Principle of superposition

Two pulses sent towards each other along a string pass straight through each other and come out unchanged.

While they overlap, the displacement at each point is the algebraic sum of the displacements each pulse alone would cause. This is the principle of superposition: y = y₁ + y₂ + … = Σ fᵢ(x − vt).

Two equal pulses, one up and one down, cancel for an instant when they exactly overlap and the string looks flat; the string's energy is then all kinetic, and the pulses reappear and move on.

Two harmonic waves of the same amplitude a, frequency and wavelength, moving the same way but differing in phase by φ, add to y = 2a cos(φ/2) sin(kx − ωt + φ/2).

The sum is again a harmonic wave in the same direction, with the same ω and k, but its amplitude is A(φ) = 2a cos(φ/2).

If φ = 0 the waves are in phase and the amplitude is 2a: constructive interference.

If φ = π, one wave is exactly opposite to the other, so they cancel at every point for all time: destructive interference.