343. The Laws of Vibrating Strings.—The relations between the vibration rate, the length, the tension and the diameter, of vibrating strings have been carefully studied with an instrument called a sonometer (Fig. 335). By this device it is found that the pitch of a vibrating string is raised one octave when its vibrating length is reduced to one-half. By determining the vibration rate of many lengths, the following law has been derived: (Law I) The rate of vibration of a string is inversely proportional to its length.
Fig. 335.—A sonometer.
Careful tests upon the change of vibration rate produced by a change of tension or pull upon the strings show that if the pull is increased four times its vibrations rate is doubled, and if it is increased nine times its rate is tripled, that is: (Law II) The vibration rates of strings are directly proportional to the square roots of their tensions.
Tests of the effects of diameter are made by taking wires of equal length and tension and of the same material but of different diameter. Suppose one is twice as thick as the other. This string has a tone an octave lower or vibrates one-half as fast as the first. Therefore: (Law III) The vibration rates of strings are inversely proportional to the diameters. These laws may be expressed by a formula n ∝√(t)/dl.
The vibration of a string is rarely a simple matter. It usually vibrates in parts at the same time that it is vibrating as a whole. The tone produced by a string vibrating as a whole is called its fundamental. The vibrating parts of a string are called loops or segments (see Fig. 336), while the points of least or no vibration are nodes. Segments are often called antinodes.
Fig. 336.—A string yielding its fundamental and its first overtone.
344. Overtones.—The quality of the tone produced by a vibrating string is affected by its vibration in parts when it is also vibrating as a whole. (See Fig. 336.) The tones produced by the vibration in parts are called overtones or partial tones. The presence of these overtones may often be detected by the sympathetic vibration of other wires near-by. What is called the first overtone is produced by a string vibrating in two parts, the second overtone by a string vibrating in three parts, the third overtone by its vibration in four parts and so on. In any overtone, the number of the parts or vibrating segments of the string is one more than the number of the overtone. For example, gently press down the key of middle C of a piano. This will leave the string free to vibrate. Now strongly strike the C an octave lower and then remove the finger from this key. The middle C string will be heard giving its tone. In like manner try E1 and G1, with C. This experiment shows that the sound of the C string contains these tones as overtones. It also illustrates sympathetic vibration.