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Excerpt from the Feynman Lectures, Volume III, Quantum Behavior:

At those places where the two waves arrive at the detector with a phase difference of $\pi$ (where they are “out of phase”) the resulting wave motion at the detector will be the difference of the two amplitudes. The waves “interfere destructively,” and we get a low value for the wave intensity.

I watched Crash Course Physics #17 to visually understood what it means for two waves to "interfere destructively". That was lovely. But Feynman suggests that the out of phase condition is reached only when the phase difference is $\pi$, specifically. Why is that?

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It is usual for a 'wave' to be modeled as a sine wave of a single frequency (and wavelength). While incoherent light is 'out of phase' with such a wave, being entirely random, a collision of two same-wavelength waves is NOT incoherent, and is said to be constructive interference where the squared-amplitude of the sum is greater than the sum of the squared-amplitudes of the waves.

This means the interference is constructive wherever phase difference is in the range of (- π/2, + π/2) and destructive where phase difference is in the range of (+ π/2, + 3 π/2). When considering interference fringes, the center of each 'fringe' which is a destructive-interference zone, is at a phase difference of π. That fringe center is the minimum, but the entire zone is, to a lesser extent, destructive interference.

Of course, any integer times 2π can be arbitrarily added to the phase, without changing the result, so there may be many zones of interference.

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Although the idea can be interpreted as such, Feynman does not say that two waves are out of phase only when the phase difference is $\pi$ (remember that $\pi$ is a unit of radians, in degrees the equivalent value is 180°).

More generally, as per Wikipedia:

The difference $\varphi(t) = \phi_G(t) - \phi_F(t)$ between the between the phases of two periodic signals $F$ and $G$ is called the phase difference of $G$ relative to $F$. At values of $t$ when the difference is zero, the two signals are said to be in phase, otherwise they are out of phase with each other.

Therefore, two waves are "out of phase" when the phase difference is non-zero.

An illustration:

Phase Shift

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