Press a finger onto a guitar fretboard and the note jumps higher. Turn the tuning peg and it rises again. Switch from a thin string to a thick wound one and the pitch drops, even though both strings span the same distance. A guitarist makes these adjustments by feel, but every one of them is a deliberate move along a single equation that governs how fast a stretched string vibrates.
This article explains why a fixed-end string vibrates at a definite frequency, derives the fundamental-frequency formula in usable form, works a numerical example for a realistic string, and clears up the mistakes that most often distort the result.
Why this calculation matters
A vibrating string is the cleanest example of a resonant system in all of physics, and the lessons it teaches carry far beyond music. The same standing-wave behavior shapes the tension members of bridges and buildings, the cables of suspension structures, transmission lines swaying in the wind, and the strings used in sensor and measurement instruments. Anywhere a slender element is stretched between two supports, it has natural frequencies, and those frequencies decide how it responds to disturbance.
For musical instruments the calculation is the design itself. The choice of string length, the tension a frame must withstand, and the linear density of each string are all set so that the instrument lands on the correct pitches. For structural cables the same formula runs in reverse: measure the vibration frequency of a stay cable and you can infer its tension without ever cutting into it. Understanding the string is the entry point to resonance everywhere.
The core method
A string fixed at both ends cannot move at those endpoints. Any vibration must therefore fit a whole number of half-wavelengths between the supports. The longest pattern that fits — a single arch with no interior still point — is the fundamental mode, and it sets the lowest and most prominent frequency the string produces.
The fundamental frequency of a string fixed at both ends is:
f = (1 / (2*L)) * sqrt(T / mu)
Here L is the vibrating length in meters, T is the tension in newtons, and mu is the linear mass density — the mass per unit length of the string, in kilograms per meter.
The structure of the formula carries the physics. The quantity sqrt(T/mu) is the speed at which a wave travels along the string. A tighter string carries waves faster; a heavier string carries them slower. The factor 1/(2*L) then converts that wave speed into a frequency, because the fundamental wavelength is exactly twice the string length. Reading the equation directly: pitch goes up when you increase tension, and pitch goes down when you make the string longer or heavier.
The higher modes follow a simple pattern. They are integer multiples of the fundamental — 2f, 3f, 4f and so on — and that harmonic series is what gives a plucked string its rich, recognizable timbre rather than a pure tone.
A worked example
Take a string with a vibrating length of L = 0.65 m, stretched to a tension of T = 80 N, with a linear density of mu = 0.001 kg/m. These are realistic values for a steel guitar string. What is its fundamental frequency?
Step 1 — find the wave speed. Compute the ratio inside the square root:
T / mu = 80 / 0.001 = 80000
sqrt(80000) = 282.8 m/s
So waves travel along this string at about 283 meters per second.
Step 2 — apply the length factor.
1 / (2*L) = 1 / (2 * 0.65) = 1 / 1.3 = 0.769 per meter
Step 3 — combine.
f = 0.769 * 282.8
f = 217.6 Hz
The string vibrates at a fundamental frequency of about 217.6 Hz, close to the A below middle C. From here the instrument's behavior follows directly. Pressing a fret shortens L and raises the pitch. Tightening the tuning peg increases T and raises it too. Choosing a heavier string raises mu and lowers the pitch — the reason the low strings of a guitar are thick and wound rather than simply slack.
Common mistakes
Confusing total mass with linear density. The mu in the formula is mass per unit length, not the total mass of the string. If you know the total mass m and the full length, divide: mu = m / length. Substituting total mass directly produces a frequency that is wrong by a large factor.
Using the wrong length. L is the freely vibrating length between the two fixed points, not the full physical length of the string. On a fretted instrument the vibrating length is the distance from the bridge to the finger, which is exactly how a player changes the note.
Mixing up units. The formula is consistent in SI: tension in newtons, linear density in kilograms per meter, length in meters. String densities are often quoted in grams per meter, and forgetting the factor of 1000 conversion is a common source of error. Convert to kilograms per meter before substituting.
Assuming tension is independent of pitch. Tension and frequency are linked, not separate dials. Tightening a string to raise its pitch increases tension at the same time, and on a real instrument that added tension is carried by the frame and the bridge. Stiff, thick strings also add a small inharmonic correction the ideal formula does not include.
Try the interactive NovaSolver calculator
Working one string by hand shows the method, but the feel for how tension, length, and density trade off comes faster when you can hear and see it change. The String Resonance & Standing Wave Simulator on NovaSolver lets you set tension, linear density, and length with sliders, then displays the fundamental frequency, wave speed, and active vibration modes while animating the standing wave and playing the actual tone.
Related calculators
- Resonance Frequency Calculator — generalize the idea of a natural frequency to springs, masses, and other oscillating systems.
- Single-Degree-of-Freedom Response — see how a resonant system reacts when it is driven near its natural frequency.
- Room Acoustics Calculator — follow the sound a vibrating string radiates into the space around it.
You can browse the rest in the acoustics tools hub.
Closing note
The vibrating string packs a great deal of physics into one short formula. A string sings because only whole numbers of half-wavelengths fit between its fixed ends, and the pitch it settles on is set by just three quantities: length, tension, and mass per unit length. Tighten it and the pitch rises; lengthen or weight it and the pitch falls. Keep linear density distinct from total mass, use the true vibrating length, stay consistent with units, and the equation f = (1/(2L))*sqrt(T/mu) will explain not just a guitar but resonance wherever a stretched element is found.
Top comments (0)