Q.Two towers on top of two hills are $40\ \text{km}$ apart. The line joining them passes $50\ \text{m}$ above a hill halfway between the towers. What is the longest wavelength of radio waves, which can be sent between the towers without appreciable diffraction effects?

Diffraction spreading of the radio beam over the aperture set by the hill's clearance limits how far apart the towers can be without losing line-of-sight — solving for the wavelength that just keeps this spreading within the actual tower separation gi …

Step 1: Identify the geometry

The two towers are $40\ \text{km}$ apart, with a hill exactly halfway between them, so the distance from each tower to the hill is:

The line joining the towers clears the hill by $a = 50\ \text{m}$ — this clearance acts as the effective aperture through which the radio beam must pass without being significantly diffracted by the hill.

Step 2: Condition for negligible diffraction

For the transmitted beam to travel to the far tower without spreading out appreciably (i.e. for line-of-sight/ray propagation to remain a good approximation over the path length $x$), the Fresnel distance for the aperture $a$ must be at least as large as the actual path length: …