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。我们似乎无法采取任何措施来影响声辐射响应。 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方向图，外壳在最低频率下发挥着重要作用。 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然高于外壳控制响应的点，则响应几乎完全取决于驱动器的尺寸。其他的影响不大。
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性控制器件的概念。
6.1 Historical Notes 6.1 历史注释
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. 重要的是要回顾喇叭和波导理论的历史发展，以便了解其演变到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长期以来对波导和喇叭的看法，但实际上并不正确。纠正这些信念会极大地限制它们对设计中当前问题的适用性。
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，我们不知道它们第一次使用的时间和地点。号角作为乐器无疑已有数百年的历史。 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. 随着留声机的出现，人们发现喇叭在放大手写笔的微小机械运动发出的声音方面发挥着至关重要的作用。喇叭几乎负责系统中的所有增益。 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，它的用途主要是一种负载或阻抗匹配机制，需要更好地将触笔的高机械阻抗与介质（空气）的非常低的机械阻抗相匹配。 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. 当人们对导管的负载特性感兴趣时，他们只需要关心声学变量在隔膜上以及因此在导管上的平均分布。 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所谓的韦伯斯特霍恩方程时所做的假设
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. 人们普遍认为该方程仅适用于平面波，因为韦伯斯特在其推导中使用了平面波假设。然而，这个方程比单独的平面波更广泛地适用。 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，对于感兴趣坐标的比例因子为 1 的任何几何体来说，它都是精确的。比例因子是所有坐标系的基本参数，如 Morse 所示。 （具有比例因子是可分离性的要求。）比例因子是已知的，并且只能针对可分离坐标系进行计算。值得注意的是，任何三个维度具有统一比例因子的坐标系在韦氏公式中都是精确的。 Specifically these coordinates are: 具体来说，这些坐标是：
all three Cartesian Coordinates 所有三个笛卡尔坐标
the axial coordinate in all Cylindrical Coordinate systems (Elliptic, Parabolic and Circular) 所有圆柱坐标系（椭圆、抛物线和圆形）中的轴坐标
the radial coordinate in the Cylindrical coordinate system 圆柱坐标系中的径向坐标
the radial coordinate in Spherical Coordinates 球坐标中的径向坐标
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（六个坐标）适用于恒定横截面的导管，这对我们来说不是很有趣。有用的是圆柱、球面和圆锥坐标系的三个径向坐标。请注意另外两项非常重要。首先，只有三个有用的坐标可以使韦伯斯特方程精确；其次，所有这三个在其他正交坐标中都有波传播。 The importance of this last attribute will become clear later on. 最后一个属性的重要性稍后将变得清晰。
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. 如果韦伯斯特方程仅在三种有用的情况下正确，那么为什么有这么多关于它的使用的文献呢？这是因为该方程仍然可以作为任何不同横截面的管道的近似值。 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. 在应用韦氏方程的几乎所有常见情况下，它都被用作喇叭形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. 还有大量关于应用韦伯斯特方程来评估这些近似解的文献。 Almost nothing has been written about when these approximations are "good" approximations and when we should be suspect of their validity. 几乎没有任何文章讨论这些近似值何时是“好的”近似值以及何时我们应该怀疑它们的有效性。
The only place where we have seen such a discussion is, once again, in Morse where they state: 我们唯一看到这样的讨论的地方是莫尔斯电码 ，其中他们指出：
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). 压力和流体速度均服从该修正的波动方程，该方程大致考虑了横截面尺寸随 的变化。只要 与 变化率的大小远小于 1（只要管缓慢“耀斑”），该方程就是一个很好的近似值。
See Morse, Methods of Theoretical Physics 参见莫尔斯，理论物理方法
See Morse, Methods of Theoretical Physics, pg. 1352 参见莫尔斯，《理论物理方法》，第 17 页。第1352章
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. 这些都是非常有限的条件，几乎被普遍忽视。让我们看看它们对众所周知的指数号角的发展意味着什么。
Restating Morse, as applied to an exponential horn, let 重述莫尔斯电码，应用于指数号角，让
we can see that, for this example, Webster's equation is "good" only so long as 我们可以看到，对于这个例子，韦伯斯特方程只有在满足以下条件时才是“好的”
the throat area 喉咙区域
the flare rate 耀斑率
The horn contour for this example is shown in Fig. 6-1. This figure shows an 本例的喇叭轮廓如图 6-1 所示。该图显示了一个
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，我们绘制了方程（6.1.3）的值作为距喉部轴向距离的函数。 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，当喇叭长度超过 时，显然违反了将韦氏方程准确应用于指数喇叭的假设（莫尔斯说远小于 1.0，但少多少是一个问题）选择。我们在这里将使用 0.5 的值视为精度的极限，在 1.0 的值下，假设完全无效。）
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，我们显示了一条与号角轮廓相切的线，该线起始于喉部中心，即喉部波前的声学中心。 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的经验法则是，任何超过该切点的轮廓都无法通过韦伯斯特方程准确描述。这个经验法则意味着对韦伯斯特方程的局限性的几何解释。 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. 一旦要求波前绕着设备上的一个点衍射，该设备将后退边界置于原始波前声学中心的阴影区域中，喇叭方程就无法预测波前。如图。 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. 图6-1表明，唯一能用韦氏方程精确表示的指数喇叭是极短的。这种长度的号角没有任何实际意义。
Further support for our "rule of thumb" comes from considering Huygens' principle and the construction of Huygen wavefronts. 对我们的“经验法则”的进一步支持来自考虑惠更斯原理和惠更斯波前的构造。 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，波前如果要保持垂直于导管的侧面（因为它们必须如此），则必须具有位于设备实际喉部前方的明显声学中心。 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. 惠更斯原理允许非衍射波前比来自声学中心的波前更平坦，但不允许波前曲率小于到声学中心的半径。稍微思考一下就会明白为什么这一定是真的。 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. 换句话说，我们的经验法则是：曲率小于喉部半径的喇叭波前一定在沿着设备的行程中的某个地方发生了衍射 - 衍射创建了一个新的声学中心，波前从该中心出现。
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，在韦氏方程精确的情况下，喇叭轮廓上永远不会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. 区域，即我们的“经验法则”永远不会被违反。号角理论的绝大多数工作都面临着其有效性的严重问题。
