# Acoustics

## Vibrations Of A String

To understand some of the fundamentals of sound production and propagation it is instructive to first consider the small vibrations of a stretched string held at both ends under tension. While these vibrations are not an example of sound, they do illustrate many of the properties of importance in acoustics as well as in the production of sound. The string may vibrate in a variety of different ways, depending upon whether it is struck or rubbed to set it in motion, and where on the string the action took place. However, its motion can be analyzed into a combination of a large number of simple motions. The simplest, called the fundamental (or the first harmonic), appears in Figure 1, which shows the outermost extensions of the string carrying out this vibration.

The second harmonic is shown in Figure 2; the third harmonic in Figure 3; and so forth (the whole set of harmonics beyond the first are called the overtones). The rate at which these vibrations take place (number of times per second the motion is repeated) is called the frequency, denoted by f (the reciprocal of the frequency, which is the time for one cycle to be competed, is called the period). A single complete vibration is normally termed a cycle, so that the frequency is usually given in cycles per second, or the equivalent modern unit, the hertz (abbreviated Hz). It is characteristic of the stretched string that the second harmonic has a frequency twice that of the fundamental; the third harmonic has a frequency three times that of the fundamental; and so forth. This is true for only a few very simple systems, with most sound-producing systems having a far more complex relationship among the harmonics.

Those points on the string which do not move are called the nodes; the maximum extension of the string (from the horizontal in the Figures) is called the amplitude, and is denoted by A in Figures 1-3. The distance one must go along the string at any instant of time to reach a section having the identical motion is called the wavelength, and is denoted by L in Figures 1-3. It can be seen that the string only contains one-half wavelength of the fundamental, that is, the wavelength of the fundamental is twice the string length. The wavelength of the second harmonic is the length of the string. The string contains one-and-one-half (3/2) wavelengths of the third harmonic, so that its wavelength is two-thirds (2/3) of the length of the string. Similar relationships hold for all the other harmonics.

If the fundamental frequency of the string is called f0, and the length of the string is l, it can be seen from the above that the product of the frequency and the wavelength of each harmonic is equal to 2f0l. The dimension of this product is a velocity (e.g., feet per second or centimeters per second); detailed analysis of the motion of the stretched string shows that this is the velocity with which a small disturbance on the string would travel down the string.