We shall leave it to the student to show the following: if we call themaximum height of the curve of ρ2 vs. ω one unit, and we ask for the width Δω of the curve, at one half the maximum height, the full width at halfthe maximum height of the curve is Δω=γ , supposing that γ is small. The resonance is sharper and sharper as the frictional effectsare made smaller and smaller.
下面的问题，留给学生去指出：对于ρ2对 ω的曲线，如果我们把其最大高度，称为一个单位，那么，我们问，在最大高度的一半处，曲线的宽度Δω，假设γ很小，在曲线最大高度的一半处，完整宽度是Δω=γ。随着摩擦效果越来越小，共振将越来越尖锐。{？}

As another measure of the width, some peopleuse a quantity Q which is defined as Q=ω0/γ . The narrower the resonance, the higher the Q : Q=1000 means a resonance whose width is only 1000 th of the frequency scale. The Q of the resonance curve shown in Fig. 23–2is 5 .
对于宽度，还有另一种测量方法，有些人，使用量Q，它被定义为 Q=ω0/γ。共振越窄， Q值越高：Q=1000，意味着共振的宽度，是频率尺度的1000分之一。图23-2所示的共振曲线，Q值为5。

The importance of the resonance phenomenonis that it occurs in many other circumstances, and so the rest of this chapterwill describe some of these other circumstances.
共振现象，之所以重要，乃是因为，它会出现在很多情形中，所以，本章剩下的部分，将会讲述，这些情形中的一部分。

23–3Electrical resonance 23-3 电共振
The simplest and broadest technicalapplications of resonance are in electricity. In the electrical world there area number of objects which can be connected to make electric circuits. These passivecircuit elements, as they are often called, are of three main types,although each one has a little bit of the other two mixed in. Before describingthem in greater detail, let us note that the whole idea of our mechanical oscillatorbeing a mass on the end of a spring is only an approximation. All the mass is notactually at the “mass”; some of the mass is in the inertia of the spring.Similarly, all of the spring is not at the “spring”; the mass itself has alittle elasticity, and although it may appear so, it is not absolutely rigid,and as it goes up and down, it flexes ever so slightly under the action of thespring pulling it. The same thing is true in electricity. There is anapproximation in which we can lump things into “circuit elements” which areassumed to have pure, ideal characteristics. It is not the proper time todiscuss that approximation here, we shall simply assume that it is true in thecircumstances.
共振的最简单和最广泛的技术应用，在电中。在电的世界中，有若干对象，可与制造电路，联系起来。这些电路，通常被称为无源电路器件，主要分三种类型，虽然每一种中，都混有其他两个的一点点。在更仔细地描述它们之前，让我们注意一下，我们整个机械振荡的想法，是一个质量，挂在弹簧一端，这个想法，实际上只是一个近似。所有的质量，实际上，并不全在那个“质量”上，有些质量，是在弹簧的惯性中，类似地，所有的弹簧，也不全在弹簧中，质量本身，也有一点弹性，虽然，它可以表现地像刚体，但当它上下运动时，它并不是绝对的刚体，而是，在弹簧拉它时，它有轻微的弯曲。在电中，存在同样的事情。有一个近似，在其中，我们可以把事物，堆积进“电路元件”中，这些元件，被假定具有纯粹的、理想的特性。在这里讨论此近似，时间并不合适；我们将只简单地假定，这种情形，是真实的。

