figure 6-20 - Polar map for a flared waveguide 图 6-20 - 扩口波导的极坐标图
polar response pattern, calculating the required mouth velocity; back propagating this contour through the waveguide to find the required throat velocity, and finally designing a phase plug that achieves this required throat velocity. 极性响应模式，计算所需的口部速度；通过波导反向传播该轮廓，找到所需的喉部速度，最后设计一个相位塞，以达到所需的喉部速度。 Second, for constant directivity we do not want a wavefront with a velocity contour which is independent of angle (radius), i.e. 其次，对于恒定指向性，我们不希望波面的速度轮廓与角度（半径）无关，即 flat, and finally, that waveguides should always be flared into the baffle and never left with an abrupt termination since this secondary diffraction is uncontrolled. 最后，波导应始终伸入障板，切勿突然终止，因为这种二次衍射是不可控的。
Returning now to the discussion that we started earlier regarding a larger waveguide angle possibly achieving a better polar response, consider the velocity profile shown in Fig.6-21. This corresponds to a larger angular coverage 现在回到我们之前开始的关于更大波导角可能实现更好极性响应的讨论，考虑图 6-21 所示的速度曲线。这相当于更大的角度覆盖范围
with a more gradual falloff of the velocity with angle (radius). The polar map is shown in Fig. 6-22. This polar response is virtually the ideal: at from . While it is not obvious how one would obtain the velocity profile shown in the figure it certainly seems possible. For instance one might make a waveguide which had absorptive boundaries rather than reflective ones. 速度随角度（半径）的变化而逐渐下降。极坐标图如图 6-22 所示。这种极地响应实际上是理想的： ， ，从 。虽然不清楚如何获得图中所示的速度曲线，但这似乎是可能的。例如，我们可以制作一个具有吸收边界而非反射边界的波导。 This would reduce the velocity profile at the edges relative to the center in a manner similar to that desired. We could, in fact, analyze this situation by developing wave functions in the waveguide which had an impedance at the boundary instead of a zero velocity condition. 这将以类似于所需的方式降低边缘相对于中心的速度曲线。事实上，我们可以通过在波导中建立波函数来分析这种情况，这种波函数在边界处具有阻抗，而不是零速度条件。 (Yet another interesting exercise for the reader.) (这对读者来说又是一个有趣的练习）。
While complex, the task described above is certainly not impossible. The point here is that while the response shown in Fig. 6-22 may seem unreachable it is theoretically possible. Experience has shown that what is possible can be achieved. 上述任务虽然复杂，但肯定不是不可能完成的。这里的重点是，虽然图 6-22 所示的响应看似遥不可及，但在理论上是可行的。经验表明，可能的事情是可以实现的。
Figure 6-22 - A polar pattern with smooth angular variation 图 6-22 - 具有平滑角度变化的极化模式
Compare the figures in this section with those in Chap. 4 for various uncontrolled sources. It is apparent that without some form of waveguide, one is forced to deal with uncontrollable and undesirable situations in regard to the polar response. 将本节中的数字与第 4 章中各种不可控声源的数字进行比较。显然，如果没有某种形式的波导，我们就不得不处理极性响应方面无法控制的不良情况。 We have clearly shown that this situation does not have to be accepted as a constraint of the design problem. Waveguides offer a means to control the directivity yielding almost any polar response that is desired. 我们已经清楚地表明，这种情况不必作为设计问题的制约因素。波导提供了一种控制指向性的方法，几乎可以产生所需的任何极性响应。
