SPACE WAVE PROPAGATION

What this figure shows. A simple side-on diagram of a single transmitting antenna of height $h$ standing on the curved surface of the Earth, with a dashed line drawn from the top of the antenna tangent to the Earth's curved surface, marked with the distance $d$ -- the maximum straight-line (line-of-sight) distance from the base of the antenna to the point where that tangent line touches the Earth's curved horizon. This is the geometric picture behind the coverage-distance formula $d=\sqrt{2Rh}$, showing that a taller antenna pushes the horizon -- and therefore the maximum reception ran …

Worked out. A transmitting antenna of height $h_1=40$ m and a receiving antenna of height $h_2=30$ m must be connected by line-of-sight communication, with the Earth's radius given as $R=6.4\times10^6$ m; the task is to find the maximum distance between them. The total distance is the sum of each antenna's own individual coverage distance, $d=d_1+d_2=\sqrt{2Rh_1}+\sqrt{2Rh_2}=\sqrt{2R}\left(\sqrt{h_1}+\sqrt{h_2}\right)$. Substituting $2R=2\times6.4\times10^6=1.28\times10^7$, so $\sqrt{2R}\approx3577.7$, and $\sqrt{40}+\sqrt{30}\approx6.32+5.48=11.80$, gives $d\approx3577.7\times11.80\approx42{,}217$ m $=42.217$ km. This shows how the maximum reliable range of a space-wave (line-of-sight) link is fixed purely by the geometry of the two antenna heights and the Earth's curvature, and grows only with the square root of each antenna's height, so doubling an ant …