Page:Beat waves.pdf/5

 On the other hand, since

equation (13) becomes

We can express the result more simply as

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

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

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