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During WWII, the Germans were using radio beacons (the "Knickebein" system) to guide their bombers into English territory. They set up two beacons, one in Kleve, a city in West Germany, and one at Stollberg Hill (North Frisia). The two radio beams intersected over Derby.

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The Knickebein system supposedly relied on line-of-sight propagation, but the distance between these locations is simply too great to allow for LOS. According to William CY Lee (page 350, equation 11.3.1), the radio horizon (i.e. the service range of the beacons) can be calculated as $R= \sqrt{2h_a} + \sqrt{2h_b}$ where $R$ is the distance in miles, $h_a$ is the aircraft altitude and $h_b$ is the ground-station antenna height in feet. If we plug in the numbers (say, the height of the beacon in Kleve (239ft) and the flight height of the German bombers (19,200ft)), we get a maximum beam range of 217,8 miles. However, Derby and Kleve are far away from each other, about 330 miles. So how could the Knickebein beams reach Derby?

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closed as off-topic by The Photon, stafusa, Jon Custer, Kyle Kanos, Cosmas Zachos Aug 23 at 19:41

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  • $\begingroup$ Check the frequency and propagation characteristics of that frequency. Remember you receive radio broadcasts when you can’t see the antenna. $\endgroup$ – Jon Custer Aug 18 at 16:23
  • $\begingroup$ The frequency was 30-33MHz. $\endgroup$ – Hepper Aug 18 at 16:37
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As pointed out by Hepper and Jon Custer, at the frequencies of operation of the system, significant bending of the radio waves will occur along with reflection off the ionosphere- making over-the-horizon transmission easy.

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  • $\begingroup$ Using the atmosphere itself as a waveguide $\endgroup$ – DakkVader Aug 18 at 17:51
  • $\begingroup$ not exactly- 30MHz has a wavelength of 10 meters, and the atmosphere is way thicker than that. $\endgroup$ – niels nielsen Aug 18 at 18:36
  • $\begingroup$ I always thought that (V)HF radio waves do not bend. $\endgroup$ – Hepper Aug 18 at 18:53
  • $\begingroup$ 30 megahertz is not vhf, and its waves do bend and bounce. the ability to bend gradually goes away above that frequency, and by the time you are at ~50 megahertz, you are in the line-of-sight regime. 100MHz is line-of-sight only. $\endgroup$ – niels nielsen Aug 18 at 19:49
  • $\begingroup$ The German Knickebein radar stations emitted guide beams. The bombers had to follow this beam in order to reach their target. Wouldn't ionospheric refraction/reflection obscure the path of the beam? $\endgroup$ – Hepper Aug 18 at 20:26

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