Page:Beat waves.pdf/4

4. We shall now consider the explicit evaluation of the formula established in $\S$3.

Considering the first of the two integrals in equation (9), let the receiver $R$ be located on the $z$-axis at a height $z_{o}$ (see Fig. 1) and let the path of the projectile be confined to $xy$-plane and further be represented by

(10)$x=x_{o}={\text{constant;}}\quad z=0$

Finally, let $P$ denote the position of the projectile at time $t$.

Fig. 1

Fig. 1

The direction cosines of the instantaneous line of sight $RP$ are seen to be

(11)${\frac {x}{\sqrt {x_{o}^{2}+y^{2}+z_{o}^{2}}}},\,{\frac {y}{\sqrt {x_{o}^{2}+y^{2}+z_{o}^{2}}}},\,{\frac {z_{o}}{\sqrt {x_{o}^{2}+y^{2}+z_{o}^{2}}}}$

while the direction cosines of the path of the projectile are $0,\,1,\,0.$ Accordingly

(12)$\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {rec,P} }=|\mathbf {V} _{t}|{\frac {y}{\sqrt {z_{o}^{2}+x_{o}^{2}+y^{2}}}}$

Hence,

(13)$\int _{t_{1}}^{t_{2}}\mathbf {V} _{t}\cdot \mathbf {l} _{\mathrm {rec,P} }dt=\int _{t_{1}}^{t_{2}}{\frac {y}{\sqrt {x_{o}^{2}+z_{o}^{2}+y^{2}}}}|\mathbf {V} _{t}|dt$