A guitar string, a flute, and an organ pipe all make sound the same fundamental way: a wave reflects back on itself inside a fixed length, and only specific frequencies fit that length without canceling themselves out. This calculator finds those frequencies β the harmonics β for a string fixed at both ends, a pipe open at both ends, or a pipe closed at one end, and shows how the closed-pipe case differs from the other two.
Boundary conditions decide which frequencies survive
A wave traveling along a string or down a pipe reflects when it hits a boundary. If the reflected wave and the original wave line up perfectly β crest matching crest, trough matching trough β they reinforce each other into a stable, stationary pattern: a standing wave. If they don't line up, the wave interferes with itself destructively and dies out. Whether a frequency lines up depends entirely on the boundary conditions at each end. A string fixed at both ends and a pipe open at both ends share the same rule: both ends must be nodes (string) or antinodes (open pipe), which only happens when the length equals a whole number of half-wavelengths β L = nΒ·Ξ»/2, giving fn = nΒ·v/(2L) for every positive integer n.
Why a closed pipe only produces odd harmonics
A pipe closed at one end has mismatched boundary conditions: the closed end must be a node (the air can't move there) and the open end must be an antinode (the air moves freely). That combination only fits a quarter-wavelength, plus any whole number of additional half-wavelengths, into the pipe's length β L = nΒ·Ξ»/4 for odd n only. Even values of n would require the open end to be a node too, which contradicts the physical boundary condition, so those frequencies simply never form. The practical result: a closed pipe's fundamental is exactly half the frequency of an identical-length open pipe or string, and it skips every even harmonic entirely β which is part of why clarinets (largely closed-pipe behavior) sound different from flutes (open-pipe behavior) even when built to a similar length.
Reading nodes, antinodes, and overtones together
Every harmonic above the fundamental is called an overtone, and each one adds another node-antinode pair along the length. The fundamental (n=1) has the simplest pattern β a single antinode at the middle of a string, or at the open end of a pipe. Real instruments rarely produce a pure single harmonic; they produce a mix dominated by the fundamental with progressively quieter overtones layered on top, and the relative strength of those overtones is a big part of what makes a violin sound different from a flute playing the exact same fundamental pitch. This calculator's Overtone Series tab shows the full stack of harmonic frequencies your system can produce, which is the starting point for understanding that richer, real-world sound.