In this lesson, you’ll learn about the kind of waves that are responsible for our experience of musical notes. These waves are called standing waves. Learn what makes them unique and how they represent the notes we hear.

What Is a Standing Wave?

Imagine plucking a string on a guitar or blowing into a whistle. In both situations, you are creating vibrations, also known as mechanical waves, within the body of the instrument. The vibrating instrument creates sound that your ears detect after energy from the vibration (in the form of a sound wave) travels through the air to your ear. The vibrations that are created within the instrument itself are known specifically as standing waves.

A standing wave is a particular kind of wave that can only be created when a wave’s motion is restricted to a finite region. To understand exactly what this means, let’s focus on a vibrating guitar string. The guitar string’s motion is restricted on both ends of the string: on the fretboard by your finger on one side and by the bridge on the other side, where it attaches to the body of the guitar. When the string is plucked, the wave reflects off each of these boundaries. The wave’s energy quickly spreads out as it repeatedly moves back and forth between the two ends of the string. Through a complicated process of interference, standing waves are produced.

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Characteristics of a Standing Wave

A standing wave gets is name from the fact that it does not appear to be traveling from one side of the string to the other. Rather, it is waving while it stands in place. Take a look at the following standing wave modeled after the string of a guitar:

An example of one standing wave

Even though the spring is only fixed in place at the ends so the entire middle can vibrate freely, you can see that there are some points on the string that are not moving at all. These stationary points on a standing wave are called nodes.

The standing wave shown has three nodes; we only count the untethered stationary points, not the endpoints, of the string where it is being held down. The points that are halfway between two adjacent nodes, where the motion of the string has the greatest deviation from its unplucked position, are called antinodes. The wave depicted has four antinodes – a standing wave always has one more antinode than its number of nodes.

Frequency and the Harmonic Series

So, why do notes played on an instrument sound ‘musical’ compared to, say, the waves that are created when you turn on a garbage disposal or knock over a box? The answer to this question is related to the frequencies of oscillation of standing waves. The frequency of any wave (which includes standing waves) is measured in units of hertz (Hz) and is defined as the number of times the wave repeats its up-and-down motion in one second. The frequency of a standing wave is directly related to its number of antinodes. Specifically, if L is the length of the string, and v is the translational speed of waves on the string, then the frequency of oscillation of the standing wave f with n antinodes is given by this formula:

Formula relating the frequency of a standing wave to its number of antinodes

Musical sounds tend to consist of a limited number of specific frequencies that are related in mathematically simple ways, whereas general ‘noise’ consists of a continuous range of frequencies. The frequencies of standing waves turn out to be limited by a simple mathematical relation.

When a string that is restricted on both ends is plucked, the frequencies of the waves that appear on the string are limited to certain values that can be calculated with the formula. The fact that the frequencies are limited is seen in the formula because the number of antinodes n can only take on integer values – 1, 2, 3, etc. – because a standing wave cannot have, say, 1.5 or 2.75 antinodes. It can only have a whole number of antinodes, and it follows that the frequencies of standing waves on a plucked string can only take on very specific values.

The set of allowed frequencies for a particular string defined by the formula above is called the harmonic series of the string. Since each allowed frequency corresponds to a vibration with a different number of antinodes, the harmonic series is usually represented in a diagram depicting the general shape of the string for each possible frequency, or harmonic. The first three harmonics, with one, two, and three antinodes, are depicted in the diagram below. As the number of antinodes n increases, so does the frequency of the wave.

First three harmonics for a string length L.

Lesson Summary

Standing waves, or waves with fixed stationary points on them, can be observed on guitar strings when plucked, and (in some form) within nearly every other musical instrument. They form when a vibration is restricted, confined to a region with boundaries or endpoints. The stationary points on the wave are called nodes, and their inverse, the points of greatest deviation from the center, are referred to as antinodes. Due to the restriction of the vibration to a particular region, the frequencies of standing waves are limited, which is the feature of musical sounds.

We can predict the base frequency of a standing wave with a particular length, wave speed, and number of antinodes using the formula:

Formula relating the frequency of a standing wave to its number of antinodes

The same standing wave is then only capable of achieving frequencies that are whole integer multiples of its base frequency. The set of allowed frequencies is called the harmonic series and is an important term that is used to discuss the physics of music.