Page:Beat waves.pdf/5

On the other hand, since

(14)$y=\int ^{t}|\mathbf {V} _{t}|dt$

equation (13) becomes

(15)$\int _{t_{1}}^{t_{2}}\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {rec,P} }dt=\int _{y_{1}}^{y_{2}}{\frac {ydy}{\sqrt {x_{o}^{2}+z_{o}^{2}+y^{2}}}}=\left({\sqrt {x_{o}^{2}+z_{o}^{2}+y_{2}^{2}}}-{\sqrt {x_{o}^{2}+z_{o}^{2}+y_{1}^{2}}}\right)$

We can express the result more simply as

(16)$\int _{t_{1}}^{t_{2}}\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {rec,P} }dt=\left(r_{2}^{\mathrm {(rec)} }-r_{1}^{\mathrm {(rec)} }\right)$

where $r_{1}^{\mathrm {(rec)} }$ and $r_{2}^{\mathrm {(rec)} }$ are the distances of the projectile from the receiver at the beginning and at the end of the interval $(t_{2}-t_{1})$.

The second integral in equation (9) can be similarly evaluated. We have

(17)$\int _{t_{1}}^{t_{2}}\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {trans,P} }dt=\left(r_{2}^{\mathrm {(trans)} }-r_{1}^{\mathrm {(trans)} }\right)$

where $r_{1}^{\mathrm {(trans)} }$ and $r_{2}^{\mathrm {(trans)} }$ are the distances of the projectile from the transmitter at the times $t_{1}$ and $t_{2}$ respectively.

Finally, combining equations (9), (16) and (17) we have

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

From this equation we see at once how the counted number of waves is directly related only to the positions of the projectiles at the beginning and at the end of the interval. Further this relation is exact.