Page:Beat waves.pdf/3

3. Before proceeding further with the interpretation of $N$ when $\delta \nu _{t}$ is a function of time we should perhaps remark that in order that the procedure may at all have an unambiguous meaning it is necessary that two conditions be satisfied. First that $|\delta \nu _{t}|$ does not change appreciably during a time of the order $1/|\delta \nu _{t}|$ and second that the interval $(t_{2}-t_{1})$ be large compared to $|\delta \nu _{t}|$. We shall assume in our further discussions that these conditions are in fact satisfied. Accordingly we can subdivide the interval $(t_{2}-t_{1})$ into a large number of subintervals $\Delta t_{1},\dots ,\Delta t_{j},\dots ,\Delta t_{n},$ such that each of these intervals are large compared to $|\delta \nu _{t}|$ but small compared to $t_{2}-t_{1}$. Under these circumstances, the number of waves counted during the interval $(t_{2}-t_{1})$ can be written as

(6)$N=\sum _{j=1}^{n}|\delta \nu _{t}(t_{j})|\Delta t_{j}$

where $t_{j}$ denotes a value of $t$ included in the interval $\Delta t_{j}$. One further remark should be made concerning equation (6) to avoid possible misunderstandings: During the interval $(t_{2}-t_{1}),\,\delta \nu _{t}$ must always remain of the same sign: otherwise the conditions we have stated will not be met. Hence, under the conditions of the problem equation (6) can be replaced by

(7)$N=\left|\sum _{j=1}^{n}\delta \nu _{t}(t_{j})\Delta t_{j}\right|$

Moreover, it is also clear that to a sufficient approximation we can replace the summation in the foregoing formula by an integral and obtain

(8)$N=\left|\int _{t_{1}}^{t_{2}}\delta \nu _{t}dt\right|$

Substituting for $\delta \nu _{t}$ from equation (2) we finally have

(9)$N=\left|{\frac {\nu _{o}}{c}}\int _{t_{1}}^{t_{2}}\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {rec,P} }dt+{\frac {\nu _{o}}{c}}\int _{t_{1}}^{t_{2}}\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {trans,P} }dt\right|$