Imagine a guitar string. Pluck it, and you hear a note. Press your finger down at a different fret — the vibrating part of the string gets shorter — and the note gets higher. Tighten the tuning peg, and the note rises again. Swap the string for a thicker one, and the pitch drops. That's the entire physics of a sonometer, stripped down to its bones.
A sonometer is simply a laboratory version of that guitar string. It's a long, hollow wooden box with a thin wire stretched tightly over two fixed bridges. You can change three things about the wire: how long the vibrating segment is (by moving a third, movable bridge), how tight the wire is (by hanging weights on one end), and what kind of wire you use (different materials or thicknesses). A small, light paper rider placed on the wire helps you see when the wire is vibrating strongly — it dances or falls off at resonance.
Note
The hollow box isn't decorative. It acts as a sounding board, amplifying the faint sound of the wire so you can hear the note clearly.
The Core Relationship: Frequency and Its Dependence
The sonometer exists to verify one central formula. For a stretched string, the fundamental frequency f (the lowest note it can produce) is:
f=2L1μT
where:
L is the vibrating length of the string (in metres)
T is the tension in the string (in newtons)
μ is the linear mass density — mass per unit length of the string (in kg/m)
This isn't a random equation. It comes from the wave equation for a string fixed at both ends. The wave speed on a string is v=T/μ, and the fundamental standing wave has a wavelength λ=2L. Since v=fλ, you get f=v/(2L)=(1/2L)T/μ.
f=2L1μT
What the Sonometer Actually Shows You
You can test each variable one at a time, keeping the others constant.
Length: Move the movable bridge to change L. Pluck the wire and find the tuning fork that matches its pitch. You'll discover that f∝1/L — halve the length, double the frequency. That's why guitar frets get closer together as you go up the neck.
Tension: Hang different weights on the end of the wire. More weight means more tension. You'll find f∝T. To double the frequency, you need four times the tension.
Linear density: Use wires of different thicknesses or materials. A thicker wire has larger μ, so f∝1/μ. Heavy strings on a piano are thick and produce low notes; thin strings produce high notes.
Watch out
A common mistake: thinking frequency is proportional to tension itself. It's proportional to the square root of tension. Doubling tension only raises frequency by a factor of about 1.414, not 2.
The fundamental frequency of a stretched string depends on its length, tension, and mass per unit length, and each dependence is stated as a separate law. …
The fundamental frequency (n) of a vibrating stretched string depends on three factors — its vibrating length L, the tension T in the string, and its mass per unit length (linear density) m — and each dependence, when the other two are held fixed, is stated as one of the three 'laws' of transverse vibration.
For a string of vibrating length L, under tension T, with mass per unit length m, the fundamental frequency of transverse vibration is:
n = (1/2L) √(T/m)
From this single formula, three separate laws are stated by varying one quantity at a time while keeping the other two constant:
Law of length: For a given tension T and given mass per unit length m, the fundamental frequency is inversely proportional to the vibrating length: n ∝ 1/L, i.e., nL = constant.
Law of tension: For a given length L and given mass per unit length m, the fundamental frequency is directly proportional to the square root of the tension: n ∝ √T, i.e., n/√T = constant.