Sonometer for Longitudinal Vibrations.-A string to produce pure longitudinal vibrations must be stretched between heavy and firmly-fixed terminals. At each end of the long wooden resonance-box are clamps of solid metal faced with lead. A scale, usually of 1 metre or more in length, is placed below the string, which can be stretched either by wrest-pins or by a weight. A leaden clamp travels along the string. It is the inertia of this weight and that of the terminals which determines the nodal points, and not their rigidity as in the case of transverse oscillations. The vibrations themselves consisting of alternate rarefaction and condensation of the elastic material closely resemble those which take place in an organ-pipe, or in the atmosphere at large when conveying sound.

The string may be excited by means of the thumb and finger dusted with powdered rosin, and moved lengthwise, or better with the point of a violin-bow acting in the same direction.

The sounds thus elicited are very pure, and always much sharper relatively to the length of the string than those given by transverse vibrations. They have the same mutual relations as these latter and their vibration-numbers are inversely proportional to the wave-lengths.

Torsional Vibrations of Strings.-Independently of the two modes of vibration given above, every string performs a third oscillation inseparable from the others. This can easily be demonstrated by hooking lightly to the middle of the string a small double ring of fine wire, in the shape of the figure 8, carrying a little paper flyer. When the string is transversely excited, whether by plucking or with a bow, the ring and flyer will begin to turn round with great rapidity, alternating frequently in the direction of rotation. This occurs whether the fundamental tone or an upper partial be elicited. It is evident that the string has a torsional motion, which it communicates with alternate direction to the enveloping ring. The torsional vibrations of strings are of little practical importance. But they unexpectedly intervened in the writer's experiments with low notes on the double-bass as detailed elsewhere. Whenever it was attempted to produce grave tones by enlarging the sectional area of the string, a limit was found beyond which the diameter of the string could not be increased, from the predominance of these torsional tones. The bow, acting at the circumference of the string, had power enough to rotate it instead of communicating purely transverse oscillations. It was consequently

necessary to adopt strings of smaller diameter and greater specific gravity.

It will thus be seen that a string vibrating in the transverse direction has impressed upon it at least four distinct motions, two in the original direction, one longitudinal, and one torsional. By accidental circumstances the number may be indefinitely increased.

Summary of String-Vibration.

I. Transverse.-Alone used in music.

Velocity = √.

1

m

Period n = 21

m

The harmonics a complete series, as in open pipes.

II. Longitudinal

Velocity =

modulus of elasticity = M

density of string

Unaffected by tension. Inversely as length.

Their pitch higher than in No. I. Vary in pitch with the material of

the string.

Harmonics a complete series.

Afford a measure of sound-velocity,

III. Torsional

Complicated in theory. Not used musically.

Vibrations of Bars or Rods are the next in simplicity to those of strings. They are of three kinds, longitudinal, torsional, and lateral. Of these the last are the most important. Although the three classes of vibrations are quite distinct in theory, yet in actual experiments it is often found impossible to excite longitudinal or torsional vibrations without the accompaniment of some measure of lateral motion. In bars of ordinary dimensions the gravest lateral motion is far graver than the gravest longitudinal or torsional motion.

Rods or bars with one end fixed can also be made to vibrate longitudinally, the pitch being inversely proportional to the length of the rod. The time of a complete vibration is that required for the sonorous pulse to run twice to and fro over the rod. The first upper partial of such a rod produces a node at one-third from its free end; the second has two nodes, the higher at one-fifth of the length from the free end, the lower bisecting the remainder of the rod. The order of the tones is that of the odd numbers 1, 3, 5, &c., thus

resembling those of a stopped diapason pipe which will be described further on.

The only instrument founded on this property of rods is but little known and rarely used, being more an acoustical curiosity than anything else. It consists of a number of deal rods, about twenty or more, standing up vertically, like the strings of a harp, from a sound-board obliquely

Fig. 5.-Longitudinal vibrations of rods.

placed below, into which their lower extremities are firmly fixed. They are excited by vertical friction with the rosined fingers. A similar instrument furnished with glass tubes instead of wooden rods is occasionally to be heard in the streets of London.

Torsional Vibrations are even of less acoustical importance than those named above. They were first, like them, investigated by Chladni, but they have hitherto contributed no instrument to music, and are chiefly of a mathematical and theoretical interest.

Lateral Vibrations of Elastic Rods, on the other hand, are of large service both theoretically and practically. They differ materially according to whether the rod is firmly fixed at one or both ends, or free at the two ends, and supported at some other point. A rod fixed at both ends behaves exactly like a string. It may vibrate in one, or in two, three, or more segments. But the rapidity of the vibrations, and the consequent pitch of the note produced, differ entirely from those of a string. Whereas the vibrations of the string rise in a simple arithemetical series, those of the rod rise as the squares of the odd numbers. When a string divides into two

segments, each of these vibrates with twice the rapidity of the whole string; but a rod does so in the ratio of 9 to 25, a rod vibrating with two nodes in the ratio of 9 to 49, one with three nodes in that of 9 to 81, and so on. A rod supported at one end gives a number of vibrations in inverse ratio to the

square of its length. It can be made to vibrate in a single piece, or with one, two, or even more nodes. The period of the gravest tone is the time occupied by a pulse travelling four times the length of the bar.

These properties of rods were utilized by Chladni in the construction of a tonometer.

Rods fixed at one end furnish several instruments of practical application, of which the Nail-fiddle, or Violon de fer,

C

seems the simplest. It is said to have originated by accident, from the note given by a common nail inserted in the wainscot when a weight was hung to it by a string. In this a bow is

drawn over a graduated series of nails, or rods, fastened by one end to a block of wood, thus setting them in vibration. In the "Jew's harp" or Guimbarde, which in various forms appears in many parts of the world as a popular instrument, the fundamental tone of the spring is modified by alterations