My six-year old granddaughter told me, "You get what you get and you don't throw a fit". That may be the most universal statement I have ever heard. Can it apply to radio communications? Even though we agree that theory and "books smarts" build a firm foundation for us to anchor our knowledge too, we must remain aware that most of this knowledge is based upon optimum and/or theoretical conditions. The fact is, you will rarely find yourself faced with the optimum conditions, especially in mobile communications. In the real world, the gap between theory and reality is cluttered with a plethora of variables, each having an effect of its own. Sometimes the effects will combine to either hamper or improve the expected results and other times one or more will just cancel out the other. Theory versus reality!!!

We often hear "line-of-sight" references made about FM radio frequencies. In reality, the calculated horizon for visual line-of-sight is not the same as the calculated horizon for electromagnetic-wave propagation. The calculated radio wave horizon goes beyond the calculated visual horizon due to a combination of direct radiation and reflected ground wave. Even though both horizons can be calculated, the calculations do not take into account the reality of our planet. That is, dust, darkness, haze, fog and solid objects from a visual point of view or, objects, electromagnetic interference, etc. from a radio wave perspective. Nonetheless, we thought this might have some interest among the radio hobbyist.

VISUAL "LINE-OF-SIGHT" CALCULATIONS

In Statute (land) Miles

Height (in feet) divided by 0.5736 = "X"
Square root of "X" = Distance in statute miles to horizon

Example
Height to center of your eye(s) = 5.5ft
5.5ft divided by 0.5736 = 9.588
Square Root of 9.588 = 3.10 statute miles, the maximum distance you could theoretically see if standing on the beach in California looking out across the Pacific Ocean.

Note: If your feet are just in the water you might want to know what the nautical miles are (?). If so, multiply statute miles by 0.86898. If you're in up to your waist, you'll need to recalculate your eye height above the water surface.

In Kilometers

Height (in centimeters) divided by 6.752 = "X"
Square root of "X" = Distance in kilometers to horizon

Example
167.6cm divided by 6.752 = 24.822
Square root of 24.822 = 4.982 km the maximum distance you could theoretically see if standing on the beach in Portugal looking out across the Atlantic Ocean.

RADIO WAVE HORIZON CALCULATIONS

Since radio transmissions involve a transmitting antenna and a receiving antenna, both need to be considered for these calculations.

In Statute (land) Miles

Whereas; H1 = TX antenna and H2 = RX antenna

Square root of H1 (in feet) x 1.415 = D1
Square root of H2 (in feet) x 1.415 = D2
D1 + D2 = Radio Horizon in Statute miles

Example
Transmitting antenna (H1) height = 40 feet
Receiving antenna (H2) height = 8 feet

Square root of 40 = 6.325 x 1.415 = 8.95 (D1)
Square root of 8 = 2.828 x 1.415 = 4.00 (D2)
8.95 (D1) + 4.00 (D2) = 12.95 statute miles (theoretical maximum distance)

So, in theory, an 8ft high antenna should be able to receive broadcasts from a 40 foot high antenna located just under 13-miles away.

In Kilometers

Whereas; H1 = TX antenna and H2 = RX antenna

Square root of H1 (in meters) x 4.124 = D1
Square root of H2 (in meters) x 4.124 = D2
D1 + D2 = Radio Horizon in Kilometers

Example
Transmitting antenna (H1) height = 15m
Receiving antenna (H2) height = 5m

Square root of 15 = 3.873 x 4.124 = 15.972 (D1)
Square root of 5 = 2.236 x 4.124 = 9.221 (D2)
15.972 (D1) + 9.221 (D2) = 25.193 kilometers (theoretical maximum distance)