A COUSTICS, that branch of Natural Philosophy which treats of the nature of Sound, and the laws of its production and propagation. It is a subject extremely curious and interesting, and has at all times excited much attention among philosophers. In treating it here, it was at first our intention to embody the original article in this Encyclopedia with that which was contributed for the last Supplement by Professor Leslie. On considering, however, that the latter embraces in itself a general outline of the whole science;—that it is distinguished, like all the other productions of that philosopher, by soundness, depth, and striking originality; and that any changes even in the arrangement might destroy its peculiar character;—we have thought it best to preserve this valuable essay entire. It accordingly forms the Second Part of this Treatise; the First being devoted to the explanation of the more elementary principles; and the Third to some details supplemental to those contained in the Second.

PART I.

In regard to the nature of sound, the slightest attention is sufficient to convince us, that this sensation is not owing to the action of any peculiar substance or power, like Heat, Light, or Electricity, but arises merely from a certain mechanical action; a sort of concussion or agitation which takes place among the bodies from which the sound is emitted. Every noise or sound with which we are acquainted is accompanied with some action of this kind. The report of a cannon, for example, produces a concussion which shakes the ground under and to a great distance around it. In the same manner the rushing of waters, the roar of the sea, the whistling of the wind as it breaks on the trees or other obstacles which oppose it; the rattling of carriages, and that infinite diversity of sounds which arises in general from the percussion of one object against another: in all these cases there is a sensible and indeed violent agitation among the bodies from which the sound proceeds. In musical sounds, which are of a softer nature, we still also observe an agitation, which is often felt communicating itself to the surrounding bodies. If, for example, we stand under or near a piano-forte when it is sounding, we feel a sensible tremor in the floor of the apartment. If we lay the finger or hand on the instrument, or touch any other, such as a violin, when it is sounding, or a bell, we feel the same sort of tremor in every part of them; and this is well observed in the case of any glass vessel, such as a tumbler or large cup. If we strike it so as to make it sound, and then touch the mouth of it with the finger, we feel a sensible tremor in the glass; and when this internal agitation is stopped, as it generally is by the contact with the finger, then the sound ceases along with it. If there be any water in the glass, the tremor is observed communicating itself to the water, and putting the whole mass in motion. This is particularly well observed when we make the glass sound by rubbing the edge of it with the finger, as is done in the musical glasses. When the sound is brought fully out, the water is violently agitated; and as the sound declines, the agitation declines along with it. Such agitation then, in the sounding body, being Acoustics. the constant accompaniment of sound, ceasing when it ceases, and again beginning when the sound recommences; and there being no other circumstance which can be discovered accompanying the sound in this manner; there can be no doubt that this is in some way or other the cause of the sensation.

But how can any mere agitation or imperceptible tremor among bodies, and these at a distance, affect the organ of hearing? It is through the medium of the surrounding air. We have seen how the tremor of the piano-forte is communicated to the floor of the apartment, and is there felt under our feet. This communication can only take place through the legs of the instrument, which form the only connection between it and the floor. Through these, therefore, it is somehow conveyed; and in the same manner every tremor of this kind creates an agitation in the surrounding air, through which it is conveyed from the centre of agitation in all directions until it reaches the ear, and there striking or agitating the organ, produces the sensation of sound. That such agitation does take place in the air, we have a striking proof in the discharge of artillery, which often produces such an agitation as to shatter the glass in the windows of adjoining houses. The ear then forms merely a sort of organ of touch, but of such exquisite sensibility that it becomes affected by the slightest agitation in the fluid atmosphere by which it is surrounded. But this fluid is continually agitated, and in a thousand different ways, by the various motions and actions which are continually going on among the bodies on the surface of the earth. Every agitation of this kind communicates itself to the surrounding atmosphere; it is by it conveyed and propagated in all directions from the centre of agitation, somewhat like what we observe in a smooth surface of water when a stone is thrown into the middle of it,—a series of little waves are observed arising and propagating themselves in concentric circles from the centre of agitation on all sides. In a similar manner every agitation in the aerial medium propagates its influence in all directions, and from these varied impressions arise all those diversities of sound which affect the ear.

This idea of the process of hearing and of the cause of sound is certainly very remarkable, and well calculated to excite our surprise and curiosity, when we think of that infinite variety which we observe; single sounds, varying in intensity from the gentlest tap to the noise and violence of an explosion; continued sounds, from the ripple of waters to the roar of the cataract; strains of melody which enchant the ear, rising from grave to acute, and falling again to the lowest in the compass; or those harsher notes which only grate by their discord. Can it be possible that all these diversities arise from agitations in the air, differing only in the manner in which they strike the ear, in the force and quickness of their action, or in the regularity of their succession? What incredible sensibility in this organ, to perceive and be moved by such imperceptible shades of gradation, and to recognise such infinite diversities in kind! The powers of the eye in discriminating the minutest shades of colour justly excite our astonishment; but each colour presents a peculiar modification of light. The powers of the ear must appear still more extraordinary, if every sensation in it arises merely from a mechanical agitation of the same nature in every case, but only differing in the force of the impulse. Yet this is really the case; and the more we examine into the subject, the more clearly does it appear demonstrable by the laws of geometry and mechanics.

That the air is necessary for the production of sound, is clearly proved by the beautiful experiment of inclosing a bell within the receiver of an air-pump, and then exhausting the air. As the process of exhaustion proceeds, the sound is observed to become continually fainter and fainter, until at last it nearly dies away altogether. The bell, however, continues to ring, at least the hammer continues to strike, and thereby to agitate the bell as before; and yet no sound is emitted, plainly because there is no medium to convey the agitation to the ear. If the sound in this experiment cannot be altogether extinguished, this will be found to arise from the impracticability of exhausting the air altogether out of the receiver, and also of insulating completely the bell from the plate of the air-pump on which it stands, the support serving to convey the sound even after the air is exhausted. A very good mode of performing this experiment is, to have a bell with a small piece of clock-work attached to it, which strikes at regular intervals, and with a tone of uniform intensity.

In order to conceive the mode in which sound is propagated through the air, let us consider what takes place when we move a series of balls ranged in a line on a table, or suspended by threads. If we strike the one end of the line by impelling a ball against it, it is only the ball at the other end which appears to be affected. This flies off from the rest, and leaves them almost stationary. The intermediate balls, therefore, serve merely to transmit the impulse from the one end to the other of the series. In the same manner it is that the agitation or impulse from which sound arises is transmitted through the air. This fluid, like every other body, consists of an infinite number of little particles; a single series of which may be represented to us by the balls in the above example. These particles are not even in contact with each other; they are separated by minute intervals, but are yet connected together by attractive and repulsive forces, which tend to retain them perpetually in equilibrium. In every case, therefore, there is in reality a chain of such particles reaching from the sounding body to the ear. The former by its agitation strikes that particle which is next it, the intermediate ones serve to convey the impression, and the last one, flying off, strikes the sentient organ of hearing. The process is exactly similar to that of impulse along a series of balls, only that in the case of the air, the intermediate particles, instead of remaining at rest, move each of them backwards and forwards by a very minute interval; the first communicating its motion to the second, the second to the third, and so on to the last; each performing a slight oscillatory movement, which advances from the beginning to the end of the series.

We now see at once the cause of a remarkable and well-known fact, that the propagation of sound is not instantaneous; it requires time to advance from the sounding body to the ear, as is daily observed and illustrated in the discharge of fire-arms. If the distance be at all considerable, a sensible interval is always observed to elapse between the flash and the report. The light flies almost instantaneously, but the report is retarded according to the distance; as is also seen in many other cases: when we observe the workmen, for example, cutting up large stones in any quarry; if we stand at a little distance, we see invariably and distinctly the blow of the hammer on the stone before the sound reaches the ear. These and other similar facts leave no doubt that sound advances only at a certain rate, and invariably requires time for its propagation; and the reason is, that each aerial particle in the chain of communication must have a certain time, minute no doubt, but still definite, to perform its oscillation, and communicate its motion to the rest; and thus the advance of the agitation and of the sound is It is not through one series of particles merely that the oscillatory motion is communicated. The sounding body having every part of it in a state of agitation, generally acts all round; but even though it were only to act in one direction, the impulse, once begun at the centre, is propagated in all directions; for though only one particle were originally affected, so intimately are they all connected together and united into a system by their mutual attractions and repulsions, that this cannot advance in any degree forwards without affecting the particles on each side: these affect what are before and around them; and thus the impulse is communicated, and diffuses itself on all sides. These lateral impressions would appear to be necessarily somewhat enfeebled, yet it is one remarkable characteristic of such oscillatory movements, that, like the vibrations of a distended cord, or the oscillations of a pendulum in a cycloid, they are all performed in the same time, however minute or however extended. The lateral impressions, therefore, though ever so feeble, are yet transmitted with the same rapidity as the direct; the sound may be weakened, and we often observe it so;—a speaker, for example, is always best heard in front; the report of a cannon is also loudest in that direction, but still the sound is heard at the very same instant all round.

It is owing to this diffusion of the agitation in all directions, the original impression being spread out, not merely in concentric circles like the little waves in a pool, but expanding continually, if we can conceive it, into a wider and wider concentric sphere,—it is owing to this that every sound decreases so rapidly as we recede from it, and at last dies away altogether in the distance. It requires a very loud sound to be heard at the distance of a mile, yet we have heard the guns of Edinburgh Castle at the distance of 20 miles; and the noise occasioned by the falls of Niagara is said to be often heard at 60 miles. That this diffusion of the agitating impression is the true cause of the diminution of the sound, is proved in a remarkable manner by confining the air on all sides, as in a tube. M. Biot, in his Traité de Physique, gives an account of some very interesting experiments made by himself in the train of cast-iron pipes used for the conducting of water into Paris, and which extended about 2860 feet, thus including in their interior a cylindrical column of air upwards of half a mile in length; at which distance a person standing at one end of the pipes, and speaking within, could be easily heard at the other. "The lowest voice," says he, "was heard at this distance so as to distinguish completely the words, and to establish a continued conversation. I wished to ascertain at how low a tone the voice ceased to become audible, and I could not reach it. Words spoken as low as when one whispers in the ear of another were heard and appreciated; so that if we wished to speak so as not to be understood, there was only one way of doing it, and that was not to speak at all." It is on this principle that depends the effect of those tubes which are now in such general use, as modes of communication between distant apartments, in houses and public offices. Hence, also, are performed many amusing tricks with statues or busts, situated in different parts of a room, answering questions, and speaking to one another; the figures being connected by tubes concealed under the walls or floor, or communicating with an apartment below, in which a speaker is stationed.

