Do any of you play any stringed musical instruments? If so, then you’re probably familiar with the term ‘harmonic’, which is just a kind of special sound created by playing in a specific way. On the violin, you need to gently press the string with a finger whilst bowing, such that you are able to feel the vibrations of the string beneath your finger. The sound created is almost flute-like and has a silvery, hollow quality to it. However, this sound can only be created at certain intervals along the string, ½, ⅓, ¼, etc. Anywhere else on the string, it sort of sounds like metal wires being dragged down a blackboard. Yeah, don’t try it. So why is this weird sound only produced at ‘nice’ fractions of the string? And how does a stringed instrument work anyway?
To all other stringed instruments out there, I apologise now (apart from to the violas, nobody likes you) as I will be focusing mainly on the violin, although I think we can assume that the principles are essentially the same. On a basic level, the violin comprises of a hollow wooden shell with four metal strings laid across it. These four strings are of different thicknesses, which changes the frequency at which they vibrate and hence the pitch. The tension of the string can also be altered by winding the string around a peg, which also results in a change in frequency and pitch. When the hair of the violin bow is dragged across one of the strings, it causes the string to vibrate at its fundamental frequency because the vibration is occurring across the entire length of the string, sort of like a skipping rope. A note is produced corresponding to the frequency of the vibration, and this is the first harmonic.
An equation for the frequency of the first harmonic has been derived:
Where f is the frequency in Hz, T is the tension in the string in N, mu is the mass per unit length of the string in kgm^-1, and l is the length of the string length in m
The wave responsible for the note is known as a standing wave. A standing wave is formed when a wave propagates along a line, in this case the string, and is then reflected back along the line in the opposite direction, with a phase change of 180 degrees. Since the two waves have the same wavelength and are a constant phase difference apart, they are coherent, which means that when they pass over each other, they interfere so that there is no net propagation of energy; the wave is ‘standing’ still. At the points either end of the string where it is fixed, there is no movement because the two waves ‘cancel’ each other out by interfering destructively. These points are called nodes. In the middle of the string, the amplitude is greatest because the waves interfere constructively, creating an antinode. Nodes and antinodes are always separated by half a wavelength (one ‘bump’) in the wave on the string.
The second harmonic can be created, as stated earlier, by lightly touching a finger half-way down the string. This effectively creates a node in the middle of the string, where the string will not oscillate. The standing wave is able to form in this circumstance because two half wavelengths are able to fit between the ends of the string and your finger. The third harmonic can be created by putting your finger a third of the way up the string, in which case three half wavelengths will be able to fit onto the string, and so on and so forth. However, due to physical constraints, it gets progressively more difficult to produce nice sounding harmonics on the violin, so generally only the first two or three harmonics are ever used.
Stationary waves can also be formed when the air column in organ pipes or wind instruments is forced to vibrate. But wind instruments are inferior, so… I won’t bother going into any detail.