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WAVEGUIDES

CONTROLLING SOUND RADIATION
控制声辐射

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。我们似乎没有什么办法影响声辐射响应。

The size and configuration determines the radiation pattern with the enclosure playing an important role at the lowest frequencies.
尺寸和配置决定了辐射模式,外壳在最低频率时起着重要作用。

If the driver is still radiating above the point where the enclosure is controlling the response then the response is pretty much completely dependent on the driver size. Little else has much of an effect.
如果驱动器的辐射仍高于箱体控制响应的点,那么响应几乎完全取决于驱动器的尺寸。其他方面几乎没有什么影响。
In this chapter, we will study the concept of a waveguide as a directionality controlling device.
在本章中,我们将研究波导作为方向控制装置的概念。

6.1 Historical Notes 6.1 历史说明

It is important to go through the historical development of horn and waveguide theory in order to understand its evolution the current level of our understanding.
我们有必要回顾一下喇叭和波导理论的历史发展,以了解其演变过程和我们目前的理解水平。

The importance of this review stems from long-standing beliefs about waveguides and horns that are not in fact correct. Correcting these beliefs creates extreme limits on their applicability to current issues in their design.
这篇评论的重要性源于人们长期以来对波导和喇叭的看法,而事实上这些看法并不正确。纠正这些观念会极大地限制它们对当前设计问题的适用性。
Horns have been around for centuries and we have no idea when or where they were first used. Horns as musical instruments are certainly centuries old.
号角已经存在了几个世纪,我们不知道它们最早是在何时何地被使用的。作为乐器的号角当然有几百年的历史。

With the advent of the phonograph, the horn was found to play a crucial role in amplifying the sound emitted from the small mechanical motions of the stylus. The horn was responsible for virtually all of the gain in the system.
随着留声机的问世,人们发现号角在放大唱针的微小机械运动所发出的声音方面起着至关重要的作用。号角负责系统中几乎所有的增益。

Its use therefore was principally one of a loading or impedance matching mechanism required to better match the high mechanical impedance of the stylus to the very low mechanical impedance of the medium - air.
因此,它主要是一种负载或阻抗匹配机制,用于更好地将测针的高机械阻抗与介质(空气)的极低机械阻抗相匹配。

The horns role as an acoustic transformer is central to the evolution of horn theory.
号角作为声学变压器的作用是号角理论发展的核心。
When one is interested in the loading properties of a conduit, they need only be concerned with the average distribution of the acoustic variables across the diaphragm and hence across the conduit.
当人们对导管的加载特性感兴趣时,只需关注隔膜上的声学变量的平均分布,从而关注导管上的声学变量的平均分布。

This is, in fact, the assumption that Webster made when he derived what is now known as Webster's Horn Equation
事实上,这就是韦伯斯特在推导出现在被称为韦伯斯特霍恩方程时所作的假设
It is commonly thought that this equation applies only to plane waves since Webster used a plane wave assumption in its derivation. However, this equation is far more broadly applicable than to plane waves alone.
人们通常认为这个方程只适用于平面波,因为韦伯斯特在推导过程中使用了平面波假设。然而,这个方程的适用范围要比平面波广泛得多。

It is actually exact for any geometry where the scale factor of the coordinate of interest is one. The scale factor is a fundamental parameter of all coordinate systems as shown in Morse .
实际上,对于相关坐标的比例因子为 1 的任何几何体来说,它都是精确的。如莫尔斯 所示,比例因子是所有坐标系的基本参数。

(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.
(比例因子是已知的,只能通过可分离坐标系计算得出。值得注意的是,在韦伯斯特的公式中,任何一个坐标系的三个维度的比例因子都是统一的,都是精确的。

Specifically these coordinates are:
具体来说,这些坐标是
  • all three Cartesian Coordinates
    所有三个直角坐标
  • the axial coordinate in all Cylindrical Coordinate systems (Elliptic, Parabolic and Circular)
    所有圆柱坐标系(椭圆坐标系、抛物线坐标系和圆坐标系)中的轴坐标
  • the radial coordinate in the Cylindrical coordinate system
    在圆柱坐标系中的径向坐标
  • the radial coordinate in Spherical Coordinates
    球面坐标中的径向坐标
  • the radial coordinate in Conical Coordinates
    圆锥坐标中的径向坐标
The first two (six coordinates) apply to conduits of constant cross section, which are not very interesting to us.
前两个坐标(六个坐标)适用于横截面恒定的导管,我们对它们并不感兴趣。

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.
有用的是圆柱坐标系、球坐标系和圆锥坐标系的三个径向坐标。还有两点极为重要。首先,只有三个有用的坐标系中的韦伯斯特方程是精确的;其次,所有这三个坐标系在其他正交坐标系中都有波传播。

The importance of this last attribute will become clear later on.
这最后一个属性的重要性稍后会变得很清楚。
If Webster's Equation is only correct in three useful situations then why is there so much literature surrounding its use? That is because the equation is still useful as an approximation to any conduit of varying cross section.
如果韦伯斯特方程只在三种有用的情况下才是正确的,那么为什么会有这么多围绕其使用的文献呢?这是因为对于任何横截面不同的导管来说,该方程仍然是有用的近似值。

In nearly all of the common cases of the application of Webster's equation it is used as an approximation to the actual wave propagation in a flared conduit.
在应用韦伯斯特方程的几乎所有常见情况下,它都被用作扩口导管中实际波传播的近似值。

There has also been a great deal of literature written about the application of Webster's equation to the evaluation of these approximate solutions.
关于应用韦伯斯特方程评估这些近似解,也有大量的文献记载。

Almost nothing has been written about when these approximations are "good" approximations and when we should be suspect of their validity.
关于这些近似值什么时候是 "好 "近似值,什么时候我们应该怀疑它们的有效性,几乎没有任何论述。
The only place where we have seen such a discussion is, once again, in Morse where they state:
我们唯一看到过这种讨论的地方,还是在莫尔斯 中,他们在那里说:
Both pressure and fluid velocity obey this modified Wave Equation, which approximately takes into account the variation of cross sectional size with . The equation is a good approximation as long as the magnitude of the rate of change of with is much smaller than unity (as long as the tube "flares" slowly).
压力和流体速度都服从这个修正的波方程,该方程近似考虑了横截面尺寸随 的变化。只要 的变化率远小于统一值(只要管道缓慢 "扩张"),该方程就是一个很好的近似值。
  1. See Morse, Methods of Theoretical Physics
    见莫尔斯:《理论物理学方法
  2. See Morse, Methods of Theoretical Physics, pg. 1352
    参见莫尔斯:《理论物理学方法》,第 1352 页。
These are very limiting conditions that have been almost universally ignored. Let us look at what they imply about the development of the well know exponential horn.
这些限制性条件几乎被普遍忽视。让我们看看它们对众所周知的指数角的发展有何影响。
Restating Morse, as applied to an exponential horn, let
重述应用于指数角的莫尔斯,让
we can see that, for this example, Webster's equation is "good" only so long as
我们可以看到,在这个例子中,韦伯斯特方程的 "好 "之处在于
the throat area
咽喉部位
the flare rate  耀斑率
The horn contour for this example is shown in Fig. 6-1. This figure shows an
该示例的喇叭轮廓如图 6-1 所示。该图显示了
Figure 6-1 - A typical exponential horn contour
图 6-1 - 典型的指数喇叭轮廓线
exponential horn contour of typical shape and length. In Fig. 6-1, we have plotted the value of Eq.(6.1.3) as a function of the axial distance from the throat.
典型形状和长度的指数角轮廓。在图 6-1 中,我们绘制了公式(6.1.3)值与喉管轴向距离的函数关系图。

