172.

We shall, as heretofore [eqn:(158)] (158), define $J$, the "intensity of the exciting vibration,"Not to be confused with the "field intensity" (field-strength) ${⋿}_z$ of the exciting vibration. as a function of the time to be the mean value of ${⋿}_z^2$ in the time interval from $t$ to $t+\tau$, where $\tau$ is taken as large compared with the time ${\displaystyle \frac{1}{\nu }}$, which is the duration of one of the periodic partial vibrations contained in the radiation, but as small as possible compared with the time $𝖳$. In this statement there is a certain indefiniteness, from which results the fact that $J$ will, in general, depend not only on $t$ but also on $\tau$. If this is the case one cannot speak of the intensity of the exciting vibration at all. For it is an essential feature of the conception of the intensity of a vibration that its value should change but unappreciably within the time required for a single vibration. (Compare above, [sect:3.] Sec. 3.) Hence we shall consider in future only those processes for which, under the conditions mentioned, there exists a mean value of ${⋿}_z^2$ depending only on $t$. We are then obliged to assume that the quantities $C_{\nu }$ in [eqn:(311)] (311) are negligible for all values of $\nu$ which are of the same order of magnitude as ${\displaystyle \frac{1}{\tau }}$ or smaller, i.e., $\nu \tau \text{ is large.}\mathrm{\backslash Label}[eqn](312)\text{upshape (312)}$

In order to calculate $J$ we now form from [eqn:(311)] (311) the value of ${⋿}_z^2$ and determine the mean value $\overline{{⋿}_z^2\mathrm{\backslash Strut}}$ of this quantity by integrating with respect to $t$ from $t$ to $t+\tau$, then dividing by $\tau$ and passing to the limit by decreasing $\tau$ sufficiently. Thus we get ${⋿}_z^2={\int }_0^{\infty }{\int }_0^{\infty }d{\nu }^{\prime }d\nu C_{{\nu }^{\prime }}{C}_{\nu }\cos (2\pi {\nu }^{\prime }t-{\theta }_{{\nu }^{\prime }})\cos (2\pi \nu t-{\theta }_{\nu }).$ If we now exchange the values of $\nu$ and ${\nu }^{\prime }$, the function under the sign of integration does not change; hence we assume ${\nu }^{\prime }>\nu$ and write: ${⋿}_z^2=2\iint d{\nu }^{\prime }d\nu C_{{\nu }^{\prime }}{C}_{\nu }\cos (2\pi {\nu }^{\prime }t-{\theta }_{{\nu }^{\prime }})\cos (2\pi \nu t-{\theta }_{\nu }),$ or

$$\begin{gathered} {⋿}_z^2=\iint d{\nu }^{\prime }d\nu C_{{\nu }^{\prime }}{C}_{\nu }\{\cos [2\pi ({\nu }^{\prime }-\nu )t-{\theta }_{{\nu }^{\prime }}+{\theta }_{\nu }] \\ +\cos [2\pi ({\nu }^{\prime }+\nu )t-{\theta }_{{\nu }^{\prime }}-{\theta }_{\nu }]\}. \end{gathered}$$

And hence

$$\begin{gathered} J=\overline{{⋿}_z^2\mathrm{\backslash Strut}}=\frac{1}{\tau }{\int }_t^{t+\tau }{⋿}_z^{2}dt \\ =\iint d{\nu }^{\prime }d\nu C_{{\nu }^{\prime }}{C}_{\nu }\backslash \{\frac{\sin \pi ({\nu }^{\prime }-\nu )\tau ·\cos [\pi ({\nu }^{\prime }-\nu )(2t+\tau )-{\theta }_{{\nu }^{\prime }}+{\theta }_{\nu }]}{\pi ({\nu }^{\prime }-\nu )\tau } \\ +\frac{\sin \pi ({\nu }^{\prime }+\nu )\tau ·\cos [\pi ({\nu }^{\prime }+\nu )(2t+\tau )-{\theta }_{{\nu }^{\prime }}-{\theta }_{\nu }]}{\pi ({\nu }^{\prime }+\nu )\tau }\backslash \}. \end{gathered}$$

If we now let $\tau$ become smaller and smaller, since $\nu \tau$ remains large, the denominator $({\nu }^{\prime }+\nu )\tau$ of the second fraction remains large under all circumstances, while that of the first fraction $({\nu }^{\prime }-\nu )\tau$ may decrease with decreasing value of $\tau$ to less than any finite value. Hence for sufficiently small values of $({\nu }^{\prime }-\nu )$ the integral reduces to $\iint d{\nu }^{\prime }d\nu C_{{\nu }^{\prime }}{C}_{\nu }\cos [2\pi ({\nu }^{\prime }-\nu )t-{\theta }_{{\nu }^{\prime }}+{\theta }_{\nu }]$ which is in fact independent of $\tau$. The remaining terms of the double integral, which correspond to larger values of ${\nu }^{\prime }-\nu$, i.e., to more rapid changes with the time, depend in general on $\tau$ and therefore must vanish, if the intensity $J$ is not to depend on $\tau$. Hence in our case on introducing as a second variable of integration instead of $\nu$ $\mu ={\nu }^{\prime }-\nu (>0)$ we have $J=\iint d\mu d\nu C_{\nu +\mu }{C}_{\nu }\cos (2\pi \mu t-{\theta }_{\nu +\mu }+{\theta }_{\nu })\mathrm{\backslash Label}[eqn](313)\text{upshape (313)}$ or

$$\begin{gathered} J\&=\int d\mu (A_{\mu }\cos 2\pi \mu t+B_{\mu }\sin 2\pi \mu t) \\ \mathrm{\backslash LeftText}where{A}_{\mu }\&=\int d\nu C_{\nu +\mu }{C}_{\nu }\cos ({\theta }_{\nu +\mu }-{\theta }_{\nu })\mathrm{\backslash Label}[eqn](314)\text{upshape (314)} \\ {B}_{\mu }\&=\int d\nu C_{\nu +\mu }{C}_{\nu }\sin ({\theta }_{\nu +\mu }-{\theta }_{\nu }). \end{gathered}$$

By this expression the intensity $J$ of the exciting vibration, if it exists at all, is expressed by a function of the time in the form of a Fourier's integral.