ments are precisely alike, except in so far as they are interfered with by friction. Regnault found that in a conduit of 108 of a metre in diameter, the report of a pistol charged with a gramme of powder ceased to be heard at the distance of 1,150 metres. In a conduit of 3m. the distance was 3,810m. In the great St. Michel sewer of 1.10m. the sound was made, by successive reflections, to traverse a distance of 10,000 metres without becoming inaudible. In an open space, each successive layer has to impart its own energy to a larger layer; hence there is continual diminution of amplitude in the vibrations as the distance from the source increases. An undulation involves the onward transference of energy; and the amount of energy which traverses, in unit time, any closed surface described about the source, must be equal to that which the source emits in unit time. The intensity therefore follows the same law as that of radiant heat, and of light, as stated above. The energy of a particle executing simple vibrations in obedience to elasticity, has been said to vary as the square of the amplitude of its vibrations; for the amplitude being redoubled, the distance worked through, and the mean working force are both doubled, so that the work done is quadrupled. At the extreme positions all is potential energy; in the middle all is kinetic energy; at intermediate points it is partly in one form and partly in the other. If we sum up the potential and also the kinetic energies of all the particles constituting a wave, we shall find the results to be equal.1

This assumption is not absolutely true; since vibration implies friction, and friction implies the generation of heat. Sonorous energy therefore diminishes more rapidly than according to the law of inverse squares, and, in becoming extinct, is converted into heat.

Mayer has devised a plan by which the intensities of two sounds of the same pitch may be directly compared. The two sounds are separated by an impervious diaphragm, and in front of each is a resonator accurately tuned to them. Each resonator is attached by caoutchouc tubes of equal length to a U-tube, in the middle of which is a branch leading to a manometric capsule.

If the resonators are at the same distance from the sounding bodies, and one be excited, the attached flame vibrates. If both are produced in the same phase and intensity they interfere completely in the tube, and the flame is stationary.

I Everett's Deschanel, p. 799.

If they be not of the same intensity, the interference will be incomplete, and the flame will vibrate. If one be then altered until the flame is again still, the intensities will be directly as the squares of their distances from the resonators. This instrument is therefore the correlative of Rumford's shadow Photometer.

Tabular Statement of Intensity.

1. Intensity inversely as square of distance.

2. Intensity directly as square of amplitude of vibrations.

3. Increases with density of medium.

4. Modified by motion of atmosphere.

5. Strengthened by proximity of sonorous body.

Intensity, force, or loudness, may be looked upon as the first characteristic of musical tone: Pitch, dependent solely on the rapidity of the vibration, is the second, and will be considered in the next chapter. Quality or character has been shown to be connected with the form of the vibration, and will be adverted to farther on.

Consonance. A remarkable property of vibratory motions is the power they possess of communicating themselves to matter in their immediate neighbourhood. Even in a mechanical view of the subject this property is evident. If two pendulums, attached to different clocks, be fastened to one board and set going, it is well known to clockmakers that one will coerce the other into a spurious synchronism, which ceases directly they are divided. A regiment of soldiers crossing a suspension bridge, if keeping step and marching order, communicates regular impulses to the fabric of the bridge, and may even cause such oscillation as to endanger the structure; the swinging of the bells in a tall tower, such as that of Magdalen College at Oxford, itself produced by a succession of small impulses conveyed to the larger mass of each bell, is farther transmitted to the elastic material of the tower, producing in it very distinct oscillatory movements. This property is even more noticeable in the swifter alternations which form a musical note. Whatever be

"Illustrations of the powerful effects of isochronism," says Lord Rayleigh (Theory of Sound, p. 61), must be within the experience of every one. They are often of importance in very different fields from any with which acoustics are concerned. For example, few things are more dangerous to a ship than to lie in the trough of the sea, under the influence of waves whose period is nearly that of her own rolling."

F

the source of sound, its effect is immediately transmitted to the particles in contact, and with an amount of force which at first seems disproportionate to its inherent energy. For although the third law of Newton respecting the equality of action and reaction must obviously be fulfilled, the elasticity of most bodies enables them to take up transmitted vibration in a very high degree. Those which possess this property in the most marked manner are called sonorous, and their responsive vibration is termed consonance. Without consonance the effect of musical sound would be slightly, if at all, appreciable, for it is by this means that its chief propagation and dispersion is effected. In the first rank as consonators stand the producers themselves. A tuning-fork is set into sympathetic vibration by another vibrating in unison with it. A string will perform the same office, and an organ pipe instantly reinforces the sound of a corresponding tuning-fork held near its open extremity. Even a jar or bottle, the cavity of which bears some definite ratio to the wave-lengths of the sounding body, answers a similar

purpose.