These figures show that the assumptions for an accurate application of Webster's Equation to an exponential horn are clearly violated for a length of the horn beyond about (Morse says much less than 1.0, but how much less is a matter of choice. We ascribe here to the use of a value of .5 as being the limit of accuracy, the assumptions being completely invalid at the value of 1.0.)
这些数字表明,当喇叭的长度超过 时,韦伯斯特方程对指数喇叭的精确应用显然违反了假定(莫尔斯说远远小于 1.0,但究竟小于多少则是一个选择问题。在此,我们认为 0.5 的值是精确度的极限,假设值为 1.0 时完全无效)。
Figure 6-1 - Plot of Eq.(6.1.3)
图 6-1 - 公式(6.1.3)的曲线图
In Fig. 6-1, we have shown a line drawn tangent to the horn contour and originating at the center of the throat, the acoustic center of the throat's wavefront.
在图 6-1 中,我们绘制了一条与喇叭轮廓相切的直线,该直线的起点是喉管中心,即喉管波面的声学中心。

A simple rule of thumb that we use is that any contour which lies past this point of tangency cannot be accurately described by Webster's equation. This rule of thumb implies a geometrical interpretation of the limitation of Webster's equation.
我们使用的一个简单经验法则是,韦伯斯特方程无法准确描述过切点的任何轮廓。这条经验法则意味着对韦伯斯特方程局限性的几何解释。

The horn equation cannot predict the wavefronts once they are required to diffract around a point along the device that places the receding boundary in the shadow zone of the acoustic center of the originating wavefronts. Fig.
一旦要求波阵面在沿装置的某一点周围衍射,将后退边界置于起源波阵面声学中心的阴影区内,喇叭方程就无法预测波阵面。图

6-1 shows that the only exponential horn which could be accurately represented by the Webster equation is extremely short. A horn of this length is of no practical interest.
6-1 表明,唯一能用韦伯斯特方程准确表示的指数角非常短。这种长度的角没有实际意义。
Further support for our "rule of thumb" comes from considering Huygens' principle and the construction of Huygen wavefronts.
考虑惠更斯原理和惠更斯波面的构造,可以进一步支持我们的 "经验法则"。

Beyond the point of tangency, wavefronts, if they are to remain perpendicular to the sides of the conduit (as they must), have to have an apparent acoustic center which is front of the actual throat of the device.
在切点之外,如果波阵面要保持与导管两侧垂直(必须如此),就必须有一个位于设备实际喉管前方的表观声学中心。

Huygen's principle allows for un-diffracted wavefronts to be flatter than those from the acoustic center, but is does not allow for a wavefront curvature to be less than the radius to the acoustic center. A little thought will show why this must be true.
惠根原理允许未衍射波面比来自声学中心的波面更平坦,但不允许波面曲率小于到声学中心的半径。只要稍加思考,就会明白为什么这一定是正确的。

So stated another way, our rule of thumb becomes: horn wavefronts with a curvature less than the radius to the throat must have diffracted somewhere along the trip down the device - the diffraction creating a new acoustic center from which wavefronts emerge.
因此,换一种说法,我们的经验法则就变成了:曲率小于喉管半径的喇叭波面,一定是在设备的某个地方发生了衍射--衍射产生了一个新的声学中心,波面就是从这个中心出现的。
It is further interesting to note that in those cases where Webster's Equation is exact, there is never a point on the contour of the horn which is in the shadow
更有趣的是,在韦伯斯特方程精确的情况下,角的轮廓上从来没有一个点是在阴影中的

zone, i.e. our "rule of thumb" is never violated. The overwhelming majority of work done in horn theory suffers from a serious question of its validity.
区,也就是说,我们的 "经验法则 "从未被违反过。绝大多数有关号角理论的工作都存在严重的有效性问题。
How one gets around this problem leads us into another line of reasoning for which there are two paths.
如何绕过这个问题,会让我们进入另一条推理思路,而这条路有两条。

We could join the exponential section of the above contour to a spherical section continuing out from the point of tangency to the acoustic center line, thus insuring that our rule of thumb was never violated. This does in fact work reasonably well, and is in common usage.
我们可以将上述等值线的指数部分与从声学中心线的切点继续向外延伸的球形部分连接起来,从而确保我们的经验法则不被违反。事实上,这种方法效果相当不错,也很常用。

However, the fact remains that the exponential section is still only an approximation and we really don't know the actual shape of the wavefront at the joining point. As we shall see later, this is a serious limitation to the joining approach.
然而,事实上指数截面仍然只是一个近似值,我们确实不知道连接点处波面的实际形状。正如我们稍后将看到的,这是连接方法的一个严重局限。
Now that we have shown that the horn equation has severe limitations in its applicability to important problems in waveguides, we will discuss how these limitations might affect the expected results of using it.
既然我们已经证明了喇叭方程在波导重要问题的适用性上存在严重的局限性,那么我们将讨论这些局限性会如何影响使用该方程的预期结果。

As we showed in Chap.3, if we know the wave shape of the wavefront as it crosses through a boundary for which we have a radiation solution (flat, spherical or cylindrical), then we can achieve a fairly accurate prediction of the directivity of this source.
正如我们在第 3 章中所展示的,如果我们知道波阵面穿过边界时的波形,而我们又有一个辐射解(平面、球面或圆柱面),那么我们就能相当准确地预测出这个波源的指向性。

However, we must know the précis magnitude and phase of the wavefront at every point in the aperture in order to do this calculation. From the above discussion, we should have serious doubts about the ability of Eq.
但是,我们必须知道孔径中每一点的波面的简要幅度和相位,才能进行计算。从上面的讨论中,我们应该对公式......的能力产生严重怀疑。

(6.1.1) to give us this information, except, of course in those limited cases where it is exact.
(6.1.1) 向我们提供这些信息,当然,在少数情况下提供准确信息除外。

The natural question to ask is: can we develop an equation or an approach which will allow us to know, with some certainty, what the magnitude and phase is at every point within the waveguide?
自然要问的问题是:我们能否制定一个方程或一种方法,让我们能够比较肯定地知道波导内每一点的振幅和相位是多少?

The answer is yes, but the price that we must pay for this precious knowledge is a substantial increase in the complexity of the equations and their solution.
答案是肯定的,但我们必须为这些宝贵的知识付出代价,那就是方程及其解法的复杂性大大增加。

6.2 Waveguide Theory
6.2 波导理论

As we discussed above, the early use of a horn was substantially different than what we are attempting to develop here. The early need for horns was as an acoustic loading devices and our interest here is in controlling source directivity.
如上文所述,号角的早期用途与我们在此尝试开发的用途大相径庭。早期对号角的需求是作为一种声学加载装置,而我们在这里感兴趣的是控制声源的指向性。

(Loading essentially became a non-issue with the almost unlimited amplifier power capability available today.) For this reason, we will adapt the terminology that a horn is a device which was developed with Webster's Equation and its approach to calculation (wherein, only the average wavefront shape and the acoustic loading is required) and a waveguide is a device whose principal use is to control the directivity.
(如今,放大器的功率几乎不受限制,负载基本上已不再是问题)。因此,我们将采用以下术语:号角是一种利用韦伯斯特方程及其计算方法开发的设备(只需计算平均波面形状和声学负载),而波导则是一种主要用于控制指向性的设备。

