§ 3. Intensity of Sound. Law of Inverse Squares
In the case of our exploding balloon the wave of sound expands on all sides, the motion produced by the explosion being thus diffused over a continually augmenting mass of air. It is perfectly manifest that this cannot occur without an enfeeblement of the motion. Take the case of a thin shell of air with a radius of one foot, reckoned from the centre of explosion. A shell of air of the same thickness, but of two feet radius, will contain four times the quantity of matter; if its radius be three feet, it will contain nine times the quantity of matter; if four feet, it will contain sixteen times the quantity of matter, and so on. Thus the quantity of matter set in motion augments as the square of the distance from the centre of explosion. The intensity or loudness of sound diminishes in the same proportion. We express this law by saying that the intensity of the sound varies inversely as the square of the distance.
Let us look at the matter in another light. The mechanical effect of a ball striking a target depends on two things—the weight of the ball, and the velocity with which it moves. The effect is proportional to the weight simply; but it is proportional to the square of the velocity. The proof of this is easy, but it belongs to ordinary mechanics rather than to our present subject. Now what is true of the cannon-ball striking a target is also true of an air-particle striking the tympanum of the ear. Fix your attention upon a particle of air as the sound-wave passes over it; it is urged from its position of rest toward a neighbor particle, first with an accelerated motion, and then with a retarded one. The force which first urges it is opposed by the resistance of the air, which finally stops the particle and causes it to recoil. At a certain point of its excursion the velocity of the particle is its maximum. The intensity of the sound is proportional to the square of this maximum velocity.
The distance through which the air-particle moves to and fro, when the sound-wave passes it, is called the amplitude of the vibration. The intensity of the sound is proportional to the square of the amplitude.
§ 4. Confinement of Sound-waves in Tubes
This weakening of the sound, according to the law of inverse squares, would not take place if the sound-wave were so confined as to prevent its lateral diffusion. By sending it through a tube with a smooth interior surface we accomplish this, and the wave thus confined may be transmitted to great distances with very little diminution of intensity. Into one end of this tin tube, fifteen feet long, I whisper in a manner quite inaudible to the people nearest to me, but a listener at the other end hears me distinctly. If a watch be placed at one end of the tube, a person at the other end hears the ticks, though nobody else does. At the distant end of the tube is now placed a lighted candle, c, Fig. 5. When the hands are clapped at this end, the flame instantly ducks down at the other. It is not quite extinguished, but it is forcibly depressed. When two books, B B′, Fig. 5, are clapped together, the candle is blown out.14 You may here observe, in a rough way, the speed with which the sound-wave is propagated. The instant the clap is heard the flame is extinguished. I do not say that the time required by the sound to travel this tube is immeasurably short, but simply that the interval is too short for your senses to appreciate it.
Fig. 5.
That it is a pulse and not a puff of air is proved by filling one end of the tube with the smoke of brown paper. On clapping the books together no trace of this smoke is ejected from the other end. The pulse has passed through both smoke and air without carrying either of them along with it.
An effective mode of throwing the propagation of a pulse through air has been devised by my assistant. The two ends of a tin tube fifteen feet long are stopped by sheet India-rubber stretched across them. At one end, e, a hammer with a spring handle rests against the India-rubber; at the other end is an arrangement for the striking of a bell, c. Drawing back the hammer e to a distance measured on the graduated circle and liberating it, the generated pulse is propagated through the tube, strikes the other end, drives away the cork termination a of the lever a b, and causes the hammer b to strike the bell. The rapidity of propagation is well illustrated here. When hydrogen (sent through the India-rubber tube H) is substituted for air the bell does not ring.
Fig. 6.
The celebrated French philosopher, Biot, observed the transmission of sound through the empty water-pipes of Paris, and found that he could hold a conversation in a low voice through an iron tube 3,120 feet in length. The lowest possible whisper, indeed, could be heard at this distance, while the firing of a pistol into one end of the tube quenched a lighted candle at the other.
§ 5. The Reflection of Sound. Resemblances to Light
The action of sound thus illustrated is exactly the same as that of light and radiant heat. They, like sound, are wave-motions. Like sound they diffuse themselves in space, diminishing in intensity according to the same law. Like sound also, light and radiant heat, when sent through a tube with a reflecting interior surface, may be conveyed to great distances with comparatively little loss. In fact, every experiment on the reflection of light has its analogy in the reflection of sound. On yonder gallery stands an electric lamp, placed close to the clock of this lecture-room. An assistant in the gallery ignites the lamp, and directs its powerful beam upon a mirror placed here behind the lecture-table. By the act of reflection the divergent beam is converted into this splendid luminous cone traced out upon the dust of the room. The point of convergence being marked and the lamp extinguished, I place my ear at that point. Here every sound-wave sent forth by the clock and reflected by the mirror is gathered up, and the ticks are heard as if they came, not from the clock, but from the mirror. Let us stop the clock, and place a watch w, Fig. 7, at the place occupied a moment ago by the electric light. At this great distance the ticking of the watch is distinctly heard. The hearing is much aided by introducing the end f of a glass funnel into the ear, the funnel here acting the part of an ear-trumpet. We know, moreover, that in optics the positions of a body and of its image are reversible. When a candle is placed at this lower focus, you see its image on the gallery above, and I have only to turn the mirror on its stand to make the image of the flame fall upon any one of the row of persons who occupy the front seat in the gallery. Removing the candle, and putting the watch, w, Fig. 8, in its place, the person on whom the light falls distinctly hears the sound. When the ear is assisted by the glass funnel, the reflected ticks of the clock in our first experiment are so powerful as to suggest the idea of something pounding against the tympanum, while the direct ticks are scarcely if at all, heard.
Fig. 7.
Fig. 8.
One of these two parabolic mirrors, n n′, Fig. 9, is placed upon the table, the other,