Waveguides do have limitations. Most notably, it is difficult to obtain wide directivity with good control. We have also not discussed how to obtain non-axisymmetric polar patterns. 波导确实有其局限性。最明显的是，很难获得良好控制的宽指向性。我们也没有讨论如何获得非轴对称极性模式。 The former problem will be discussed in later chapters when we talk about arrays and the latter problem is really not so difficult. 前一个问题将在后面几章讨论数组时讨论，而后一个问题其实并不难。 A study of the separable coordinate systems highlights the Ellipsoidal (ES) Coordinates as having an elliptical mouth and these devices would yield a non-axi-symmetric polar pattern. 对可分离坐标系的研究表明，椭球（ES）坐标系具有椭圆口，这些装置会产生非轴对称的极化模式。 Using them is really not much different than using the OS, except that the boundaries are defined by an elliptic equation which is not so widely know. 使用它们实际上与使用操作系统没有太大区别，只是边界是由一个椭圆方程定义的，而这个方程并不广为人知。 Not much would likely be gleaned from doing an analysis of these devices (so we will leave that task to the interested reader), although, as we pointed out in Chap. 2, the wavefunctions in these coordinates are not known. 尽管我们在第 2 章中指出，这些坐标中的波函数并不为人所知，但对这些装置进行分析可能并不能获得太多信息（因此我们将把这项任务留给感兴趣的读者）。 One final point here is that the ES Coordinates require an elliptical throat, which of course is easy to achieve in the phasing plug design. This is one more reason why the phasing plug design must be part of the waveguide and not the driver. 最后一点是，ES 坐标要求椭圆形喉管，这在相位插头设计中当然很容易实现。这也是为什么相位插头设计必须是波导的一部分而不是驱动器的原因之一。
6.7 Diffraction Horns 6.7 衍射喇叭
It would not be fair to leave this chapter without mentioning the horn design which has dominated the marketplace for so many years. 在结束本章之前，不提一下多年来在市场上占据主导地位的喇叭设计是不公平的。 Diffraction horns are devices which obtain their control by diffracting the wavefront and then constraining it to some confined angle via a sort of conical contour. 衍射角是一种通过衍射波阵面，然后通过某种锥形轮廓将波阵面限制在某个限定角度内来实现控制的装置。
A typical layout of a diffraction horn is shown in Fig. 6-23. This drawing shows a top view and a side view in cross section. The side view is basically the top view swung through an arc of the desired vertical polar pattern. 衍射喇叭的典型布局如图 6-23 所示。该图显示了横截面的俯视图和侧视图。侧视图基本上是俯视图通过所需的垂直极化模式的弧线旋转而成。 Many variations on this construction are possible, but this drawing shows the device in it simplest embodiment. 这种结构可以有多种变化，但本图示出的是最简单的装置。
From Chap. 4 we know that when the radiating surface is small compared to the wavelength, then the polar response is wide (recall the transform of a Rect 从第 4 章中我们可以知道，当辐射表面与波长相比很小时，极性响应就会很宽（回忆一下 Rect
Figure 6-23 - Typical layout of a diffraction horn 图 6-23 - 典型的衍射喇叭布局
function, which widens as the function narrows.). A diffraction device usually works on the horizontal and vertical axis as separate designs, thereby allowing for different patterns in the two directions. 衍射装置通常在水平轴和垂直轴上分别进行设计，从而在两个方向上形成不同的图案。）衍射装置通常在水平轴和垂直轴上分别进行设计，从而在两个方向上形成不同的图案。 The throat is initially an exponential horn owing to the fact that most compression drivers have exponential sections in their phasing plugs. 由于大多数压缩驱动器的相位塞中都有指数部分，因此喉管最初是一个指数喇叭。 It initially expands in only one dimension - usually the dimension with the narrower pattern, the other dimension remains constant at a value chosen so that it is no wider than a half wavelength at the highest frequency of control. 它最初只在一个维度上扩展--通常是图案较窄的维度，另一个维度则保持恒定值，以便在最高控制频率时，其宽度不超过半波长。
At some point this initial horn is terminated. The exact location, distance from the diaphragm, is not critical and is usually chosen as a compromise for convenient forming of the horn and to set the desired distance that we will discuss below. 在某一点上，最初的喇叭会被终止。确切的位置（与振膜的距离）并不重要，通常是为了方便形成喇叭和设定我们将在下文讨论的理想距离而折中选择的。 The diffraction slot is usually an abrupt slope discontinuity although it can also be rounded - which tends to smooth out response ripples caused by the diffraction and reflection at this junction. 衍射槽通常是一个突兀的斜坡不连续性，但也可以是圆形的--这往往会平滑该交界处的衍射和反射所产生的响应波纹。 