126 HAND-BOOK OF ACOUSTICS.
next simplest form of vibration is with, six nodes, the next with eight, and so on; an odd number of nodes never being produced. The corresponding vibration rates are as follows:—
Practically, however, owing among other things, to unavoidable irregularities in the casting, no church bell ever has a single fundamental, or only one series of overtones. To this fact is due, the well-known difficulty in ascertaining the precise pitch of such a bell; the discords and throbbings that are heard, even in the best sounding bells, when the listener is close to them, may be put down to the same cause. There are no absolute points of rest in a vibrating bell, for the fundamental is never produced alone ; but it is easy to explore the surface of the sounding bell with a light ball suspended from a thread, and thus find the places of least and greatest motion, the ball being violently dashed away from the latter.
These, in the form of side drums, bass drums, and kettle drums, are used in the orchestra rather to mark the rhythm, than to produce a musical sound; although it is true, the last mentioned are approximately tuned to two or three notes of the 16 foot octave.
They may be studied by exciting them sympathetically, by means of organ pipes, and analysing the vibration forms produced, by scattering sand over them, as in the case of plates. The nodal lines are circles and diameters, or combinations of these.
Rods vibrating longitudinally and (1) free or (2) fixed at both ends are analogous with open organ pipes: their vibration numbers are inversely as their lengths, and they give the complete series of partial tones.
When (3) fixed at one end only, they are analogous with stopped organ pipes and give only the odd series of partials.
In (1) and (2) the time required for a complete vibration is the same as the time taken by a pulse to move along the whole length of the rod, and back again ; in (3) the time required for a complete vibration is twice this time. These facts being known, it is possible to determine the velocities of sound in various solids, just as a