In regard to the actual velocity with which the impulse of sound advances, it appears, from the most accurate experiments on the discharge of pieces of ordnance, and marking the interval between the flash and the report, at Acoustics, a distance carefully measured, that in ordinary circumstances this amounts to no less than 1130 feet each second, which is nearly equal to the velocity of a cannon ball the moment it issues from the piece. This last is very speedily retarded by the resistance of the air; but sound advances with undiminished velocity. Hence it will travel a mile in a little more than four seconds and a half, or 12½ miles per minute. On this depends an easy method of determining in many cases our distance from objects, and which may often prove useful, particularly in military operations. We have only to observe, in seconds, the interval between the flash and report of the cannon or musket, and allow 4½ seconds to every mile, or 1130 feet to every second. Thus, in a house in Lothian Street, directly opposite to the castle of Edinburgh, we have frequently observed a sensible interval elapse, as the sound of the guns travelled across the intermediate valley, we think about 2º or more; and the distance in a straight line is about 760 yards, or a little less than half a mile. In the same manner, by observing the interval between the flash of lightning and the thunder, we can tell the distance of the point where the electric discharge takes place.

It is remarkable also, that all kinds of sound, strong or weak, acute or grave, advance with the same velocity; and this arises from the circumstance already noticed, that all the oscillatory movements in the air, however minute or however extended, are performed each in the very same interval of time. This effect was distinctly proved in the experiments made by Biot in the cast-iron pipes already noticed, by playing different airs on the flute at one of the extremities of the tube. Now, it is well known that a musical air is adapted to a certain measure or time, which regulates very nicely the intervals between the successive notes; consequently, if any of these were propagated more rapidly or more slowly than others, by the time they reached the ear these would have been confounded with what preceded or followed them; and the air would have appeared quite altered, in place of which it was uniformly regular, and in its natural time; whence it clearly followed, that all sounds are propagated with equal velocity.

The above view of the propagation of sound explains Cause of at once the remarkable phenomenon of the echo, which echoes arises in every case from obstacles opposing the progress of sound. The agitation in the air, however, though interrupted by such obstacles, is not destroyed: each aerial particle which strikes against the opposing surface is reflected from it like an elastic ball which strikes against any wall or table. The sound is thus reflected at an angle equal to the angle of incidence; and it is when a number of these reflected impressions are thrown back to the point whence the original sound issues, by the configuration of the opposing obstacles, as so frequently happens among rocks, walls, &c. that an echo is produced.

Hitherto we have considered the air only as the vehicle Air not the of sound, and, without doubt, it is the grand medium of only vehi its transmission. Other bodies, however, convey it in a similar manner, and some of them even with much greater cule of a rapidity and force. In the liquid element this is proved by the acuteness which fishes display to sounds made in the air, and by many experiments. Professor Robison related, that with his head plunged under water, he could hear the sound of a bell, rung also in the water, at the distance of 1200 feet. We have already seen how the tremor of the piano-forte is communicated to the floor of the apartment; and many other familiar facts show clearly that sound is transmitted through the most solid bodies. How readily do we hear from one apartment of a Acoustics. house, or from one floor to another. The scratch of a pin is easily heard from one end of a log to another. A well-known but striking experiment in illustration of the transmission of sound is, to suspend any sonorous body, as a bell, a glass, a silver spoon, or a tuning fork, from a thread, and putting with the finger the extremities of the thread one in each ear;—if the body be then struck against any obstacle, the apparent loudness and depth of the sound are quite surprising. Again, if we shut the ears altogether, we yet feel very sensibly the impression of any sound conveyed through the mouth, the teeth, or the head:—if we put a small stick or rod in the mouth, and touch with the other extremity a watch lying on the table, the beats will become quite audible, though the ears be actually shut. Every noise in the mouth or among the teeth is conveyed internally to the ear in the same manner. Sound, therefore, is transmitted through liquids and solids, as well as through the air; and indeed, when we consider that the former are quite similarly constituted with the air, being composed of an infinite number of little particles, combined into a system by the same species of attractive and repulsive forces, it is noway surprising that an impulse communicated to any of these bodies should in like manner be diffused throughout the mass; and this must be by the same species of internal oscillations among the particles.

Such being the nature and propagation of sound, and its actual velocity as determined by experiment, it has long been the study of philosophers to reconcile these effects with those physical properties of the air and of liquid bodies, which are known from other circumstances, and can be calculated by the laws of mechanics, aided by the powers of mathematical analysis. This subject is investigated in the following Part; written, as already mentioned, by Professor Leslie.

PART II.

The doctrine of sound is unquestionably the most subtle and abstruse in the whole range of physical science. It has given occasion, in recent times, to much controversy and discussion, and has eventually called forth all the mighty resources of a refined and elaborate calculus. Yet an evident obscurity still remains to overcloud the subject. The discrepancies between theory and observation have been made entirely to disappear from astronomy, which has at last attained a degree of perfection befitting the sublimity of the science. But some latent suspicions pervade the structure of acoustics, sufficient to disturb that feeling of confidence which is calculated to invigorate our pursuits.

The impression of sound is conveyed by means of a certain tremor or internal agitation, which shoots, with more or less celerity and force, through any substance, whether solid or fluid. Nor is it requisite that the conducting medium should belong to the class of bodies which are commonly denominated elastic. In fact, all bodies whatever, in the minute and sudden alterations of their form, exert a perfect elasticity, and only seem to want this energy when they undergo such great changes that their component particles take a new set or arrangement, which prevents the full effect of re-action.

It is not every kind of tremulous motion, however, that will excite the sensation of sound. A certain degree of force and frequency in the pulsations appears always necessary to affect our sense of hearing. Yet the impression of sound is not confined to the mere external organ: the auditory nerves have a considerable expansion, and sympathize with those of taste and of smell. The only inlet of vision is by that very narrow aperture, the pupil of the eye; but the reception of sound partakes more of the character of the general sense of feeling, which, though most vivid at the extremities of the fingers, is likewise diffused over the whole surface of the body. The intimation of the ear is accordingly assisted by the consent of the palate, the teeth, and the nostrils. Fishes hear very acutely under water, though the organ itself lies so concealed in the head as to have long escaped the diligence of anatomists.

It was formerly supposed, that the transmission of impulse through a solid body is perfectly instantaneous. This formed, indeed, one of the Cartesian tenets, which Newton himself has tacitly admitted. But accurate observations have since proved, that motion is always really progressive, and propagated in succession. Professor Leslie has shown that the darting of impact through any substance, whether hard or soft, is accomplished by the agency of the same interior mechanism as that of sound, and has furnished the method of calculating, in some of the more difficult cases, the celerity of transmission.

All bodies may be considered as composed of physical points, without any sensible magnitude, but connected together by a system of mutual attraction and repulsion. When those integrant particles are compressed by external violence, a repulsive force is exerted to regain their first position; or if they be dilated, a corresponding attraction now draws them back to their neutral site of equilibrium. We may further presume, that in solids these constituent forces are confined to the proximate particles only, but that in the case of liquids or other fluids they embrace the particles in their near vicinity, and include a sphere of action varying in its extent. Hence, the former suffer disruption, without bending or giving way to powerful pressure; while the latter, acting by a sympathetic union, gently recede and take a new arrangement. In fact, the attribute of hardness applied to body is only a relative, and not an absolute quality; in the inferior degrees it relapses into softness, and softness again passes through interminable shades to the most yielding fluidity. The application of heat, by enlarging the system of internal connection, generally promotes softness, and heightens the degree of fluidity itself. The effect is conspicuous in the increased flow from a capillary syphon, when kept warm. But even liquids, when struck with a blow so rapid and sudden as to preclude the sympathy of their adjacent molecules, will assume all the character of the hardest substances. This fact has a familiar illustration in the play of duck and drake; but it is beautifully exemplified in the successive rebounds made by cannon-shot, from the surface of the sea.

In confirmation of the remark, we may quote a very singular and curious circumstance, mentioned by travellers, relative to the method of catching fish, which is successfully practised in some of the more northern countries. The hardy peasant, when the smaller lakes and rives of Lapland or Siberia are completely frozen over, as soon as he observes, through the clear ice, a fish, perhaps at a considerable depth, but lying close to the bottom, strikes a smart blow against the firm surface, and the impulse sent through the vertical column of water instantly stuns or kills his prey, which he draws up by a large hook let down through the hole just made in the ice.

If we conceive a conducting substance to be struck at one extremity, the proximate particles, yielding at first to the impulsion, will again expand themselves, like the recoil of a spring, and press against the next particles in the chain. The vibratory commotion will thus be conveyed, by a successive transfer of impressions, along the whole series of physical points. Analogous also to the Austistic oscillations of a spring or a pendulum, this multitude of concatenated internal pulses, whatever be the force or extent of agitation, will constantly be performed in the same instants of time. The celerity of transmission must depend on the elasticity of the medium compared with its gravity. This estimate is most readily obtained by determining what may be called the modulus of elasticity, or the height of a column of the same density as the conducting substance, whose weight would measure that elasticity; or, to speak more precisely, that the thousandth part of such a column, for instance, should be equivalent to the repulsive force corresponding to a condensation of one thousandth part in the vibrating body. It may be demonstrated from the principles of dynamics, that the celerity of the transmission of impulse or sound through any medium is equal to what a falling body would acquire in falling through half the height of the modulus of elasticity. Hence this celerity for each second will be expressed in English feet by multiplying the square root of half the modulus by 8, or by extracting the square root of the modulus multiplied by 32.

