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Its possible for that to happen. After all ham’s use the moon for “moon bounce” communications on a variety of frequencies.
Or as the more modern terminology puts it…”Earth-Moon-Earth” communications.
That FM signal would be incredibly weak, so weak that it would take an incredibly high gain dish of appropriate wavelength to dig it out of the noise. And it would depend on how far that signal traveled prior to its “bounce” and the distance back.
The signal will weaken as distance increases…and again further weakened by the “bounce” as some of that already weak signal will be absorbed by whatever its bouncing off of, as well as “scattered” by the surface where the signal bounced, and then attenuated even more by the return trip distance.
I cant remember where I read it but there was a report about signals being picked up from Jupiter when NASA discovered a very long “something” hanging around the gas giant. After independent researchers analyzed the sounds, it was determined that what they picked up were reflected signals that originated from Earth back in the late 1930’s. The signals apparently were from a rally that was broadcast during the Nazi regime as the audio after it was analyzed said something about “The language of mankind”, which the language heard was in fact German.
I put this one in the “tall tail big fish” story category. But who knows….according to estimates, signals sent out back in the 40’s were passing Neptune only hours after they were sent out. No telling how far those signals have traveled or if they “bounced” in some other direction from the various stuff out there in the Ort cloud (outer solar system asteroid belt), or if those signals continue to travel deep into inter stellar space.
“Space..the area between the ears….this is the voyage of the tiny flea signal “Boobieprize”….its long hard struggle….to reach new worlds…to resonate alien antenna arrays….to boldly go where no flea signal has gone before”!!!