Establish a relation between maximum distance of broadcast and height of antenna in space wave propagation.

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Solution

$P Q = h_{T} (h e i g h t o f a n t e n n a)$
$P S = P T = d$
From right angled $Δ O T Q$ , we get
$O Q^{2} = O T^{2} + Q T^{2}$
Here $O Q = R + h_{T}$
$Q T = P T = d$
$O T = R = R a d i u s o f e a r t h$
$(R + h_{t})^{2} = R^{2} + d^{2}$
$R^{2} + h_{t}^{2} + 2 R h_{T} = R^{2} + d^{2}$
$d^{2} = h_{T}^{2} + 2 R h_{T}$
$d^{2} = 2 R h_{T} [1 + \frac{h_{T}}{2 R}]$
$\frac{h_{T}}{2 R} << 1$
$d^{2} = 2 h_{T} R$
$d = \sqrt{2 h_{T} R}$
the maximum distance $d_{M}$ , between Transmitting & receiving antennas is given by
$d = d_{T} + d_{R}$
$d = \sqrt{2 R h_{T}} + \sqrt{2 R h_{R}}$
$d_{T}$ =horizon for transmitting antenna
$d_{R}$ =horizon for receiving antenna
$h_{T}$ =height of transmitter
$h_{R}$ =height ofreceiver
$d$ =maximum distance of broadcast

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