When waves overlap, they create new patterns that neither wave contains alone.
Two pebbles in still water send out ripples that overlap. Where crests align, water rises higher than either wave; where a crest meets a trough, they cancel. The bright/dark band pattern isn't in either wave — it emerges from overlap.
This is interference, from one rule: when two waves coincide, their amplitudes add (superposition). Simple arithmetic, complex patterns.
Two point sources emit circular waves. Red is crest, blue is trough, white is zero. Aligned crests give intense red; crest-meets-trough gives white. The stable bands are the interference pattern.
Drag the sources (circles) to reposition them.
Figure 1. Point sources emitting circular waves. Red = positive amplitude, blue = negative, white = zero. The stable bands of constructive and destructive interference emerge from pure superposition. Try a five-source array to see how added sources sharpen the forward beam, or the double slit to recover the canonical Young's experiment pattern.
Bright bands are hyperbolas of constant path difference. On a bright band, waves arrive in phase; on a dark band, they arrive half a wavelength apart and cancel. More sources, higher frequency, or wider separation produce finer patterns.
Two waves of equal frequency traveling in opposite directions lock into a standing wave. Nodes are points that never move; antinodes oscillate at maximum amplitude.
Figure 2. Two traveling waves of equal frequency moving in opposite directions produce a standing wave. The dots mark nodes (stationary points) and triangles mark antinodes (maximum displacement). Dashed curves are the stationary envelope that bounds the oscillation.
Standing waves are everywhere — guitar strings, air in a flute. Fixed endpoints force nodes at both ends, constraining wavelengths to integer half-wave fits. These are the natural frequencies (harmonics).
On a 2D vibrating plate, sand bounces off vibrating regions and collects at nodal lines. The geometric figures were first studied by Ernst Chladni in the 18th century.
Higher modes have more nodal lines, dividing the plate into smaller regions. Sand (bright pixels) accumulates where the amplitude is near zero.
Figure 3. Chladni patterns on a vibrating square plate. Each mode (m,n) corresponds to a distinct frequency. Sand (bright lines) collects at the nodal lines where the plate stays still. Higher modes produce finer, more intricate geometry.
Chladni patterns were first demonstrated to Napoleon in 1809. Today, the same physics governs the modes of vibrating guitar strings, the resonances of concert halls, and the structure of electron orbitals in atoms.