Stationary Waves, Resonance and Harmonics in Strings and Pipes

1One string, many notes

An inanga player plucks strings stretched over a hollow wooden trough. A plucked string does not vibrate in any shape it likes: it settles into a few patterns that fit exactly between its fixed ends, and each pattern has its own frequency. Wind instruments, from a flute to a blown bottle, do the same with a column of air. These patterns are stationary waves, and the frequencies they allow decide the notes an instrument can play.

2Stationary waves: a reminder from S5

When two waves of the same frequency and amplitude travel in opposite directions, for example a wave and its reflection, they superpose to give a stationary wave:

Points where $\sin kx=0\backslash sin\; kx\; =\; 0$ never move: these are nodes (N). Halfway between them the amplitude is $2A2A$: antinodes (A). Adjacent nodes are $\frac{\lambda }{2}\backslash frac\{\backslash lambda\}\{2\}$ apart, and a node and the next antinode are $\frac{\lambda }{4}\backslash frac\{\backslash lambda\}\{4\}$ apart. No energy is carried along; it moves back and forth within each loop.

3Strings fixed at both ends

Both ends are nodes, so a whole number of half-wavelengths must fit on the string: $L=n\frac{{\lambda }_n}{2}L\; =\; n\backslash frac\{\backslash lambda\_n\}\{2\}$. The speed of a transverse wave on a string of tension $TT$ and mass per unit length $\mu \backslash mu$ is $v=\sqrt{\frac{T}{\mu }}v\; =\; \backslash sqrt\{\backslash frac\{T\}\{\backslash mu\}\}$, so

Figure 3.1: The first three modes of a string. Solid and dashed curves show the string at its two extreme positions.

A string can vibrate in all its harmonics at once. The fundamental ($n=1n\; =\; 1$) sets the pitch; the mixture of higher harmonics sets the quality. The formula gives the laws of a vibrating string: for the fundamental, $f\propto \frac{1}{L}f\; \backslash propto\; \backslash frac\{1\}\{L\}$ (fixed $TT$ and $\mu \backslash mu$), $f\propto \sqrt{T}f\; \backslash propto\; \backslash sqrt\{T\}$ (fixed $LL$ and $\mu \backslash mu$) and $f\propto \frac{1}{\sqrt{\mu }}f\; \backslash propto\; \backslash frac\{1\}\{\backslash sqrt\{\backslash mu\}\}$ (fixed $LL$ and $TT$). Guitarists press a string against a fret to shorten it; a thicker string (larger $\mu \backslash mu$) gives the low notes.

4Air columns in pipes

In a pipe the stationary wave is a sound wave, and we describe it 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. (Remember lesson 1: where the displacement is zero the pressure change is greatest, so a displacement node is a pressure antinode.)

Figure 3.2: Displacement envelopes of the first three modes in each kind of pipe.

A closed pipe has a fundamental one octave lower than an open pipe of the same length, and its lack of even harmonics gives it a hollow quality.

5The end correction

The air just outside an open end also moves, so the antinode forms slightly beyond the end. The effective length is $L+eL\; +\; e$, where the end correction $e\approx 0.6re\; \backslash approx\; 0.6r$ for a pipe of radius $rr$. For a closed pipe, $L+e=\frac{\lambda }{4}L\; +\; e\; =\; \backslash frac\{\backslash lambda\}\{4\}$ for the first resonance.

6Experiment 3.1: the speed of sound with a resonance tube

Apparatus: a tube open at both ends, about 1 m long and 3 to 4 cm wide; a tall cylinder of water to stand it in; a clamp; tuning forks of known frequency; a rubber pad; a metre rule.

Figure 3.3: The water closes the bottom of the air column. Raising the tube lengthens the column.

1. Stand the tube in the water with only a short length of air column. Strike a 512 Hz fork on the rubber pad and hold it, prongs vertical, just above the open end.
2. Raise the tube slowly until the sound swells to its loudest: the first resonance. Measure the air column $L_{1}L\_1$ from the top of the tube to the water surface. Repeat, approaching the position from above as well as from below, and average.
3. Keep raising the tube to find the second resonance, $L_{2}L\_2$.

Analysis: the air column resonates at $\frac{1}{4}\lambda \backslash frac\{1\}\{4\}\backslash lambda$ and then $\frac{3}{4}\lambda \backslash frac\{3\}\{4\}\backslash lambda$, each measured from the antinode a distance $ee$ above the tube:

Subtracting removes $ee$: $L_2-L_1=\frac{\lambda }{2}L\_2\; -\; L\_1\; =\; \backslash frac\{\backslash lambda\}\{2\}$, so $\lambda =2(L_2-L_1)\backslash lambda\; =\; 2(L\_2\; -\; L\_1)$ and $v=2f(L_2-L_1)v\; =\; 2f(L\_2\; -\; L\_1)$. Then $e=\frac{\lambda }{4}-L_1=\frac{L_2-3L_1}{2}e\; =\; \backslash frac\{\backslash lambda\}\{4\}\; -\; L\_1\; =\; \backslash frac\{L\_2\; -\; 3L\_1\}\{2\}$.