After the diffraction, slot the device flares in both dimensions and is usually straight sided at this point (although some curvature is often found to be useful.) The angles of the sides are set to the desired coverage pattern, where the line-of-sight rule-of-thumb is the design concept. 在衍射之后，设备在两个维度上都会出现槽口，此时通常是直边（尽管通常会发现一些弯曲是有用的）。 The diffraction slot can be either curved (as shown) or flat depending on the design. This aspect makes little difference except that the reflected wave, which we will talk about later, is not as coherent in the flat slot as it would be in the curved one. 根据设计的不同，衍射槽可以是弯曲的（如图所示），也可以是扁平的。除了我们稍后将讨论的反射波在平面衍射槽中的相干性不如在曲面衍射槽中的相干性好之外，其他方面几乎没有区别。 This tends to spread the reflection ripples slightly. 这往往会使反射波纹略微扩散。
The value of basically sets the upper limit of horizontal control while will set the lower limit of this control. The same is true for the horizontal dimension where controls the highest frequency and the lower frequency. It should be apparent that the design does not allow for independent control over all of these variables. It is this latter factor and the need to compromise that leads to the wide variations seen for this design. 的值基本上设定了水平控制的上限，而 则设定了该控制的下限。水平维度也是如此， 控制最高频率， 控制最低频率。显而易见，设计不允许对所有这些变量进行独立控制。正是后一个因素和折衷的需要导致了这种设计的巨大差异。
The throat of a compression driver is usually round (although a rectangular phasing plug would allow for a greater flexibility in the design compromise) and so the initial section usually transitions from round to square in some manner. 压缩驱动器的喉管通常是圆形的（尽管矩形相位塞可以在设计折衷时提供更大的灵活性），因此初始部分通常以某种方式从圆形过渡到方形。 The final horn contours are usually flared to a certain extent because this has been found to be advantageous. Flaring into the baffle is also seen and not seen depending on the designer. 最终的喇叭轮廓通常会有一定程度的外扩，因为人们发现这样做很有好处。根据设计者的不同，在障板上也会出现或不出现喇叭口外扩的情况。
The problem with a diffraction horn is the large amount of energy reflected from the diffraction slot. There is (must be) a large impedance mismatch at this slot in order for the device to work, i.e. in order for there to be sufficient diffraction to work with. 衍射喇叭的问题在于衍射槽反射的大量能量。为了使设备工作，即为了有足够的衍射效果，该槽存在（必须存在）较大的阻抗失配。 This impedance mismatch will reflect a great deal of the incident wavefront back down the device. A standing wave results, which is evident in both the electrical impedance of the driver as well as the frequency response of the system. 这种阻抗失配会将大量入射波面反射回设备。这样就会产生驻波，这在驱动器的电阻抗和系统的频率响应中都很明显。
A second problem with the diffraction device is the ambiguity of its acoustic center. That is because there are actually two. One is at the throat of the device and the other at the diffraction slot. 衍射装置的第二个问题是其声学中心的模糊性。这是因为实际上有两个。一个位于装置的喉部，另一个位于衍射槽。 The plane in which the sound wave has been diffracted will have the diffraction slot as its acoustic center and the other plane 声波被衍射的平面将以衍射槽为声学中心，而另一个平面
will have the acoustic center at the throat. At angles not in one of these planes the acoustic center is ambiguous. This causes problems with arraying these devices, because, as we shall see, the location of the acoustic center is the crucial point in array performance. 则声学中心位于喉部。如果角度不在其中一个平面上，声学中心就会模糊不清。这就给这些设备的阵列带来了问题，因为我们将会看到，声学中心的位置是阵列性能的关键点。