The weight and density of the consonant body do not necessarily prevent its acting as a propagator of sound if its modulus of elasticity be high. Lead or clay for instance deaden sound by their inertness, while steel and glass convey

it with the utmost facility. But bodies of lighter character and less dense molecular construction, such as the softer woods, are obviously the fittest for this function. It is to the

highly resonant structure of pine-wood that the predominant tone of the violin family especially is due.

Generally speaking, reinforcement in sound is correlative with the power of producing it. All sounding bodies reinforce, but some have been divided off into what Clerk-Maxwell terms distributors. Others have the power of singling out particular sounds for reinforcement. If, for example, the dampers be lifted off a piano and the voice be used in its neighbourhood, it will be heard to sing out loudly with a humming tone the notes which have been spoken on. The same effect occurs with drums and tuning-forks: even the flat crown of a hat responds by vibration sensible to the touch when loud noises occur in proximity to it.

This power of singling out sounds has been utilized by Helmholtz for the analysis of musical notes in making Resonators. They originally had external membranes, but he afterwards found that the tympanic membrane, or drum of the ear itself, could be used for the same purpose, by making the resonant cavity of a particular size, such that, itself speaking a certain note, it will single out that note from all others, and reinforce it vigorously.

The simplest method, however, of demonstrating resonance is to take a tall jar or tube and hold over it a sounding body, such as a tuning-fork. As long as the fork and the cavity of the jar are in no definite relation to one another, the sound is unaltered. But if, by gradually pouring in water, we alter the depth of the cavity, a point is suddenly reached at which the note starts out with exceeding clearness. It will then be found that the length of the column of air in the tube bears an exact proportion to the wave-length of the vibrations emitted by the fork, usually that of one to four. The reason of this is obvious. At each vibration of the fork, a wave of condensation travels down the tube, is reflected from the bottom, and returns to find it in the same phase as when it started. It thus superposes its own motion upon that of the fork, and by a succession of such actions reinforces the sound. In so doing, however, the fork has the additional labour imposed upon it of setting in motion the contained particles of air as well as its own, and therefore comes sooner to rest than when vibrating independently. An effective experiment is produced by combining a sonorous bell with a resonant cavity of variable dimensions. A source of sound may also act upon a tuning-fork by consonance. If two forks in

1 See Chapter V.

accurate unison be placed at some distance from one another, and one be excited, the other immediately begins to sound with vigour, and if the first be damped, it may be again set in motion by the continuance of derived vibration established in the second. Tuning-forks and other sonorous bodies, such as glass or metal vases, often commence sounding spontaneously when a musical instrument, an organ or harmonium, is played on in the same room; even the glass windows of a church are apt to take up the note of a particular pedal pipe to the exclusion of those in its immediate neighbourhood.

Helmholtz has shown that a stretched string may be made to perform the office of a resonator. "If a sounding tuningfork have its stem placed on a string, and it be moved so near the bridge that one of the proper tones of the section of string lying between the fork and the bridge is the same as that of the tuning-fork, the string begins to vibrate strongly, and conducts the tone of the tuning-fork with great power to the sounding-board and surrounding air; whereas the tone is scarcely if at all heard as long as the section is not in unison with the tone of the fork."

A simple apparatus was used by Savart to show the influence of jars or boxes in strengthening sound. Close to a source of sound, such as a bell or tuning-fork, was placed a hollow cylinder, closed at its farther end by a moveable bottom, by means of which its capacity could be increased or diminished. This was supported on a sliding rest, so that its open end could be brought near or removed from the vibrating body. The bell or fork being excited by a rosined bow, the cavity of the resonator was altered until it coincided in pitch with it, and immediately the sound, originally feeble, and all but inaudible, became distinct and loud. The loudness could then be varied to any extent by moving the open end of the consonating cavity into closer proximity to the source of sound. Helmholtz has utilized the latter phenomenon in his synthetical reproduction of compound vowel-qualities. Koenig has improved on the original form of resonator by introducing a slide such as that named above, by which the same instrument may be made to reinforce several notes.

Theory of Resonators. In a pipe closed at one end, we have a mass of air vibrating in certain definite periods peculiar to itself, in more or less complete independence of the external atmosphere. If the air beyond the open end were

I Helmholtz, Sensations of Tone, Ellis's translation, p. 88.