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LOS distance: 125 m and 9 m antennas

Short answer

Using d = 4.12(√ht+√hr) km with 4/3 earth radius, ht = 125 m, hr = 9 m: √125 = 11.18, √9 = 3, so d = 4.12 × 14.18 ≈ 58.4 km.

The antenna of a TV transmitter is located at a height of 125 m above ground level. Calculate the distance up to which the LOS communication is possible if the height of receiving antenna is to be 9 m.

For space-wave (line-of-sight) communication, the maximum distance is the sum of the radio horizons of the two antennas. With standard atmospheric refraction, the effective earth radius is kR with k = 4/3.

Given: ht = 125 m, hr = 9 m, R = 6370 km, k = 4/3.

Formula:

d = √(2kR·ht) + √(2kR·hr)
With kR = 8493 km and h in metres:
d (km) = 4.12 (√ht + √hr)

Calculation:

√ht = √125 = 11.180
√hr = √9   = 3.000
d   = 4.12 × (11.180 + 3.000)
    = 4.12 × 14.180
    = 58.42 km

Separately: transmitter horizon = 4.12 × 11.180 = 46.06 km, receiver horizon = 4.12 × 3 = 12.36 km.

(If refraction is ignored, d = 3.57(√ht + √hr) = 50.62 km.)

Answer: LOS communication is possible up to about 58.4 km (taking standard 4/3 earth refraction).

✓ Answer checked by Claude Opus 5.5 (high effort)

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