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Tropospheric range: 100 ft and 50 ft antennas

Short answer

Using d (miles) = √(2ht) + √(2hr) with ht = 100 ft, hr = 50 ft: √200 ≈ 14.14, √100 = 10, total 24.14 miles (38.9 km).

Find the maximum range of tropospheric transmission for which the transmitting antenna height is 100 ft and receiving antenna is 50 ft.

The maximum range of tropospheric (space-wave) transmission is the sum of the radio horizons of the two antennas. With standard atmospheric refraction (effective earth radius 4/3 R), the horizon in miles for a height in feet is:

d (miles) = √(2 ht) + √(2 hr)     (h in ft)

Given: ht = 100 ft, hr = 50 ft.

√(2 × 100) = √200 = 14.142 miles
√(2 × 50)  = √100 = 10.000 miles
d = 14.142 + 10.000 = 24.142 miles
d = 24.142 × 1.609 = 38.85 km

Check in metric units: ht = 30.48 m, hr = 15.24 m, d = 4.12(√30.48 + √15.24) = 4.12(5.521 + 3.904) = 38.83 km (same, within rounding).

Answer: maximum range ≈ 24.1 miles ≈ 38.9 km.

✓ Answer checked by Claude Opus 5.5 (high effort)

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