Fig. 23–4.The three passive circuitelements. 图23-4 三个无源电路元件（电容，电阻，感应线圈）

The three main kinds of circuit elementsare the following. The first is called a capacitor (Fig. 23–4); anexample is two plane metallic plates spaced a very small distance apart by aninsulating material. When the plates are charged there is a certain voltagedifference, that is, a certain difference in potential, between them. The samedifference of potential appears between the terminals A and B , because if there were any difference along the connecting wire,electricity would flow right away. So there is a certain voltage difference Vbetween the plates if there is a certain electric charge +q and −q on them, respectively. Between the plates there will be a certainelectric field; we have even found a formula for it (Chapters 13and 14):
三个主要的电路元件如下。第一个，被称为电容器（图23-4）；一个例子就是，两个平的金属板，中间距离很小，由隔离材料分开。当板子被充电后，就有一定的电压差，也就是说，在板子之间，电势上，有一定的差别。在终端A和 B上，会出现同样的电势差，因为，如果在连接线上，有任何差别的话，电立即就会流走。所以，如果在两个板子上，分别有一定的电荷+q 和−q，那么，其间就有一定的电压差 V。在板子之间，将会有一定的电场；我们甚至为它找到了一个公式（13章，14章）：
V=σd/ϵ0=qd/ϵ0A, (23.14)
where d is the spacing and A is the area of the plates. Note that the potential difference is alinear function of the charge. If we do not have parallel plates, but insulatedelectrodes which are of any other shape, the difference in potential is stillprecisely proportional to the charge, but the constant of proportionality maynot be so easy to compute. However, all we need to know is that the potentialdifference across a capacitor is proportional to the charge: V=q/C; the proportionality constant is 1/C , where C is the capacitance of the object.
这里，d是板子间的距离，A是板子的面积。注意，电势差是电荷的一个线性函数。如果我们所拥有的，不是平行的板子，而是隔离的、任意形状的电极，那么，电势差，就仍精确地正比于电荷，但是，比例常数，就不是那么容易计算了。然而，我们根本想知道的，是一个电容器的电势差，正比于电荷：V=q/C；比例常数是1/C，这里C是对象的电容量。

The second kind of circuit element iscalled a resistor; it offers resistance to the flow of electricalcurrent. It turns out that metallic wires and many other substances resist theflow of electricity in this manner: if there is a voltage difference across a pieceof some substance, there exists an electric current I=dq/dtthat is proportional to the electric voltage difference:
第二类电路元件，被称为电阻，对于电流的流动，它提供了阻力。结果就是，金属线和很多其他材料，以下面的方式，阻碍着电的流动：如果沿着某种材料，有电压差，那么，就有电流I=dq/dt，正比于电压差：
V=RI=Rdq/dt (23.15)
The proportionality coefficient is called the resistance R. This relationship may already be familiar to you; it is Ohm’s law.
比例系数，被称为电阻R。这个关系，你可能已经熟悉了；它就是欧姆规律。

If we think of the charge q on a capacitor as being analogous to the displacement x of a mechanical system, we see that the current, I=dq/dt, is analogous to velocity, 1/C is analogous to a spring constant k , and R is analogous to the resistive coefficient c=mγ in Eq. (23.6).Now it is very interesting that there exists another circuit element which isthe analog of mass! This is a coil which builds up a magnetic fieldwithin itself when there is a current in it. A changing magnetic fielddevelops in the coil a voltage that is proportional to dI/dt(this is how a transformer works, in fact). The magnetic field isproportional to a current, and the induced voltage (so-called) in such a coilis proportional to the rate of change of the current:
如果我们把电容器上的电量q，认为是可类比于机械系统中的位移x，我们看到，电流I=dq/dt，就可类比于矢速，1/C可类比于弹簧常数 k, R可类比于方程（23.6）中的电阻系数c=mγ。现在，很有趣的就是，还有另外一个电路元件，是质量的类比！这是一个线圈，当它里面有电流时，就会在自身中产生磁场。一个改变着的磁场，会在线圈中，开发出电压，它正比于dI/dt（事实上，这就是变压器如何工作的）。磁场正比于电流，在这种线圈中的（所谓的）感应电压，正比于电流的变化率：
V=LdI/dt=Ld2q/dt2.(23.16)
The coefficient L is the self-inductance, and is analogous to the mass in amechanical oscillating circuit.
系数L，是自感系数，可类比于机械震荡回路中的质量。