How one gets around this problem leads us into another line of reasoning for which there are two paths. 如何解决这个问题将我们引向另一条推理路线，其中有两条路径。 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. 我们可以将上述轮廓的指数部分连接到从声学中心线的切点继续延伸的球形部分，从而确保我们的经验法则永远不会被违反。事实上，这确实工作得相当好，并且很常见。 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，事实是指数部分仍然只是一个近似值，我们确实不知道连接点处波前的实际形状。正如我们稍后将看到的，这是对连接方法的严重限制。
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. 现在我们已经表明，喇叭方程在解决波导中的重要问题时具有严重的局限性，我们将讨论这些限制如何影响使用它的预期结果。 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 章中所示，如果我们知道波前穿过我们有辐射解（平面、球形或圆柱形）的边界时的波形，那么我们就可以相当准确地预测波前的方向性。这个来源。 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. 然而，为了进行此计算，我们必须知道孔径中每个点的波前的精确幅度和相位。从上面的讨论来看，我们应该对式（1）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，这些信息是准确的。 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? 自然要问的问题是：我们能否开发一个方程或方法，使我们能够在一定程度上确定波导内每个点的幅度和相位？ 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的代价是方程及其解的复杂性大幅增加。
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. 正如我们上面所讨论的，喇叭的早期使用与我们在这里尝试开发的有很大不同。早期对喇叭的需求是作为声学负载设备，我们的兴趣是控制源方向性。 (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. （随着当今几乎无限的放大器功率能力，负载基本上不再是问题。）因此，我们将采用这样的术语：喇叭是一种利用韦伯斯特方程及其计算方法开发的设备（其中，仅需要平均波前形状和声学负载），波导是一种主要用途是控制方向性的设备。 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. 波导设计遵循我们将在以下部分中开发的路线，即使用韦伯斯特方程的喇叭设计。波导的声学载荷通常可以轻松计算，但并非总是如此。 However, since loading is not a central concern for us this limitation is not significant. 然而，由于加载不是我们关心的中心问题，所以这个限制并不重要。
See Geddes "Waveguide Theory" and "Waveguide Theory Revisited", JAES. 参见 Geddes“波导理论”和“波导理论重访”，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 从前面的章节中我们知道，只要我们能够应用适当的边界条件，波动方程总是准确的。这只能发生在一组有限的坐标系中的一个中。以球形管道的简单情况为例，如图 6-2 所示。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. 这里的导管在 中是对称的，并且 速度在某些 处为零。现在我们不必担心该管道沿其轴的终端，并简单地假设它是半无限的。换句话说，它的喉咙是有限的，但嘴巴却是无限的。
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. 我们可以使用完整的波动方程或方程。 （6.1.1）在这个例子中是因为它的简单性。我们已经讨论了喇叭方法，所以让我们使用我们在第 3 章中开发的完整波动方程。我们知道以下解同时适用于球坐标系和韦氏方程。
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. 从这个解决方案中我们可以立即看出，波动方程方法的一个方面是韦氏方法中所没有的。也就是说，我们从波动方程知道波数 是 和 中另外两个方程的耦合常数。我们可以排除与 的耦合，因为边界条件没有 变化，但我们不能简单地假设 依赖性，那么当波沿设备传播时，波前也会存在 依赖性。这与韦伯斯特的方法明显不同，因为韦伯斯特方程不允许任何 依赖性。实际上，这种限制比我们上面讨论的耀斑率引起的误差要严重得多。
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，考虑分别使用韦氏方程和声波导方法来评估“圆锥形”喇叭与球形波导（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 霍恩理论在喉部产生阻抗，但它没有产生有关设备内任何位置的波前的幅度和相位的信息。假设振幅均匀
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. 该设备意味着韦伯斯特的方法预测与喇叭边界正交的某个（未知）表面上的每个点都有相同的值。当然，我们可以认为（在某些情况下是正确的）波前在每个点都必须是球形截面。 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. 但如果该设备在喉咙处被平面波馈送，会发生什么呢？使用韦伯斯特提供给我们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 现在考虑替代波导方法。来自 Sec. 3.4 第49页我们知道径向坐标的解是
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 如果驱动波前是球形的，则只有单模 将被激发。在这种情况下，对于霍恩方程或波动方程，我们会得到沿设备传播的波的相同答案
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. 其中 ，因为在这种情况下它始终处于 方向。然而，我们也知道 存在无数其他可能性。只有波动方程方法在其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. 当喉部由与喉部相同半径的球形部分不重合的波前驱动时，则波前可以在沿着装置传播之前扩展成一系列可容许的波前。 Since we have solutions for all of these waves, we can develop the final solution as a sum over these various wave orders. 由于我们对所有这些波都有解，因此我们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 考虑喉部的平面波激励。众所周知，平面波可以展开为无限系列的球面波。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，勒让德多项式与左侧项给出的权重因子形成展开式。等式。 (6.2.7) 适用于平面压力波——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. 如果我们在喉部有一个平面速度源（即扁平活塞），那么我们必须将其与喉部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. 这实际上并不太困难，只是这种方法还有另一个问题，所以我们将把这个讨论留到以后再解决我们当前问题的解决方案。
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，勒让德多项式，如方程所示。 (6.2.7)，不符合我们波导的边界条件，即它们不具有零斜率
See Morse, Any text 查看莫尔斯电码，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. 在波导的壁上， 对于所有 。正常的勒让德多项式必须具有一个分离常数 ，由于周期性，它是一个整数。新函数不再要求该常量为整数。 （为什么？）比较图 6-3 中的两个图。该图的右侧显示了 弧上的正常勒让德多项式。左侧显示了满足波导壁边界条件的修改勒让德多项式（由于新的分离常数 而修改）。新的多项式可用于扩展喉部的任何轴对称源 - 它们形成一个完整的正交集。值得注意的是图 6-3 与第 73 页的图 4-3 的相似之处。
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，通过利用波动方程的完整机制，我们可以匹配位于波导喉部的任何速度分布，只要该波导位于可分离坐标系之一的坐标表面上（尽管我们有但只看了一个很简单的）。 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，这种波前匹配不能通过使用韦氏喇叭方程来完成；该表述中不存在这样做的机制。 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，我们可以强制喉部波前与喇叭的最低阶模式相匹配，然后喇叭方程实际上就会是准确的。
Figure 6-3 - Normal Legendre Polynomials (right) for waveguide (left) 图 6-3 - 波导（左）的正态勒让德多项式（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! 问题在于，没有任何源的速度分布与具有精确喇叭解决方案的三个几何形状中的任何一个相匹配！
In order to continue we have a choice of three alternatives: 为了继续，我们可以选择三种替代方案：
accurately use the horn equation with unrealistic sources 准确地使用喇叭方程与不切实际的来源
