Short answers, pulled from the story.
The phase of a wave is an angle-like quantity, measured in degrees or radians, that represents how far a periodic signal has traveled through its cycle at a given moment. It increases by 360 degrees, or 2 pi radians, each time the signal completes one full period. The numeric value depends on the chosen start point of the cycle.
Constructive interference occurs when two signals have a phase difference of zero, so their values reinforce each other at every instant. Destructive interference occurs when the phase difference is 180 degrees, placing the signals in antiphase so their values are always opposite in sign and they can cancel each other out.
Two sinusoidal signals are said to be in quadrature when their phase difference is a quarter of a full cycle, equal to 90 degrees or pi over two radians. This condition appears, for example, in the in-phase and quadrature components of a composite communications signal, and in the relationship between voltage and current in certain electrical circuits.
Phase comparison connects two signals to a two-channel oscilloscope and observes how the phase relationship between them changes over time. When the frequencies are identical, both signals appear stationary on the display. When the frequencies differ, the test signal drifts, and measuring the rate of that drift reveals the frequency offset between the two signals.
When a player holds a single long note on a Native American flute, different harmonic components of that note reach dominance at different points in the phase cycle. The varying phase relationships among those harmonics create the characteristic warble, and the effect is directly visible on a spectrogram of the recorded sound.
The cosine function is phase-shifted by positive 90 degrees relative to the sine function. This fixed offset means that for two sinusoidal signals with the same frequency, one based on sine and one on cosine, they are always in quadrature.