Question

Derive an expression for the maximum line of sight distance between two antennas for space wave propagation.

Answer 1

 

Maximum line of sight distance:

The transmitting and receiving antennae are shown below.

Consider only the transmitter antenna. From the geometry of the figure, we get

${\displaystyle {(R+h_r)}^2=R^2+d_r^2}$

${\displaystyle \therefore R^2+2h_{r}R+h_r^2=R^2+d_r^2}$

But, we know that ${\displaystyle h_r^2\text{<<}2h_{r}R}$

∴ ${\displaystyle R^2+2h_{r}R=R^2+d_r^2}$

${\displaystyle \therefore d_r^2=2h_{r}R}$

 ${\displaystyle \therefore d_r=\sqrt{2h_{r}R}}$

This is the distance between the transmitter and horizon.

Similarly, for the receiver, we have ${\displaystyle d_r=\sqrt{2h_{r}R}}$

Hence, the maximum line-of-sight distance between the transmitter and the receiver is

${\displaystyle d=d_r+d_R=\sqrt{2R{h}_r}+\sqrt{2R{h}_R}}$