Suppose we make a circuit in which we haveconnected the three circuit elements in series (Fig. 23–5);then the voltage across the whole thing from 1 to 2 is the work done in carrying a charge through, and it consists of thesum of several pieces: across the inductor, VL=Ld2q/dt2; across the resistance, VR=Rdq/dt ; across the capacitor, VC=q/C . The sum of these is equal to the applied voltage, V :
假设，我们建一个回路（图23-5），在其中，我们把三个电路元件，串连起来；那么，从1到2、沿着整个回路的电压，就是载着电荷通过其中所做的功，它由以下几部分的和组成：电感上的电压VL=Ld2q/dt2；电阻上的电压VR=Rdq/dt；电容上的电压VC=q/C。它们之和，就等于外加电压V :
Ld2q/dt2 + Rdq/dt+ q/C= V(t). (23.17)

Now we see that this equation is exactly the same as the mechanicalequation (23.6),and of course it can be solved in exactly the same manner. We suppose that V(t)is oscillatory: we are driving the circuit with a generator with apure sine wave oscillation. Then we can write our V(t) as a complex V^ with the understanding that it must be ultimately multiplied by eiωt, and the real part taken in order to find the true V . Likewise, the charge q can thus be analyzed, and then in exactly the same manner as inEq. (23.8)we write the corresponding equation: the second derivative of q is (iω)2q ; the first derivative is (iω)q . Thus Eq. (23.17)translates to
现在，我们看到，这个方程，与力学方程（23.6），几乎一样，当然，解它所用的方式，也几乎一样。我们假设，V(t)是震荡的：我们用一个发电机，来驱动电路，发电机拥有一个纯正弦波的震荡。因此，我们就可以把我们的V(t) ，写作V^，按我们的理解，它最终要乘以eiωt，取其实部，以找出真正的V。同样，电荷q，也可如此分析，然后，按照方程（23.8）中同样的方式，我们可以写出相应的方程：q的二阶导数，是(iω)2q ；一阶导数，是 (iω)q。这样，方程（23。17）就变为：
or
which we can write in the form
我们可把它写成如下形式：
(23.18)
where ω20=1/LC and γ=R/L. It is exactly the same denominator as we had in the mechanical case,with exactly the same resonance properties! The correspondence between the electricaland mechanical cases is outlined in Table 23–1.
这里，ω20=1/LCand γ=R/L。我们在力学情况中，所拥有的分母，与它几乎一样：拥有几乎一样的共振属性！电与力情况中的相应，概述在表（23-1）中。
表23-1

We must mention a small technical point. Inthe electrical literature, a different notation is used. (From one field toanother, the subject is not really any different, but the way of writing thenotations is often different.) First, j is commonly used instead of i in electrical engineering, to denote （-1）1/2 . (After all, i must be the current!) Also, the engineers would rather have arelationship between V^ and I^ than between V^ and q^ , just because they are more used to it that way. Thus, since I=dq/dt=iωq, we can just substitute I^/iω for q^ and get
有一个小的技术要点，我们要提一下。在电学资料中，用到了一个不同的表示法（从一个领域到另一个领域，主题确实并没有任何不同，但是，写表示法的方式，通常会不同）。首先，在电工程中，j通常被用来替换 i，以表示（-1）1/2（毕竟，i应该是电流！），另外呢，工程师宁肯喜欢V^ 和I^之间的关系，也不喜欢V^ 和 q^之间的关系，这只是因为，他们习惯了这种方式。这样，由于I=dq/dt=iωq，我们可以用I^/iω，替换q^，就得到：
(23.19)
Another way is to rewrite Eq. (23.17),so that it looks more familiar; one often sees it written this way:
另外一种方式，就是重写方程（23.17），这样，它看起来，就更熟悉些了；我们通常看到的，是这种方式：
(23.20)

