Dear L.D. Edmonds

Your reasoning is insightful, and the confusion arises from how random phases interact statistically rather than deterministically. The key mistake in concluding perfect destructive interference is assuming that every contribution to the field has an exact counterpart that is precisely 180 degrees out of phase at every point in space and time. In reality, the phases are randomly distributed, meaning that while some contributions may cancel, others will reinforce each other.

Here’s why the interference term averages to zero without leading to total cancellation:

A useful analogy is tossing coins: if you flip a large number of fair coins, you expect roughly equal heads and tails, but you wouldn’t expect exactly half at every moment. Similarly, random phases ensure that interference terms cancel on average, but not at every instant or location.

Thank you Len Leonid Mizrah . You understood the question perfectly and gave an answer that I understand.

The vector fields of the random contributions from many different sources add up as a random walk, as described by Einstein. The reason a grain of pollen slowly moves around when randomly hit by air molecules is the same reason that random noise adds up to a finite noise signal and that incoherent light adds up to a finite brightness. The distance or amplitude of the sum is proportional to the square root of the number of contributions (if they are of equal size), not to the number of contributions, so the square of this (the intensity) is proportional to the number of contributions. The average power of the sum of incoherent signals increases as the addition of the powers.

Thank you Malcolm White . Another good answer. I think I got the idea now. One prediction comes from what might be called a low-order approximation, while another prediction comes from what might be called a higher-order approximation. For either case, the intensity from many sources with random phases is much less than what the intensity would be if the sources were all in phase. (This is consistent with a sum of squares of amplitudes being much less than the square of a sum of amplitudes when all amplitudes are positive numbers). A low-order approximation is complete cancellation. This actually does produce one correct prediction, that the resulting intensity divided by what the intensity would be if all sources were in phase goes to zero as the number of sources increases. But we need a higher-order approximation if we want to do better than say that the resulting intensity is negligible compared to what the intensity would be if the sources were all in phase. It can be shown that this higher-order approximation is to set the resulting intensity equal to the sum of intensities of all sources. I thank you for mentioning random walk because that inspired me to look it up in the Feynman Lectures where I found the mathematics behind that higher-order approximation.