use approximate solutions for real sources, or 使用真实源的近似解，或者
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 获得实际来源的精确解决方案，但仅限于使用一些可以进行精确分析的规定几何图形（可分离的坐标系）
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. 我们对第一个选择不感兴趣。第二个选择可能是可行的，我们稍后会研究该替代方案，但现在我们将选择第三个选项以获得准确的答案，稍后我们可以将其用作与近似解决方案的比较。 We will also get a better understanding of the nature of the exact solution. 我们还将更好地理解精确解的本质。
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源将具有不止一种传播模式，因为该源需要与喉部的波导配合。 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. 通过在一系列修改的勒让德多项式中展开源速度，我们可以确定波导的各种模式的贡献。我们在第 2 节中看到了一个这样的例子。 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. 第 83 页的 4.5，我们根据一组平面孔径模式扩展了球面波前。我们现在正在做相反的事情，即根据一组有限角球面模式来扩展孔径中的平面波。 The two processes are completely analogous albeit reversed. 这两个过程完全相似，尽管方向相反。
Each of the waveguide modes will propagate with a different phase and amplitude which can readily be calculated as 每个波导模式将以不同的相位和幅度传播，可以很容易地计算为
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. 需要针对每个波导专门确定特征值 ，因为它们随壁的角度而变化。函数 与我们之前见过的球形 Hankel 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，尽管详细信息超出了本文的范围并在其他地方进行了介绍 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 节中看到的，模式辐射（在当前情况下传播）的效率随频率而变化。图 6-4 显示了 球形波导中前三个模式的模态阻抗。从这张图中，我们可以准确地看到喇叭理论和波导理论的差异。
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. 霍恩理论仅产生该图中所示的实线，这与该最低阶模式的波导计算完全相同。在 之上（其中 是波导喉部的半径），第一模式开始切入。
5. See Zhang, Computation of Special Functions 5.参见张，特殊函数的计算
Figure 6-4 - Real and imaginary parts of the modal impedances for a 图 6-4 - 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). 低于该频率，韦氏喇叭理论和波导理论将产生几乎相同的结果（微小差异是由于高阶模式的有限虚部造成的）。 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. 高于该频率，第一模式（这对于驱动球形波导的活塞源来说非常重要）比零阶模式对波前具有更大的比例影响，并且很难忽略以获得准确的结果。 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. 正是在这个区域（第一模态的切入点上方），号角理论存在严重缺陷。随着频率的升高以及更多模式的介入，其有效性逐渐变差。波导理论的精确度与计算其模式的值一样高。 This is a significant difference in accuracy for a large directivity controlling device. 对于大型方向性控制装置来说，这是精度上的显着差异。
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. 现在我们已经了解了为什么波导理论在高频方向性控制装置中比喇叭理论更可取，我们将研究波导理论直接适用的各种可分离坐标系，以确定哪些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. 我们已经讨论了几个可分离坐标系，但下面是完整的 11 个坐标系以及纯零阶模式喉部所需的源类型的表格。
Name
Coordinate
Source
Aperture
Source
Curvature
Mouth
Curvature
Rectangular
any
rectangle
flat
flat
Circular Cylinder
radial
rectangle
cylindrical
cylindrical
Elliptic Cylinder
radial
rectangle
Flat
cylindrical
Parabolic Cylinder
none
Spherical
radial
circular
spherical
spherical
Conical
radial
elliptical
spherical
spherical
Parabolic
none
Prolate Spheroidal
radial
rectangle
cylindrical
spherical
Oblate Spheroidal
radial
circular
flat
spherical
Ellipsoidal
radial
elliptical
flat
spherical
Paraboloidal
none
Table 6.1: Useful waveguides for Separable Coordinates 表 6.1：适用于可分离坐标的波导
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). 所有有用的坐标都是径向的，并且所有的嘴孔都与喉部孔相同（未示出）。源孔径可以是矩形或椭圆形（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. 嘴部曲率（辐射波前）只能是球形或圆柱形，平坦的则没什么意义。最后一个特征是我们在第 1 章中研究几何形状的主要原因。 4. 如果孔径是圆形的，则物理设备必须是轴对称的。 The wave propagation need not be axi-symmetric, however. We will not look into this possibility since it is rather unusual in practice. 然而，波传播不必是轴对称的。我们不会研究这种可能性，因为这在实践中相当不寻常。
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. 还可以结合波导来创建新设备。例如，长椭球 (PS) 波导在其喉部采用方柱形弯曲波前，这正是椭圆柱波导所产生的。 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. 作为这两个坐标系中的两个波导的组合而创建的波导将采用方形平面波前作为输入并产生方形球形波前。
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 有趣的是，只有当输入和输出波前曲率保持不变时，喇叭方程才是精确的。我们的意思是喉部和喉部的波前半径中心的位置
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. 嘴在空间中不动。这正是具有统一比例因子的含义。不幸的是，不具有统一比例因子的几何形状明显更难以分析——这是我们为获得更高精度的波导方法必须付出的代价。 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. 我们已经详细研究了球形波导，现在我们将研究基于扁球体 (OS) 坐标系的波导，以便将其特性与我们已经研究过的特性进行比较和对比。
6.4 The Oblate Spheroidal Waveguide 6.4 扁球面波导
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. 与前面几节一样，我们需要对 OS 波导进行的第一个计算是确定该坐标系中的波函数（或本征函数）。 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的情况不同，OS 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). 这些没有统一比例因子的坐标系的独特之处在于；即使方程在空间坐标中分离，它们仍然通过分离常数（波数或时间坐标）耦合。 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.) 迄今为止，我们在研究的任何问题中都没有遇到过这种复杂情况。径向坐标和角坐标中的波函数都取决于一个公共参数 ，其中 是焦间分离距离（见图1-第 6 页上的 2）。 （我们必须小心，不要将这个 （非斜体粗体）与同一个字母的波速 混淆。 的使用是历史性的，作者没有足够的特权来改变它。）
The separated Wave Equation in OS Coordinates is OS坐标中分离的波动方程为
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，我们已经通过假设波导周围轴对称波传播并设置 的值（ 是 的传统常数来简化这些方程） b2> 坐标并在大多数关于 OS 和 PS 波函数的文本中找到。关于 OS 波函数的所有已发布信息均假定 具有周期性，这与我们此处所需的边界条件不同。我们必须在壁上应用零 速度（零梯度）边界条件，就像我们在上一节中所做的那样。不幸的是，没有任何可用的已发布的表和子例程可以应用于我们的问题。