A waveguides design is along the lines that we will develop in the following sections, a horns design using Webster's equation. The acoustic loading of a waveguide can usually be calculated without much trouble, but not always.
波导的设计思路与我们将在以下章节中介绍的使用韦伯斯特方程进行喇叭设计的思路一致。波导的声学负载通常可以轻松计算出来,但并非总是如此。

However, since loading is not a central concern for us this limitation is not significant.
不过,由于负载并不是我们关注的核心问题,因此这一限制并不重要。
  1. See Geddes "Waveguide Theory" and "Waveguide Theory Revisited", JAES.
    见 Geddes "波导理论 "和 "波导理论重温",JAES。
We know from earlier chapters that the Wave Equation is always accurate so long as we can apply the proper boundary conditions.
通过前面几章的学习,我们知道,只要能应用适当的边界条件,波方程总是准确的。

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
这种情况只能在有限的坐标系中发生。以图 6-2 所示的球形几何中的导管为例。边界条件
Figure 6-2 - A simple hornwaveguide example
图 6-2 - 一个简单的喇叭波导示例
here are that the conduit is symmetric in , and has a velocity which is zero at some . For now we will not worry about the terminations of this conduit along its axis and simply assume that it is semi-infinite. In other words it has a finite throat, but a mouth at infinity.
在这里,导管在 上是对称的,并且有一个 速度,该速度在某个 处为零。现在,我们暂且不考虑导管沿其轴线的终端,只假设它是半无限的。换句话说,它的喉部是有限的,但在无穷远处有一个口。
We can use either the full Wave Equation, or Eq. (6.1.1) in this example because of its simplicity.
在这个例子中,我们既可以使用完整的波方程,也可以使用公式 (6.1.1),因为它比较简单。

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.
我们已经讨论过喇叭口方法,因此让我们使用第 3 章中的完整波方程。 我们知道,以下解法既适用于球面坐标系,也适用于韦伯斯特方程。
We can immediately see from this solution that there is one aspect to the Wave Equation approach that is not present in Webster's approach. That is, we know from the Wave Equation that the wave number is a coupling constant to two other equations in and . We can exclude the coupling to since there is no variation in the boundary conditions, but we cannot simply make the assumption that there is no variation of the waves in . If there is a dependence of the wavefront at the throat (or any point for that matter) then there will be a dependence in the wave as it is propagates down the device. This is significantly different from Webster's approach since Webster's equation does not allow for any dependence. This limitation is actually far more significant than the errors due to the flare rate that we discussed above.
从这一解法中我们可以立即看出,波方程方法有一个方面是韦伯斯特方法所不具备的。也就是说,我们从波浪方程中知道,波数 中另外两个方程的耦合常数。我们可以排除与 的耦合,因为 的边界条件没有变化,但我们不能简单地假设 中的波没有变化。如果喉部(或任何一点)的波面与 有关,那么波在设备上向下传播时就会与 有关。这与韦伯斯特的方法大相径庭,因为韦伯斯特方程不允许任何 相关性。这一限制实际上比我们上面讨论的耀斑率所造成的误差要大得多。
As an example, consider evaluating a "conical" horn versus a Spherical waveguide (both are as shown in Fig. 6-2) using Webster's equation and the acoustic waveguide approaches respectively.
例如,考虑分别使用韦伯斯特方程和声波导方法评估 "锥形 "喇叭和球形波导(两者如图 6-2 所示)。

Horn theory yields an impedance at the throat, but it yields no information about the amplitude and phase of the wavefront anywhere within the device. The assumption of uniform amplitude across
喇叭理论可以得出喉管处的阻抗,但无法得出设备内任何位置的波面振幅和相位信息。假定整个波面的振幅是均匀的

the device means that Webster's approach predicts the same value at every point on some (unknown) surface which is orthogonal to the horn boundaries. Of course we could argue (correct in some cases) that the wavefront must be a spherical section at every point.
该装置意味着韦伯斯特方法预测了与喇叭边界正交的某个(未知)表面上每一点的相同值。当然,我们也可以认为(在某些情况下是正确的),波阵面在每一点上都必须是一个球形截面。

But what happens if the device is fed at the throat with a plane wave? There is simply no way to answer this question with the tools available to us from Webster.
但是,如果在设备的喉部输入平面波,会发生什么情况呢?韦伯斯特提供的工具根本无法回答这个问题。
Consider now the alternate waveguide approach. From Sec. 3.4 on page 49 we know that the solutions for the radial coordinate are
现在考虑另一种波导法。根据第 49 页第 3.4 节,我们知道径向坐标的解法为
If the driving wavefront is of a spherical shape, then only a single mode will be excited. In that case we get the same answer for a wave propagating down the device for either the Horn Equation or the Wave Equation
如果驱动波面呈球形,则只会激发单模 。在这种情况下,对于沿设备向下传播的波,无论是喇叭方程还是波方程,我们都能得到相同的答案
where , since it is always in the direction in this case. However, we also know that there are an infinite number of other possibilities where . Only the Wave Equation approach offers up this added flexibility in its application.
,因为在这种情况下,它总是朝着 方向。不过,我们也知道,在 的情况下,还有其他无限的可能性。只有波方程方法在应用中提供了这种额外的灵活性。
When the throat is driven by a wavefront which does not coincide with a spherical section of the same radius as the throat, then the wavefront can be expanded into a series of admissible wavefronts prior to propagation down the device.
当喉管受到与喉管半径相同的球面截面不重合的波阵面驱动时,波阵面就会在沿设备向下传播之前扩展成一系列可接受的波阵面。

Since we have solutions for all of these waves, we can develop the final solution as a sum over these various wave orders.
由于我们有所有这些波的解,因此可以将最终解作为这些不同波阶的总和。
Consider a plane wave excitation at the throat. It is well know that a plane wave can be expanded into an infinite series of spherical waves. This series is
考虑在喉管处激发平面波。众所周知,平面波可以扩展成无限的球面波系列。这个系列是
Basically, the Legendre Polynomials form an expansion with the weighting factors given by the terms to the left of them. Eq. (6.2.7) is applicable to a plane pressure wave - a scalar function.
基本上,Legendre 多项式形成了一个展开式,其左侧项给出了加权因子。公式 (6.2.7) 适用于平面压力波--一个标量函数。

If we had a planar velocity source at the throat (i.e., a flat piston) then we would have to match this to the radial and angular velocities at the throat.
如果喉管处有一个平面速度源(即平面活塞),那么我们就必须将其与喉管处的径向速度和角速度相匹配。

