Now, it is very interesting to show thatthe addition of the two fields is in fact a vector addition. We havejust checked it for up and down motion, but let us check two nonparalleldirections. First, we restore S1 and S2to the same phase; that is, they are again moving together. But now we turn S1through 90∘ , as shownin Fig. 28–4.Now we should have at point 1 the sum of two effects, one of which is verticaland the other horizontal. The electric field is the vector sum of these twoin-phase signals—they are both strong at the same time and go through zerotogether; the total field should be a signal R at 45∘ . If we turn D to get the maximum noise, it should be atabout 45∘ , and notvertical. And if we turn it at right angles to that direction, we should getzero, which is easy to measure. Indeed, we observe just such behavior!
现在，我们来看一个非常有趣的事情：两个场相加，其实就是两个矢量相加。对于上下运动，我们已经检查了这一点。但是，让我们检查两个非平行方向的运动。首先，我们把S1和S2，恢复到同样的相位，也就是说，它们又一起运动了。但是，现在我们把S1转动了90度，如图28-4所示。现在，我们在点1，应该有它们两个的效果之和，其中之一，是垂直的，另一个，是水平的。这两个信号，是同相位的--它们同时变强、同时经过零，它们的矢量和，就是电场；总的场，应该是在45度处的一个信号R。如果我们转动D，以得到最大的噪音，那么，应该大约是在45度方向，而不是垂直方向。如果我们把它转一个直角，成为垂直的，我们就应该得到零，这很容易测量。确实，我们观察到的，正是如此。

Fig. 28–4.Illustration of the vectorcharacter of the combination of sources. 图28-4 两个源，组合在一起，其矢量特性之图示。

Now, how about the retardation? How can wedemonstrate that the signal is retarded? We could, with a great deal ofequipment, measure the time at which it arrives, but there is another, verysimple way. Referring again to Fig. 28–3,suppose that S1 and S2 are in phase.They are both shaking together, and they produce equal electric fields atpoint 1 . But suppose we go to a certain place 2 which is closerto S2 and farther from S1 . Then,in accordance with the principle that the acceleration should be retarded by anamount equal to r/c , if the retardations are not equal, thesignals are no longer in phase. Thus it should be possible to find a positionat which the distances of D from S1 and S2differ by some amount Δ , in such a manner that there is no net signal. 现在，迟滞又如何呢？我们如何演证，信号被迟滞了呢？我们可以用大量的设备，来测量信号到达的时间，但是，还有另外一个非常简单的方法。参考图28-3，假设S1与S2是同相位的。它们在一起摇动，在点1，产生的电场相等。但是，假设我们到某确定位置2，它离近S2，离S1远。因此，依据原理，加速度应该被迟滞一个量 r/c，如果迟滞不相等，那么，信号就不再是一个相位的了。这样，应该可能找到一个位置，在那里，D到S1的距离，与D到S2的距离，相差一个量 Δ，在这种方式下，净的信号，就没有了。That is, the distance Δ is to be the distance light goes inone-half an oscillation of the generator. We may go still further, and find apoint where the difference is greater by a whole cycle; that is to say, thesignal from the first antenna reaches point 3 with a delay in time that isgreater than that of the second antenna by just the length of time it takes forthe electric current to oscillate once, and therefore the two electric fieldsproduced at 3 are in phase again. At point 3 the signal is strongagain.
也就是说，距离Δ就是，光在发电机的震荡的半周期中，所走的距离。我们还可以走的更远，找到一个点，在那里，要相差一个完整的周期；也就是说，从第一个天线到点3，有一个时间延迟，从第二个天线到点3，又有一个时间延迟，两者之差，可以让电流，震荡一次，因此，在点3所产生的两个电场，就又是同相的了。在点三，信号又变强了。

This completes our discussion of theexperimental verification of some of the important features of Eq. (28.6).Of course we have not really checked the 1/r variation of theelectric field strength, or the fact that there is also a magnetic field thatgoes along with the electric field. To do so would require rather sophisticatedtechniques and would hardly add to our understanding at this point. In anycase, we have checked those features that are of the greatest importance forour later applications, and we shall come back to study some of the otherproperties of electromagnetic waves next year.
方程（28.6）有一些重要的特性，对于其中一些，我们做了实验性的验证，现在，已经讨论完毕。当然，我们并没真正去检查：电场强度随着1/r的变化，或者如下事实：跟电场在一起的，还有一个磁场。要做这些，要求相当老练成熟的技术，且在此时，也很难进一步加深我们对物理的理解。无论如何，我们已经检查的那些特性，对我们后面的应用来说，具有最重要的意义。明年，我们将回来学习：电磁波的另外一些属性。