Mr Leslie has pointed out a very simple method for ascertaining the modulus of elasticity in the case of solid rods or planks, by observing, when they are laid in a horizontal position, with their ends resting against two props, the swag or curvature which they take. By an experiment of this kind, he found that Memel fir had a modulus equal to 671,625 feet. Wherefore, an impulse would shoot through the substance of a deal-board with the velocity of 4,636 feet each second, or about four times the rapidity of sound. Professor Chladni, who has thrown so much curious light on the convoluted curves formed by vibrations spreading along the surface of solid bodies, inferred, from a very different procedure,—from the musical note which a bar of the substance emits when struck,—the celerity of the transmission of sound through iron and glass, which he reckoned for both at 17,500 feet, or above three miles each second, being more than fifteen times swifter than the ordinary communication through the atmosphere.

The rate with which the tremor of sound is transmitted through cast-iron, was very lately ascertained, from actual experiment, by the ingenious M. Biot. This philosopher availed himself of the opportunity of the laying of a system of iron pipes, to convey water to Paris. These pipes were about eight feet each in length, connected together by narrow leaden rings. A bell being suspended within the cavity, at one end of the train of pipes, on striking the clapper at the same instant against the side of the bell and against the internal surface of the pipe, two distinct sounds were successively heard by an observer stationed at the other extremity. In these observations M. Biot was often assisted by the late M. Malus, who has, too soon for the progress of science, been hurried away by death, after having opened the delicate discovery of the polarization of light. With a train of iron pipes of 2550 feet, or nearly half a mile in length, the interval between the two sounds was found, from a mean of two hundred trials, to be 279 seconds. But the transmission of sound through the internal column of air would have taken 2·5 seconds; which leaves 29" for the rapidity of the tremor conducted through the cast-iron. From other more direct trials, it was concluded that the exact interval of time during which the sound performed its passage through the substance of the train of pipes, amounted only to 26·100th parts of a second; being ten to twelve times less than the ordinary transmission through the atmosphere.

Except the observations of M. Hassenfratz, in the famous subterranean quarries which extend under almost the whole of Paris, we are not acquainted with any attempts that have been made to measure the elasticity of stone or brick. Yet sound is conveyed through these materials with great effect. The rattling of a carriage on the street spreads a very sensible tremor along the most solid buildings and the stateliest edifices. If a large stone be rubbed against the outside of the wall of a house, it will occasion within doors a strange rumbling noise. A miner will strike his pick against the side of a long gallery, when he wishes to give intimation to his companion, who listens at the other extremity. But stones or bricks, without being directly excited, may yet form a part of the chain which transmits sound, by receiving the tremulous impressions from the air on the one side, and delivering them again to that fluid on the other. We all know how easily the voice is heard through a thin partition. The mode of obstructing the passage of sound is, either to employ very thick masonry, or to interrupt the facility of communication and transfer, by means of subdivisions opposed. Hence another distinct use of lath and plaster. Experiments on the elasticity of stones and other articles of building are not only curious, but of real importance; for, in many cases, their efficient strength must depend on their fitness to resist incidental impressions. This consideration is peculiarly necessary in selecting and combining the materials employed in the construction of bridges.

Respecting the elasticity of water and other liquids, Supposed our information is more satisfactory and complete. It was long held as an axiom, that the substance of water is absolutely incompressible. Yet the experiments on which this belief was grounded would, if weighed attentively, point to an opposite inference. On such a subject it were idle to cite Lord Bacon, whose credulity and ignorance of mathematical science betrayed him so often into false or shallow conclusions. The philosophers of the Florentine academy del Cimento tried the compression of water in three different ways, which are described in the account of their experiments printed in 1661. 1. Having provided two glass tubes terminated by hollow balls, they filled the one partly and the other to excess with pure water, and joined the tubes hermetically, so as to form one piece. Then applying heat to the first ball till the water boiled, they forced its vapour to press against the column in the other stem. But no contraction of the fluid took place, though a copper ball was afterwards substituted; and when the action of the heat was still farther urged, the tube at last burst with violence. 2. Into a glass tube, immediately above six pounds of water, they introduced eighty pounds of quicksilver, without causing any diminution of volume. 3. Their most noted experiment was, having filled a hollow silver ball with water by a small hole, afterwards soldered accurately, to give it a few smart blows with a hammer; when, far from suffering compression, the water was seen to ooze or spurt from the pores, as they imagined, of the silver.

Mr Boyle, whose practice it was generally to repeat the more striking experiments made on the Continent, had a round tin or pewter vessel filled carefully with water, and tightly plugged: the blow of a wooden mallet beat it flat, but on piercing the tin with the point of a small nail, the confined water instantly sprung to the height of two or three feet. About the year 1752, Dr Peter Shaw, who read public lectures in London, exhibited a stout copper ball of four inches in diameter, and filled with water by a small orifice, into which a screw was fitted, and forced to enter by turning an iron arm or lever: the globe was partly opened by this enormous Acoustics. squeeze, and the water spouted from the crevice as from a fountain.

These experiments all concur to show that water is capable of sustaining an immense pressure without undergoing any very sensible contraction; but they prove, at the same time, the actual existence of such a contraction, since the projecting of the water, after a crack has once begun in the vessel that confines it, could only proceed from the evolution of an internal repulsive force. Divers, accordingly, at considerable depths under water, hear distinctly the collision of two stones, or the remote ringing of a bell. Authentic instances are mentioned of sounds being transmitted audibly more than two miles through that fluid.

The compressibility of water was first demonstrated by the ingenious Mr Canton in 1762, by a very simple and conclusive experiment. To a glass ball of rather more than an inch and half in diameter, he joined hermetically a tube about four inches long, and having a bore equal to the hundredth part of an inch. The relative capacity of this ball and of the stem he ascertained by introducing mercury, and weighing nicely its separate portions. The stem was then marked by the edge of a file into divisions, corresponding each to the hundred thousandth part of the whole capacity of the ball. This instrument was now filled with distilled water, carefully purged of its adhering air, and placed under the receiver of a pneumatic machine: on producing an exhaustion, the water appeared constantly to swell, rising four divisions and three-fifths in the stem, or a space nearly equal to the mercurial expansion corresponding to half a degree of heat on Fahrenheit's scale. In a condensing engine, the water sunk just as much, for each additional pressure of an atmosphere,—the bulb remaining always at the same temperature, or at the fiftieth degree of Fahrenheit. Since the stem was left open, the pressure exerted by the air, both on the inside and the outside of the instrument, must in all cases have been precisely the same; and consequently, the glass had no disposition to alter its figure, and modify the results. The contraction or expansion produced was, therefore, confined wholly to the body of water and to the thin shell of glass, of which indeed the influence might be rejected as insignificant. It was hence decided that the purest water suffers a visible concentration, or a diminution of its volume, under a powerful compression. But, in the course of his experiments, Mr Canton observed a curious circumstance, that water is more compressible in cold than in warm weather. Thus, the contraction, under a single incumbent atmosphere, amounted to 4-9 divisions when the thermometer stood at 34°, but was only 4-4 divisions when the heat rose to 64°. This singular fact might afford room for speculation; but it were better, in the mean while, to repeat the experiment again with more delicacy, and on a greater scale.

The compression of some other fluids was likewise measured in the same way. The contraction, under the weight of an atmosphere, and at the ordinary temperature, amounted, in millionth parts of the entire capacity of the ball, to sixty-six with alcohol, to forty-eight with olive oil, to forty with sea-water, and only to three when mercury was opposed. We may therefore estimate, in round numbers, the modulus of elasticity belonging to those different substances as under:

Alcohol.....................580,000 English feet. Distilled water.............700,000 Olive oil....................730,000 Sea-water...................780,000 Mercury.....................800,000

In liquids of so distinct a nature, we should have expected a greater diversity in their elastic power; nor is it easy to Acoust. conceive on what conditions or habits that quality actually depends. The elasticity of a body, like its other constitutional properties, may result from the peculiar internal structure, or the arrangement of the integral molecules.

Some experiments on the compressibility of water have been since performed with more striking effect, but not equally exempt from all objections. In 1779, Professor Zimmerman of Brunswick printed a short account of some Zimm trials made by him and Abich, director of the salt-mines, man. with a press of a particular construction, consisting of a tight cylinder of very thick brass, with a piston nicely fitted, to be pushed down by means of a long lever, at whose extremity different weights were appended. Rainwater being introduced into the cavity, was subjected to an enormous pressure, equivalent to that of 313 atmospheres, and had its volume then diminished between one thirty-fifth and one thirty-sixth part. This quantity gives, for the effect of a single incumbent atmosphere, a condensation amounting to seventy-five millionth parts, instead of forty-six, as found by Mr Canton. The excess was no doubt owing to the distention of the brass cylinder, which, with all its strength and solidity, would yet partially yield to the action of such prodigious force. This circumstance renders the experiment somewhat unsatisfactory, and the influence of friction must likewise affect the accuracy of the calculation.

The effect of such distention is easily witnessed in the case of glass. If a large bulb of a thermometer be suddenly squeezed between the finger and the thumb, the mercury will start up in the stem perhaps several degrees, and will again sink as quickly after the pressure is removed. To prevent any derangement from communication of heat, the hand may be covered with a thick glove. But the fact can be shown in a less exceptionable way: Let a mercural thermometer, with a large bulb and a long stem, be first held upright, and then immediately inverted; between these two positions the column of mercury will descend through a visible space. This apparent change of volume has been hastily supposed by some experimenters to mark the compressibility of mercury, which could not be sensible but under the action of a column of incomparably greater height.