This would not actually be too difficult, except that there is yet another problem with this approach, so we will leave this discussion for later in order to address the solution to our current problem.
这实际上并不难,只是这种方法还有另一个问题,所以我们将留待以后再讨论,以便解决目前的问题。
Unfortunately, the Legendre Polynomials, as shown in Eq. (6.2.7), do not fit the boundary conditions of our waveguide, that is, they do not have a zero slope
不幸的是,如公式 (6.2.7) 所示,Legendre 多项式不符合我们波导的边界条件,即它们的斜率不是零
  1. See Morse, Any text 见莫尔斯,任何文本
    at the walls of the waveguide, for all . The normal Legendre Polynomials must have a separation constant which is an integer because of periodicity. The new functions can no longer require this constant to be an integer. (Why?) Compare the two plots in Fig. 6-3. The right side of the plot shows the normal Legendre Polynomials over a arc. The left side shows the Modified Legendre Polynomials (modified because of the new separation constant ) that meet the boundary conditions at the walls of the waveguide. The new polynomials can be used to expand any axi-symmetric source at the throat - they form a complete orthogonal set. It is worth noting the similarity of Fig. 6-3 with Fig.4-3 on page 73.
    在波导壁上, ,所有 。由于周期性的原因,普通 Legendre 多项式必须有一个整数的分离常数 。新函数不再要求该常数为整数。(为什么?)比较图 6-3 中的两幅图。图的右侧显示的是 弧线上的普通 Legendre 多项式。左侧显示的是符合波导壁边界条件的修正 Legendre 多项式(由于使用了新的分离常数 )。新的多项式可用于扩展喉部的任何轴对称声源--它们构成了一个完整的正交集合。值得注意的是,图 6-3 与第 73 页的图 4-3 非常相似。
Thus far we have seen that by utilizing the full machinery of the Wave Equation we can match any velocity distribution placed at the throat of a waveguide so long as this waveguide lies along a coordinate surface of one of the separable coordinate systems (although we have as yet only looked at a very simple one).
到目前为止,我们已经看到,通过利用波方程的全部机制,我们可以匹配波导喉部的任何速度分布,只要该波导位于一个可分离坐标系的坐标面上(尽管我们还只是研究了一个非常简单的坐标系)。

We have also seen that this wavefront matching cannot be accomplished by using Webster's horn equation; the machinery to do so just does not exist in that formulation.
我们还看到,这种波面匹配无法通过韦伯斯特的角方程来实现,因为在该方程中不存在这样的机制。

Of course we could force the throat wavefront to match the lowest order mode of the horn and then the horn equation would be accurate, exact in fact.
当然,我们可以迫使喉管波面与喇叭的低阶模式相匹配,这样喇叭方程就会准确无误。
Figure 6-3 - Normal Legendre Polynomials (right) for waveguide (left)
图 6-3 - 波导(左)的法线 Legendre 多项式(右)。
The problem is that there are no sources which have a velocity profile that matches any of the three geometries which have an exact horn solution!
问题是,没有任何信号源的速度曲线与三种几何图形中的任何一种相匹配,而这三种几何图形都有精确的喇叭解!
In order to continue we have a choice of three alternatives:
为了继续前进,我们有三种选择:
  • accurately use the horn equation with unrealistic sources
    用不现实的声源准确地使用喇叭方程
  • use approximate solutions for real sources, or
    使用真实源的近似解,或
  • obtain exact solutions for realistic sources but restricting ourselves to the use of a few prescribed geometries (separable coordinate systems) for which an exact analysis is possible
    为现实源获取精确解,但仅限于使用少数几个规定的几何图形(可分离坐标系),对这些几何图形可以进行精确分析
The first choice is not of interest to us. The second choice may be workable and we will investigate that alternative later, but for now we will choose the third option in order to get exact answers, which we can use later as a comparison to the approximate solutions.
我们对第一种选择不感兴趣。第二种选择可能可行,我们稍后会研究这种选择,但现在我们会选择第三种选择,以便得到精确的答案,稍后我们可以用它与近似解进行比较。

We will also get a better understanding of the nature of the exact solution.
我们还将更好地了解精确解的性质。
Returning to the above example, we can see that a planar source at the throat of a Spherical waveguide will have more than a single mode of propagation due to the required fitting of this source to the waveguide at the throat.
回到上面的例子,我们可以看到,球形波导喉部的平面声源将不止一种传播模式,这是因为该声源需要与喉部的波导相匹配。

By expanding the source velocity in a series of modified Legendre Polynomials, we can determine the contribution of the various modes of the waveguide. We saw an example of this in Sec.
通过在一系列修正的 Legendre 多项式中展开源速度,我们可以确定波导中各种模式的贡献。我们在第二章中看到了一个例子。

4.5 on page 83 where we expanded a spherical wavefront in terms of a set of plane aperture modes. We are now doing the reverse, namely, expanding a plane wave in an aperture in terms of a set of finite angular spherical modes.
在第 83 页的 4.5 节中,我们用一组平面孔径模来扩展球面波。现在我们反过来,用一组有限角球面模来扩展孔径中的平面波。

The two processes are completely analogous albeit reversed.
这两个过程虽然相反,但完全类似。
Each of the waveguide modes will propagate with a different phase and amplitude which can readily be calculated as
每种波导模式都会以不同的相位和振幅传播,其计算公式为
The eigenvalues need to be determined specifically for each waveguide since they vary with the angle of the walls. The function is the same Spherical Hankel Function (of the second kind - outgoing) that we have seen before except that now these functions have a non-integer order.
由于特征值 会随波导壁角度的变化而变化,因此需要根据每个波导的具体情况来确定。函数 与我们之前看到的球形汉克尔函数(第二类--传出)相同,只是现在这些函数具有非整数阶。

The calculation of these functions is not difficult although the details are beyond the scope of this text and covered elsewhere 5 .
这些函数的计算并不困难,尽管细节超出了本文的范围,在其他地方也有涉及5。
As we saw in Sec.4.2, the modes radiate (propagate in the current case) with efficiencies which vary with frequency. Fig. 6-4 shows the modal impedances for the first three modes in a Spherical waveguide. From this figure, we can see precisely the differences in horn theory and waveguide theory.
正如我们在第 4.2 节中所看到的,这些模式的辐射(在当前情况下是传播)效率随频率而变化。图 6-4 显示了 球形波导中前三个模态的模态阻抗。从图中我们可以清楚地看到喇叭理论和波导理论的不同之处。
Horn theory yields only the solid line shown in this figure, which is exactly the same as the waveguide calculation for this lowest order mode. Above (where is the radius of the waveguide's throat), the first mode begins to cut-in.
喇叭理论只得到了图中所示的实线,这与波导对这一最低阶模式的计算结果完全相同。在 以上(其中 为波导喉部半径),第一个模式开始切入。

5. See Zhang, Computation of Special Functions
5.见 Zhang,《特殊函数的计算》。

Figure 6-4 - Real and imaginary parts of the modal impedances for a
图 6-4 - a 的模态阻抗的实部和虚部
Below this frequency both the horn theory of Webster and waveguide theory will yield nearly identical results (a small difference is due to the finite imaginary part of the higher order modes).
在此频率以下,韦伯斯特的号角理论和波导理论将得出几乎相同的结果(由于高阶模式的有限虚部,两者之间存在微小差异)。

Above this frequency the first mode (which would be quite significant for a piston source driving a Spherical waveguide) has an even greater proportional effect on the wavefront than the zero order mode, and could hardly be ignored for accurate results.
在此频率以上,第一种模式(对于驱动球形波导的活塞源来说非常重要)对波面的比例影响甚至大于零阶模式,因此很难忽略它以获得准确的结果。

It is in this region (above cut-in of the first mode) that horn theory has serious shortcomings. Its validity becomes progressively worse as the frequency goes up and even more modes cut-in.
正是在这一区域(第一模切入以上),喇叭理论存在严重缺陷。随着频率的升高和更多模式的切入,喇叭理论的有效性会逐渐减弱。

Waveguide theory remains accurate to as high a value as one cares to calculate its modes. This is a significant difference in accuracy for a large directivity controlling device.
波导理论在计算其模式时,其精确度仍可达到所需的最高值。对于大型指向性控制设备来说,这在精确度上是一个很大的差别。
Now that we have seen why waveguide theory is preferable to horn theory for high frequency directivity controlling devices, we will investigate the various separable coordinate systems for which waveguide theory is directly applicable in order to determine which ones have useful geometries.
既然我们已经了解了在高频指向性控制设备方面波导理论优于喇叭理论的原因,那么我们将研究波导理论可直接适用的各种可分离坐标系,以确定哪些坐标系具有有用的几何形状。