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特定边界条件。为此，我们将使用称为“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。这意味着仅考虑 的偶数值。从 和 开始，我们“射击”到点 ，其中 是波导的设计角度，并且强制这些函数的斜率在该点处为零的边界条件。这里需要澄清一点：设计角度不一定是设备的“覆盖”角度。 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. 稍后将更详细地研究这个问题，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 ). 在球形情况下，特征值为 ，它们是 。还知道，特征值会随着 以恒定速率增加而减小。使用这些特征值的近似值作为起始值，我们通过从一个边界“射击”到另一个边界来计算任何给定值 的精确特征值。调整特征值，直到两个端点的边界条件达到满意的匹配。一旦我们有了特征值，我们只需使用标准数值积分来编译角度函数 （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! 使用从上述计算中计算出的特征值，我们还可以对从 开始、斜率为零和任意值的径向函数进行数值积分。可以证明，这将给出径向函数的正确形式，但不是正确的缩放。径向函数必须进行缩放以匹配大 的球面汉克尔函数，因为两个函数的大小必须在距源较远的距离处渐近地彼此接近。这个过程对于我们来说并不算太难，但是对于计算机来说却是一个巨大的工作量！
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 处的扁椭球角波函数。由于频率（c 值）相对较低，这些函数类似于球形波导的相应波函数。 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，几乎所有形状的具有相似耀斑率的波导的行为几乎相同，耀斑率是唯一重要的方面 - 即它设置了低频的位置
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. 重要传输的最大频率通常称为截止。该图中值得注意的是，最低阶模式始终与球形波导的角度无关，开始相对于 OS 波导中的 变得弯曲。这意味着，即使我们用扁平活塞驱动 OS 波导（完美匹配该波导的孔径要求），它仍然会产生更高阶模式。随着频率的增加，这种影响变得更大。
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 处 情况下实部和虚部的 OS 坐标波导径向函数。 （球面径向波函数 - 第二类球面 Hankel 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. 由于原点处接近奇点，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（来自 Wronskian，如下所示），但必须确定原点处的值，该值产生适当的渐近缩放。 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. 这些函数的收敛要求在原点处找到该值的精度非常高 - 对于 处的 模式，大约有 15 个有效数字。这使得几乎不可能在计算机上使用标准迭代技术计算较高模态阶数的小值 的这些函数。为了计算模态辐射阻抗，通常需要波函数的虚部。我们很快就会找到解决这种情况的方法。
Figure 6-6 - The OS radial wavefunctions for a waveguide at 图 6-6 - 处 波导的 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 显示了与上图中相同的波导但值为 的角波函数。在这里我们看到最低阶波函数相对于喉部的平坦孔径变得更加弯曲。这意味着当该孔径由平坦波前驱动时，将存在大量的 模式。
Figure 6-7 - Angular wave functions for OS waveguide at 图 6-7 - OS 波导在 处的角波函数
Fig. 6-8 shows the radial wave functions for the case at . 图 6-8 显示了 情况下 处的径向波函数。
We can now compute the modal impedances for the radiation modes as follows. 我们现在可以计算辐射模式的模态阻抗，如下所示。
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，找到原点处所需的值，以在高 处产生正确的渐近值。
Note that the Wronskian (a characteristic of all PDE's) for the OS Coordinates is 请注意，操作系统坐标的 Wronskian（所有偏微分方程的特征）为
which when evaluated at yields 当在 处评估时，会产生
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 利用这个已知的第二类径向波函数在原点的斜率，我们可以使用普通的积分来计算该函数的某个大值 。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
Figure 6-8 - The radial wave functions for a waveguide at 图 6-8 - 波导在 处的径向波函数
starting value until the two functions match amplitudes at these higher radial values. 起始值，直到两个函数在这些较高的径向值处匹配幅度。
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) 这是一项巨大的工作量，结果是众所周知的 。我们不会详细说明这些计算的细节，但我们会展示结果。图 6-9 显示了前两种模式的 波导的模态阻抗。第三种模式 不会出现在该图的比例上。因此，只有前两种模式对于该波导在感兴趣的带宽上是重要的。第二种模式 ，以 值进入图像，对应于频率（对于一英寸半径的喉部）
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- 高于此值，二阶波模传播时的影响力将等于或大于“平面”波模 。在 附近，该波导将变得严重依赖于控制喉部波前的组件的具体配置（相位塞等）。下面关于 波前喉部孔径处的几何形状并不重要，因为只有最低阶模式 - 基本上是速度的平均值
See Geddes, "Acoustic Waveguide Theory Revisited", JAES 参见 Geddes，“声学波导理论重温”，JAES
ity of wavefront across the aperture - will propagate. At low frequencies, the details of the throat wavefront are irrelevant. 穿过孔径的波前的强度 - 将传播。在低频下，喉部波前的细节是无关紧要的。
The mode in the OS waveguide exhibits an impedance characteristic which is similar to that for a simple Spherical waveguide or conical horn. OS 波导中的 模式表现出与简单球形波导或圆锥形喇叭相似的阻抗特性。 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. 然而，我们在比较时必须小心，因为尽管 OS 波导的阻抗和传输特性与球形波导相似，但仍然存在显着差异。
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。 （这些是垂直于由嘴定义的球面的速度幅度。）这些速度取决于频率和角度 。请注意，速度在中心变得更大，并且在大约 之后，这种效应随着频率而迅速增加，此时第二模式变得显着。这种口处的速度分布计算是波导理论和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. 波导理论预测，即使在喉部均匀速度分布驱动的情况下，设备口部的波前幅度也会发生显着变化。霍恩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。当我们增加波导覆盖角度时会发生一些事情。首先，模式在 的较低值处切入，如图 6-11 所示。现在，第二个模式很重要，高于 ，我们可以看到第三个模式 将成为设备通带的一个因素。这
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. 角度增加的下一个方面是，随着角度的增大，波函数 的变化程度也更大。所有这些效应加起来导致波前速度更大地聚焦到口腔中心。我们尚未表明这是好还是坏，但重要的是要注意其影响。
If instead of driving the throat with a wavefront of constant amplitude we taper this amplitude as shown in Fig. 如果我们不是用恒定振幅的波前驱动喉部，而是逐渐减小该振幅，如图 1 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. 如图6-12所示，那么最终效果将是在口处产生一种速度分布，该速度分布比用平坦喉部速度分布 供给的速度分布均匀得多。这是一个重要的结果，因为它意味着可以通过操纵喉部的速度分布来更好地控制波导的声辐射覆盖范围。霍恩理论永远无法预测到这个结果。 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定相插头主要是为了创建平坦的速度分布而设计的，因为号角理论没有足够的复杂性来考虑其他任何事情。 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. 这一结果对定相插头设计的影响在于，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. 将来肯定会制造定相插头，以使驱动波导的设备更好地适应波导本身的要求。定相插头是设计问题中的变量，而不是固定组件。
