Last updated: 2026-06-15

Standing Wave Harmonics

A guitar string and an organ pipe both make sound for the same reason: a wave trapped in a fixed length can only vibrate at certain special frequencies. These are the harmonics, and they set the pitch and the character of almost every musical instrument. This calculator finds the harmonic frequencies and wavelengths from the wave speed, the length and the boundary conditions.

Why only certain frequencies

A wave confined between two ends reflects back and forth. Most frequencies interfere with themselves and die away, but a few fit the length perfectly and reinforce into a stable standing-wave pattern. Which ones fit depends on what the ends demand — a fixed or closed end forces a node (no motion), while a free or open end allows an antinode (maximum motion).

The formulas

For a string fixed at both ends, or a pipe open at both ends, every whole-number harmonic exists:

For a pipe closed at one end, only the odd harmonics survive:

Here $v$ is the wave speed in the medium and $L$ is the length. The lowest harmonic is the fundamental, which sets the pitch.

Worked example

A pipe open at both ends, length $L = 0.5\ \text{m}$, with sound travelling at $v = 343\ \text{m/s}$:

Close one end and the fundamental drops to $\dfrac{343}{4 \times 0.5} = 171.5\ \text{Hz}$ — an octave lower for the same length, which is why a stopped pipe sounds so much deeper than an open one.

Fundamental, overtones and timbre

The fundamental is usually the loudest harmonic and fixes the note you hear. The higher harmonics, or overtones, sound at the same time; their relative strengths are what make a violin, a flute and a human voice singing the same note all sound completely different. Tone is largely a question of which harmonics are present and how strong each one is.

What changes the pitch

Anything that changes $v$ or $L$ retunes the instrument. Shortening the vibrating length raises every frequency — that is what fretting a guitar string or covering a flute hole does. On a string the speed depends on tension and thickness, so tightening a string raises its pitch. In a pipe the speed is the speed of sound, which rises with temperature, which is why wind instruments drift sharp as they warm up.