6.3 Waveguide Geometries
6.3 波导几何形状

We have already discussed several of the separable coordinate systems, but below is a table of the complete set of 11 along with the type of source that is required at the throat for a pure zero order mode.
我们已经讨论过几种可分离坐标系,下面是完整的 11 种坐标系以及纯零阶模式喉部所需的声源类型表。
Name 名称 Coordinate 协调
 光圈来源
Source
Aperture
 源曲率
Source
Curvature
 口腔弧度
Mouth
Curvature
Rectangular 矩形 any 任何 rectangle 矩形 flat 单位 flat 单位
Circular Cylinder 圆柱形 radial 迳向 rectangle 矩形 cylindrical 外圆 cylindrical 外圆
Elliptic Cylinder 椭圆柱 radial 迳向 rectangle 矩形 Flat 扁平 cylindrical 外圆
Parabolic Cylinder 抛物线圆柱体 none 
Spherical 球形 radial 迳向 circular 通报 spherical 球形 spherical 球形
Conical 锥形 radial 迳向 elliptical 椭圆形 spherical 球形 spherical 球形
Parabolic 抛物线 none 
Prolate Spheroidal 扁球形 radial 迳向 rectangle 矩形 cylindrical 外圆 spherical 球形
Oblate Spheroidal 扁球形 radial 迳向 circular 通报 flat 单位 spherical 球形
Ellipsoidal 椭圆体 radial 迳向 elliptical 椭圆形 flat 单位 spherical 球形
Paraboloidal 抛物面 none 
Table 6.1: Useful waveguides for Separable Coordinates
表 6.1:可分离坐标的有用波导
All of the useful coordinates are radial and all of the mouth apertures are the same as the throat apertures (not shown). The source apertures are either rectangular or elliptical (circular being a special case of elliptical).
所有有用坐标都是径向坐标,所有口孔都与喉孔相同(未显示)。声源孔径为矩形或椭圆形(圆形是椭圆形的特例)。

The mouth curvatures (radiation wavefronts) can only be spherical or cylindrical, flat being of little interest. This last feature is the main reason why we studied the geometries that we did in Chap. 4.
口部曲率(辐射波面)只能是球形或圆柱形,扁平的并不重要。这也是我们在第 4 章中研究几何图形的主要原因。

If the apertures are circular then the physical device must be axi-symmetric. The wave propagation need not be axi-symmetric, however. We will not look into this possibility since it is rather unusual in practice.
如果孔径是圆形的,那么物理装置必须是轴对称的。然而,波的传播不一定是轴对称的。我们将不研究这种可能性,因为它在实践中并不常见。
It is also possible to combine waveguides to create new devices. For example, the Prolate Spheroidal (PS) waveguide takes a square cylindrically curved wavefront at its throat, which is exactly what an Elliptic Cylinder waveguide produces.
还可以将波导组合起来,创造出新的设备。例如,Prolate Spheroidal(PS)波导在其喉部产生方形圆柱弯曲波面,而这正是椭圆圆柱波导所产生的波面。

A waveguide created as a combination of two waveguides in these two coordinate systems would take a square flat wavefront as input and produce a square spherical one.
由这两个坐标系中的两个波导组合而成的波导,将以方形平面波面作为输入,并产生一个方形球面波面。
It is interesting to note that the horn equation is only exact when the input and output wavefront curvatures remain unchanged. By this we mean that the location of the center of radius of the wavefront for both the throat and the
值得注意的是,只有当输入和输出波面曲率保持不变时,喇叭方程才是精确的。这意味着波面半径中心的位置对于喉部和

mouth does not move in space. This is exactly what it means to have a unity scale factor. Unfortunately, geometries that do not have unity scale factors are significantly more difficult to analyze - the price that we must pay for the higher accuracy of the waveguide approach.
嘴巴在空间中不会移动。这正是比例系数为一的含义。遗憾的是,不具有统一比例系数的几何图形分析起来要困难得多,这也是我们必须为波导法的高精度付出的代价。

We have already looked at a Spherical waveguide in some detail and now we will investigate a waveguide that is based on the Oblate Spheroidal (OS) coordinate system in order to compare and contrast its characteristics with those that we have already studied.
我们已经对球面波导进行了较为详细的研究,现在我们将研究一种基于扁球面 (OS) 坐标系的波导,以便将其特性与我们已经研究过的波导进行比较和对比。

6.4 The Oblate Spheroidal Waveguide
6.4 橢圓球面波導

Proceeding as in the previous sections, the first calculations that we need to do for an OS waveguide are to determine the wave functions (or Eigenfunctions) in this coordinate system.
与前几节一样,我们需要对 OS 波导进行的第一项计算是确定该坐标系中的波函数(或特征函数)。

Unlike the previous case of the Spherical Wave Equation, the wave functions for the OS coordinate system are not as readily available.
与之前的球面波方程不同,操作系统坐标系的波函数并不容易获得。

The unique thing about those coordinate systems that do not have unity scale factors is; even though the equations separate in the spatial coordinates they remain coupled through the separation constants (the wavenumber or time coordinate).
这些坐标系没有统一的比例因子,其独特之处在于:即使方程在空间坐标中分离,它们仍然通过分离常数(波长或时间坐标)耦合在一起。

We have not encountered this complication in any of the problems that we have studied thus far. The wave functions in both the radial coordinate and the angular coordinate will be found to depend on a common parameter , where is the inter-focal separation distance (See Fig.1-2 on pg. 6). (We must be careful not to confuse this (non-italic-bold) with the wave velocity , of the same letter. The use of is historical and the authors do not feel privileged enough to change it.)
在我们迄今研究的问题中,还没有遇到过这种复杂情况。径向坐标和角坐标的波函数都将取决于一个共同的参数 ,其中 是焦距间距(见第 6 页图 1-2)。(我们必须注意不要将 (非大写加粗)与同一字母的波速 混淆。 的使用是有历史渊源的,作者认为没有足够的特权来改变它)。
The separated Wave Equation in OS Coordinates is
OS 坐标中的分离波方程为
and 
For brevity, we have already simplified these equations by assuming axi-symmetric wave propagation around the waveguide and set the value of ( is the traditional constant for the coordinate and found in most texts on OS and PS wave functions). All of the published information on the OS wave functions assumes periodicity in , which is a different boundary condition than what we require here. We must apply a zero velocity (zero gradient) boundary condition at the walls just as we did in the previous section. Unfortunately, none of the published tables and subroutines which are readily available can be applied to our problem.
为了简洁起见,我们已经简化了这些方程,假定波导周围的波是轴对称传播的,并设置了 的值 ( 坐标的传统常数,可在大多数有关 OS 和 PS 波函数的文章中找到)。所有出版的 OS 波函数资料都假定 具有周期性,这与我们这里所要求的边界条件不同。我们必须在壁面上应用 速度为零(梯度为零)的边界条件,就像上一节所做的那样。遗憾的是,已出版的表格和子程序都无法应用于我们的问题。
We will need to revise the techniques used in the published literature and apply them to the specific boundary conditions for our particular problem. To do this, we will use a differential equation solution technique known as "shooting."
我们需要修改已发表文献中使用的技术,并将其应用于我们特定问题的具体边界条件。为此,我们将使用一种称为 "射击 "的微分方程求解技术。
We must first calculate the Eigenvalues for the boundary conditions of our problem. These boundary conditions are
我们必须首先计算问题边界条件的特征值 。这些边界条件是
and 
The first condition allows us to only consider functions which are symmetric about . This means that only even values of will be considered. By starting at and we "shoot" to the point , where is the design angle of the waveguide, and enforce a boundary condition on the slope of these functions to be zero at that point. A point of clarification here: the design angle is not necessarily the "coverage" angle of the device.
第一个条件允许我们只考虑关于 对称的函数。这意味着只考虑 的偶数值。从 开始,我们 "射击 "到 点,其中 是波导的设计角,并在该点强制执行这些函数斜率为零的边界条件。这里有一点需要说明:设计角并不一定是设备的 "覆盖 "角。