It would be most desirable to institute a new set of observations on the condensation of different substances, by means of Bramah's hydraulic press, which is a far more Bramah perfect machine, and scarcely subject at all to the dis-hydraul disturbance of friction. Having once ascertained the dis-pres- tention of the metallic cavity from pressure, it would be hence easy to correct all the other results. This mode of experimenting promises also the important advantage of enabling us to determine, with ease, the compressibility of solids themselves. It would only be required to give those bodies a cylindrical form nearly adapted to the cavity, and to fill up the interstice with water, or rather with mercury. The contraction which the thin sheet of fluid would undergo, being deducted from the whole contraction, would exhibit the contraction suffered by the solid nucleus.

From all these investigations we may gather, that an Celerity impulse, or a sonorous tremor, would shoot through a body sound of fresh water with the velocity of about 4,475 feet each second, being four times swifter than the ordinary flight water, of sound in the atmosphere. Through the waters of the ocean, the transmission of sound would be still more rapid, by a seventeenth part. It hence follows, that a violent commotion, excited under that vast mass, might reach from pole to pole in the space of three hours and twenty minutes. Acoustics. The swell of the sea is accordingly always observed to precede the coming storm. The shocks of the famous earthquake at Lisbon, in 1755, were partially felt at very distant points of the ocean, as far even as the West Indies, but after a considerable interval of time.

Respecting the power of ice to conduct sound, we possess not sufficient data for the solution of the problem. The Danish philosophers are indeed said to have lately performed experiments of this kind on a very extensive scale, along the frozen surface of the Baltic. We are not acquainted with the precise results; but it seems probable, from various analogies, that ice has nearly the same faculty of transmission as water itself. If a heavy blow be struck against any part of the frozen surface of a large pool or lake, a person standing at a wide distance from the spot will feel, under foot, a very sensible tremor, some considerable time before the noise conveyed through the atmosphere has reached his ear. It is asserted, that the savage tribes who rove on the icy steppes of Tartary can readily distinguish, from afar, the approach of cavalry, by applying their head close to the frozen surface of the ground.

But the proper and ordinary vehicle of sound is our atmosphere. Aristotle, deriving his information probably from the tenets of the Pythagorean school, seems to have acquired tolerably just notions of the nature of sound and of the theory of harmonics. The language of that philosopher was so much corrupted, however, and disguised by ignorant transcribers, that Galileo, who not only studied music as a science, but practised it as a delightful art, may be fairly allowed to have rediscovered those general doctrines. Mersenne and Kircher afterwards made a variety of most ingenious experiments, which, though rather overlooked at the time, tended greatly to extend the science of harmony. But it was reserved for the genius of Newton to sketch out the true theory of sound. In his Principia he explained the origin of aerial pulses, and, by a fine application of dynamics, conducted with his usual sagacity, he succeeded in calculating their celerity of transmission. The solution which he has given of this intricate problem is far, however, from being unexceptionable in the form and mode of reasoning. Instead of attempting to embrace all the conditions affecting the problem, in a differential equation, for which, indeed, his fluxionary calculus was not yet far enough advanced, he proceeds less boldly, and only arrives at the conclusion by an indirect process and a sort of compensation of errors. His investigation of the progress of sound through the air is chiefly drawn from the analogy of the motion of waves along the surface of water. This comparison greatly assists our conceptions, but it fails in a variety of essential points. Newton further assumed the rising and subsiding of waves to be a reciprocating motion, similar to that of the oscillations of a fluid contained in a wide and long tube, with its ends turned upwards. On this supposition, it was not difficult to prove, that those alternating movements would correspond to the vibrations of a pendulum of half the length of the tube. Transferring the inference, therefore, to the undulations of a fluid, it followed, that the space between two consecutive waves would be described during the sweep of a pendulum having a length equal to this interval. But the conclusion does not very well accord with the phenomena. That a wave travels with a velocity as the square root of its breadth, may be nearly true; and that its reciprocating motions, whatever be the height, are all performed in the same time, is a necessary consequence of the great principle in dynamics first pointed out by Huygens and Hooke,—that when the effort to restore equilibrium is proportioned to the quantity of displacement, the alternations of figure are constantly isochronous. But the velocity of the undulating progression, as calculated from those principles, will not be found to correspond with actual observation. Newton was apparently sensible of this disagreement, and would consider his proposition as only an approximation to the truth; assigning as the cause of discrepancy, that the particles of water do not rise and fall perpendicularly, but rather describe arcs of a circle. The great defect of the hypothesis, however, consisted in supposing all the parts of a wave to rise up and sink together in the same spot. The fact is, that the fore part of a wave is always in the act of ascending, while the hinder part of it is as constantly subsiding; which combined but contrary movements, without actually transferring any portion of the water, give an appearance of progressive advance to the swell.

In extending this theory to the propagation of sound, Newton was, on the whole, more successful. It resulted from his investigation, that the aerial pulses fly uniformly, spreading themselves equally on every side, and with a celerity equal to what would be acquired by a body in falling through half the height of the modulus of the air's elasticity. This modulus, or the altitude of a column of air, of uniform density, and whose pressure would be equivalent to the ordinary elasticity of that fluid, was computed in the first edition of the Principia, which came out in 1687, on the supposition that water is 850 times denser than air, mercury 13 1/3 times denser than water, and that the mean height of the barometer is thirty English inches. The modulus of elasticity, or the height of an equiponderant column of air, was therefore estimated at 29,042 feet, which gave 968 feet each second for the celerity of the transmission of sound through the atmosphere. In the next edition, which did not appear till twenty-six years thereafter, the computation of the modulus was somewhat altered, but certainly not rendered more correct. Assuming the same standard of barometric height as before, and supposing mercury to be 13 1/3 times heavier than water, and water 870 times heavier than air, the modulus would be 29,725 feet, to which the corresponding velocity of sound is 979 feet in the second.

In these successive estimates, there is perhaps betrayed some desire to magnify the result, yet without nearly approaching to the amount of actual observation. Dr Derham had recently determined, from repeated trials made with care, that the ordinary flight of sound is at the rate of 1142 feet each second; and Newton endeavoured, by some very strained hypotheses, to accommodate his calculation to this correct measure. 1. He supposes the particles of air to be perfectly solid spherules, whose diameter is the ninth part of their mutual distance. Sound, being instantaneously communicated through these, would thus have its velocity increased by one-ninth, or 109 feet, or brought up to 1088 feet in the second. 2. He next assumes, that the particles of vapour concealed in the air, and augmenting the common elasticity without partaking of the impression of sound, amount to a tenth part of the whole. This would increase the celerity of the sonorous pulse in the subduplicate ratio of 10 to 11, or as 20 to 21 nearly, and consequently advance the last measure from 1088 to 1142 feet.

But these random and fanciful conjectures hardly require any serious consideration. What may be the size of the ultimate particles of air, or whether they have any sensible magnitude at all, we are utterly without the means of determining. There appears no limit, indeed, to the degree of condensation of which the air is capable, but what proceeds from the imperfection of the engines employed for that purpose. Nay, supposing so large a proportion of ab- Acoustics. solute matter to exist in the composition of our atmosphere, it really would not affect the result, since the transit of sound, as we have shown, is necessarily progressive, even through the most solid substance. To this principle there could be no exception, unless the particles of air were held to be mere atoms, incapable of further subdivision,—in short, without actual magnitude, and therefore bearing no relation whatever to the space in which they float. The second hypothesis advanced is still more insufficient to rectify the general conclusion. That moisture, in its latent or gaseous form, is united with the air, will be granted; but it by no means constitutes so notable a share of the fluid as Newton has assumed, scarcely exceeding, at the ordinary temperature, perhaps the five-hundredth part of the whole weight. But this diffuse vapour could not in the least derange the original calculation; for, being always combined with the air, the measure of elasticity assigned by experiment was really that of the compound fluid which forms our atmosphere.

Rectified calculation of the velocity of sound. We are now enabled, by the help of more perfect data, to rectify the modulus of atmospheric elasticity, or the height of a homogeneous and equiponderant column of the fluid. From the observations made with barometrical measurements, it appears that such a column, exerting a pressure equivalent to the elasticity of the air, has, at the limit of freezing water, an altitude of 26,060 feet, and consequently, that the modulus would, at an ordinary temperature of 62° by Fahrenheit, amount to 27,800 feet. This corrected estimate gives only 943 feet each second for the celerity of sound. And since the elasticity of the medium is exactly proportioned to its density, the result is the same, whatever be the rarefaction or condensation of the air, so long as its temperature continues unaltered. The flight of sound is hence as rapid near the surface as in the higher regions of the atmosphere. It is the conjunction of heat alone that will increase the celerity of transmission, by augmenting the elasticity of the medium without adding to its weight. The acceleration thus produced must amount to rather more than one foot in the second for each degree by Fahrenheit's scale. Such a difference ought to be perceptible under the torrid zone.

Rate of transmission through different gases. But the rate of the transmission of sound must vary in different gases, after the inverse subduplicate ratio of their densities. Thus, through carbolic gas, the communication of the tremor would be about one-third slower than ordinary; but through hydrogen gas, which is twelve times more elastic than common air, the flight would very nearly exceed three and a half times the usual rapidity. An admixture of this gas with the atmosphere would, therefore, greatly accelerate the transmission of sound. The joint combination of heat and moisture, by heightening the elasticity of the air, must likewise produce a similar effect.

These inferences are confirmed by observation, as far as it extends. The velocity of sound was determined with considerable accuracy, and on a great scale, by Cassini and Maraldi, while employed in conducting the trigonometrical survey of France. During the winter of the years 1738 and 1739, these astronomers repeatedly discharged, at night, when the air was calm and the temperature uniform, a small piece of ordnance, from their station on Mont Martre, above Paris, and measured the time that elapsed between the flash and the report, as observed from their signal tower at Montlehey, at the distance of about eighteen miles. The mean of numerous trials gave 1130 feet for the velocity of the transmission of sound.

