Superposition and Interference

When two or more waves meet at a point, the resultant displacement is the $\text{vector sum}$ of the individual displacements.

Two crests meeting give a larger crest — $\text{constructive}$ superposition. A crest meeting an equal trough gives zero displacement — $\text{destructive}$ superposition.

The waves pass through each other unchanged; superposition describes only what happens at the point where they overlap.

Two waves from separate sources reaching a point have travelled different distances. That $\text{path difference}$ determines their phase difference on arrival.

If the path difference is a whole number of wavelengths, $n\lambda$, the waves arrive in phase and interfere constructively. If it is an odd number of half-wavelengths, $(n+\frac{1}{2})\lambda$, they arrive in antiphase and interfere destructively.

This assumes the sources themselves are in phase. If they start in antiphase, the conditions swap.

A stable interference pattern needs $\text{coherent}$ sources: the same frequency and a constant phase difference. Two separate bulbs are not coherent, so their patterns shift too quickly to see. Splitting light from one source through two slits guarantees coherence.

In $\text{Young’s double-slit}$ experiment, light through two narrow slits a distance $a$ apart produces bright and dark fringes on a screen a distance $D$ away. The fringe spacing $x$ satisfies $\lambda =ax/D$, valid when $D\gg a$.

The result was historically decisive: particles could not produce dark fringes where two beams overlap, so the pattern established light as a wave.

A $\text{diffraction grating}$ has many closely spaced slits. Light from adjacent slits reaching a maximum at angle $\theta$ has path difference $d\sin \theta$, where $d$ is the slit spacing; constructive interference requires that to be a whole number of wavelengths.

With so many slits contributing, the maxima are extremely sharp and bright, and everything between is effectively dark — which makes gratings far better than double slits for measuring wavelength precisely.

Gratings are specified in lines per millimetre: $d=1/N$. The highest order visible is the largest $n$ with $\sin \theta \le 1$.