This issue will be investigated in more detail later on, but for now, it is important to note that here refers to the physical angle of the walls of the waveguide.
稍后我们将对这一问题进行更详细的研究,但现在需要注意的是,这里 指的是波导壁的物理角度。
The Eigenvalues are known for , from the spherical case, and they are . It is also known that the Eigenvalues will decrease as increases at a constant rate. Using these approximations to the Eigenvalues as starting values, we calculate the exact Eigenvalues for any given value of by "shooting" from one boundary to the other. The Eigenvalue is adjusted until a satisfactory match has been achieved between the boundary conditions at the two end points. Once we have the Eigenvalue we simply use standard numerical integration to compile the angular functions (where ).
根据球面情况, 的特征值是已知的,它们是 。此外,我们还知道,随着 以恒定速率增加,特征值也会减小。利用这些特征值的近似值作为起始值,我们通过从一个边界到另一个边界的 "射击",计算出 的任何给定值的精确特征值。对特征值进行调整,直到两个端点的边界条件达到令人满意的匹配为止。得到特征值后,我们只需使用标准数值积分来编制角度函数 (其中 )。
Using the Eigenvalues calculated from the above calculations we can also numerically integrate the radial functions starting at with a slope of zero and an arbitrary value. It can be shown that this will give the correct form for the radial functions, but not the correct scaling. The radial functions must be scaled to match the Spherical Hankel Functions for large , because the magnitudes of both functions must asymptotically approach each other at large distances from the source. This process is not too difficult for us to contemplate, but it is a heck of a lot of work for the computer!
利用上述计算得出的特征值,我们还可以对径向函数进行数值积分,从 开始,斜率为零,取任意值。可以看出,这将给出径向函数的正确形式,但不是正确的缩放。必须对径向函数进行缩放,使其与球面汉克尔函数(Spherical Hankel Functions for large )相匹配,因为在距离源很远的地方,这两个函数的大小必须渐近地相互接近。这个过程对我们来说并不难,但对计算机来说却是一项艰巨的工作!
Fig. 6-5 shows the Oblate Spheroidal angular wave functions at for a waveguide. These functions are similar to the corresponding wave functions for a Spherical waveguide due to the relatively low frequency (c value).
图 6-5 显示了 波导在 处的 Oblate Spheroidal 角波函数。由于频率(c 值)相对较低,这些函数与球面波导的相应波函数相似。

This similarity between the functions supports our contention that at low frequencies nearly all shapes of waveguides with similar flare rates act just about the same, the flare rate being the only aspect of importance - i.e. it sets the location of the low-
函数之间的这种相似性支持了我们的论点,即在低频下,几乎所有具有相似耀斑率的波导形状的作用都差不多,耀斑率是唯一重要的方面,即它决定了低频的位置。

6. See Press, Numerical Recipes, Chap. 11
6.参见 Press, Numerical Recipes, Chap.

Figure 6-5- Angular wave functions for case,
图 6-5- 情况下的角波函数、
est frequency of significant transmission usually called cutoff. Of note in this figure is the fact that the lowest order mode, which is always independent of angle for a Spherical waveguide, is beginning to become curved with respect to in the OS waveguide. This means that even if we drive an OS waveguide with a flat piston, one which perfectly matches the aperture requirements for this waveguide, it will still generate higher order modes. This effect becomes greater as the frequency increases.
在球面波导中,最低阶模式始终与角度无关,而在 OS 波导中,最低阶模式开始变得弯曲。图中值得注意的是,对于球形波导来说,最低阶模式总是与角度无关,而在 OS 波导中,最低阶模式开始变得相对于 而弯曲。这意味着,即使我们用完全符合该波导孔径要求的扁平活塞驱动 OS 波导,它仍会产生高阶模式。频率越高,这种影响越大。
Fig. 6-6 shows the OS coordinate waveguide radial functions for both the real and imaginary parts for the case at . (The Spherical radial wave function - Spherical Hankel Function of the second kind - is also shown in these plots for reference.)
图 6-6 显示了 情况下 的 OS 坐标波导径向函数的实部和虚部(球面径向波函数--第二类球面汉克尔函数--也显示在这些图中,以供参考)。
The imaginary parts of the radial wave functions can be very difficult to develop owing to the near singularity at the origin.
由于原点处的奇异性,径向波函数的虚部可能很难展开。

The slope of these functions is known (from the Wronskian as shown below) but the values at the origin, which yields the proper asymptotic scaling, must be determined.
这些函数的斜率是已知的(如下图所示,来自弗伦斯基函数),但必须确定原点处的值,这样才能得到适当的渐近比例。

Convergence of these functions requires a very high degree of accuracy in finding this value at the origin - about 15 significant digits for the mode at . This makes it almost impossible to calculate these functions on a computer using standard iterative techniques for small values of at the higher modal orders. The imaginary parts of the wave functions are usually required in order to calculate the modal radiation impedances. We will see a way around this situation shortly.
这些函数的收敛需要非常高的精度,才能找到原点处的这个值--对于 的模态 ,需要约 15 个有效位数。因此,对于模态阶数较高的 的较小值,几乎不可能使用标准迭代技术在计算机上计算这些函数。为了计算模态辐射阻抗,通常需要波函数的虚部。我们很快就会看到解决这种情况的方法。

Figure 6-6 - The OS radial wavefunctions for a waveguide at
图 6-6 - 波导在以下位置的 OS 径向波函数
Fig.6-7 shows the angular wave functions for the same waveguide as in the previous figure but at a value of . Here we see that lowest order wave function is becoming even more curved relative to the flat aperture at the throat. This means that there will be a significant amount of the mode present when this aperture is driven by a flat wavefront.
图 6-7 显示了与上图相同波导的角波函数,但波导值为 。在这里我们可以看到,相对于喉部的平面孔径,最低阶波函数变得更加弯曲。这意味着,当该孔径由平坦波面驱动时,将存在大量的 模式。
Figure 6-7 - Angular wave functions for OS waveguide at
图 6-7 - OS 波导的角度波函数,位于
Fig. 6-8 shows the radial wave functions for the case at .
图 6-8 显示了 情况下的径向波函数,
We can now compute the modal impedances for the radiation modes as follows.
现在我们可以计算出辐射模式的模态阻抗如下。
  • Calculate the radial wave functions finding the value required at the origin to yield the correct asymptotic values at high .
    计算径向波函数,找出原点所需的值,以便在高 时得出正确的渐近值。
  • Note that the Wronskian (a characteristic of all PDE's) for the OS Coordinates is
    请注意,OS 坐标的 Wronskian(所有 PDE 的特征)为
which when evaluated at yields
进行求值,得到
  • Using this know slope for the radial wave function of the second kind at the origin, we can use ordinary integration to calculate the function to some large value of . Once again we compare the magnitude of this function to that of the Spherical Bessel Functions and iterate the
    利用原点处第二类径向波函数的已知斜率,我们可以使用普通积分来计算函数到 的某个大值。我们再次将该函数的大小与球面贝塞尔函数的大小进行比较,并迭代出