At any rate, we find the relation (23.19)between voltage V^ and current I^ which is just the same as (23.18)except divided by iω , and that produces Eq. (23.19).The quantity R+iωL+1/iωC is a complex number, and is used so much in electrical engineeringthat it has a name: it is called the complex impedance, Z^ . Thus we can write V^=Z^I^ . The reason that the engineers like to do this is that they learnedsomething when they were young: V=RI for resistances, when they only knew about resistances and DC. Nowthey have become more educated and have ac circuits, so they want the equationto look the same. Thus they write V^=Z^I^ , the only difference being that the resistance is replaced by a morecomplicated thing, a complex quantity. So they insist that they cannot use whateveryone else in the world uses for imaginary numbers, they have to use a jfor that; it is a miracle that they did not insist also that theletter Z be an R ! (Then they get into trouble when they talk about current densities,for which they also use j . The difficulties of science are to a large extent the difficultiesof notations, the units, and all the other artificialities which are inventedby man, not by nature.)
无论如何，我们发现，在电压V^和电流I^之间的关系（23.19），与（23.18）正好一样，除了被除以iω，这就产生了方程（23.19）。量R+iωL+1/iωC，是一个复数，电工程中，被大量使用，以至于它有了一个名字：被称为复数阻抗。这样，我们就可以写V^=Z^I^。工程师们喜欢这样做，原因就是，当他们年轻时，学到了一些东西，对于电阻，V=RI，那时，他们只了解电阻和直流电。现在，他们受得更教育更多，了解了交流电路，于是，他们想让教育看上去，是一样的。这样他们就写，V^=Z^I^，唯一的区别，就是电阻，被一个更复杂的东西，代替了，即一个复数量。所以，尽管对于虚数，全世界的人，都在使用i，但他们坚持，他们不能使用它，他们必须使用 j，来代替它；他们并没有坚持，字母 Z应该是R，这真是一个奇迹！（因此，当他们谈论电流密度时，还会用j，这样，他们就会陷入麻烦。科学的很多困难，在很大程度上，是表示法、单位、和其他人造东西的困难，因为，它们是由人发明的，而不是由自然。）

23–4Resonance in nature 22-4 自然中的共振
Although we have discussed the electricalcase in detail, we could also bring up case after case in many fields, and showexactly how the resonance equation is the same. There are many circumstances innature in which something is “oscillating” and in which the resonancephenomenon occurs. We said that in an earlier chapter; let us now demonstrateit. If we walk around our study, pulling books off the shelves and simplylooking through them to find an example of a curve that corresponds toFig. 23–2and comes from the same equation, what do we find? Just to demonstrate the widerange obtained by taking the smallest possible sample, it takes only five orsix books to produce quite a series of phenomena which show resonances.
虽然，我们已经仔细地讨论了电的情况，对于很多领域中的情况，我们也可以一个一个地讨论，然后，准确地指出，共振方程，是如何一样的。在自然中，有很多情形，在其中，有“某物”在震荡，且在其中，也发生了共振现象。在较早的一章中，我们说过此事；现在，让我们演证之。如果我们绕着我们的书房转，把书从书架上拿下来，浏览它们，去找一个曲线的例子，该曲线，与图23-2相应，也是来自同一个方程，我们能找到什么呢？通过取最小可能的样本，来演证能广泛获得的东西，只需五到六本书，就会找到一个现象的序列，它会显示出共振。

Fig. 23–6.Response of the atmosphere toexternal excitation. a is the required response if the atmospheric S2 -tide is of gravitational origin; peak amplification is 100:1 . b is derived from observed magnification and phase of M2-tide. [Munk and MacDonald, “Rotation of the Earth,” CambridgeUniversity Press (1960)]
图23-6 大气对外部激发所产生的反应。如果大气的S2潮汐，是重力起源的，a就是所要求的反应；峰值放大是100:1。b从观测到的M2潮汐的放大值和相位出发，所导出的。[Munkhe MacDonald,“地球的旋转”，剑桥大学出版社（1960）]

