Page:Beat waves.pdf/2

 2. The wave train reflected from a moving projectile has the form

where $$\nu_o$$ is the incident frequency (with respect to a stationary observer) and

where $$c$$ denotes the velocity of light, $$\mathbf{V}_t$$ the velocity of the projectile, $$\mathbf{l}_{\mathrm{rec,P}}$$ and $$\mathbf{l}_{\mathrm{trans,P}}$$ unit vectors in the directions pointing from the receiver, respectively, transmitter, to the instantaneous position $$P$$ of the projectile.

Let us first consider the result of combining the reflected wave train with an incident wave train of the form

The result will be a beat phenomenon which can be represented (apart from additive factors) by

If $$\delta\nu_t$$ had been a constant then the constant beat frequency $$|\delta\nu_t|$$ can be determined by counting the number of waves $$N$$ in a specified interval of time $$(t_2-t_1)$$. For, then

However, in practice $$\delta\nu_t$$ will be a function of time. This dependence of $$\delta\nu_t$$ on time arises principally from the geometry of the situation which makes $$\mathbf{l}_{\mathrm{rec,P}}$$ and $$\mathbf{l}_{\mathrm{trans,P}}$$ vary with time during the intervals considered. To a lesser extent the variation of $$\delta\nu_t$$ may also result from the variation of $$|\mathbf{V}_t|$$ along the path of the projectile. The question now arises as to how best we can interpret the number of waves $$N$$ counted during a specified interval of time. As we shall see presently the obvious way of regarding $$N/(t_2-t_1)$$ as an "average" frequency does not provide the most satisfactory or indeed, even the most convenient one for the practical reduction of the observations.