Resonance Frequency Calculator

A string or pipe resonates when its length equals a whole number of half-wavelengths (or quarter-wavelengths for a closed pipe). These are standing wave modes, and each produces a distinct resonant frequency called a harmonic.

Vibrating string (n-th harmonic): f_n = (n / 2L) * sqrt(T / mu)

L = string length in meters, T = tension in newtons, mu = linear mass density in kg/m, n = harmonic number (1 = fundamental, 2 = first overtone, etc.). A guitar string tuned to 440 Hz with L = 0.65 m and mu = 0.0004 kg/m has tension T = f^2 * 4 * L^2 * mu = 440^2 * 4 * 0.65^2 * 0.0004 = about 131 N. That is a fairly typical working tension for a single steel string.

The fundamental (n=1) is the loudest partial. Higher harmonics add timbre, which is why a violin and a flute playing the same note sound different. Their harmonic content differs.

Acoustic pipe:

• Open at both ends: f_n = n * v / (2L), all harmonics present (n = 1, 2, 3…)
• Closed at one end: f_n = (2n - 1) * v / (4L), odd harmonics only (n = 1, 2, 3…)

v = speed of sound (343 m/s in air at 20 C, adjust for other temperatures or gases). A flute is approximately open-open; a clarinet behaves as closed-open below the register key (which is why it overblows at the 12th, a 12th above the fundamental, rather than an octave).

Wavelength: lambda = v / f for pipes, or lambda = 2L / n for strings.