What is your favorite musical instrument? How do you play it? Do you pluck a string, like a guitar? Do you blow through it, like a flute? Do you hit it, like a drum? All musical instruments work by making standing waves. Each instrument has a unique sound because of the special waves made in it. These waves could be in the strings of a guitar or violin. They could also be in the skin of a drum or a tube of air in a trumpet. These waves are picked up by the air and later reach your ear as sound.

In Grade 10, you learned about standing waves and boundary conditions. We saw a rope that was:

We also saw a pipe that was:

String and wind instruments are good examples of standing waves on strings and pipes.

One way to describe standing waves is to count nodes. Recall that a node is a point on a string that does not move as the wave changes. The anti-nodes are the highest and lowest points on the wave. There is a node at each end of a fixed string. There is also a node at the closed end of a pipe. But an open end of a pipe has an anti-node.

What causes a standing wave? There are incident and reflected waves traveling back and forth on our string or pipe. For some frequencies, these waves combine in just the right way so that the whole wave appears to be standing still. These special cases are called harmonic frequencies, or harmonics. They depend on the length and material of the medium.

Definition 1: Harmonic

A harmonic frequency is a frequency at which standing waves can be made in a particular object or on a particular instrument.

Let us look at a basic "instrument": a string pulled tight and fixed at both ends. When you pluck the string, you hear a certain pitch. This pitch is made by a certain frequency. What causes the string to emit sounds at this pitch?

You have learned that the frequency of a standing wave depends on the length of the wave. The wavelength depends on the nodes and anti-nodes. The longest wave that can "fit" on the string is shown in Figure 1. This is called the fundamental or natural frequency of the string. The string has nodes at both ends. The wavelength of the fundamental is twice the length of the string.

Now put your finger on the center of the string. Hold it down gently and pluck it. The standing wave now has a node in the middle of the string. There are three nodes. We can fit a whole wave between the ends of the string. This means the wavelength is equal to the length of the string. This wave is called the first harmonic. As we add more nodes, we find the second harmonic, third harmonic, and so on. We must keep the nodes equally spaced or we will lose our standing wave.

Figure 1: Harmonics on a string fixed at both ends.

You should have found this formula:

Here, nn is the number of nodes. LL is the length of the string. The frequency ff is:

Here, vv is the velocity of the wave. This may seem confusing. The wave is a standing wave, so how can it have a velocity? But one standing wave is made up of many waves that travel back and forth on the string. Each of these waves has the same velocity. This speed depends on the mass and tension of the string.

Guitar

Guitars use strings with high tension. The length, tension and mass of the strings affect the pitches you hear. High tension and short strings make high frequencies; low tension and long strings make low frequencies. When a string is first plucked, it vibrates at many frequencies. All of these except the harmonics are quickly filtered out. The harmonics make up the tone we hear.

The body of a guitar acts as a large wooden soundboard. Here is how a soundboard works: the body picks up the vibrations of the strings. It then passes these vibrations to the air. A sound hole allows the soundboard of the guitar to vibrate more freely. It also helps sound waves to get out of the body.

The neck of the guitar has thin metal bumps on it called frets. Pressing a string against a fret shortens the length of that string. This raises the natural frequency and the pitch of that string.

Most guitars use an "equal tempered" tuning of 12 notes per octave. A 6 string guitar has a range of 4 1212 octaves with pitches from 82.407 Hz (low E) to 2093 kHz (high C). Harmonics may reach over 20 kHz, in the inaudible range.

Figure 12

Piano

Let us look at another stringed instrument: the piano. The piano has strings that you cannot see. When a key is pressed, a felt-tipped hammer hits a string inside the piano. The pitch depends on the length, tension and mass of the string. But there are many more strings than keys on a piano. This is because the short and thin strings are not as loud as the long and heavy strings. To make up for this, the higher keys have groups of two to four strings each.

The soundboard in a piano is a large cast iron plate. It picks up vibrations from the strings. This heavy plate can withstand over 200 tons of pressure from string tension! Its mass also allows the piano to sustain notes for long periods of time.

The piano has a wide frequency range, from 27,5 Hz (low A) to 4186,0 Hz (upper C). But these are just the fundamental frequencies. A piano plays complex, rich tones with over 20 harmonics per note. Some of these are out of the range of human hearing. Very low piano notes can be heard mostly because of their higher harmonics.

Figure 13

A wind instrument is an instrument that is usually made with a pipe or thin tube. Examples of wind instruments are recorders, clarinets, flutes, organs etc.

When one plays a wind instrument, the air that is pushed through the pipe vibrates and standing waves are formed. Just like with strings, the wavelengths of the standing waves will depend on the length of the pipe and whether it is open or closed at each end. Let's consider each of the following situations:

If you blow across a small hole in a pipe or reed, it makes a sound. If both ends are open, standing waves will form according to Figure 14. You will notice that there is an anti-node at each end. In the next activity you will find how this affects the wavelengths.

Figure 14: Harmonics in a pipe open at both ends.

The formula is different because there are more anti-nodes than nodes. The right formula is:

Here, nn is still the number of nodes.

Exercise 2: The Organ Pipe

An open organ pipe is 0,853 m long. The speed of sound in air is 345 m.s-1-1. Can this pipe play middle C? (Middle C has a frequency of about 262 Hz)

Figure 25

Solution

The main frequency of a note is the fundamental frequency. The fundamental frequency of the open pipe has one node.

1. Step 1. To find the frequency we will use the equation: :
   f = v λ f = v λ

   (5)

   We need to find the wavelength first.

   λ = 2 L n = 2 ( 0 , 853 ) 1 = 1 , 706 m λ = 2 L n = 2 ( 0 , 853 ) 1 = 1 , 706 m

   (6)
2. Step 2. Now we can calculate the frequency: :
   f = v λ = 345 1 , 706 = 202 Hz f = v λ = 345 1 , 706 = 202 Hz

   (7)

   This is lower than 262 Hz, so this pipe will not play middle C. We will need a shorter pipe for a higher pitch.

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Now let's look at a pipe that is open on one end and closed on the other. This pipe has a node at one end and an antinode at the other. An example of a musical instrument that has a node at one end and an antinode at the other is a clarinet. In the activity you will find out how the wavelengths are affected.

Figure 26: Harmonics in a pipe open at one end.

The right formula for this pipe is:

A long wavelength has a low frequency and low pitch. If you took your pipe from the last example and covered one end, you should hear a much lower note! Also, the wavelengths of the harmonics for this tube are not integer multiples of each other.

Resonance is the tendency of a system to vibrate at a maximum amplitude at the natural frequency of the system.

Resonance takes place when a system is made to vibrate at its natural frequency as a result of vibrations that are received from another source of the same frequency. In the following investigation you will measure the speed of sound using resonance.

Figure 37

From the investigation you will notice that the column of air will make a sound at a certain length. This is where resonance takes place.

Figure 38

In the sound chapter, we referred to the quality of sound as its tone. What makes the tone of a note played on an instrument? When you pluck a string or vibrate air in a tube, you hear mostly the fundamental frequency. Higher harmonics are present, but are fainter. These are called overtones. The tone of a note depends on its mixture of overtones. Different instruments have different mixtures of overtones. This is why the same note sounds different on a flute and a piano.

Let us see how overtones can change the shape of a wave:

Figure 39: The quality of a tone depends on its mixture of harmonics.

The resultant waveform is very different from the fundamental frequency. Even though the two waves have the same main frequency, they do not sound the same!