ON THE VIBRATIONS OF STRINGS. 87
The following simple experiment will illustrate the important part played by the sound-board or its substitute, in stringed instruments. Fasten one end of a string, 3 or 4 feet in length, to a heavy weight, and, holding the other end in one hand, let the weight hang freely. On plucking or bowing the string, scarcely any sound will be heard. Now attach the free end of the string to the peg at the left hand of the Sonometer (fig. 23), and let the weight hang freely over the pulley at the right hand. If the string be now plucked or bowed, a loud sound will be emitted.
We now proceed to study the conditions which determine pitch, quality, and intensity, in stringed instruments. As the tones produced by such instruments are rarely or never simple, it will be understood, that in investigating the laws relating to pitch, it is the pitch of the fundamental tone alone, that is considered.
If T denote the tension of a stretched string, and M its mass, it may be shown mathematically, that the velocity V, with which a transverse vibration will travel along it, will be—
and if L denote the length of the string, it is evident that — is the time required for it to execute one complete vibration. Therefore if N denotes the number of vibrations the string performs in one second
Substituting the above value of V, we get
From this formula we may deduce the following laws:—
(1). The tension of the string remaining the same, N (the vibration number) varies inversely as the length of the string.
(2). Other things remaining the same, ^/"varies inversely as the diameter of the string.
(3). Other things remaining constant, N varies directly as the square root of the tension—that is, of the stretching force or weight.
(4). Other things remaining constant, ^/"varies inversely as the square root of the density or weight of the string.
These statements can also be verified experimentally, without recourse to mathematics, by means of the Sonometer described in