这是用户在 2024-3-29 7:35 为 https://app.immersivetranslate.com/pdf-pro/9188d091-b7d3-4ddb-b8af-b8ff0c974ab3 保存的双语快照页面,由 沉浸式翻译 提供双语支持。了解如何保存?
2024_03_28_2e20da4a038e6bb4a2c9g

WAVEGUIDES  波导

CONTROLLING SOUND RADIATION
控制声音辐射

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 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。
  1. See Morse, Methods of Theoretical Physics
    参见莫尔斯,理论物理方法
  2. 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
Figure 6-1 - A typical exponential horn contour
图 6-1 - 典型的指数喇叭轮廓
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。)
Figure 6-1 - Plot of Eq.(6.1.3)
图 6-1 - 方程 (6.1.3) 的绘图
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。
  1. 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 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
  1. 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( 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 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

6. See Press, Numerical Recipes, Chap. 11
6. 参见《新闻》,《数字食谱》,第 1 章。 11

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 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 - 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)
Figure 6-9 - impedance (real part) for waveguide
图 6-9 - 波导的阻抗(实部)
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
  1. 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 6-10 - Mouth radial velocity amplitude
图 6-10 口径向速度幅值
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:
  1. 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。

6.5 Approximate Numerical Calculations 9,10
6.5 近似数值计算9,10

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