Page:Beat waves.pdf/7

6. In view of our formula for the number of waves counted during a given interval of time, it appears that the most convenient arrangement for the purposes of the practical reduction of the observations is that indicated in Figure 2. Specifically, we arrange the receiver, the transmitter and the gun in such a way that the path of the projectile is confined to a plane which bisects the line joining the centers of the receiver and the transmitter orthogonally. For under these circumstances

(23)$r^{\mathrm {(rec)} }=r^{\mathrm {(trans)} }=r\quad {\text{(say)}}$

Fig. 2

Fig. 2

for all points on the trajectory and equation (23) now reduces to

(24)$N\pm \delta \nu _{o}(t_{2}-t_{1})=2{\frac {\nu _{o}}{c}}|[r_{2}-r_{1}]|$

7. We have seen how the number of waves counted on a record of the beat phenomenon during a certain interval of time $(t_{2}-t_{1})$ enables us to derive only the position of the projectile at the time $t_{2}$ from a knowledge of its position at the time $t_{1}$. Since the position of the projectile at the instant of firing can be assumed known we should in principle be able to follow the projectile in its trajectory. This is precisely the basic information which is provided by the observations. It is to be particularly noted that in this statement as to what exactly the observations provide us we have had no occasion to take any reference to an