Page:Beat waves.pdf/2

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

(1)$a\cos \left[2\pi (\nu _{o}+\delta \nu _{t})t+\varepsilon \right]$

where $\nu _{o}$ is the incident frequency (with respect to a stationary observer) and

(2)$\delta \nu _{t}=-{\frac {\nu _{o}}{c}}\mathbf {V} _{t}\cdot (\mathbf {l} _{\mathrm {rec,P} }+\mathbf {l} _{\mathrm {trans,P} })$

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

(3)$a'\cos[2\pi \nu _{o}t+\varepsilon ']$

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

(4)$A\cos[2\pi \delta \nu _{t}t+\varepsilon _{1}]$

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

(5)$N=(t_{2}-t_{1})|\delta \nu |$

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.