Let us consider two plane waves of same amplitude and frequency but having 180 degrees phase difference.Both the waves are having energy E in them which is directly proportional to the square of the amplitude.So the total energy is two times E.Now if both waves interfere,the amplitude of resulting wave is zero and the energy in the resulting wave is zero.What happend to the energy in both plane waves
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$\begingroup$ Or physics.stackexchange.com/questions/55318/… and many others. $\endgroup$– ProfRobCommented Dec 9, 2019 at 7:52
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A "perfect" plane wave does not exist. It would have an infinite energy. What really exist in the physical world are spherical outgoing waves coming from sources. Two sources cannot be at the exact same point. Consider two antennas emitting radio waves. Also, if one tries to put the two antennas too close to each other, they will act each on the other and the energy one tries to put in each antenna using whatever wave generator one uses will act on the other one and cause havoc. Your wave generators will not work as expected.
So take two identical wave generators each feeding an antenna, and far enough of each other not to interact (which means at least a few wavelengths away). Each produces a spherical wave.
Now put yourself very, very far from both. The waves you receive are almost plane waves. Depending on the direction you are standing, the relative phase between the two sources varies. In some places they will interfere negatively, and you indeed get zero energy there. But in some places they will interfere positively. There the amplitude will add up, and you'll get twice the amplitude you'd receive from each antenna separately. The total energy there is thus four times what you'd expect from one antenna. And of course there will be all the other possible values between zero and four times the energy of a single antenna.
The point is that provided the two antennas are far away from each other that the generators feed to them do not hinder each other, the average between all these values, over all the directions, will be the mean of the extreme values zero and four (it can be proven), thus it is twice the energy of a single antenna. Which is what you expect: there are indeed two antennas.
