5.10 Harmonics

You have probably heard the word harmonics, or harmony before, though probably in a musical context rather than a physics one. Harmonics are indeed important in music - and we shall find out why. However, it is first necessary to discuss standing waves to a greater extent than was covered in section 5.3.

A standing wave is made up of nodes and antinodes. Nodes are points on a standing wave that do not oscillate. In practical terms, you could consider these points to be fixed. An antinode is the point on a standing wave that oscillates the most, and therefore, cannot be fixed. Between two nodes, there is always an antinode, and vice versa.

Now let’s consider a guitar string, which is fixed at both ends. What is the simplest arrangement of nodes and antinodes that would be possible for the string to vibrate at? The answer for this and any other string with fixed ends is two nodes, the two fixed ends, with an antinode in the middle. . This is illustrated below.

As you can see, the shape of the string is similar to the shape of half of a wave. In fact, this is exactly what it is. If the length of the string is known, the wavelength of the standing wave can be found; it is twice the length of the string.

The velocity of a standing wave, for any given set of conditions, is constant, and is fixed by the equation

Where T is the tension in the string in Newtons, and m is the mass per unit length in kilograms per meter. We will not go into the derivation of this equation, as it is not important in the waves unit.

Because the velocity of a standing wave is constant in any given situation, and because the wavelength can be found from a simple calculation, one more important information can also be found: the frequency. Using equation 5.1, we know

The frequency of the simplest standing wave is also particularly important in harmonics, and is known as the fundamental frequency. There is no frequency lower than the fundamental frequency that can be reached for a given set of conditions (e.g. a given string).

Let us move onto the second simplest standing wave that it is possible to create in a string. This is a series of three nodes, with two antinodes between, shown below:

In this case, the wavelength is the actual length of the string, as you can see fron the illustration. Comparing the frequencies of the two strings, we find (where L is the length of the string)

for the first standing wave,

for the second standing wave,

As you can see, the frequency of the second standing wave is twice the fundamental frequency for the string. Perhaps you will now have guessed that this trend continues, and in fact, it does. The third possible standing wave on a string has a frequency three times the fundamental frequency. Standing waves that have a frequency which is an integer multiple of the fundamental frequency of their system are known as harmonics. The fundamental frequency oscillation is called the first harmonic, and all futher harmonics are preceded by a number which is the integer multiple of the fundamental frequency. Taking the second wave we considered, because its frequency is twice that of the fundamental frequency of the string, the second wave is known as the second harmonic. Some physicists and also musicians refer to harmonics as overtones, where the first overtone is equal to the second harmonic, the second overtone is equal to the third harmonic, and so on. In this case, the fundamental frequency is not an overtone

When two notes (frequencies) are played by an instrument, if one is a harmonic of the other, the similar waves will result in constructive interference. The new wave will be of the fundamental frequency, but to our ears, this just makes the notes sound louder and richer.