While diffraction horns have served the market well, they are now basically obsolete. The wavefronts in these devices are complex and can only be analyzed with complex numerical codes like FEA. 虽然衍射喇叭在市场上发挥了很好的作用，但现在基本上已经过时了。这些设备的波面非常复杂，只能通过复杂的数值代码（如有限元分析）进行分析。 At their best, their performance can be as good as a well designed waveguide and at their worst, they can be a disaster. This, of course, is only our opinion, but we hope that the reader will recognize that the results shown in this chapter support such an opinion. 在最好的情况下，它们的性能可以与精心设计的波导一样好，而在最坏的情况下，它们可能是一场灾难。当然，这只是我们的看法，但我们希望读者能够认识到，本章所展示的结果支持这种看法。
6.8 Summary 6.8 小结
A long and complex chapter the subject matter presented here is none the less of crucial importance to the design of loudspeaker systems. We showed that we must seriously question the use of the Horn Equation for waveguide design as it is inapplicable. 本章篇幅较长，内容复杂，但对于扬声器系统的设计却至关重要。我们表明，我们必须认真质疑在波导设计中使用 "号角方程"，因为它并不适用。 However, the fact remains that some horns designed from this equation have worked well. From our results, we can see that the diffraction that occurs in an exponential horn (for example) could yield exactly the right mouth wavefront, but if it did it would be purely a coincidence! 然而，事实是，根据这一方程设计的一些喇叭效果很好。从我们的结果可以看出，指数型喇叭（例如）发生的衍射可能会产生完全正确的喇叭口波面，但如果真的发生了，那也纯属巧合！ There is no way that one could design such a successes from horn theory. 人们不可能根据喇叭理论设计出这样的成功。
We have attempted to point out that controlling the polar response of the system is possible, albeit not easy. We will see in later chapters that directivity becomes a major component in the design of loudspeaker systems once one includes the room effects in this design. 我们试图指出，控制扬声器系统的极性响应是可能的，尽管并非易事。我们将在后面的章节中看到，一旦将房间效应纳入扬声器系统的设计中，指向性将成为扬声器系统设计中的一个重要组成部分。 This is true for both the large and small venues, but for different reasons. In either case, it just does not seem reasonable to accept as fact that loudspeaker systems have to have directivity properties which are uncontrolled, or omni-directional. 大型和小型场馆都是如此，但原因不同。无论是哪种情况，将扬声器系统必须具有不受控制的指向性或全向性作为事实似乎都是不合理的。 (Omni-directional being, in our opinion, not really controlled, but simply the acceptance of the notion that nothing else is possible.) (在我们看来，全方位并不是真正意义上的控制，而仅仅是接受了 "别无他法 "的观念）。
It is true that the recent trend towards smaller and smaller loudspeakers does not allow for directivity control to any appreciable degree. 诚然，近年来扬声器的体积越来越小，无法实现明显的指向性控制。 Directivity control below the frequency where the dimensions of the enclosure become comparable to the wavelength of the sound cannot be accomplished without substantial amounts of signal processing and multiple drivers - an expensive proposition and not one that we are likely to see with wide availability in the near future. 如果没有大量的信号处理和多个驱动器，就无法实现频率以下的指向性控制，因为在这个频率下，箱体的尺寸与声音的波长相当。 While the idea that enclosures need to be large for good low frequency response is certainly true, we can see that there is now another reason for larger enclosures - the ability to control the directivity/power response of the loudspeaker system. 虽然 "为了获得良好的低频响应，箱体必须较大 "的观点是正确的，但我们现在可以看到，箱体较大还有另一个原因--能够控制扬声器系统的指向性/功率响应。 The authors hope that good sound quality demands will prevail and that we will see a return to the "substantial" sized loudspeakers of the past in the modern applications of home theatre and other applications where sound quality is a primary consideration. 作者希望良好的音质要求能够得到满足，在家庭影院和其他以音质为首要考虑因素的现代应用中，我们将看到过去 "大 "尺寸扬声器的回归。
See Mapes-Riordan, "Horn Modeling with Conical and Cylindrical Transmission line Elements", JAES. 参见 Mapes-Riordan，"使用锥形和圆柱形传输线元件进行喇叭建模"，JAES。
See Putland “Every One-Parameter acoustic Field Obeys Webster's Horn Equation", JAES. 见 Putland "每个单参数声场都服从韦伯斯特角方程"，JAES。
