Hard

Vibrating Strings

Simple Explanation

A stretched string, when plucked, vibrates and produces a stationary wave pattern. Its fundamental (lowest, and usually loudest) frequency depends on the string's length, the tension applied, and how much mass it has per unit length.

Why Do We Need It?

This relationship is exactly how stringed instruments (guitars, violins, pianos) work — musicians tune and play notes by controlling these very factors: string length (fretting), tension (tuning pegs), and string thickness (mass per unit length).

Formula

Fundamental Frequency of a Vibrating String

f = (1/2L)√(T/μ)

The lowest (fundamental) frequency at which a stretched string vibrates depends on its length, the tension applied, and its mass per unit length.

f: — Fundamental frequency, in Hz
L: — Length of the string, in metres
T: — Tension in the string, in newtons
μ: — Mass per unit length of the string (linear density), in kg/m

When to use it: Use to find the fundamental (lowest) frequency a string produces, or to see how changing tension, length, or thickness affects pitch.

Worked Example

Finding a string's fundamental frequency

A 0.65 m string has tension 80 N and mass per unit length 0.005 kg/m. Find its fundamental frequency.

Why Does This Work?

A shorter string, a tighter (higher tension) string, or a lighter (lower μ) string all vibrate faster, producing a higher frequency — this matches the formula's structure directly: f increases as L decreases, as T increases, or as μ decreases.

Real-Life Example

Tuning a guitar string

Turning a guitar's tuning peg tightens (increases tension in) a string, raising its pitch.

Increasing tension T directly increases the fundamental frequency (f ∝ √T), raising the pitch — this is exactly the mechanism guitarists use every time they tune a string.

Practice

A guitarist presses a fret, shortening the vibrating length of a string. What happens to the pitch?

Hard

Common mistake

Assuming only length affects a string's pitch — tension and mass per unit length (string thickness) matter equally, which is why guitar strings of different thicknesses are tuned to different notes even at the same length and similar tension.

Quick Review

f = (1/2L)√(T/μ): frequency depends on length, tension, and mass per unit length.
Shorter length, higher tension, or lower mass per unit length → higher frequency (pitch).
This explains fretting (length), tuning (tension), and string gauge (μ) on stringed instruments.