At this point, it would be a good idea to review the key aspects of the waveguide theory developed in this chapter: 此时，最好回顾一下本章中发展的波导理论的关键方面：
See Geddes, "Acoustic Waveguide Theory - Revisited", JAES. 参见 Geddes，“声波导理论 - 重温”，JAES。
Figure 6-12 - Proposed throat velocity distribution for a flatter velocity distribution at the mouth 图 6-12 - 为实现口腔处较平坦的速度分布而建议的喉部速度分布
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）都具有高阶模式。号角理论既没有预测也没有能力处理这种情况，这一事实是该理论的严重失败。
The wavefront geometry (magnitude and phase distribution) at the throat of the waveguide is critical to its performance at higher frequencies. 波导喉部的波前几何形状（幅度和相位分布）对其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. 出于所有实际目的，几乎所有波导/喇叭设备的负载方面都是相同的。辐射的总包围立体角实际上是影响载荷的唯一因素。
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. 喇叭理论仅适用于波导的低频方面 - 远低于第一模式切入，即使如此，它也没有给出关于波前形状在口处的指示。
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. 现在我们将回到我们在第 2 节中进行的讨论。 6.2 关于评估波导的近似方法，该方法不符合可分离坐标系，但保留了波导理论中获得可接受结果所必需的那些特征。
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：我们能否找到一种方法进行近似数值计算，同时仍保留波导理论的主要特征？ 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，并且必须能够针对输入（喉部）处的任何给定分布预测输出（嘴部）处的实际波前分布。
The way to do this is a modification of the obvious technique of breaking a waveguide down into a series of finite spherical sections. 做到这一点的方法是对将波导分解成一系列有限球形部分的明显技术进行修改。 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. 这是一项旧技术，但我们将添加一个新功能 - 我们将跟踪所有模式，包括高阶模式，因为它们在元素中进展。 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. 我们只会开发和概述这项技术，因为时间和空间限制不允许我们展示其应用示例。要对这项技术及其影响进行彻底的研究需要太多的篇幅。 We will disclose the techniques and leave it to the reader to develop the applications. 我们将公开这些技术并留给读者开发应用程序。
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. 考虑图 6-13 所示的几何形状。波导分为四个部分，每个部分都是球坐标中圆锥体的一部分。沿着该波导计算一阶模式是微不足道的。 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 矩阵相乘即可生成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 - 分成多个部分的简单波导和单个结的细节
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. 从该图的右侧可以看出，在各部分之间的每个交汇处，波前不是连续的 - 这些波的半径必须在每个交汇处的两个部分之间发生变化。 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. 为了使该波前从一个部分传播到下一个部分，我们必须通过在第二部分中创建更高阶模式来匹配波前。波前需要这些高阶模式才能在两个部分的交界处匹配。 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. 然后我们将有两个（或更多）模式，它们必须通过以下每个部分传播。同样的情况会发生在每个结点处，从而不断增加波前的高阶模式含量。
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. 不发生这种高阶模式创建的唯一方法是存在一个不在空间中移动的波前半径源点，而不是在每个部分中变化的波前半径源点。 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. 现在显而易见的是，任何波前半径的原点位置发生变化的波导都需要存在更高阶模式来解决这种变化的原点。 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，这种变化的半径位置正是坐标比例因子所解释的，也是方程变得如此复杂的原因。这是查看我们在前面几节中详细阐述的结果的另一种方式。
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) 波的整个概念因认识到只能存在三个波导而支离破碎，其中没有一个是我们感兴趣的，其中可以存在真正的 行为，并且那么只有当我们用不存在的来源喂养它们时。无论我们如何尝试最小化它们，所有其他几何形状和源配置都将具有更高阶模式。
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，我们不能通过制作更多的部分来规避这个问题，从而减少部分之间的变化，从而导致在每个连接处创建更少的高阶模式。 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. 高阶模式分量的较小值乘以增加的结点数量，得到与我们上面描述的完全相同的结果。 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. 根本无法回避这样的结论：为了准确地进行所有波导计算，我们必须包括高阶模式的存在，否则它们就会存在严重缺陷。
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. 认识到这一点，我们可以庆幸我们已经开发了处理这种复杂情况的机制，即 T 矩阵。通过向 矩阵添加两个维度（对于我们想要计算的每个高阶模式），我们可以适应这种新的复杂性。我们将需要处理更大的矩阵（ 表示两种模式， 表示三种模式，依此类推），但我们将让计算机来处理该问题。
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 节中使用的技术相同的技术，可以轻松地从球面波方程导出双模球面单元的 T 矩阵，只是现在我们在球面坐标中使用两种模态。
Without belaboring the details in the derivation of the T-matrix for this problem, the results are 无需详细讨论该问题 T 矩阵的推导细节，结果为
the modal pressure at junction 交界点 处的模态压力
the modal velocity at junction 交汇点 处的模态速度
mode number 模式编号
section-junction number, section has junctions and 节-路口号，节 有路口 和
and where 以及哪里
the Spherical Hankel Function of the first kind of order the Spherical Hankel Function of the second kind of order 第一类阶的球汉克尔函数 第二类阶的球汉克尔函数
the Eigenvalue for mode with the specific angle of section 模式 的特征值与特定截面角度
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 部分 入口孔径的半径
the radius to the exit aperture for section 部分 的出口孔径的半径
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——最好留给计算机来解决。在最低阶模式的情况下，它可以被大大简化，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，最好将矩阵保留为式（1）所示的形式。 (6.5.14) 同上。波函数可以作为对子程序的单个调用来进行
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. 与正确的论点保持一致。事实上，每种模式只需要进行两次调用，因为子例程返回四个值，即第一类型和第二类型的函数的值及其导数，所有这些都是必需的。
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. 应用该技术的过程是从喉部开始，根据适合该部分的角模式计算在该位置发现的波前的模态贡献。 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，使用上面所示的矩阵将这些模式传播到下一个结点，此时波前（模态波前之和）再次根据适合下一部分的角模式进行扩展。 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，我们需要知道作为截面角度函数的特征值。