About the same time, Condamine, who was sent with the other academicians to ascertain the length of a degree in Peru, took an opportunity of likewise measuring the celerity of sound, at two very different points. He found Acoust this was 1175 feet on the sultry plain of Cayenne, and only 1120 feet on the frozen heights of Quito. It was obvious, therefore, that the rarefaction of the air in those lofty regions had in no degree affected the result. Compared with what had been observed in France, the velocity of the aerial pulses was somewhat diminished at Quito by the prevailing cold, but was, on the other hand, considerably augmented by the excessive heat and moisture which oppress Cayenne.

But the difference, amounting indeed to one-fifth of the whole, between the velocity of sound as deduced from theory, and as determined by actual experiment, still appeared very perplexing. This want of congruity was the more felt, since the Newtonian system of gravitation, after maintaining a long struggle with the adherents of the Cartesian philosophy, had at last obtained the undisputed possession of the Continent. Its triumph was insured by the admirable dissertations on the subject of tides, transmitted to the Academy of Sciences at Paris in the year 1740, when our celebrated countryman Maclaurin had the honour of sharing the prize with Euler and Daniel Bernoulli. The law of attraction received, indeed, a temporary shock a few years afterwards, from the result of the investigation which Clairaut first gave of the lunar inequalities; but, on resuming his analysis of the problem, and computing the values of the smaller terms of the formula, that great geometer obtained, in 1752, a final product, exactly conformable to the best astronomical observations; and the solidity of the Newtonian system was henceforth placed on the firmest foundation.

It was therefore peculiarly desirable to examine likewise the justness of the hydrodynamical conclusions of Newton. The propositions concerning the propagation of sound were perhaps justly considered as the most obscure part of the whole Principia. Some of the first-rate mathematicians abroad, particularly D'Alembert and John Bernoulli, declared their utter inability to comprehend such intricate and disjointed demonstrations. At last the problem of sonorous pulses was attacked directly, and in its full extent, by the late Count Lagrange, whose investigation, although at a ripe age, will be lamented as a most severe loss to mathematical science. That illustrious geometer shone forth at once like a meteor, and before he had completed his twenty-third year he gave a rigorous and profound analysis of the propagation of sound through the atmosphere, in the first volume of the Turin Memoirs, which appeared in 1759. "He pointed out some mistakes that even Newton had committed in the reasoning; but mistakes which, by a happy compensation of errors, did not affect essentially the results. Advancing from these discussions, he assigned the dynamical conditions of undulation, which, after the proper limitations, were reduced to an equation involving partial differences of the second order. But this refined branch of analysis, invented by D'Alembert and Euler, is still so imperfect, that, in order to integrate the final expression, it had become requisite to omit the higher powers of the differentials. Yet after all this display of accurate research and skillful adaptation of symbols, followed by a lax and incomplete calculus, the same conclusion was obtained as that which Newton had derived chiefly from the force of analogy and sagacity of observation; and philosophers were thus obliged to submit, and to content themselves with recording the variance between theory and experiment in regard to the celerity of sound, or with referring that discrepancy to some extraneous influence." (Edinb. Review, vol. xv. p. 431.)

M. Poisson, one of those interesting men whose native Acoustics. genius has surmounted all the obstacles of fortune, very lately attempted a more complete analysis of the propagation of sound, in the Papers of the Polytechnic School. The final equation is more fully expressed, and its integration is pushed some few steps farther; but still the result is precisely the same as before. The skill and precaution displayed in framing the conditions of the problem are afterwards mostly abandoned in the various simplifications adopted to arrive at the conclusion.

A very ingenious and apparently satisfactory method of reconciling theory with observation, in the estimate of the transmission of sound, was not long since suggested by the celebrated Count Laplace. If the heat contained in air had, at every state of the density, been united constantly after the same proportion, the elasticity resulting from the infusion of this subtle and highly distensible element would invariably accord with what observation assigns to the compound aerial fluid. But the capacity of air, or its aptitude to retain heat, varies with its internal condition; being increased by rarefaction, and proportionally diminished by condensation. When air is compressed, therefore, it liberates a portion of its heat; and when it undergoes dilatation, it becomes disposed to abstract more heat from the adjoining bodies. Till the equilibrium of heat is restored, the air will be sensibly warmer after each act of compression, and colder when suffered to dilate. If the shock given to a portion of air be very sudden and violent, the quantity of heat evolved from it is profuse and powerful. On this principle, M. Mollet, member of the academy of Lyons, led by some facts noticed by artists who manufactured windguns, first constructed, in 1804, the curious instrument for producing fire by the rapid condensation of air confined in a tube. But such evolution of heat must besides augment the elasticity of the air, as the contrary abstraction of it will, in a like degree, diminish that force. At every sudden alteration of density, therefore, a new power is infused, which had not entered into the ordinary undisturbed estimate of the air's elasticity. Consequently, from this consideration alone, the aerial pulses must shoot with some greater celerity than calculation assigns, because the particles of air, which are suddenly condensed, have their elasticity further augmented by the portion of heat evolved, while the corresponding particles, which are simultaneously dilated, have their disposition to contract likewise increased, by the momentary prevalence of cold.

The principle advanced by Laplace must therefore have a real operation, tending to reconcile the calculated velocity of sound with that which is deduced from experiment. The only question is, how far its influence could actually extend. But, according to the formula given in Leslie's Elements of Geometry, p. 495, a condensation equal to the 90th part of the volume of air would occasion the extrication of one degree of heat by Fahrenheit's scale. Now, since each degree of heat enlarges the bulk or augments the elasticity of the air by the 450th part, it follows, that the heat, extricated by sudden impulse, will communicate to the air a momentary additional spring, amounting to one-fifth of the whole elastic force. Wherefore the celerity of sound would, by that influence, be increased in the subduplicate ratio of five to six, or nearly as 21 to 23; which gives an addition of only 90 feet each second to the whole quantity, bringing it up to 1033 feet. The correction is thus insufficient, not amounting to half of the discrepancy which it was its object to reconcile.

It may be suspected, therefore, that some inaccuracy or omission infects the investigation itself. Till the integral calculus has arrived at much greater perfection, it will often be requisite for the analyst, in the solution of Acoustical problems, to descend from his elevation, and seek to simplify the differential expressions by a sober and judicious application of the principles of physics. "Imagine a string of particles, or physical points, A, B, C, D, E, F, &c. in a state of rest, or mutual balance. If A were pushed nearer to B, and then suddenly abandoned, it would recoil with a motion exactly similar to the oscillation of a pendulum. The time of this relapse might easily be determined, from a comparison of the force of gravity with that of elasticity, or from the number of particles contained in a column of equipoise. The minute interval between the adjacent particles, being now divided by the duration of each fit of contraction, will give the velocity with which the vibratory influence shoots along the chain of communication. This simple investigation leads still to the same result as before. But it proceeds on assumptions which are evidently incorrect; for it supposes the pulses to follow each other in accurate succession, every contraction terminating as the next begins. Since the particles, however, do not exist in a state of insulation, while B repels A, it must likewise press against C; and C, in its turn, must gradually affect D. Before the contraction of A and B is completed, that of B and C is therefore partially performed; and this anticipated influence may even extend to the remoter particles. Nor is the system of mutual action, at all materially disturbed by such anticipations. Each pulsation is performed in the same way as if it were quite detached; only the succeeding one is partly accomplished before the regular period of its commencement. 'The velocity of aerial undulation is in this way much accelerated.' (Edinb. Rev. vol. xv. p. 433.)

Each successive movement among the particles may be viewed as produced by a force not regularly decreasing, but partaking of the uniformity which obtains in projection. Hence the velocity of sound is intermediate between that derived from theory and that with which air would rush into a vacuum. But the arithmetical mean between 943 and 1334 feet is 1138\frac{1}{2}, and the geometrical is 1121\frac{1}{2} feet; neither of which differs much from 1130 feet, the quantity determined by actual experiment.

After the last correction, however, proposed by M. Modifica- Laplace, for adjusting theory with observation relative to the celerity of the transmission of sound, the difference required will not perhaps be regarded such as longer to present any theory serious obstacle; especially when the coincidence appears closer than what generally attends the theoretical deductions concerning the motions of fluids. The remaining difficulties affecting the subject refer chiefly to the way in which the aerial pulses are propagated, and the modifications which they are afterwards capable of receiving.

1. No sensation is ever excited, unless the impression made upon our organs be repeated or continued during a certain short space of time. On this principle depends strength of the whole success of the juggler, who contrives to change the situation of the various objects before us with a rapidity exceeding the ordinary exercise of sight or touch. A brand whirled swiftly round the head gives all the appearance of a circle of fire; and if one presses very hard an ivory ball between his fingers, he will seem still to feel it for several instants after it has been withdrawn. To excite the sensation of sound, it is requisite that the aerial pulses should have a certain force and duration. According to some observations, the ear is not affected at all, unless the tremulous impulse communicated to the tympanum lasts during the tenth part of a second. Every pulsation of a more transient kind is lost absolutely and completely to our organ of hearing.

On the other hand, the impression of sound is not pro- Acoustics. longed beyond the time of its actual production. If it were otherwise, indeed, all sounds would degenerate into indistinct noises; and articulate discourse, which distinguishes man from the lower animals, and constitutes the charm of social life, would have been utterly impossible. This fact, so obvious, and yet so important, shows indisputably that the propagation of sonorous pulses through the atmosphere is not, in all its circumstances, analogous to the succession of waves on the surface of water. These undulations continue long afterwards to rise and spread from the centre of their production. The pulsations of the air, no doubt, likewise survive their excitement; but such of them as succeed the first impulsion must not have the force and character of those which are directly shot through the fluid. What is the precise discrimination between these different pulses, we are not enabled from mere theory to determine. But such a distinction must undoubtedly exist, otherwise indeed all discourse would continue to fill the ear with a monotonous hum, or an indistinct muttering. It would be difficult to institute conclusive experiments on this subject, yet collateral researches might be devised which could not fail to guide our inquiry.

