Page:Beat waves.pdf/6

According to equation (18), we can derive in a straightforward manner the position of the projectile at the end of any specified interval of time from a knowledge of the initial position of the projectile (i.e., $r_{1}^{\mathrm {(rec)} }$ and $r_{1}^{\mathrm {(trans)} }$), the geometry of the arrangement and the counted number of beats during the interval.

5. Now, in practice the number of beat waves in (4) are not directly counted. Instead, this beat wave is again combined with another wave train of the form

(19)$B\cos(2\pi \delta \nu _{o}t+\varepsilon _{2})$

where $\delta \nu _{o}$ is some constant frequency and which is very close to the frequencies $\delta \nu _{t}$ ordinarily encountered. It is however necessary that in any given interval during which the beats are counted, $(\delta \nu _{o}-\delta \nu _{t})$ does not change sign.

The superposition of the wave trains (4) and (19) will again result in a beat phenomenon which can be represented by a wave train of the form

(20)$C\cos \left[2\pi (\delta \nu _{o}-|\delta \nu _{t}|)t+\varepsilon _{3}\right]$

The number of waves $N$ which will be counted in this wave train (20) in an interval $(t_{2}-t_{1})$ will now be given by (cf. eq. [8])

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

or, since $\delta \nu _{o}$ is a constant $(>0)$, we have

(22)$N=\left|\left\{\delta \nu _{o}(t_{2}-t_{1})-{\frac {\nu _{o}}{c}}\left|\left[r_{2}^{\mathrm {(rec)} }+r_{2}^{\mathrm {(trans)} }-r_{1}^{\mathrm {(rec)} }-r_{1}^{\mathrm {(trans)} }\right]\right|\right\}\right|$

This equation can be expressed alternatively in the form

(23)$N\pm \delta \nu _{o}(t_{2}-t_{1})={\frac {\nu _{o}}{c}}\left|\left[r_{2}^{\mathrm {(rec)} }+r_{2}^{\mathrm {(trans)} }-r_{1}^{\mathrm {(rec)} }-r_{1}^{\mathrm {(trans)} }\right]\right|$

an equation which can be treated in exactly the same way as equation (18).