Superposition and Interference

The principle of superposition — when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements.

The waves pass through one another unchanged. After overlapping they continue exactly as before, which is why two conversations can cross a room without becoming mixed.

Constructive interference — the waves arrive in phase, so the displacements add and the resultant amplitude is a maximum.

Destructive interference — the waves arrive in antiphase, so the displacements subtract. If the amplitudes are equal the resultant is zero.

Path difference — the difference in the distances travelled by the two waves from their sources to the point.

Constructive when the path difference is a whole number of wavelengths: nλ

Destructive when the path difference is an odd number of half wavelengths: (n + ½)λ

Phase difference — path difference expressed as an angle. A path difference of one wavelength is a phase difference of 2π radians.

Coherence — two sources are coherent if they have the same frequency and a constant phase difference between them.

Why coherence is required — if the phase difference between the sources varied with time, the positions of the maxima and minima would move about, and the pattern would smear into uniform illumination too quickly for anything to be seen. The pattern would still exist at each instant; it simply would not be stable.

Example: this is why two separate lamps never produce visible interference fringes, even if they emit the same colour. Each emits light in short random bursts, so their phase difference changes millions of times a second. Splitting light from a single source, as Young did, guarantees the two paths stay in step.

Amplitude matters too — for complete cancellation the two waves must have equal amplitudes as well as opposite phase. Unequal amplitudes give a minimum that is not zero.