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 这些特征值可以使用上一节中相同的拍摄技术来计算。然后用一个简单的方程拟合数据。精确值（圆圈）和拟合方程均如图 6-14 所示。该拟合方程为
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. 已知特征值后，在每个部分中生成“允许的”角度函数 就很容易了。为了找到每个模式在每个交汇处的贡献，我们必须选择一个表面，在该表面上我们可以将两组函数展开 - 传入部分中的函数到传出部分中的函数。 The simplest surface on which to do this is a planar one through the junction. 最简单的表面是通过连接处的平面。
Expanding the velocity function , as shown in Fig. 6-13, at junction 展开速度函数 ，如图6-13所示，在交汇处
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 由式(6.5.13)获得。我们可以计算球面波在平盘上连接点 上的复速度分布。我们需要稍微修改我们在第 2 节中找到的值。第 75 页上的 4.2 将高阶模式的存在解释为
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 的虚拟顶点到连接点 的长度
This equation represents the complex normal velocity distribution across the waveguide at junction . 该方程表示结点 处波导的复法向速度分布。
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 我们现在需要将此速度函数扩展到 部分的角度模式。输入到部分 的模态速度贡献将是
the angle of the nth section 第n段的角度
angle of section 截面角度
The calculation procedure then becomes: 那么计算过程就变成：
calculate the modal contributions at the throat 计算喉部的模态贡献
propagate these modes to junction 1 using Eq. (6.5.13) 使用方程式将这些模式传播到结点 1。 (6.5.13)
using Eq.(6.5.17), find the angular velocity 利用式（6.5.17）求出角速度
find the normal velocity on the matching disk from Eq. (6.5.18) 从等式中找到匹配盘 上的法向速度。 (6.5.18)
expand in terms of the new angular functions in section 1 from Eq.(6.5.19) 根据方程（6.5.19）第 1 节中的新角度函数扩展
loop through all sections to end junction - the mouth 循环穿过所有部分直至结束连接处 - 嘴
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前开始，然后回到喉咙，以确定该表面上的波前应该是什么，从而在口腔处实现我们想要的波前。这只是一个采取的问题
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. 从喉咙到嘴巴的矩阵的逆。问题是我们还没有说过在特定情况下嘴波前应该是什么。我们研究了分析给定设备的问题，但没有研究寻找最佳设备的问题。 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. 在后面的章节中我们会发现整个问题可以逆向解决，即我们可以指定所需的极响应场，计算实现该场所需的嘴部速度分布，最后逆向工作以找到实现该场所需的喉部分布所需的极性响应。 With any luck we could design a phasing/amplitude plug for the driver to give us the throat velocity that we want. 如果幸运的话，我们可以为驱动器设计一个定相/振幅插头，以提供我们想要的喉部速度。
6.6 Treatment of Mouth Diffraction 6.6 口腔衍射的治疗
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. 到目前为止，我们还没有讨论设备口处波导波前的衍射。由于波导不可能具有无限长度，因此波导总是有一个终端。 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. 我们需要了解嘴衍射将如何影响极性响应。然而，首先，我们应该了解我们的目标极性响应需要是什么，以便知道我们想要的口腔速度。 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. 显而易见的问题是“我们想要什么样的极地反应？”不会在这里解决；可能没有单一的答案。然而，我们将尝试在室内声学章节中阐明这个主题。 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，我们简单地假设我们想要的是某个标称角度的可控辐射方向图，例如 ，因为这是我们迄今为止在示例中使用的角度。
It may not be obvious that we can work the polar radiation problem backwards. 我们能否逆向解决极地辐射问题可能并不明显。 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，给定所需的极响应，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、球面所需的速度分布。 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. 该过程是将所需场扩展为其基本辐射模式，然后计算（反向）这些模式在源处的振幅和相位，以产生所需的辐射方向图。
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. 知道这个逆向问题是一个变换，让我们考虑一下我们应该预期的问题的一些特征。最重要的是声音测量中众所周知的能量-时间权衡的 k 空间等效项。 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. 极性响应越受限、越窄，我们应该预期源速度必须越宽才能实现这一结果。这意味着，由于我们的源的大小是有限的，我们可以很容易地要求一个给定源无法实现的极坐标模式。 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. 如果我们想要在波导口以及喉部实现平滑的速度分布，我们应该预期极性响应随角度的逐渐变化。 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，其中更尖锐的截止需要滤波器中更多的系数 - 本质上是“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很多辐射模式，但我们已经知道，波导内将有三种或最多四种模式可供使用，并且只能高于
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。显然，我们希望实现的最好结果是相当低阶的极性响应，即响应随角度的逐渐变化。
As an example of the characteristics that we have been talking about, consider the unrealistic polar response where the pressure is 作为我们一直在讨论的特征的一个例子，考虑不切实际的极地响应，其中压力为
We can calculate the required radiation modes as follows. From Chap. 3 we know 我们可以如下计算所需的辐射模式。来自章。 3 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 到测量表面的距离
from which, using orthogonality for the Legendre Polynomials and a know function , we will find 从中，使用勒让德多项式的正交性和已知函数 ，我们会发现
where 在哪里
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.) 如果我们只对远场方向性函数感兴趣，那么方程中的分母项： (6.6.21) 构成一个复常数，除了因子 之外，每个模态都相同，该因子来自汉克尔函数的大参数近似。任何独立于 的项都会简单地缩放系数。由于我们将研究归一化方向性函数，因此这些常数并不重要。 （如果我们想在某个时刻查看频率响应，那么我们当然需要保留这些术语。）
Finally, we can simplify Eq. (6.6.22) to get 最后，我们可以简化方程。 (6.6.22) 得到
Once we have these velocity modes the velocity distribution in the mouth will be 一旦我们有了这些速度模式，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 显示了计算得到的实现所需极坐标模式所需的嘴部速度剖面。
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，它们对于不同的 值（计算中的模式数量）并不稳定。随着在速度分布中添加越来越多的项，速度分布变得不稳定 - 广泛的振荡
