String Frequency Calculator (Standing Waves & Harmonics)
By Santaji GadepatilinUpdated November 3, 2026
A string fixed at both ends vibrates in standing waves whose wavelengths fit a whole number of half-waves into its length. This string frequency calculator uses f = (n ÷ 2L) √(T ÷ μ) to find the fundamental and harmonic frequencies, the wave speed and the wavelength — useful for musical instruments, cables and wire-tension checks.
How to use the string frequency calculator
Enter the vibrating length (nut to bridge on an instrument), the tension, the string’s linear density in grams per metre and the harmonic number. The calculator returns the wave speed, the frequency of that harmonic and its wavelength. To find the tension for a target note, enter the frequency and clear the tension.
String frequency formula
v = √(T ÷ μ) · fn = n v ÷ (2L) · λn = 2L ÷ n
Example
A steel guitar string, 648 mm long, 0.4 g/m, tensioned to 70 N:
- Wave speed v = √(70 ÷ 0.0004) = 418.3 m/s
- Fundamental f₁ = 1 ÷ (2 × 0.648) × 418.3 = 322.8 Hz (close to the E₄ note at 329.6 Hz)
Harmonics of the example string
| Harmonic n | Frequency | Wavelength | Nodes between ends |
|---|---|---|---|
| 1 | 322.8 Hz | 1.296 m | 0 |
| 2 | 645.6 Hz | 0.648 m | 1 |
| 3 | 968.4 Hz | 0.432 m | 2 |
| 4 | 1291.1 Hz | 0.324 m | 3 |
| 5 | 1613.9 Hz | 0.259 m | 4 |
| 6 | 1936.7 Hz | 0.216 m | 5 |
| 7 | 2259.5 Hz | 0.185 m | 6 |
| 8 | 2582.3 Hz | 0.162 m | 7 |
Notes
- Stiff, thick strings are slightly inharmonic: their overtones sit a little above exact multiples of f₁.
- The same equation estimates the tension in a stay cable or wire rope from its measured natural frequency.
- Two strings tuned slightly apart produce audible beats — see the beat frequency calculator.
Frequently asked questions
What is the formula for the frequency of a string?
f = (n ÷ 2L) × √(T ÷ μ), where L is the vibrating length, T the tension in newtons, μ the mass per metre in kg/m and n the harmonic number (1 for the fundamental).
How does tension change the pitch?
Frequency is proportional to the square root of tension, so raising the pitch by an octave (double the frequency) needs four times the tension.
What are harmonics?
A string can vibrate with 1, 2, 3… half-wavelengths along its length. These modes have frequencies n × f₁ and together give an instrument its tone. Harmonic n has n − 1 nodes between the ends.
How do I find the tension needed for a note?
Enter the length, the linear density and the target frequency, then clear the tension field. The calculator solves T = μ (2Lf ÷ n)².