Physics-first music theory
Physics-first music theory
A standing wave is a vibration pattern that forms when two identical waves travel in opposite directions through the same medium and interfere with each other. Unlike a traveling wave that moves from one place to another, a standing wave oscillates in place — some points vibrate with maximum amplitude while others remain permanently still.
🎯 Simple version: Pluck a guitar string. The wave bounces back and forth between the two ends. The bouncing waves add up and create a pattern that stands still — some spots vibrate a lot, others don’t move at all. The simplest pattern has one big hump. The next has two humps. Each pattern vibrates faster than the last.
When a wave hits a fixed boundary — the nut or bridge of a guitar string, the closed end of a pipe — it reflects back. The reflected wave has the same frequency, wavelength, and amplitude as the original, but travels in the opposite direction.
Mathematically, the incident and reflected waves are:
y₁ = A × sin(kx - ωt) (traveling right)
y₂ = A × sin(kx + ωt) (traveling left, after reflection)
Their superposition (sum) gives:
y = y₁ + y₂ = 2A × sin(kx) × cos(ωt)
This is the key result. The spatial part sin(kx) and the temporal part cos(ωt) are separated — the shape of the wave doesn’t travel. Every point on the string oscillates up and down in place, with an amplitude determined by its position.
A standing wave on a string of length L (fixed at both ends) can only exist at specific wavelengths — those where the boundary conditions are satisfied (zero displacement at both ends). The allowed wavelengths are:
λₙ = 2L / n for n = 1, 2, 3, ...
Each value of n is called a mode:
| Mode | Wavelength | Frequency | Pattern |
|---|---|---|---|
| n = 1 (fundamental) | 2L | f₁ = v / 2L | One half-wave: one antinode |
| n = 2 | L | 2f₁ | Two half-waves: two antinodes |
| n = 3 | 2L/3 | 3f₁ | Three half-waves: three antinodes |
| n = N | 2L/N | Nf₁ | N half-waves: N antinodes |
Nodes are points of zero displacement — they never move. A mode-N standing wave has N + 1 nodes (including the two fixed endpoints). Antinodes are the points of maximum displacement, located halfway between adjacent nodes.
The frequency of mode N is exactly N times the fundamental frequency:
fₙ = n × f₁ = n × v / 2L
This is the same integer-multiple relationship that defines the harmonic series. When a string vibrates, it doesn’t vibrate in just one mode — it vibrates in many modes simultaneously. The relative amplitudes of the modes determine the sound’s timbre. The fundamental (mode 1) determines the perceived pitch; the higher modes are the overtones that give the sound its character.
This is not a coincidence. The harmonic series is the set of standing wave modes. The physics of bounded vibrating systems produces integer-frequency-multiple patterns automatically.
The standing wave patterns above assume fixed ends — both endpoints are forced to remain at zero displacement. This is the case for:
But not all boundaries are fixed. An open end (like the open end of a flute or organ pipe) creates an antinode instead of a node — the air is free to vibrate with maximum displacement there. This changes which modes are allowed:
| Boundary type | Allowed modes | Examples |
|---|---|---|
| Both ends fixed | All integer modes: 1, 2, 3, 4, … | Strings, closed-closed pipes |
| One open, one closed | Odd modes only: 1, 3, 5, 7, … | Clarinet, closed organ pipe |
| Both ends open | All integer modes: 1, 2, 3, 4, … | Flute, open organ pipe |
The “one open, one closed” case is particularly interesting — only odd-numbered harmonics are present. This gives instruments like the clarinet their characteristic hollow timbre (missing even harmonics). Details of how pipe geometry shapes instrument sound are in instrument-physics.md.
| PhizMusic | Western | Other Systems |
|---|---|---|
| Mode N | Nth harmonic / Nth partial | Eigenmode, normal mode (physics) |
| Node | Node | Knotenpunkt (German acoustics) |
| Antinode | Antinode, belly | Schwingungsbauch (German acoustics) |
| Fundamental (mode 1) | Fundamental, first harmonic | Grundton (German), 基音 (Chinese: jīyīn) |
| Standing wave | Standing wave, stationary wave | Stehende Welle (German), 定常波 (Japanese: teijōha) |
| Boundary condition | Boundary condition | Randbedingung (German) |