The waves from a radio station can reach a home receiver by two paths. One is a

Question

The waves from a radio station can reach a home receiver by two paths. One is a straight-line path from transmitter to home, a distance of 32.0 km. The second path is by reflection from the ionosphere (a layer of ionized air molecules high in the atmosphere). Assume this reflection takes place at a point midway between receiver and transmitter and that the wavelength broadcast by the radio station is 360 m. Find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected beams. (Assume that no phase changes occur on reflection.)

____ km

Explanation / Answer

The minimum height would refer to the first destructive interference fringe, when the path difference D(ionosphere) - D(direct) = 0.5 x the wavelength.
So the signal that bounced off the ionosphere would travel 114m further than the direct signal (32.0 km + 0.5*360m = 32.180 km)

So, using trigonometry, draw a right angled triangle with base = 32km/2 = 16 km, and a hypotenuse of 32.18 m/2= 16.09 km.
Then, using Pythagoras' theorem, h = sqrt(16.09^2 - 16^2) = 1.699 km.

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