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. 扩张。这是可以理解的，因为高阶项试图适应极坐标响应中的急剧不连续性。然后，对于此处所示的特定情况，会出现另一个问题，因为这些速度模式与频率相关。 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，我们必须选择其中一条曲线。我们可以尝试在不同的频率下匹配不同的配置文件，但我们很快就会发现这是不可行的。 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. 可以合理地期望能够通过波导及其相位塞的正确设计来创建近似频率独立的口波前（或者如果它遵循我们在第 6.4 节中看到的自然模态变化，则可能有轻微的频率依赖性），但是假设我们可以按照规定的方式具有特定的频率依赖性，如上图所示，这是不合理的。
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. 另一个问题是速度分布一直围绕球体，这在实践中是无法实现的。即使可以实现，它仍然是不可取的。我们的速度控制的限制，即给定长度的 设备的嘴尺寸，将迫使我们以相同的角度终止速度剖面。终止速度分布的方法对于实现我们期望的结果至关重要。
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 中选择一个配置文件，在 终止它，并重新计算一组新的修改后的 。然后我们可以使用这组新的系数来计算预期的极坐标模式。突然终止口腔速度的响应极坐标图如图 6-16 所示。这里的波导是一个长度为 的 器件。嘴的宽度约为 ，位于直径为 的外壳（球体）中，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. 我们需要再次详细阐述这个情节的一些特征。此压力响应的求和中使用了 50 种模式。请记住，这些是比波导或其口大得多的球体的模式。 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. 这意味着解决方案的收敛需要更多的模式。通过改变这些图所使用的模式数量并注意图变化的位置（即以什么频率），我们可以估计在任何频率下获得准确结果所需的模式数量。 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所示的结果估计良好到大约 。我们可以看到在这个频率附近响应发生了明显的变化。大约 处的特殊响应。是真实的。
Figure 6-16 - The polar response map for a waveguide with an abruptly terminated mouth 图 6-16 - 具有突然终止口的波导的极性响应图
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，这些极坐标图都在轴上标准化为 级别。这提出了一个重要的观点，也许我们应该等到讨论室内声学后再提出，但我们将在这里讨论它。 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. 始终可以以电子方式均衡沿着这些极坐标图之一中的任何水平线的任何扬声器系统的响应。我们几乎总是沿着轴线这样做。 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. 然而，重要的是要认识到，不可能以电子方式校正整个极地地图。这是根本不可能做到的。这里的含义是，电子设备在解决扬声器问题方面只能发挥这么大的作用。剩下的就看设计师了。
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 记住速度极坐标响应的变换关系的基础知识，也许设计一个不会像所使用的“阶跃响应”极坐标模式那样随角度突然变化的极坐标模式会更合乎逻辑。
above. We will find it convenient to deal with polar response functions of the following form 多于。我们会发现处理以下形式的极坐标响应函数很方便
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. 值会在 附近给出 点，如图 6-17 所示。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）。
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. 该指数极坐标响应的计算速度分布如图 6-18 所示。该曲线是使用求和中的 20 种模式创建的。从 曲线可以清楚地看出逆计算的不稳定性。在某些模式下，所有曲线都变得如此复杂。在这里我们可以看到，如果曲线不是振荡的，那么它将与所有过度曲线相同。这很有吸引力，因为现在选择哪条曲线作为我们的目标分布已经不存在任何歧义了。 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. 速度完全环绕源这一事实仍然是一个问题。否则，该速度分布似乎在所有频率下都是可实现的。
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 有趣的是，图 6-18 中的速度分布远远超出了球体上的 点，这与球面“视线”方向性的既定原理形成鲜明对比来源。产生 覆盖模式所需的宽角速度表明使用的可能性
Figure 6-18 - The source velocity magnitude for a smooth polar pattern 图 6-18 - 平滑极坐标模式的源速度大小
a much wider waveguide angle to yield an improved narrower angle response, an option that we will be developing later. 更宽的波导角度以产生改进的窄角度响应，这是我们稍后将开发的选项。
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. 如果我们以突然的方式在挡板处终止波导，我们会发现我们不会得到比图 6-16 中看到的更好的极性响应。这意味着突然终止源球面的速度分布不是我们想要做的事情。 There may be a better way. 可能有更好的方法。
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. 如果我们通过从波导主体到挡板应用大半径（“大”是我们稍后将定义的东西）将波导张开到挡板中，而不是突然终止，那么我们会发现我们可以得到更符合我们喜好的极性反应。 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. 不幸的是，将波导张开到挡板中并不是一个精确的数学描述。出于本文的目的，我们假设耀斑具有以逐渐减小口部速度的功能，如图 6-19 所示。该图的横坐标是一个角度，但它也可以被认为是从嘴的中心到外壳的假设球形表面的外边缘的半径，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坐标响应图如图 6-20 所示。此响应几乎是理想的，从大约 及以上的 （ 点）几乎恒定的覆盖范围。
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，有几点需要注意。首先，通过指定所需的波导设计来向后进行波导设计是可能且合理的。
Figure 6-19-Velocity distribution for flared waveguide 图6-19-喇叭形波导的速度分布
Figure 6-20 - Polar map for a flared waveguide 图 6-20 - 喇叭形波导的极坐标图
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，计算所需的口腔速度；通过波导反向传播该轮廓以找到所需的喉部速度，最后设计一个相位塞来实现所需的喉部速度。 Second, for constant directivity we do not want a wavefront with a velocity contour which is independent of angle (radius), i.e. 其次，对于恒定的方向性，我们不希望波前的速度轮廓与角度（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. 平坦，最后，波导应始终张开进入挡板，并且永远不会突然终止，因为二次衍射不受控制。
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的波导角可能实现更好的极性响应的讨论，考虑图 6-21 中所示的速度分布。这对应于更大的角度覆盖
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）逐渐衰减。极坐标图如图6-22所示。这种极性响应实际上是理想的： 位于 处，来自 。虽然尚不清楚如何获得图中所示的速度剖面，但这似乎是可能的。例如，人们可能会制造一种具有吸收边界而不是反射边界的波导。 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. 这将以类似于期望的方式减小边缘处相对于中心的速度分布。事实上，我们可以通过在波导中开发波函数来分析这种情况，波导在边界处具有阻抗而不是零速度条件。 (Yet another interesting exercise for the reader.) （这对读者来说是另一个有趣的练习。）
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. 虽然复杂，但上述任务当然不是不可能的。这里的要点是，虽然图 6-22 中所示的响应可能看起来无法达到，但理论上是可能的。经验表明，凡是可能的事情都是可以实现的。
Figure 6-22 - A polar pattern with smooth angular variation 图 6-22 - 具有平滑角度变化的极坐标模式
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 1 章中的数字进行比较。 4 针对各种不受控制的来源。显然，如果没有某种形式的波导，人们就被迫应对极性响应方面的不可控和不期望的情况。 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，这种情况不必被视为设计问题的约束。波导提供了一种控制方向性的方法，几乎可以产生任何所需的极性响应。
