§ 3 Longitudinal Vibrations of Rods Fixed at One End: Musical Instruments Formed on This Principle
Rods and bars with one end fixed are also capable of vibrating longitudinally. A smooth wooden or metal rod, for example, with one of its ends fixed in a vise, yields a musical note, when the resined fingers are passed along it. When such a note yields its lowest note, it simply elongates and shortens in quick alternation; there is, then, no node upon the rod. The pitch of the note is Fig. 83. inversely proportional to the length of the rod. This follows necessarily from the fact that the time of a complete vibration is the time required for the sonorous pulse to run twice to and fro over the rod. The first overtone of a rod, fixed at one end, corresponds to its division by a node at a point one-third of its length from its free end. Its second overtone corresponds to a division by two nodes, the highest of which is at a point one-fifth of the length of the rod from its free end, the remainder of the rod being divided into two equal parts by the second node. In Fig. 83, a and b, c and d, e and f, are shown the conditions of the rod answering to its first three modes of vibration: the nodes, Fig. 84. as before, are marked by dotted lines, the arrows in the respective cases marking the direction of the pulses.
The order of the tones of a rod fixed at one end and vibrating longitudinally is that of the odd numbers 1, 3, 5, 7, etc. It is easy to see that this must be the case. For the time of vibration of c or d is that of the segment above the dotted line: and the length of this segment being only one-third that of the whole rod, its vibrations must be three times as rapid. The time of vibration in e or f is also that of its highest segment, and as this segment is one-fifth of the length of the whole rod, its vibrations must be five times as rapid. Thus the order of the tones must be that of the odd numbers.
Before you, Fig. 84, is a musical instrument, the sounds of which are due to the longitudinal vibrations of a number of deal rods of different lengths. Passing the resined fingers over the rods in succession, a series of notes of varying pitch is obtained. An expert performer might render the tones of this instrument very pleasant to you.