Page:Beat waves.pdf/8

"average" velocity. This may sound paradoxical since the Doppler principle utilized in the method is related to the velocity of the projectile during its motion. But the origin of the general belief that this method (based as it is on the Doppler principle) must be expected to yield velocities directly can be readily traced:

Suppose that the projectile is fired along the line joining the transmitter and the receiver. Then the counted number of waves $N$ during $(t_{2}-t_{1})$ will give us $\nu _{o}(x_{2}-x_{1})/c$. If it is now further supposed that the projectile moves with a constant velocity $V$ during the interval under consideration then $x_{2}-x_{1}=V(t_{2}-t_{1})$. Accordingly in this case $N$ is directly related to $V$. However, the situation in general is not as simple and as we have seen we can in principle derive only $\left[r_{2}^{\mathrm {(rec)} }+r_{2}^{\mathrm {(trans)} }-r_{1}^{\mathrm {(rec)} }-r_{1}^{\mathrm {(trans)} }\right]$. The question now arises as to how from a knowledge of this quantity we can derive an average velocity for relatively small intervals $(t_{2}-t_{1})$.

Considering now for the sake of simplicity, the geometrical disposition of the apparatus as indicated in Fig. 2, the quantity directly deduced from the observations is

(25)$\Delta r={\sqrt {r_{o}^{2}+x_{2}^{2}}}-{\sqrt {r_{o}^{2}+x_{1}^{2}}}$

where we have used $r_{o}$ to denote the shortest distance from the mid point joining the centers of the receiver and the transmitter to the path of the projectile. We shall now suppose that $r_{o}$ is small compared to either $x_{2}$ or $x_{1}$. Then, since

(26)$\Delta r=(x_{2}-x_{1})\left[1-{\frac {r_{o}^{2}}{2x_{2}x_{1}}}+O(r_{o}^{4})\right]$

we have to a sufficient approximation

(27)$\Delta r=(x_{2}-x_{1})\left(1-{\frac {r_{o}^{2}}{2x_{2}x_{1}}}\right)$

Setting

(28)$x_{2}-x_{1}={\bar {V}}(t_{2}-t_{1})$