What standing waves are and how they are produced

• Standing waves are particular kinds of oscillations of a medium in which energy does not propagate from one point to another, as happens for travelling waves, but stays distributed in a way that does not change in time.
• In particular, there are places in space where there is no oscillation (nodes), and others where there is always the maximum oscillation (antinodes). These places do not change in time.

Looking at the picture alongside, we notice that the oscillation in black is stationary. Indeed, each node, marked by a small black disc, stays fixed in time, and so does each antinode, which lies halfway between two consecutive nodes. In red and green, instead, travelling waves are shown, whose nodes (and antinodes) move at constant speed to the right and to the left respectively. We observe that the standing wave can be obtained by adding the two travelling waves. To understand the mechanism of formation of standing waves in more detail, see the page reflection in strings.

Two consecutive nodes are half a wavelength apart. On a string fixed at both ends, the ends are necessarily nodes: a standing wave can therefore form only if the length L of the string contains a whole number n of half wavelengths, ${\displaystyle L=n\lambda /2}$. Since ${\displaystyle f=v/\lambda }$, the only possible frequencies are

1

({{{2}}})

that is, the fundamental frequency ${\displaystyle f_1=v/2L}$ and its whole-number multiples. These are the natural frequencies of the string (see normal modes of a string). In pipes the condition is similar, but depends on whether the ends are open or closed (see standing waves in one dimension).

The formation of standing waves in vibrating elements is of fundamental importance for the production of sound, and is therefore necessary (though not sufficient) for a vibrating system to become a musical instrument. The simplest example illustrating this fact is that of sound tubes. A flute, for example, is a tube about 60 cm long. A single sound pulse therefore takes less than two thousandths of a second to travel along it, after which it meets an end or a side hole. There, part of the energy is transmitted to the outside air, and part is reflected back into the tube, which it now travels backwards for another two thousandths of a second until it meets the other end of the pipe (see the animations in reflection in strings, interference between reflected waves, impedance matching). Of course, in such a short time a sound pulse cannot be perceived by the ear as a sound. To obtain a definite sound sensation from it, therefore, a stable oscillation must build up in the tube after hundreds or thousands of reflections back and forth.

At this point, then, to determine whether our tube is suitable to become a musical instrument, we must answer two questions:

1. can oscillations of any frequency propagate along the pipe?
2. Is the oscillation obtained from the superposition of these travelling waves stable?

The first question is answered by studying the properties of waveguides. We find that in a pipe the plane wave, which travels along the axis, propagates at any frequency, while the transverse modes (waves bouncing between the walls) propagate only above a minimum frequency, inversely proportional to the diameter of the tube, called the cutoff frequency (see waveguides in more depth). This property is not a limitation, at least for narrow instruments such as the flute, since the cutoff frequency (about 10 kHz for a tube 19 mm in diameter) is much higher than the notes played. It becomes very relevant, instead, in instruments ending with a large bell.

The second question is the one that interests us most here: given that every frequency in the audible acoustic range can propagate back and forth along the pipe, will it be possible to set up a stable oscillation after the wave has travelled back and forth along the pipe thousands of times?

Well, standing waves are precisely the only waves that are "stable" in the long run. Indeed, by superposing travelling waves with frequencies different from the natural frequencies of the pipe, which are precisely the frequencies of the standing waves, one obtains an unstable oscillation, which cannot produce over time the sensation of a definite, constant tone, although in very many cases it contributes substantially to determining the timbre of the instrument, characterizing its transients at the attack, at the release, and in the passages between the different resonances.

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Further reading and links

• String instruments and wind instruments, that is: Standing waves in one-dimensional systems
• Percussion, that is, Standing waves in two-dimensional systems, Snare drum, Timpani
• Architectural acoustics: Standing waves in three-dimensional systems
• Questions and answers on standing waves.
• Related questions and answers:
  • Seeing that a stationary wave does not propagate, how is it a wave?