2. But another defect in the analogy between waves and sonorous pulses is, that the latter, without affecting to spread equally, are capable of acquiring a superior force or tendency in some given direction. Certain unconfined sounds, indeed, are diffused uniformly on every side. Thus, the noise of the explosion of a powder-mill is heard, and often dreadfully felt, at a great distance all round the scene of disaster. But the report of a cannon, though audible in every direction, appears invariably loudest in the quarter to which the engine is pointed. On this principle, a seaman, when he seeks to be heard more audibly, or at a greater distance, is accustomed, if no other help occurs, to apply his spread hands on each side of his mouth, and thus check or diminish the waste of sound by its lateral dispersion. For the same reason, the bent and projecting circular piece annexed to the farther end of a speaking-trumpet is of most decided use, in assisting to give direction to the flight of the aerial pulses.

3. The theory of undulatory movements furnishes some elucidation, but no adequate explication, of the augmented effect of sound in the direction of a lateral barrier. The extension of such an obstacle might appear to check merely the spread and consequent attenuation of the sonorous pulses; but the great accumulation of impulse always occurs, on either side, at the extremity of the advancing wave. By what system of interior forces this effect is produced, it would be difficult satisfactorily to explain. Yet we perceive something analogous in the swell which runs along the margin of a pool, and in the billow which, flowing from the open sea, heaves against the sides of a projecting mole.

It is hence that sound is made to sweep with such intensity over the smooth surface of a long wall or of an extended gallery. An elliptical figure, though of manifest advantage, is not really essential to a whispering gallery; for the point of sonorous concentration is found beyond the true catoptrical focus, and much nearer to the wall. A fact of the same kind is well ascertained—that sounds are always heard the most audibly, and at the greatest distance, in a level open country, or still better on the smooth surface of a vast lake, or of the ocean itself. The roaring of the cannon in certain naval engagements has been noticed at points so very remote from the scene of action, as might seem, if not perfectly authenticated, to be altogether incredible. On the other hand, again, sound is enfeebled and dissipated sooner in alpine regions. Thus, the traveller, roving at some height above a valley, accurs, with uncommon clearness, perhaps a huntsman on the brow of the opposite mountain, and while he watches every flash, yet can he scarcely hear the report of the fowling-piece.

On a similar principle, we would explain the operation of the ear-trumpet, which affords such relief to one of the most cheerless maladies that can afflict humanity. The wide mouth of that instrument, it is well known, is turned to catch the stream of sound; the extent of pulsation is gradually contracted as the tide advances; and the same quantity of impulse being probably maintained, the vibratory energy is intensely accumulated at the narrow extremity, where it strikes the cavity of the ear. A trumpet of this form might, in many cases, be found very advantageous, not only for remedying the defects of the organ of hearing, but for assisting the observer to collect feeble and distant sounds. Even an umbrella held close behind the head, with its concavity fronting the sonorous pulses, will, it has been alleged, sensibly heighten their impression.

4. To explain legitimately the reflection of sound, would require some modifications in the theory of atmospheric modification. Each obstructing point is certainly not the centre of a new system of pulses; for, in many cases, this would occasion unutterable confusion. Nor can the excitement of sound be supposed to dart in straight lines, or to perform the same accurate reflection as the rays of light. In fact, neither smoothness nor exact regularity of surface is required for the production of an echo. A range of buildings, a row of tall trees, a ridge of rocks, or a chain of heights, will, in certain positions, reflect sound with clear and audible effect. It follows, therefore, that the reflection must be formed, not at the immediate surface of those obstacles, which could occasion only an irregular dispersion, but at some boundary at a small distance, and running parallel to the mean direction of the whole barrier. We may conceive the tide of sound accumulating where it stops, and investing the opposite surface like an atmosphere, till a repulsion is exerted, which again rolls it back. What seems to constitute the perfection of an echo is, that the sum of the distances of every point of the reflecting surface from the person who speaks, and from him who listens, should be the same. When this disposition obtains, all the reflected sounds must reach the ear in due succession, without being intermingled or confused.

We may observe, that echoes are often confounded with the mere resonance occasioned by vibrations excited among the obstacles themselves. In a large empty room, with its naked floor, and walls, and benches, the voice quickly throws the whole into a tremulous commotion, and seems drowned in the ringing prolonged sound which is produced; nor does this unpleasant effect cease, until the spectators have occupied the benches, filled the hall, and obstructed by their weight the vibration of the floor. What is called the deadening of sound, consists in merely checking or preventing the disturbance of extraneous tremor. For this purpose, the floor is covered with carpets, and the walls lined with wainscot or hangings. Such barriers, we have seen, would not, by their yielding quality, blunt or obstruct the formation of echoes. Their only effect is, to muffle the elastic surfaces which they cover.

The performance of the speaking-trumpet has generally been referred to the concentrated reflection of sound. Some authors have carried the hypothesis even so far as to investigate, from mathematical principles, the best figure of that instrument. Much labour and great ingenuity have been utterly wasted in this fruitless attempt. Acoustics. Kircher proposed the tube to be shaped like a truncated parabolic conoid, the mouth-piece occupying the focus; and he concludes that all the rays of sound would, by reflection from such a surface, be sent forward exactly in parallel lines. Other philosophers have imagined, from a fanciful analogy to the property of ivory balls, that the figure described by the revolution of the logarithmic curve about its absciss would be the most proper for the speaking-trumpet. M. Lambert, of the Berlin academy, whose genius and originality were both of the first order, has given a solution still different. But it would be idle to recite the various attempts which have ended in no practical result.

The true physical explication of the speaking-trumpet was first given, as far as we know, in the course of an incidental remark by Professor Leslie, in his Experimental Inquiry into the Nature and Propagation of Heat. "In the case of articulate sounds," says he, "the confining of the air does not affect the pitch of voice, but it augments the degree of intonation. The lateral flow being checked, that fugacious medium receives a more condensed and vigorous impulsion. As the breath then escapes more slowly from the mouth, it waits and bears a fuller stroke from the organs of speech. But the speaking-trumpet is only an extension of the same principle. Its performance does certainly not depend upon any supposed repercussion of sound; repeated echoes might divide, but could not augment the quantity of impulse. In reality, however, neither the shape of the instrument, nor the kind of material of which it is made, seems to be of much consequence. Nor can we admit that the speaking-trumpet possesses any peculiar power of collecting sound in one direction; for it is distinctly audible on all sides, and is perhaps not much louder in front, comparatively, than the simple unassisted voice. The tube, by its length and narrowness, detains the efflux of air, and has the same effect as if it diminished the volatility of that fluid, or increased its density. The organs of articulation strike with concentrated force; and the pulses, so vigorously thus excited, are, from the reflected form of the aperture, finally enabled to escape, and to spread themselves along the atmosphere. To speak through a trumpet costs a very sensible effort, and soon fatigues and exhausts a person. This observation singularly confirms the justness of the theory which I have now brought forward."

Nearly about the same time, this theory was confirmed by some ingenious experiments made by M. Hassenfratz, at Paris. His method of estimating the power of a speaking-trumpet consisted in fixing a small watch in the mouth-piece, and observing at what distance the beats ceased to be distinctly audible. He found that the effects were precisely the same with a trumpet of tinned iron, whether used in its naked form, or after it was tightly bound with linen to prevent any vibration of the metal. Nor could there be the smallest reflection of sound from the internal surface of the tube, for the beating of the watch was heard exactly at the same distance after the whole of the inside had been lined with woollen cloth. These simple experiments prove decisively that the performance of the speaking-trumpet depends principally on the intenser pulsation which is excited in the column of confined air. In the same way, sound is prodigiously augmented in a long narrow passage. If a musket be fired within the gallery of a mine, the explosion heard in a remote corner will have the loudness and character of thunder.

The progressive motion of sound furnishes the explanation of various remarkable facts and striking phenomena. Thus, to a person standing at some distance, and directly in front of a long file of musketry, the general discharge will appear as a single collected sound, the numerous reports all reaching his ear nearly at the same instant. But one stationed at the end of the line will hear only a prolonged rolling noise, not unlike a running fire; because the distinct sounds, from the different distances which they have to travel, will arrive in a continued succession. Hence, likewise, the tremendous rumbling noise of distant thunder, which is not produced, as many have supposed, by the repetition of echoes. In certain situations, indeed, and particularly in hilly tracts, echoes may no doubt contribute to augment the general effect; but their ordinary influence seems to be really insignificant, since it should cause the same modification of sound in the explosion of a cannon, which is essentially different, however, from the muttering and crash of thunder. This lengthened and varied noise must yet be the production of a moment. The rapidity of lightning surpasses conception, and the prolongation of the sound which follows it is owing to the various distances of the chain of points which emit the sonorous impressions. The electrical influence darts with immeasurable swiftness from cloud to cloud, till perhaps it strikes at last into the ground. But from every point of this tortuous path distinct pulses of sound are transmitted, which consequently reach the ear at very different intervals. Sometimes they arrive intermingled, and give the sensation of a violent crash; at other times they seem suspended, and form a sort of pause. It would not be very difficult in any case to imagine the zig-zag track which the lightning must pursue in order to produce a given protracted rumbling noise. The duration of each peal of thunder will evidently be shortened if it chance to shoot athwart, but must continue the longest when it runs in the line of the spectator. As the distance of thunder is estimated by allowing somewhat more than a mile for every five seconds that elapse between the flash and the beginning of the report, so the space traversed by the lightning, if its general direction were known, might be computed by the same rule, from the endurance of the sound.

We will not enter at present on that branch of acoustics which treats of the doctrine of harmony; but a few scattered remarks may trace the general outline of the subject. A musical note, far from being only a repetition of the same simple sound, should be considered as the conjunction of subordinate sounds reiterated at proportional intervals. The sweetness of this compound effect or tone appears to depend on the frequent recurrence of interior union. The secondary sounds which naturally and invariably accompany the fundamental note are repeated only two, three, or four times faster; nor does the science of music admit of any proportions but what arise from the limited combinations of those very simple numbers. Harmony, again, is created by an artificial union of different notes, analogous to the natural composition of tone.

