Laws of Vibrating Strings & the Sonometer

Violinists slide their fingers to shorten a string; tuners twist pegs to change its tension; bass strings are made thick. Each trick is one of three simple laws.

Three laws#

From $\nu = \dfrac1{2L}\sqrt{\dfrac F\mu}$ for a string fixed at both ends:

The sonometer#

A sonometer is a hollow wooden sound box with two fixed bridges at its ends. An auxiliary wire is welded tight between them. An experimental wire is fixed at one end, runs over the other bridge and carries a hanger for weights, which set its tension. Pairs of movable bridges under each wire mark out the vibrating length. Wires can be swapped to change μ.

Strike a tuning fork on a rubber pad and press its stem on the box: the box passes the fork’s vibration to the wire. If the fork’s frequency equals a natural frequency of the wire, large standing waves build up: the fork is in unison with the wire.

Spotting resonance

• Paper rider: a small folded paper on the middle of the wire. In the fundamental mode the middle is an antinode, and the rider shakes violently and usually flies off. (It also sits on an antinode in the 3rd harmonic, but shakes less.)
• Beats: sound the fork and pluck the wire together. If the frequencies are close but not equal, the sound swells and fades periodically (beats, next chapter). Adjust the length until the beats disappear.

Testing the law of length

Load the experimental wire (say 3–4 kg), then for each of several tuning forks slide the bridges until the wire resonates in its fundamental. The product νL should come out the same each time.

Testing the laws of tension and mass

Here the experimental wire’s length is fixed, so its frequency changes in a continuous way as you vary the tension (or swap wires). Tuning forks only come in a few fixed frequencies, so instead use the auxiliary wire as an adjustable source: pluck it and slide its bridges until it resonates with the experimental wire, and note its length l′. Since its frequency ∝ 1/l′:

• Law of tension: ν ∝ √F means $l'\sqrt F$ stays constant as you change the load.
• Law of mass: ν ∝ 1/√μ means $l'/\sqrt\mu$ stays constant as you change wires (μ found by weighing a known length).