§ 1 Transverse Vibrations of a Rod Fixed at Both Ends
Our last chapter was devoted to the transverse vibrations of strings. This one I propose devoting to the transverse vibrations of rods, plates, and bells, commencing with the case of a rod fixed at both ends. Its modes of vibration are exactly those of a string. It vibrates as a whole, and can also divide itself into two, three, four, or more vibrating parts. But, for a reason to be immediately assigned, the laws which regulate the pitch of the successive notes are entirely different in the two cases. Thus, when a string divides into two equal parts, each of its halves vibrates with twice the rapidity of the whole; while, in the case of the rod, each of its halves vibrates with nearly three times the rapidity of the whole. With greater strictness, the ratio of the two rates of vibration is as 9 is to 25, or as the square of 3 to the square of 5. In Fig. 52, a a′, c c′, b b′, d d′, are sketched the first four modes of vibration of a rod fixed at both ends: the successive rates of vibration, in the four cases bear to each other the following relation:
| Number of nodes | 0 | 1 | 2 | 3 |
| Number of vibrations | 9 | 25 | 49 | 81 |
the last row of figures being the squares of the odd numbers 3, 5, 7, 9.
In the case of a string, the vibrations are maintained by a tension externally applied; in the case of a rod, the vibrations are maintained by the elasticity of the rod itself. The modes of division are in both cases the same, but the forces brought into play are different, and hence also the successive rates of vibration.