All tones are produced by the regular vibrations either of solid substances or of confined air itself. Strings of gut or of metal are most generally used; but small plates or pillars of wood, of glass, or even of stone, will answer the same purpose, forming the singular instrument called staccata or harmonica. In these cases, the quality of the vibrations depends on the joint influence of a variety of circumstances; not only on the length of the fibres, but on their thickness, their elasticity, their density, and the degree of tension to which they are subjected. The motion of a musical stretched chord was first investigated by the very ingenious Dr Brook Taylor, Acoustics, though his solution has been since proved to be incomplete. At the same time, in fact, that the whole chord oscillates, its simpler portions, the half, the third, and the fourth of its length, actually perform a set of intermediate vibrations.

Wind-instruments produce their effect by the vibrations of a column of air confined at one end, and either open or shut at the other. These vibrations are determined merely by the length of the sounding column. Yet interior and subordinate vibrations are found to co-exist with the fundamental one. The whole column spontaneously divides itself into portions equal to the half, the third, or the fourth of its longitudinal extent. We shall more easily conceive these longitudinal vibrations, by observing the contractions and expansions of a long and very elastic string, to the end of which a ball is attached. A spiral spring shows still better the repeated stretching and recoil. If struck suddenly at the one end, it will exhibit not only a total vibration, but likewise partial ones, winding vermicularly along the chain of elastic rings.

But when the air is struck with uncommon force, the subordinate vibrations become predominant, and yield the clearest and loudest tones. This we perceive in the dying sounds of a bell, which rise by one or two octaves, and expire in the shrillest note. On such a very narrow foundation—on the variable force with which it is blown—rests the whole performance of the bugle-horn, whose compass is extremely small, consisting only of the simplest notes. In other wind-instruments, the several notes are caused by the different lengths of the tube, or by the various positions of the holes made in its side.

The longitudinal vibrations of a column of air, contained within a tube open at both ends, are powerfully excited, and very loud and clear tones produced, by the inflammation of a streamlet of hydrogen gas. This curious experiment was made first in Germany, and appears indeed to have been scarcely known, or at least noticed in other countries. Yet it is most easily performed, and will be considered as amusing, if not instructive. A phial, having a long narrow glass pipe fitted to its neck, being partly filled with dilute sulphuric acid, a few bits of zinc are dropt into the liquid. As the decomposition of the water embodied with the acid now proceeds, the hydrogen gas thus generated flows regularly from the aperture, and is capable of catching fire, and of burning for some considerable time, with a small yet steady round flame. This very simple arrangement, frequently styled the philosophic lamp, is in reality of the same nature with the combination, on a large scale, of the gas lights. A glass tube being passed over the exit-pipe, the burning speck at its point instantly shoots into an elongated flame, and creates a continued sharp and brilliant musical sound. This effect is not owing to any vibrations of the tube itself, for it is nowise altered by tying a handkerchief tightly about the glass, or even by substituting a cylinder of paper. The tremor excited in the column of air is therefore the sole cause of the incessant tone, which only varies by a change in the place of the flame, or a partial obstruction applied at the end of the tube. But still it is not easy to conceive how the mere burning of a jet of hydrogen gas within the cavity should produce such powerful vibrations. The exciting force must necessarily act by starts, and not uniformly. The length of the flame might seem to prove, that the hydrogen gas is not consumed or converted into aqueous vapour as fast as it issues from the aperture. A jet of it catches instantaneous fire, but is immediately followed by another, the succession of inflamed portions being so rapid as entirely to escape the keenness of sight. The column of air contained within the tube would thus be agitated by a series of incessant strokes or sudden expansions.

The singular fact now described had occurred incidentally to the writer of this article in the course of his earliest experiments; and he has often thought since, that, on the same principle, an organ might be constructed, which would have a very curious and pleasing effect. A vertical motion of the glass tubes, and the partial shutting or opening of their upper ends, would occasion a considerable variety of notes. By passing the hydrogen gas over different metals, the flame would be made to assume various colours. The apparatus might work by a spontaneous mechanism; and while the eye was gratified by the display of rich and vivid tints, the ear would be charmed with strains of new and melodious symphony.

(P.)

PART III.

We shall here add some further explanations in regard to the nature of different sounds, and particularly of musical sounds.

Nothing appears more surprising than that variety of sounds which different bodies emit when excited either by percussion or by any other method. If we strike, for example, upon a log of wood, or a table, or a book, we obtain nothing but a harsh sort of noise, which ceases almost the moment it is emitted. Whatever shape we form the wood into, it makes hardly any difference. The same thing takes place with other bodies, such as metal, or glass, or earthen ware, &c. when these are in large masses; but how different is the case if we form them into rods or slips, into thin plates, or, still better, into cylindrical or hemispherical vessels, as cups, tumblers, bells, cymbals, &c. These, when struck, invariably emit a sound much more prolonged, and in general highly musical or grateful to the ear. What, then, is the cause of this remarkable distinction? All that we can observe generally in regard to these sonorous bodies when sounding is, that the whole body is agitated by an internal tremor, which continues so long as the sound can be heard. But if we examine them in their simplest forms, some very remarkable circumstances are brought to light. Take, for example, a slip of metal or glass, fix it firmly at one extremity, and then strike the other; the body, as in other cases, emits a prolonged and musical note, and becomes agitated with the usual sonorous tremor. This is particularly well observed in a common tuning fork. But the nature of this agitation is of a very simple kind. The slip merely oscillates backwards and forwards with great rapidity on the extremity by which it is fixed. Each of these oscillations, as the slip strikes the surrounding air, must produce a distinct sound; but these impressions following each other in very rapid succession, the ear can distinguish nothing but a single continued and prolonged note. The same thing takes place if we fix the slip at both ends, provided it be long enough and thin enough to oscillate in the middle; and this is the case with all kinds of distended wires or strings. These are fixed at the two ends, and when they are struck, as with the hammer of the key of the piano-forte, or drawn aside by the hand and then abandoned, they continue to vibrate for a long time, and to emit tones varying in gravity or acuteness, but all highly musical. One remarkable circumstance regarding the vibrations of these slips or chords is, that they are all performed in the very same time, however minute or however extended. They vary in different slips or chords, according to their length, tension, and other circumstances; but in the same slip or chord they are all alike. When any chord or slip is drawn out of the straight line and then abandoned, the oscillations are wide at first, but Acoustics continually diminish in extent till they cease altogether, as at Plate I. Fig. 1. Now the smallest of these vibrations takes just as long time in its performance as the largest; and the latter, again, however wide, is performed just as quickly as the former. The reason is, that the force of elasticity, which produces the vibrations, increases as the chord is drawn farther from the straight line, and thus accelerates the wide vibrations in proportion to their extent, as can easily be demonstrated. The consequence is, that the successive impressions of sound which constitute the continued and prolonged note, all follow each other at the same interval; and the regularity with which these impressions successively fall on the ear appears to be one essential ingredient in that melody or agreeable effect which the sound produces. This is proved by a very simple experiment. If we sound one of the strings of a violin or violoncello, it emits for a long time the musical note which belongs to it; but if, while it is sounding, we move the finger quickly along it from the bottom upwards, so as to shorten continually the length of the string, to accelerate thereby the vibrations, and thus alter the regularity of their succession, the sound continues, but all sort of melody is gone. Some philosophers, and in particular Dr Robison, have thought that this regular succession of sonorous impressions is the only source of melody; but this is far from being the case, else how could we account for those diversities in the sweetness of different tones, produced by different instruments or from different sources? What can be more regular in succession than the impressions of sound from a distant water-fall? And accordingly the effect is soft and agreeable in a high degree, yet how inferior in melody to the tone of a fine bell, or of an organ, or the notes of a distant bugle. Another remarkable circumstance regarding the vibrations of these slips or chords is, that the sound is not emitted from the string itself so much as from the mass with which it communicates. If, for example, we strike a tuning fork gently, so as to make it vibrate, it emits a sound so feeble that it cannot be heard until we bring it close up to the ear; but if, when it is vibrating, we press the extremity on a table, or a book, or any similar substance, the sound becomes perfectly audible. In the same manner a distended wire, if quite insulated, hardly emits an audible tone; but set it on a board, or, which is still better, on a hollow box, and then the tones are loud, deep, and highly musical. Many authors who have treated this subject ascribe the sound to the wire or chord striking the air at every vibration. This, however, is but an imperfect view of what really takes place. The tones arising from this source are in general perfectly inaudible, and the use of the wire is rather to excite the vibrations in the mass with which it is connected. This is the true sonorous body; and in this manner even wood, which appears when struck the most tuneless of all substances, is yet made to emit tones of the most exquisite sweetness; as appears in the violin and violoncello, harp, and indeed in all stringed instruments. The wood appears incapable of having a continued series of vibrations excited in it by percussion; but the strings produce this effect in a very remarkable manner, and which has hardly received that attention from writers on Acoustics which it deserves. They put the whole mass of the wood into a tremor, they excite its vibrations, and they determine and govern their frequency and the regularity of their succession. These vibrations over the whole surface of the wood communicate an infinite multitude of similar vibrations in the air, all which reaching the ear at the same instant, produce those powerful and audible tones which we observe.