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. 波导确实有局限性。最值得注意的是，很难通过良好的控制获得宽的方向性。我们也没有讨论如何获得非轴对称极性图案。 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讨论数组时讨论，而后一个问题其实并不那么困难。 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强调，椭圆 (ES) 坐标具有椭圆口，这些设备将产生非轴对称极坐标图案。 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，只是边界是由椭圆方程定义的，而椭圆方程并不广为人知。 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. 不过，正如我们在第 1 章中指出的那样，对这些设备进行分析可能不会收集到太多信息（因此我们将把这项任务留给感兴趣的读者）。如图2所示，这些坐标中的波函数是未知的。 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 坐标需要椭圆形喉部，这在定相插头设计中当然很容易实现。这也是为什么定相插头设计必须是波导而不是驱动器的一部分的另一个原因。
6.7 Diffraction Horns 6.7 衍射角
It would not be fair to leave this chapter without mentioning the horn design which has dominated the marketplace for so many years. 如果在本章结束时不提及多年来主导市场的喇叭设计，那是不公平的。 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. 衍射喇叭是通过衍射波前然后通过某种圆锥形轮廓将其限制在某个有限角度来获得控制的装置。
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。侧视图基本上是通过所需垂直极性图案的弧线摆动的顶视图。 Many variations on this construction are possible, but this drawing shows the device in it simplest embodiment. 该结构可以有多种变化，但该图显示了该装置最简单的实施例。
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 来自章。 4 我们知道，当辐射表面与波长相比较小时，则极性响应很宽（回想一下 Rect 的变换）
Figure 6-23 - Typical layout of a diffraction horn 图 6-23 - 衍射喇叭的典型布局
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. 函数，随着函数变窄而变宽。）。衍射装置通常在水平轴和垂直轴上作为单独的设计工作，从而允许在两个方向上产生不同的图案。 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定相插头中都有指数部分，喉部最初是指数号角。 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. 它最初仅在一个维度上扩展——通常是具有较窄图案的维度，另一个维度在选定的值上保持恒定，以便它在最高控制频率下不宽于半波长。
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. 在某个时刻，这个初始喇叭会终止。确切的位置、距振膜的距离并不重要，通常被选择作为方便形成喇叭并设置我们将在下面讨论的所需距离的折衷方案。 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. 衍射槽通常是一个陡峭的斜坡不连续性，尽管它也可以是圆形的 - 这往往会平滑由该连接处的衍射和反射引起的响应波纹。 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. 衍射后，在两个维度上开槽设备耀斑，并且此时通常是直边（尽管经常发现一些曲率是有用的。）侧面的角度设置为所需的覆盖图案，其中线视觉经验法则是设计理念。 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，衍射槽可以是弯曲的（如图所示）或平坦的。这方面没有什么区别，只是反射波（我们稍后将讨论）在平槽中不像在弯曲槽中那样相干。 This tends to spread the reflection ripples slightly. 这往往会稍微扩散反射波纹。
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. 的值基本上设置水平控制的上限，而 将设置该控制的下限。对于水平维度也是如此，其中 控制最高频率， 控制较低频率。显然，该设计不允许对所有这些变量进行独立控制。正是后一个因素和妥协的需要导致了该设计的广泛变化。
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. 压缩驱动器的喉部通常是圆形的（尽管矩形定相插头可以在设计折衷中提供更大的灵活性），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张开到一定程度，因为这已被发现是有利的。根据设计者的不同，挡板上的喇叭形也可见或不可见。
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. 衍射喇叭的问题是从衍射槽反射的大量能量。为了使设备能够工作，即为了有足够的衍射来工作，该槽处（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. 这种阻抗不匹配会将大量的入射波前反射回设备。会产生驻波，这在驱动器的电阻抗以及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。那是因为实际上有两个。一个位于设备的喉部，另一个位于衍射槽处。 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，另一个平面为声学中心
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. 声学中心位于喉咙。在不在这些平面之一内的角度处，声学中心是不明确的。这会导致这些设备的排列出现问题，因为正如我们将看到的，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表现良好，但现在基本上已经过时了。这些设备中的波前非常复杂，只能使用 FEA 等复杂的数字代码进行分析。 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，它们可能会是一场灾难。当然，这只是我们的观点，但我们希望读者认识到本章中显示的结果支持这样的观点。
6.8 Summary 6.8 总结
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. 这是一个漫长而复杂的章节，这里介绍的主题对于扬声器系统的设计仍然至关重要。我们表明，我们必须认真质疑霍恩方程在波导设计中的使用，因为它不适用。 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，事实仍然是，根据这个方程设计的一些喇叭效果很好。从我们的结果中，我们可以看到（例如）指数喇叭中发生的衍射可以产生完全正确的嘴部波前，但如果确实如此，那将纯粹是巧合！ There is no way that one could design such a successes from horn theory. 凭号角理论不可能设计出如此成功的产品。
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. 我们试图指出控制系统的极性响应是可能的，尽管并不容易。我们将在后面的章节中看到，一旦设计中包含了房间效果，指向性就成为扬声器系统设计中的一个主要组成部分。 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. 无论是大型场馆还是小型场馆，都是如此，但原因不同。无论哪种情况，接受扬声器系统必须具有不受控制或全向的指向性这一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，全方位的存在并没有真正受到控制，而只是接受“没有其他事情是可能的”这一观念。）
It is true that the recent trend towards smaller and smaller loudspeakers does not allow for directivity control to any appreciable degree. 确实，最近扬声器越来越小的趋势不允许进行任何明显程度的方向性控制。 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，则无法实现频率以下的方向性控制，在该频率下，外壳的尺寸变得与声音的波长相当，这是一个昂贵的提议，而且我们在不久的将来可能不会看到广泛的可用性。未来。 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而需要较大的音箱的想法确实是正确的，但我们可以看到现在需要较大音箱的另一个原因 - 控制扬声器系统的方向性/功率响应的能力。 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需求能够盛行，并且我们将在家庭影院和其他以音质为首要考虑因素的现代应用中看到过去“大”尺寸扬声器的回归。
See Mapes-Riordan, "Horn Modeling with Conical and Cylindrical Transmission line Elements", JAES. 请参阅 Mapes-Riordan，“使用圆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. 参见普特兰“每个单参数声场都服从韦氏霍恩方程”，JAES。
