Standing Waves and Harmonics

A traveling wave carries a pattern through space. A standing wave is different: the pattern seems to stay in place while the medium oscillates. This happens when two matching waves move through the same region in opposite directions, such as an outgoing wave and its reflection from a boundary.

Where the two waves always cancel, the medium stays still; those points are nodes. Where the two waves reinforce most strongly, the medium has the largest oscillation; those points are antinodes. The boundary conditions determine which node-antinode patterns can fit.

Reflection and Superposition

Imagine sending a pulse down a string. At a fixed end, the reflected pulse comes back inverted. If the string is driven periodically, the outgoing and reflected waves overlap again and again. At most frequencies the shape is messy, but at special frequencies a stable standing wave pattern can form.

For two equal sinusoidal waves traveling in opposite directions, their sum can be expressed as:

The key feature is that the spatial and temporal variables can be separated. Each point oscillates in time, but the node locations are fixed in space.

Waves on a String

A boundary condition is a rule the wave must obey at the ends of the system. A string clamped at both ends must have zero displacement at both ends, so both ends are nodes. That requirement allows only wavelengths that fit an integer number of half-wavelengths (i.e. crests/troughs) inside the length $L$:

Here $n = 1, 2, 3, \ldots$ labels the harmonic. The first harmonic is the fundamental. The second harmonic has twice the frequency, the third has three times the frequency, and so on.

For a stretched string, the wave speed can be estimated from the tension $T$ and linear mass density $\mu$:

A tighter string raises the wave speed and therefore raises the harmonic frequencies. A heavier string lowers the wave speed and lowers the frequencies. Note that even though we are discussing standing waves, the wave speed still has an effect.

Air Columns

Air columns also support standing waves. You can describe these as pressure waves, but it is often easiest to draw the displacement of the air. At an open end, air can move freely, so the open end is a displacement antinode. At a closed end, air cannot move through the wall, so the closed end is a displacement node. Note that displacement in this context refers to a correlated motion of ~$10^{23}$ molecules which are surprisingly speedy as individuals.

An open-open air column follows the same wavelength and frequency sequence as a fixed-fixed string:

An open-closed air column fits one quarter-wavelength in the fundamental mode, then only odd multiples after that:

This is why an ideal closed-open pipe supports the 1st, 3rd, 5th, and higher odd harmonics, but skips the even ones. One bookkeeping note: pressure nodes and antinodes are reversed compared with displacement nodes and antinodes.

As you explore the standing-wave animations below, try this sequence:

1. Start with the guitar string and step through the first few harmonics.
2. Switch to the flute pipe and compare the endpoint labels.
3. Switch to the clarinet pipe and notice how the ladder jumps through odd harmonics.
4. Change the length and wave speed separately to see how each one shifts the whole set of frequencies.

Music Connections

Musical pitch is defined by the fundamental frequency, but the higher harmonics are just as important. Real instruments produce a variety of harmonics at different strengths. Those extra components help shape timbre, which is why a violin, flute, and clarinet can play the same pitch but still sound different.

Tuning an instrument means changing the allowed frequencies. A guitarist changes the vibrating length by pressing a fret, and changes tension by tuning the string. A wind player changes the effective air-column length with holes, valves, slides, or keys. In every case, the sound produced comes from the underlying standing wave patterns.