Figure 6-8 - The radial wave functions for a waveguide at
图 6-8 - 波导在以下位置的径向波函数

starting value until the two functions match amplitudes at these higher radial values.
直到两个函数在这些较高径向值上的振幅相匹配。
This is a tremendous amount of work, and the results are known . We will not elaborate on the details of these calculations, but we will show the results. Fig. 6-9 shows the modal impedances of the waveguide for the first two modes. The third mode would not appear on this graph's scale. Thus only the first two modes are of significance for this waveguide over the bandwidth of interest. The second mode, , enters the picture at a value of , corresponding to a frequency of, (for a one-inch radius throat)
这是一项艰巨的工作,其结果已在 上公布。我们将不再详述这些计算的细节,但会展示计算结果。图 6-9 显示了 波导前两种模式的模态阻抗。第三个模式 不会出现在该图的比例尺上。因此,在所关注的带宽范围内,只有前两种模式对该波导具有重要意义。第二种模式 的值为 ,对应的频率为(对于半径为一英寸的喉管)。
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-
在此值以上,二阶模式的传播影响将等于或大于 "平面 "波模式, 。在 附近,这种波导将在很大程度上取决于控制喉部波面的部件(相位塞等)的具体配置。在 以下,喉部孔径处的波面几何形状并不重要,因为只有最低阶模式--基本上是速度--的平均值--才是最重要的。
  1. See Geddes, "Acoustic Waveguide Theory Revisited", JAES
    见 Geddes,"声波导理论再探",JAES

    ity of wavefront across the aperture - will propagate. At low frequencies, the details of the throat wavefront are irrelevant.
    在低频情况下,喉管波面的细节并不重要。在低频情况下,喉管波面的细节无关紧要。
The mode in the OS waveguide exhibits an impedance characteristic which is similar to that for a simple Spherical waveguide or conical horn.
模式在 OS 波导中的阻抗特性与简单球形波导或锥形喇叭的阻抗特性相似。

We must be careful in this comparison however, because even though the impedance and transfer characteristics for the OS waveguide are similar to those for a Spherical waveguide there are still significant differences.
不过,我们在进行比较时必须小心谨慎,因为尽管 OS 波导的阻抗和传输特性与球形波导相似,但两者之间仍存在显著差异。
Fig. 6-10 shows the velocity distribution at the mouth for the waveguide. (These are the velocity amplitudes normal to the spherical surface defined by the mouth.) These velocities are dependent on both the frequency and the angle . Note that the velocity gets greater at the center, and that this effect increases rapidly with frequency after about where the second mode is becoming significant. This velocity distribution calculation at the mouth is one of the most important distinctions between waveguide theory and horn theory.
图 6-10 显示了 波导口处的速度分布。(这些是波导口球面法线方向的速度振幅)。这些速度取决于频率和角度 。请注意,中心位置的速度会变大,而且在 之后,这种影响会随着频率的增加而迅速增大,此时第二种模式会变得非常重要。口部的速度分布计算是波导理论与喇叭理论之间最重要的区别之一。

Waveguide theory predicts a significant variation of the wavefront amplitudes across the mouth of the device even when driven by a uniform velocity distribution at the throat. Horn theory can only predict amplitudes which are independent of angle, which is clearly incorrect.
根据波导理论的预测,即使在喉部的速度分布均匀的情况下,整个装置口的波面振幅也会有明显的变化。喇叭口理论只能预测与角度无关的振幅,这显然是不正确的。
Consider now a waveguide. Several things happen when we increase the waveguide coverage angle. First, the modes cut in at a lower value of as shown in Fig. 6-11. The second mode is now significant, above about and we can see that the third mode will be a factor in the passband of the device. The
现在考虑一个 波导。当我们增大波导覆盖角时,会发生几种情况。首先,如图 6-11 所示,模式会在 的较低值处截止。现在,第二种模式在 以上非常重要,我们可以看到,第三种模式 将成为器件通带的一个因素。图 6-11
Figure 6-10 - Mouth radial velocity amplitude
图 6-10 - 口径径向速度振幅
Figure 6-11 - The impedances for the waveguide
图 6-11 - 波导的阻抗
next aspect of the angle increase is that the wave functions vary in to a greater extent with the larger angle. All of these effects add up to cause an even greater focusing of the wavefront velocities towards the center of the mouth. We have not yet shown whether this is good or bad, but it is important to note the effect.
角度增大的另一个方面是,随着角度的增大,波函数在 上的变化幅度也会增大。所有这些影响加在一起,会使波面速度更加集中于口腔中心。我们还没有证明这是好是坏,但注意到这种影响是很重要的。
If instead of driving the throat with a wavefront of constant amplitude we taper this amplitude as shown in Fig.
如果我们不使用恒定振幅的波阵面来驱动喉管,而是将振幅变细,如图所示。

6-12, then the net effect will be to create a distribution of the velocity at the mouth which has a far more uniform distribution than one fed with a flat throat velocity distribution . This is an important result, for it means that better control of the sound radiation coverage of a waveguide can be achieved by manipulating the velocity distribution at the throat. Horn theory could never have predicted this result.
6-12,那么净效果将是在口部产生一个速度分布,其分布比一个平坦的喉部速度分布 要均匀得多。这是一个重要的结果,因为它意味着通过操纵喉部的速度分布,可以更好地控制波导的声辐射覆盖范围。喇叭理论从未预测到这一结果。

Phasing plugs in compression drivers in common use today are principally designed to create a flat velocity distribution, because horn theory did not have the sophistication to consider anything else.
目前常用的压缩驱动器中的相位塞主要是为了产生平坦的速度分布,因为当时的号角理论还不成熟,无法考虑其他因素。

The implications of this result to the phasing plug design is that, in essence, it must be part of the waveguide design and not part of the compression driver design.
这一结果对相位塞设计的影响是,从本质上讲,它必须是波导设计的一部分,而不是压缩驱动器设计的一部分。

In the future phasing plugs will certainly be made to better adapt the device driving the waveguide to the requirements of the waveguide itself. The phasing plug is a variable in the design problem, not a fixed component.
未来,相位调节插头的制造肯定会使波导驱动装置更好地适应波导本身的要求。相位调节插头是设计问题中的一个变量,而不是一个固定部件。
At this point, it would be a good idea to review the key aspects of the waveguide theory developed in this chapter:
在此,我们不妨回顾一下本章所阐述的波导理论的主要方面:
  1. See Geddes, "Acoustic Waveguide Theory - Revisited", JAES.
    见 Geddes,"Acoustic Waveguide Theory - Revisited",JAES。
Figure 6-12 - Proposed throat velocity distribution for a flatter velocity distribution at the mouth
图 6-12 - 建议的喉部速度分布,使河口处的速度分布更平缓
  • All waveguides (as well as horns) have higher order modes. The fact that horn theory neither predicts nor is able to deal with this situation is a serious failing of the theory.
    所有波导(以及喇叭)都有高阶模式。喇叭理论既不能预测也不能处理这种情况,这是理论的严重缺陷。
  • The wavefront geometry (magnitude and phase distribution) at the throat of the waveguide is critical to its performance at higher frequencies.
    波导喉部的波面几何形状(幅度和相位分布)对波导在较高频率下的性能至关重要。
  • The loading aspects of nearly all waveguide/horn devices is, for all practical purposes, the same. The total encompassed solid angle of radiation is really the only factor influencing the loading.
    实际上,几乎所有波导/喇叭装置的负载都是一样的。总的辐射实体角是影响负载的唯一因素。
  • Horn theory is adequate only for the low frequency aspects of waveguides - well below the first mode cut-in and even then it gives no indication as to what the wavefront shape is at the mouth.
    喇叭理论仅适用于波导的低频方面--远低于第一模切入点,即便如此,它也无法说明波导口处的波面形状。

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,我们将回到第 6.2 节中关于波导评估近似方法的讨论,该方法不符合可分离坐标系,但保留了波导理论中对可接受结果所必需的特征。
The question is: can we find a way to do an approximate numerical calculation while still retaining the main features of waveguide theory?
问题是:我们能否找到一种既能进行近似数值计算,又能保留波导理论主要特征的方法?

