Resonance in Sound Waves

Have you ever seen a video of a trained singer breaking a glass with only their voice? What about a video of a large bridge swaying wildly in the wind? This must be due to some clever editing, right? Not quite! These effects are indeed possible due to the effects of a phenomenon called resonance. In nature, everything tends to vibrate, some objects more than others. If an external force increases the energy of these vibrations, we say that it has achieved resonance. In this article, we will discuss resonance in sound waves and learn more about how the talented singer can break a glass with only their voice.

Definition of Resonance

When a guitar string is plucked, it vibrates with its natural frequency. This vibration causes a vibration in the surrounding air molecules which we perceive as sound.

Let's imagine that we have strings of a variety of different lengths. We can perform an experiment to see which of our new strings, when plucked, causes our original string to vibrate the most in response. As you may have guessed, the new string which has the same length as the original is going to be the string that elicits the strongest response in the original string. Specifically, the amplitude of the oscillations of the string that are produced in response to the waves produced by the plucked string is largest when the length of the plucked string is the same as the original string. This effect is called resonance and is the same effect that allows well-trained singers to break glass with their voices.

Definition of Resonance in Sound Waves

For sound waves, resonance occurs when incoming sound waves acting on an oscillating system amplify the oscillations when the frequency of the incoming sound waves is close to or the same as the natural frequency of the oscillating frequency. You can think of this as the definition of resonance in sound waves.

In the case of the singer that can break a wine glass with their voice, the frequency of sound waves from their voice will match the natural frequency with which the glass tends to vibrate. You'll notice that when you strike a wine glass with a solid object it will ring at a particular pitch. The particular pitch that you hear corresponds to a particular frequency at which the glass is oscillating. The vibration of the glass increases in amplitude and if this new amplitude is great enough, the glass shatters. The frequency that is responsible for this effect is called the resonant frequency. A similar effect can be achieved if the singer is replaced by a tuning fork of the correct resonant frequency.

Causes of Resonance in Sound Waves

We have discussed the concept of resonance but to understand it better we must discuss exactly how resonance occurs. Resonance is caused by the vibrations of standing waves. We will discuss how these standing waves can be formed on strings under tension and in hollow pipes.

Standing Waves on Strings

Standing waves, also known as stationary waves, are the waves generated when two waves of equal amplitude and frequency moving in opposite directions interfere to form a pattern. Waves on a guitar string are examples of standing waves. When plucked, a guitar string vibrates and creates a wave pulse that travels along the string to a fixed end of the guitar. The wave then reflects and travels back along the string. If the string is plucked a second time a second wave pulse is generated which will overlap and interfere with the reflected wave. This interference can produce a pattern which is the standing wave. Imagine the image below to be that of standing waves on a guitar string.

Standing waves that can and cannot occur, Wikimedia Commons CC BY-SA 3.0

The string cannot vibrate at the fixed ends and these are referred to as nodes. Nodes are areas of zero amplitude. Areas of maximum vibration are called antinodes. Note that standing waves like the ones on the right-hand side of the diagram cannot occur because the guitar string cannot vibrate outside the fixed ends of the guitar.

Standing Waves in Pipes

We can use our imagination to think of the diagram above as a closed pipe. That is, as a hollow pipe that is sealed at both ends. The wave generated is now a sound wave produced by a speaker. Instead of a string, the vibration is produced in air molecules. Again, air molecules at the closed ends of the pipe cannot vibrate and so the ends form nodes. In between successive nodes are the positions of maximum amplitude, which are antinodes. If the pipe were, instead, open at both ends, the air molecules at the ends will vibrate with maximum amplitude, i.e. antinodes would form as shown in the figure below.

Standing sound wave in a hollow pipe that is open at both ends, Vaia Originals

Examples of Resonance in Sound Waves

Guitar Strings

We will consider the cases of sound waves created by waves on a string and sound waves traveling in a hollow pipe. On guitars, strings of different lengths and under different tensions are plucked to create musical notes of different pitches in the strings. These vibrations in the strings cause sound waves in the air surrounding them, which we perceive as music. The frequencies corresponding to different notes are created by resonance. The figure below is an illustration of a guitar string vibrating with a resonant frequency after being plucked.

A guitar string vibrating with a resonant frequency after being plucked, - Vaia Originals

Closed Pipes

Pipe organs send compressed air into long, hollow pipes. The air column vibrates when air is pumped into it. Standing waves are set up in the pipe when the driving frequency of the keyboard note matches one of the standing wave frequencies in the pipe. These frequencies are hence the resonant frequencies of the pipe. The pipe itself may be closed at both ends, open at one end and closed at the other, or open at both ends. The type of pipe will determine the frequency that will be produced. The frequency with which the air column vibrates will then determine the note of the sound wave heard. The figure below is an example of a sound wave of a resonant frequency in a pipe closed at both ends.

Sound waves vibrating at a resonant frequency in a closed pipe, Vaia Originals

The Frequency of Resonance in Sound Waves

Resonant Frequencies of a Vibrating String

A guitar string is an example of a vibrating string that is fixed at both ends. When the string is plucked, there are certain specific frequencies with which it can vibrate. A driving frequency is used to achieve these frequencies and, since these vibrations are amplified, this is an example of resonance according to the definition of resonance in sound waves. The standing waves formed have resonant frequencies that depend on the mass of the string \(m\), its length \(L\), and the tension in the string \(T\),

$$f_n=\frac{nv}{2L}=\frac{n\sqrt{T/\mu}}{2L}$$

since

$$v=\frac{T}{\mu}$$

where \(f_n\) denotes the frequency of the \(n^{\mathrm{th}}\) resonant frequency, \(v\) is the speed of the wave on the string and \(\mu\) is the mass per unit length of the string. The figure below illustrates the first three resonant frequencies/harmonics for a vibrating string of length \(L\), that is, \(n=1\), \(n=2\) and \(n=3\).

The lowest resonant frequency \((n=1)\) is called the fundamental frequency and all frequencies higher than this are referred to as overtones.

Resonant Frequencies of a Closed Pipe

If a standing wave pattern is set up using sound waves in a hollow closed pipe, we can find the resonant frequencies just as we did for the waves on a string. A pipe organ uses this phenomenon to create sound waves of different notes. A driving frequency, created using the organ’s keyboard, matches one of the natural standing wave frequencies in the pipe and the resulting sound wave is amplified, which gives the pipe organ a clear, loud sound. Pipe organs have many different pipes of different lengths to create the resonance of different notes.

The resonant frequencies \(f_n\) of a closed pipe can be calculated as follows

$$f_n=\frac{nv}{4L}$$

for the \(n^{th}\) resonant frequency, where the speed of sound in the pipe is \(v\), and \(L\) is the length of the pipe. The figure below illustrates the first three resonant frequencies/harmonics for a vibrating string, that is, \(n=1\), \(n=3\) and \(n=3\).

The first three resonant frequencies/harmonics forstanding waves in a closed pipe of length \(L\), Vaia Originals