The first two are from mechanics, the first on alarge scale: the atmosphere of the whole earth. If the atmosphere, which wesuppose surrounds the earth evenly on all sides, is pulled to one side by themoon or, rather, squashed prolate into a double tide, and if we could then letit go, it would go sloshing up and down; it is an oscillator. This oscillatoris driven by the moon, which is effectively revolving about the earth;any one component of the force, say in the x -direction, has a cosine component, and so the response of the earth’satmosphere to the tidal pull of the moon is that of an oscillator. The expectedresponse of the atmosphere is shown in Fig. 23–6,curve b (curve a is another theoretical curve under discussion in the book from whichthis is taken out of context). 前两个来自力学，第一个是大尺度的：整个地球的大气。我们假设大气，在各方面，均匀地包围着地球；如果大气，被月亮拉向一边，或者，毋宁说，被拉扁成双潮汐，如果我们让它这样下去，它就会上下涨落；这就是一个震荡。这个震荡，由月亮所驱动，月亮绕着地球，有效地转动着；力的任何一个分量，比如说x方向的，都有一个余弦分量，所以，月亮对地球大气的吸引，是潮汐性的，大气对此的反应，就是一个震荡器的反应。期待的大气的反应，如图23-6，曲线b所示（曲线a，在书中所讨论的另外一个理论上的曲线，我们就是从此书中，取出这个曲线的）。

Now one might think that we only have onepoint on this resonance curve, since we only have the one frequency,corresponding to the rotation of the earth under the moon, which occurs at aperiod of 12.42 hours—12 hours for the earth (the tide is a double bump), plus a little morebecause the moon is going around. 现在，有人可能会想，在这个共振曲线上，由于我们只有一个点，对于这个反应，只有这条共振曲线，由于地球的转动，是在月亮下，相应于这个转动，我们只有一个频率，它出现的周期，是12.42小时--12个小时，这是对地球而言的（潮汐是两次隆起的），因为月亮是在绕着转，所以，再加上一点儿时间。But from the size of the atmospheric tides, and from the phase,the amount of delay, we can get both ρ and θ . From those we can get ω0 and γ , and thus draw the entire curve! This is an example of very poor science.From two numbers we obtain two numbers, and from those two numbers we draw abeautiful curve, which of course goes through the very point that determined thecurve! It is of no use unless we can measure something else, and in thecase of geophysics that is often very difficult.但是，从大气潮汐的尺寸，从相位，即延迟的量{？}，我们可以得到ρ和 θ。从这些，我们又可以得到ω0和 γ，这样，就可画出完整的曲线。这个例子中，科学很贫乏。从两个数字，我们得到两个数字，从这两个数字，我们画出了漂亮的曲线，这个曲线，当然会通过那个规定了它的点！除非我们可以测量其他的东西，否则这没有什么用处，而在地理学的情况下，这种测量，通常又非常困难。But in this particular case there is another thing which we can showtheoretically must have the same timing as the natural frequency ω0: that is, if someone disturbed the atmosphere, it would oscillate withthe frequency ω0 . Now there was such a sharp disturbance in 1883; theKrakatoa volcano exploded and half the island blew off, and it made such aterrific explosion in the atmosphere that the period of oscillation of theatmosphere could be measured. It came out to 10+1/2 hours. The ω0 obtained from Fig. 23–6 comesout 10 hours and 20 minutes, so there we have at least one check on the reality ofour understanding of the atmospheric tides.但是，在此具体情况下，还有另外的一件事情，我们可以理论上指出，该事情，应该与自然频率ω0，有着同样的时序（timing）：那就是，如果某人，扰动了大气，那么，大气将会震荡，频率为ω0。现在，例子来了，1883年，就有这样一个急剧的扰动；Krakatoa火山爆发了，半个岛被炸飞，在大气中所产生的爆炸，如此巨大，以至于，大气震荡的周期，可以测量。结果就是，10+1/2小时。从图23-6中，得到的ω0，是10小时和20分钟，所以，我们对于大气潮汐的理解，是否真实，至少有了一个验证。