Why does an AM radio channel require any bandwidth at all? Why can’t it just transmit on a single, precise frequency?
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The only way to transmit on a single, precise frequency is to emit a perfect sine wave of constant amplitude. If the signal changes in amplitude, as required by AM radio transmission, then the signal is a no longer a single frequency, but a mixture of frequencies. The more rapidly the amplitude changes, the wider the range of frequencies that are mixed to produce the final signal.
The math that shows how this works is complex, but the result is simple. If you want to modulate an AM carrier wave with a 10Hz signal, you will need at least 10Hz of bandwidth. (Note that the simplest way to amplitude modulate a carrier wave with a 10Hz signal will use of 20 Hz of bandwidth.)
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