Fig. 29–1.The electric field Edue to a positive charge whose retarded acceleration is a′ . 图29-1 电场 E，可归于一个正电荷，该电荷被迟滞的加速度是a′。
We have already physically analyzed themeaning of formula (28.6) quite satisfactorily, but there are a few pointsto be made about it mathematically. In the first place, if a charge isaccelerating up and down along a line, in a motion of very small amplitude, thefield at some angle θ from the axis of the motion is in a directionat right angles to the line of sight and in the plane containing both theacceleration and the line of sight (Fig. 29–1). Ifthe distance is called r , then at time t the electricfield has the magnitude
对于公式（28.6）的意义，我们已经从物理上，进行了分析，非常令人满意，但是，从数学上，还要再说几点。首先，如果一个电荷，正在沿着一条直线，被加速，而在做上下运动，且运动的幅度，非常小，那么，在运动的轴，成θ角的方向上，形成一个电场，其方向，垂直于视线的角度，且其所在平面，包含着加速度和视线（图29-1）。如果距离被称为r，那么，在时间t，电场大小就是：
(29.1)
where a(t−r/c)is the acceleration at the time (t−r/c) , called the retardedacceleration.
这里，a(t−r/c) ，就是在时间(t−r/c)的加速度，被称为迟滞的加速度。

Stated another way: if we add a littletime Δt , we can restore a(t−r/c) toits former value by adding a little distance Δr=cΔt. That is, as time goes on the field moves as a wave outward from the source.That is the reason why we sometimes say light is propagated as waves. It isequivalent to saying that the field is delayed, or to saying that the electricfield is moving outward as time goes on.
换种方式说：如果我们增加一个小的时间Δt，那么，通过增加一个小的距离Δr=cΔt，我们可以把a(t−r/c)，重建到其前一个值。也就是说，随着时间的过去，场从源处，作为一个波，向外移动。这也就是为什么我们有时候会说:光是以波的形式在传播。它等于是说：场被迟滞了，或者说：随着时间的过去，电场向外移动。

An interesting special case is that wherethe charge q is moving up and down in an oscillatory manner. Thecase which we studied experimentally in the last chapter was one in which thedisplacement x at any time t was equal to a certainconstant x0 , the magnitude of the oscillation,times cosωt . Then the acceleration is
一种有趣的特殊情况，就是电荷q，以一种的震荡的方式，上下移动。我们上一章，以实验的方式，研究过此情况，它就是，任意时间t的位移x，等于某个常数x0、即震荡的的大小，乘以cosωt。因此，加速度就是：
a=−ω2x0cosωt=a0cosωt, (29.2)
where a0 is the maximumacceleration, −ω2x0 . Putting this formulainto (29.1),we find
这里，a0是最大加速度，−ω2x0。把这个公式，代入（29.1），我们发现：
(29.3)
Now, ignoring the angle θ andthe constant factors, let us see what that looks like as a function of positionor as a function of time.
现在，忽略角度θ和常数因子，让我们看看，作为位置的函数，或作为时间的函数，它看上去像什么。

29–2Energy of radiation 29-2 辐射的能量
First of all, at any particular moment orin any particular place, the strength of the field varies inversely as thedistance r , as we mentioned previously. Now we must point out thatthe energy content of a wave, or the energy effects that such anelectric field can have, are proportional to the square of the field,because if, for instance, we have some kind of a charge or an oscillator in theelectric field, then if we let the field act on the oscillator, it makes itmove. If this is a linear oscillator, the acceleration, velocity, anddisplacement produced by the electric field acting on the charge are allproportional to the field. So the kinetic energy which is developed in thecharge is proportional to the square of the field. So we shall take itthat the energy that a field can deliver to a system is proportional somehow tothe square of the field.

首先，正如我们以前所说，在任何具体的瞬间、或任何具体位置，场的强度，都与距离r，成反比。现在，我们要指出，一个波的能量内容，或这样一个电场所能有的能量效果，是正比于场的平方的，因为，例如，如果在电场中，我们有某种电荷或一个振荡器，那么，如果我们让场作用于此振荡器，那么，它会让它运动。如果这是一个线性振荡器，那么，通过电场作用于电荷而产生的加速度、速度、和位移，就正比于场。于是，在电荷中发展出来的动能，就正比于场的平方。所以，我们将认同：一个场可以给一个系统传递的能量，在某种程度上，就正比于场的平方。

Fig. 29–4.The energy flowing within thecone OABCD is independent of the distance r at which itis measured. 图29-4 在圆锥形中流动的能量，独立于测量能量时的距离 r。

This means that the energy that the source can deliverdecreases as we get farther away; in fact, it varies inversely as the squareof the distance. But that has a very simple interpretation: if we wanted topick up all the energy we could from the wave in a certain cone at adistance r1 (Fig. 29–4), andwe do the same at another distance r2 , we find that theamount of energy per unit area at any one place goes inversely as the squareof r , but the area of the surface intercepted by the cone goes directlyas the square of r . So the energy that we can take out of the wavewithin a given conical angle is the same, no matter how far away we are! Inparticular, the total energy that we could take out of the whole wave byputting absorbing oscillators all around is a certain fixed amount. 这就意味着，当我们走的较远时，源可以传送的能量，就会降低；事实上，它与距离的平方，成反比。但是，这有一个非常简单的解释：如果在某个圆锥形中，我们想在距离r1，把我们所能从波中收集到的能量，全收起来，然后，在另外一个距离r2，我们想做同样的事，那么，我们就发现，每单位面积的能量的量，在任何一个位置，都与距离的平方，成反比，但是，圆锥截面的面积，也与距离 r的平方成反比。所以，在一个被给与的圆锥角中，不论我们的距离有多远，我们能从波中收集到的能量，是一样的！尤其是，通过四处放置吸收性的振荡器，我们能从整个波中取得的总的能量，是某个固定的量。

