Harmonics in strings and air columns

Standing waves on a string

When a string fixed at both ends is plucked, waves travel to each end, reflect and overlap, and a standing wave forms (page 3D). The fixed ends cannot move, so each end must be a node. Only patterns that fit a whole number of half-wavelengths into the length $L$ are possible:

$L = \dfrac{n\lambda}{2} \quad \Rightarrow \quad \lambda_n = \dfrac{2L}{n}, \quad n = 1, 2, 3, \ldots$

Using $f = v/\lambda$, the allowed frequencies are

$f_n = \dfrac{nv}{2L}$

The lowest frequency, $n = 1$, is the fundamental (first harmonic). The others are whole-number multiples of it and are called the second, third, ... harmonics. A plucked string vibrates in many harmonics at once.

The wave speed $v$ on a string depends on the tension and on the mass per unit length: tighter and lighter strings carry waves faster. So a string's pitch can be raised by

• shortening it (pressing a fret): $f \propto 1/L$,
• tightening it (turning the tuning peg): larger $v$,
• using a thinner, lighter string: larger $v$.

(The exact relationship $v = \sqrt{T/\mu}$, with tension $T$ and mass per unit length $\mu$, is beyond the syllabus, but the trends are not.)

Standing waves in air columns

Blow across the top of a bottle or into a flute and the air inside resonates. Sound waves travel up and down the tube and reflect at both ends, even at an open end, and a standing wave forms in the air. It is usual to describe these waves by the displacement of the air:

• At a closed end the air cannot move: a displacement node.
• At an open end the air moves freely: a displacement antinode.

Pipe open at both ends (flute, many organ pipes): an antinode at each end, just like the string's nodes, so

$L = \dfrac{n\lambda}{2}, \qquad f_n = \dfrac{nv}{2L}, \qquad n = 1, 2, 3, \ldots$

Pipe closed at one end (clarinet roughly, a bottle, the resonance tube below): a node at the closed end and an antinode at the open end. The shortest fit is a quarter wavelength, and after that only odd numbers of quarter wavelengths fit:

$L = \dfrac{n\lambda}{4}, \qquad f_n = \dfrac{nv}{4L}, \qquad n = 1, 3, 5, \ldots$

So a closed pipe has a fundamental half that of an open pipe of the same length, and it is missing all the even harmonics. That missing set is part of why a clarinet sounds "hollow" next to a flute.

The antinode at an open end actually sits slightly outside the tube. For a tube of radius $r$, the effective length is about $L + 0.6r$; this end correction matters in accurate experiments.

Why instruments sound different

A flute and a violin can play the same note (the same fundamental) but still sound different. The difference, called timbre or tone quality, comes from the mix of harmonics: which ones are present and how strong each is. The waveform on an oscilloscope is the superposition of all of them, so each instrument has its own shape. The yidaki (didjeridu), which you met in Year 9, is a pipe closed at one end by the player's lips, which is why its overtones are mostly odd harmonics; players shape them with their mouth and voice.

Measuring the speed of sound with a resonance tube