Waves do something that solid objects never do: they pass straight through each other and add up while they overlap. This is called superposition, and it produces two of the most striking effects in acoustics.
Beats and Standing Waves
Interference you can see and hearBeats: two tones, one pulse
Play two frequencies that are close but not identical — say 330 Hz and 326 Hz — and you do not hear two tones. You hear one tone whose loudness swells and fades four times a second.
That pulse rate is simply the difference between the two frequencies:
f_beat = |f₁ − f₂|
Mathematically, adding the two sine waves gives a fast carrier wave multiplied by a slow envelope — the envelope is what your ear hears as the pulsing. Watch the dashed envelope curve in the simulation: the waveform fills it, and the envelope breathes at exactly the beat rate.
This is how musicians tune instruments by ear. Out of tune, you hear beating; as the two strings converge, the beats slow down; when they match, the beating stops entirely.
Standing waves: motion that stands still
Switch to the standing wave tab. Send a wave down a string fixed at both ends and it reflects back and interferes with itself. At most frequencies the result is a mess — but at certain frequencies the pattern locks into place.
The wave appears to stop travelling. Some points, the nodes, never move at all. Between them, the antinodes swing with maximum amplitude. The string can only do this when its length holds a whole number of half-wavelengths:
λ = 2L / n
Increase the harmonic number n and you add a node and an antinode, exactly as a guitarist does when they lightly touch a string to force a higher harmonic.
Try this
- Set the beat rate to 0 Hz. The two frequencies are identical, the envelope becomes flat, and the beating disappears.
- Set it to around 2 Hz and count the pulses — you should get two per second.
- In the standing wave tab, step n from 1 to 6 and count the nodes. It is always n + 1, including the two fixed ends.
Why it matters
Every musical instrument is a standing wave device: strings, air columns, drum heads. The same mathematics describes electron orbitals in atoms and the resonant modes of a bridge — which is precisely why engineers care about a structure’s natural frequencies.