Vibrating strings: harmonics, laws of strings and the sonometer

Pluck a guitar string and it vibrates at a frequency set by its length, its tension and its thickness. Tighten it, and the note rises. Shorten it with a finger on a fret, and the note rises again. This lesson explains why.

Learning objectives

By the end of this lesson you should be able to:

• explain how stationary waves form on a stretched string
• find the frequencies of the fundamental and harmonics of a string
• state the laws of vibration of strings and use $f = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$
• describe the sonometer and its use

Stationary waves on a string

A string fixed at both ends (a guitar string, a sonometer wire) is plucked. Waves travel to each fixed end, reflect and superpose with the oncoming waves, setting up a stationary wave. The fixed ends cannot move, so they are always nodes.

Figure 1. The fundamental and the second harmonic of a stretched string. WASSCE Tutor diagram.

Modes of vibration

A string can vibrate at all whole-number multiples of its fundamental frequency, which is why a plucked string gives a rich sound.

Speed of waves on a string

The speed of transverse waves on a string depends on the tension $T$ and the mass per unit length $\mu$ (kg m⁻¹):

$$v = \sqrt{\dfrac{T}{\mu}}$$

So the fundamental frequency is

$$f_0 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$$

The laws of vibration of strings

For a stretched string vibrating in its fundamental mode:

1. Length law: $f \propto \dfrac{1}{L}$ (tension and mass per unit length constant) — shortening the string raises the pitch.
2. Tension law: $f \propto \sqrt{T}$ (length and mass per unit length constant) — tightening the string raises the pitch.
3. Mass law: $f \propto \dfrac{1}{\sqrt{\mu}}$ (length and tension constant) — thicker, heavier strings give lower notes.

The sonometer

A sonometer is a hollow wooden box with a wire stretched over two movable bridges. The tension is set by hanging masses from one end of the wire (over a pulley). The box amplifies the sound.

Using it to verify the length law:

1. Keep the tension constant. Strike a tuning fork of known frequency and hold its stem on the box.
2. Place a small paper rider on the wire midway between the bridges. Move a bridge until the paper rider is thrown off: the wire is resonating with the fork (same frequency).
3. Measure the vibrating length $L$ between the bridges.
4. Repeat with forks of different frequencies $f$.
5. Plot $f$ against $\dfrac{1}{L}$: a straight line through the origin verifies $f \propto \dfrac{1}{L}$.

Key points

• A string fixed at both ends has nodes at the ends; fundamental $L = \lambda/2$, $f_0 = v/2L$.
• Harmonics are whole-number multiples of $f_0$.
• $v = \sqrt{T/\mu}$; $f_0 = \dfrac{1}{2L}\sqrt{T/\mu}$.
• Laws: $f \propto 1/L$, $f \propto \sqrt{T}$, $f \propto 1/\sqrt{\mu}$.

WASSCE examination focus

Quick check

1. What is the wavelength of the fundamental of a string 0.5 m long?
2. If the length of a vibrating string is halved (same tension), what happens to its frequency?
3. Which harmonic has three loops?

Answers: 1. $\lambda = 2L = 1.0$ m. 2. It doubles. 3. The third harmonic.