Clearly, any new technique must include the possibility for higher order modes, and it must be able to predict the actual wavefront distributions at the output (mouth) for any given distribution at the input (throat).
显然,任何新技术都必须包括高阶模式的可能性,而且必须能够预测输入端(喉管)的任何给定分布在输出端(口)的实际波面分布。
The way to do this is a modification of the obvious technique of breaking a waveguide down into a series of finite spherical sections.
这种方法是对将波导分解成一系列有限球形截面的显而易见的技术进行修改。

This is an old technique but we will add one new feature - we will track all of the modes, including the higher order ones, as they progress through the elements.
这是一项老技术,但我们将增加一项新功能--我们将跟踪所有模式,包括高阶模式,因为它们在元素中不断变化。

We will only develop and outline this technique because time and space constraints will not allow us to show an example of its application. To do a thorough study of this technique and its implications would require far too much space.
由于时间和篇幅所限,我们无法举例说明这一技术的应用,因此我们将只对其进行阐 述和概述。如果要对这一技术及其影响进行深入研究,将需要太多的篇幅。

We will disclose the techniques and leave it to the reader to develop the applications.
我们将披露相关技术,并让读者自行开发应用。
Consider the geometry shown in Fig. 6-13. The waveguide is broken into four sections where each section is a section of a cone in Spherical Coordinates. Calculation of the first order mode down this waveguide is trivial.
请看图 6-13 所示的几何图形。波导被分成四段,每段都是球面坐标中锥体的一段。计算该波导的一阶模式非常简单。

It is done by simply multiplying together the T-matrices for each section to yield the composite matrix which represents the whole waveguide.
只需将每个部分的 T 矩阵相乘,即可得到代表整个波导的复合矩阵。

The problem with this approach is that it ignores the presence and propagation of higher order modes within the waveguide, just as horn theory does.
这种方法的问题在于,它忽略了波导内高阶模式的存在和传播,就像喇叭理论一样。

Figure 6-13 - A simple waveguide broken into sections and a detail of a single junction
图 6-13 - 分成几个部分的简单波导和单个结点的细节
It can be seen in the right hand side of this figure that at each junction between the sections, the wavefronts are not contiguous - the radius of these waves must changes between the two sections at each junction.
从图中右侧可以看到,在每个部分之间的交界处,波面并不连续--在每个交界处的两个部分之间,这些波的半径必须发生变化。

In order for this wavefront to propagate from one section into the next section we must match the wavefronts by creating higher order mode modes in the second section. These higher order modes are required for the wavefronts to match at the junction of the two sections.
为了让波面从一个部分传播到下一个部分,我们必须在第二个部分产生高阶模式,从而匹配波面。这些高阶模式是波面在两段交界处匹配的必要条件。

We will then have two (or more) modes which must be propagated through each of the following sections. This same situation will occur at each and every junction, thus continually increasing the higher order mode content of the wavefront.
这样,我们就会有两个(或更多)模式必须通过下面的每个部分传播。每一个交界处都会出现同样的情况,从而不断增加波面的高阶模式含量。
The only way for this higher order mode creation to not occur would be for there to be a wavefront radius source point which did not move in space, as opposed to one that is changing in each section.
不产生这种高阶模式的唯一方法是,波面半径源点在空间中不移动,而不是在每个截面中不断变化。

It is now readily apparent that any waveguide which has a changing location of the origin for the wavefront radius will require the presence of higher order modes to account for this changing origin.
现在很明显,任何波导,如果波面半径的原点位置不断变化,就需要有高阶模式来解释这种原点的变化。

In separable coordinates this changing radius location is exactly what the coordinate scale factors account for and exactly why the equations have become so much more complicated. This is another way of looking at the results that we elaborated on in the previous sections.
在可分离坐标中,这种半径位置的变化正是坐标比例因子所要考虑的,也正是方程变得如此复杂的原因。这也是我们在前几节中阐述的结果的另一种理解方式。
The entire concept of one-parameter (1P) waves is thus shattered by the realization that there can only ever be three waveguides, none of which are of interest to us, in which there can be true behavior and then only if we feed them with non-existent sources. All other geometries and source configurations will have higher order modes no matter how we attempt to minimize them.
因此,单参数(1P)波的整个概念都被打破了,因为我们发现只有三种波导(我们对其中的任何一种都不感兴趣)可能存在真正的 行为,而且只有当我们向它们馈入不存在的源时才会出现这种行为。所有其他几何形状和声源配置都会产生高阶模式,无论我们如何试图将其最小化。
Furthermore, we cannot circumvent this problem by making more sections and thus a smaller change between sections, resulting in less higher order mode creation at each junction.
此外,我们无法通过增加分段数来规避这一问题,因为分段数越多,分段之间的变化就越小,从而导致每个交界处产生的高阶模式越少。

The smaller values of the higher order mode components are multiplied many more times by the increased number of junctions, resulting in the exact same result that we have described above.
高阶模式成分的较小值与增加的连接点数量相乘,结果与上述结果完全相同。

There is simply no way around the conclusion that, in order to be accurate all waveguide calculations, we must include the presence of higher order modes or they are seriously flawed.
我们无法回避这样一个结论:为了保证所有波导计算的准确性,我们必须将高阶模式的存在包括在内,否则计算就会存在严重缺陷。
With this realization in mind we can be thankful that we have developed the machinery to deal with this complication, namely the T-matrix. By adding two more dimensions to the -matrices - for each higher order mode that we want to calculate - we can accommodate this new complication. We will have larger matrices to deal with ( for two modes, for three modes and so on), but we'll let the computer deal with that problem.
有鉴于此,我们可以庆幸我们已经开发出了处理这一复杂问题的机制,即 T 矩阵。通过为 - 矩阵增加两个维度--针对我们想要计算的每一个高阶模式--我们就可以适应这一新的复杂性。我们将需要处理更大的矩阵( 用于两种模式, 用于三种模式,依此类推),但我们会让计算机来处理这个问题。
The T-matrix for the two mode spherical element is easily derived from the Spherical Wave Equations by using techniques identical to those we used in Sec.5.2 on page 95, only now we are using two modes in Spherical Coordinates.
通过使用与第 95 页第 5.2 节中相同的技术,可以很容易地从球面波方程中推导出二模球面元素的 T 矩阵,只是现在我们使用的是球面坐标中的两种模式。
Without belaboring the details in the derivation of the T-matrix for this problem, the results are
无需赘述推导该问题 T 矩阵的细节,结果如下