So the fact that the amplitude of Evaries as 1/r is the same as saying that there is an energy fluxwhich is never lost, an energy which goes on and on, spreading over a greaterand greater effective area. Thus we see that after a charge has oscillated, ithas lost some energy which it can never recover; the energy keeps going fartherand farther away without diminution. So if we are far enough away that ourbasic approximation is good enough, the charge cannot recover the energy whichhas been, as we say, radiated away. Of course the energy still existssomewhere, and is available to be picked up by other systems. We shall studythis energy “loss” further in Chapter 32.
所以，E的振幅，随着1/r而变化这一事实，就等于是说，有一个永不减少的能量流，这个能量，不断地往前走，扩展到越来越大的范围。这样，我们就看到，一个电荷，震荡之后，会失去一些能量，这些能量，永远也不会恢复；此能量，一直往前走，越走越远，没有减少。所以，如果我们足够远，且我们的基础假设是足够的好，那么，就正如我们所说，电荷不能恢复它所辐射出去的能量。当然，能量仍将会在某处存在，且可以通过其他的系统，被收集起来。在32章，我们将进一步研究这种能量“损失”。

Let us now consider more carefully how thewave (29.3)varies as a function of time at a given place, and as a function of position ata given time. Again we ignore the 1/r variation and the constants.
现在，让我们再次仔细地考虑波（29.3），在一个被给予的地方，作为时间的函数，是如何变化的，以及，在一个被给予的时间，作为位置的函数，是如何变化的。我们还是忽略1/r的变化、和常数。

29–3Sinusoidal waves 29-3 正弦波
First let us fix the position r, and watch the field as a function of time. It is oscillatory at the angularfrequency ω . The angular frequency ω can be defined asthe rate of change of phase with time (radians per second). We havealready studied such a thing, so it should be quite familiar to us by now. The periodis the time needed for one oscillation, one complete cycle, and we have workedthat out too; it is 2π/ω , because ω times theperiod is one cycle of the cosine.
首先，让我们固定位置r，而把场，看做时间的函数。它以角频率ω，在震荡。角频率ω可被定义为：相位随着时间的变化率（弧度每秒）。我们已经研究过这种事情，所以现在，对它我们应该很熟了。周期，就是一次震荡、即一个完整循环，所需要的时间，这个我们也已经得到了；它就是2π/ω，因为，ω乘以周期，就是余弦函数的一个完整循环。

Now we introduce a new quantity which isused a great deal in physics. This has to do with the opposite situation, inwhich we fix t and look at the wave as a function of distance r. Of course we notice that, as a function of r , the wave (29.3)is also oscillatory. That is, aside from 1/r , which we are ignoring, wesee that E oscillates as we change the position. So, in analogy with ω, we can define a quantity called the wave number, symbolized as k. This is defined as the rate of change of phase with distance (radiansper meter). That is, as we move in space at a fixed time, the phase changes.
现在，我们引入一个新量，它在物理学中，被大量使用。这就需要与一个相反的情况打交道，在此情况中，我们固定 t，而把波，看做r的函数。当然，我们注意到，波（29.3），作为r的函数，也是震荡的。也就是说，除了1/r，我们要继续忽略外，我们看到，当我们改变位置时，E在震荡，所以，根据与ω的类比，我们可以定义一个量，它被称为波数，符号为 k。这被定义为：相位随着距离的变化率（弧度每秒）。也就是说，当在固定的时间中，我们在空间移动时，相位改变了。

There is another quantity that correspondsto the period, and we might call it the period in space, but it is usuallycalled the wavelength, symbolized λ . The wavelength is thedistance occupied by one complete cycle. It is easy to see, then, that thewavelength is 2π/k , because k times thewavelength would be the number of radians that the whole thing changes, beingthe product of the rate of change of the radians per meter, times the number ofmeters, and we must make a 2π change for one cycle. So kλ=2πis exactly analogous to ωt0=2π .
与周期相应的，还有另外一个量，我们可以称其为空间的周期，但通常，它被称为波长，符号为λ。波长就是一个完整循环，所占的距离。因此，很容易看到，波长就是 2π/k，因为， k乘以波长，就是整个事物改变的弧度的数目，就是每米弧度的改变率，乘以米的数目，对于一个循环，我们应该使其变化，为一个 2π。所以，kλ=2π，可完全准确地类比于ωt0=2π。