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

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: : otherwise the conditions we have stated will not be met. Hence, under the conditions of the problem equation (6) can be replaced by

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

Substituting for $$\delta\nu_t$$ from equation (2) we finally have