alt Hacker NewsIt shouldn't, under the assumptions they are opening up with. Inverse square law twice is appropriate for an approximation that the reflection is entirely diffuse because you assume that the diffuse reflection is uniform in all directions away from the illuminated surface. A single inverse square law is appropriate (with both distances added to each other before squaring) under the approximation that the reflection is specular.
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tonyarkles • today at 3:37 AM
I’m guessing it’s because the detector is then working with power, not amplitude. The whole 10 log vs 20 log thing with dB.
That only holds in the near field of the reflecting surface, which is roughly (it's a fuzzy boundary so who cares about a 2 pi here or there) where the ratio of distance to width of the surface is less than the ratio of the width of the surface to the wavelength. For wavelengths around 10^-2 m and surfaces around 10^0 m, that means distances within 10^2 m. Much further than that and the reflected wave is spreading out and falling off as 1/r^2. A radar working at ranges of tens or hundreds of kilometers is firmly in the far field.
There's still a strong angular dependence in the reflection amplitude; 1/r^2 fall-off does not imply isotropic distribution.