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Suppose we have two radio waveforms each has amplitude of 1, then the total power is 2.

Suppose these two waveforms add up some where constructively, then the amplitude become 2, and the total power is 4, so where is the extra power comes from?

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2 Answers 2

Because you have two stations they are separated by some distance, which means that for every region of constructive interference there is an equal sized area of destructive interference (well, the math is more involved, but let it go for now).

In short this means that the average power is roughly $(4 + 0)/2$ (again, pretending the math is trivial) or $2$ times the power from one station which is just what you would expect...

There is no "extra" power, the existing power has just been redistributed.

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I agree with dmckee's answer, but I would like to add that a different situation is also possible, as far as I know: if the two sources of radiation ("stations" in dmckee's answer) are close to each other (compared to the wavelength of the radiation), they can affect each other, so, compared to the case where they radiate independently (and each of them radiates power $P_0$), the power they radiate together can be higher or lower than $2 P_0$, depending on the relative phase of radiation of the two sources. In such a case, the extra energy comes from the power supplies of the sources.

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