Hong Life CalculatorsUnderstanding the string vibration and standing wave calculator — concept and principle
A string fixed at both ends, like a guitar, violin or piano string, vibrates only at certain frequencies. This calculator finds the frequency of any harmonic from the string's vibrating length, its tension and its mass per unit length, and also gives the speed of waves along the string. It is used to choose string gauges, estimate the tension needed for a target note and explain why tightening a string raises its pitch.
How it is calculated
- Wave speed: v = √(T / μ)
- Frequency of harmonic n: fₙ = (n / 2L) · √(T / μ) = n · v / (2L)
L is the vibrating length in metres, T the tension in newtons, μ the linear density, and n = 1, 2, 3 … the harmonic number (n = 1 is the fundamental). The calculator takes μ in g/m and divides by 1000 to get kg/m, so the units come out in m/s and Hz.
The principle
A transverse wave on a stretched string travels at √(T/μ): more tension means a stronger restoring force, more mass means more inertia. When waves reflect from the two fixed ends, they interfere and form a standing wave. Because both ends must stay still (nodes), the string can only hold a whole number of half-wavelengths: L = n · λ/2, so λₙ = 2L/n. Combining with f = v/λ gives fₙ = n·v/(2L). These relations are known as Mersenne's laws: frequency is inversely proportional to length and proportional to the square root of tension, and inversely proportional to the square root of linear density.
Worked example
- Inputs: L = 0.65 m, T = 70 N, μ = 1.1 g/m = 0.0011 kg/m, n = 1.
- T / μ = 70 / 0.0011 ≈ 63,636.4 m²/s².
- Wave speed v = √63,636.4 ≈ 252.26 m/s.
- f₁ = 1 × 252.26 / (2 × 0.65) = 252.26 / 1.3 ≈ 194.05 Hz (194.0480689).
This is close to G3 (about 196.0 Hz in standard tuning).
| String length (m) | Frequency (Hz) |
|---|---|
| 0.163 | 776.1922756 |
| 0.325 | 388.0961378 |
| 0.487 | 258.7307585 |
| 0.65 | 194.0480689 |
| 0.812 | 155.2384551 |
| 0.975 | 129.3653793 |
| 1.3 | 97.02403444 |
How to read the result
Higher harmonics are exact integer multiples of the fundamental in this ideal model: n = 2 gives about 388.1 Hz, n = 3 about 582.1 Hz. To raise the pitch by one octave you need four times the tension, or half the length, which is why frets work by shortening the string. Because frequency depends on the square root of tension, a 1% change in tension shifts the pitch by only about 0.5%, roughly 8.6 cents.
Common mistakes
- Entering μ in kg/m when the field expects g/m, which makes the result about 31.6 times too high or low.
- Using the total string length instead of the vibrating length between the nut and the saddle (or bridge).
- Entering tension in kilograms-force; convert with 1 kgf ≈ 9.81 N.
Frequently asked questions
How do I find the tension for a target note?
Rearrange the formula: T = μ · (2Lf / n)². For the fundamental, T = μ(2Lf)².
Why do bass strings have wound wire?
Winding adds mass (higher μ) without making the string much stiffer, so it can reach low notes at a playable tension and length.
Sources: M. Mersenne, Harmonie universelle (1636) · N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, Springer