How can two waves add up to nothing?
When two waves meet, they simply add up: at every point, the result is the sum of the two swings. This is called superposition, and it is the most important idea of this module. It will come back in every module of the course, all the way to quantum algorithms.
If the two waves are in phase, crest on crest, they reinforce each other: the sum is twice as tall. This is constructive interference. If one is shifted by half a wavelength, crest on trough, they cancel exactly: destructive interference. In between, you get everything in between.
Noise-cancelling headphones use exactly this: a microphone listens to the noise and the headphones play the same sound shifted by half a wavelength, so the two cancel in your ear.
Two equal waves are shifted by exactly one whole wavelength. What do you get? A shift of one whole wavelength puts crest on crest again: the waves are back in phase and reinforce each other. Only half-wavelength shifts (½, 1½, 2½…) cancel.
One detail matters a lot for what comes next. The brightness of light does not follow the height of the wave but its square. Two waves in phase have twice the height, so four times the brightness; two waves out of phase give zero. Interference moves energy around: dark here, extra bright there.
Two waves A sin(x) and A sin(x + φ) add up to a wave of amplitude 2A cos(φ/2): φ is the phase shift.
φ = 0 gives 2A (constructive); φ = π, half a wavelength, gives 0 (destructive).
Brightness goes as the square of the amplitude: 4A^2 cos^2(φ/2). The same cos^2(φ/2) appears in the lab's interferometer, one photon at a time.
Waves add up: in phase they reinforce, half a wavelength apart they cancel. This is interference, the engine of everything that follows.