Let us now consider the nature of those remarkable distinctions of musical sounds into grave and acute, on Acoustics, the combination of which depends the whole charm of music. Every one knows, that in stringed instruments Grave and acute this depends entirely on the length and tension of the string, as is well observed in the violin. By shortening the strings with the fingers, we obtain notes always more and more acute; and by tightening the strings with the pins at their extremities, we produce the same effect. In different strings, also, another circumstance has a sensible effect, namely, the thickness or mass of the string: the greater this is, the sound is the more grave, and the less the more acute. All these effects are very clearly observed, and proved with mathematical accuracy, by means of an instrument termed the monochord or sonometer; which is an apparatus contrived for stretching different strings and causing them to sound by vibration, with props or bridges to vary their lengths, and weights suspended to vary and measure the different tensions, the vibrations of the strings being communicated to a hollow box of thin elastic wood, to augment the sound. Fig. 2. represents the instrument in an upright form, which is best adapted for experiment with tension, because the weight can be applied directly under the string without friction. Fig. 3. is a horizontal form: the weight here is applied over a pulley, and therefore the tension cannot be so very accurately measured. It answers, however, better for some other kinds of experiments, in regard to the length of the strings and the subordinate vibrations. Whichever of these forms of the instrument be used, we obtain invariably the same results. The more weight we apply to stretch any of the chords, the more acute does the sound become. If we lengthen the strings again in any degree, the more grave does the sound become; and the same takes place if we substitute a heavy for a light string. It is very remarkable, however, that if we examine these effects carefully, and apply the aid of mathematical investigation, they all resolve into one general law. Adding to the weight and tension on the strings, shortening their length and diminishing their weight, have all the same common effect; namely, to quicken their vibrations. It is on the single circumstance, therefore, of the quickness or slowness of the vibrations of the strings, in whatever way it is produced, that the acuteness or gravity of the tones depends. This, then, is the great law on which all the gradations of musical notes depend: quicken the vibrations of the chord by any means, and we are sure to sharpen the tone in the very same proportion: make the vibrations slower in any way, and in the same proportion does the tone become more grave. The above circumstances, however, do not all alter the rate of vibration in the same degree. If we shorten or lengthen the string, the vibrations are quickened or slackened, and consequently the tones made more acute or grave in the very same proportion; that is, a string half the length of another just makes double the number of vibrations in the same time, and one one-third of the length three times; one double the length, again, makes half the number of vibrations, and one triple the length one-third of the number. The effect of the tension, again, and the weight of the strings, follows a different law. If we load one string with double the weight of another, it does not double the number of vibrations: it requires four times the weight to do this, and nine times the weight to triple the number; the quickness being in all cases in proportion to the square root of the weight. The effect of the weight of the string follows a similar law; one string requiring four times the mass of another to make its vibrations in half the time, and nine times this to make them in one-third of the time. So that if we denote the length of the string by L, the weight Acoustics. of a lineal inch by W, and the weight or force of tension by T, then the number of vibrations, N, in a given time will be in proportion to \( \frac{\sqrt{T}}{L\sqrt{W}} \).

On this law depends the construction of all kinds of stringed instruments, and the adjustment of the length, magnitude, and tension of the strings, to produce the tones that we wish in the instrument, without overstraining the materials, or making them any way liable to go out of tune. Accurate experiments, however, are wanting to apply with success the above formula to actual practice. On the above principle, also, of the velocity of the vibrations regulating the tone of every sound, depends entirely the system of our musical scale, as well as all those combinations of sound which constitute concord or harmony in music. Every one knows, that if with any sound we strike its octave at the same time, the two coalesce so completely together as to form almost a single continued note, which in music forms the most perfect concord. Now the octave above is emitted from that string whose vibrations are exactly double in number to the original note. The vibrations of the latter, therefore, must coincide completely with the former at every other vibration. The impression of both these reaches the ear at the very same instant, and thus the two sounds are blended into one. The intermediate sound of the octave which intervenes between these is hardly perceived, owing to the rapidity of their succession; and as it also recurs at regular intervals, it does not interrupt the harmony, but rather contributes to give fulness and richness to the combined tone. The same thing takes place, though in a less degree, with the double, the triple, and other octaves, and with all the different sounds which form concords together; and the more frequently the two sounds unite, the more perfect in every case is the concord. When they do not unite but at sensible intervals, there is then produced at each of these a distinct rise in the sound, which becomes suddenly louder from the union of the two, and then diminishes as they begin to separate, producing a succession of swells, or beats, as they are termed in music. The intervals between these beats are filled up by the vibrations of the two sounds occurring, first one and then the other; and as they form on the whole a very irregular succession of impressions, they produce a jarring effect, which grates on the ear. These combined sounds are hence termed discords. All this will be readily understood by an example or two, and a figure. Take first the fundamental tone and its octave; let the vibrations of each which produce the continued sound be represented by dots, as at Fig. 4; and when the two sounds concur, let this be denoted by dots larger in size: then the succession of impressions will be as at Fig. 5, where we have a regular succession of beats, recurring in such rapid succession that the ear can only distinguish a single sound. The octave also intervenes, but quite regularly, and also in such rapid succession, that it seems rather to heighten than to mar the general harmony. In the same manner, Fig. 6. represents the union of the fundamental sound and its double octave; which is the same system, only the beats are at wider intervals, and the intermediate octaves double in number. Take now two sounds, the number of whose vibrations are to each other as 2 to 3; that is, suppose one of them performs its vibrations in two very small parts of a second, and the other in three. If we take the annexed scale to represent the parts of a second, we easily form the series of sounds and of beats at Fig. 7. We have still, it will be seen, a regular succession of beats, but the intermediate single sounds do not recur at regular intervals.

We have first an interval of two parts of a second; then we have three sounds succeeding each other during the other three parts; and then an interval of two parts between that and the beat. There is obviously, therefore, a tendency to discord; but the whole series occurs in such rapid succession that the effect on the ear is quite harmonious. Take now the proportion of 5 to 7. Here, as at Fig. 8, we have a regular succession of beats, but we have no fewer than ten intermediate single sounds, and these succeeding each other in a very irregular manner. We have all the elements of discord, therefore; and accordingly the union of these two sounds would be sure to grate on the ear, unless they were of so high a pitch that the beats should become inaudible. These would then produce, as before, a continued sound; and the effect of the intermediate ones would be in some degree, but not altogether, lost: we might have a concord, but not a very perfect one. It hence appears obvious why the union of some notes, which on a grave key are discordant, become harmonious by raising sufficiently the pitch. In the same manner, in every other case the greater the intervals between the successive conjunctions of the sounds, and the more irregular the succession of the intermediate impression, the greater will be the discord.

Such are the effects of combined sounds; but it is extremely remarkable that every single sound, besides its fundamental note, yields also spontaneously various others of the most perfect concord with it, but all higher, and each rising successively in pitch, so that they have hence received the name of Acute Harmonics. This curious and important fact, first noticed by Galileo, to whom also we are indebted for the theory of musical strings, has since excited much attention among succeeding philosophers, as well as among those skilled in the practice of music. In stringed instruments, it is best observed in the low notes of the piano-forte or violoncello. These were formerly thought to emit one simple uniform tone. An attentive ear, however, as is now universally allowed by musicians, can actually discover three others besides the original, viz. the octave above, the twelfth, and the double octave. This was first distinctly proved by Rameau; and the experiment is easily tried by striking one of the low keys, and withdrawing the finger briskly: then, after the fundamental note has ceased, the three shriller ones will be heard. The octave being in some measure blended with the original note, is not so easily perceived, except by an ear habituated to the minutest discrimination of sound. But with ordinary attention the twelfth and the double octave are heard distinctly.

Nothing in the whole science of Acoustics has given rise to so much speculation and varied discussion as this singular property of musical strings. It would appear at first sight to overturn the whole doctrine we have been laying down regarding the effect of the rapidity of the vibrations in regulating the tone; for here we have the same string yielding at once a diversity of notes, varying in a wide gradation from grave to acute. Can it be possible that the string, besides one principal series of vibrations from end to end, may at the same time execute vibrations among its parts, throwing itself, as in Fig. 9, into two, three, or four subdivisions; and the whole string, as it oscillates between its extremities, serving as a movable axis on which such partial vibrations may be at the same time performed? This appeared at first so extraordinary and wild a conjecture, that it was long ere it could meet with any serious attention among philosophers, far less be admitted as an accurate view of what really takes place. It was rather ascribed to something unconnected with the string; some referring the production of these harmonic Acoustics notes to the structure of the ear, and the mode in which it receives the impression of the fundamental note; others, as the celebrated Lagrange, (to whom, along with our countryman Taylor, and Bernoulli, Euler, and D'Alembert, we owe the whole theory of the curves assumed by the vibration of musical strings,) supposed that they arose from sympathetic vibrations excited in the different bodies adjacent to the string, but without attempting to give any reason why these affections should always excite notes more acute than the fundamental. The possibility of these partial vibrations, however, was clearly proved by Daniel Bernoulli; and, however curious it may seem, the fact has now been established by many conclusive experiments and observations. That strings are capable of vibrating in this way, we may be convinced, from the following considerations. If we take any string, and withdraw it from the straight line by applying the finger, not to the centre, but nearer to one of the ends, so as to bring it into the position A B C, Fig. 10, then it will on the other side assume the position A D C, in every way the reverse of the first, as is easily proved by experiment. In general, the vibrations are so quick that they cannot be perceived by the eye; but if the finger be held opposite to B, or in any other part between A and C, it will be found that the vibration is nowhere so great as at D. In the same manner, if we apply both fingers in opposite directions, and at equal distances from the extremities, so as to draw the string into the position A B D C, the centre E remaining unchanged, then, on the other side, it will assume the position A b E d C; and as the central point E continues all the time immovable, it is evident that the whole string will continue vibrating partially, in the very same manner as if it had been composed of two strings, each half the length, and fixed at E. This point E is hence generally termed a node, or nodal point. In the same way, by applying the moving forces at other different points, the string may be made to perform three, four, or any greater number of partial vibrations, in relation to so many intermediate nodal points, Fig. 12; and it is not difficult to conceive, therefore, how, by the application of certain forces, the whole string may be performing its oscillations from end to end, while at the same time it may be agitated internally by two, three, or more partial vibrations; and this is proved by actual experiment. If we take a piece of common twine about fifteen or twenty feet long, and stretch it between two fixed points rather loosely—if we then cause it to vibrate by the middle, it merely makes the ordinary vibrations; but if we apply our force to withdraw it from the straight line near one of the extremities, we can then see distinctly, besides its principal vibrations, a series of subordinate ones going on throughout its whole length. It is remarkable also that strings of this kind are extremely susceptible of these subordinate vibrations. When the string is vibrating regularly between its extremities, the slightest deranging circumstance is sufficient to excite and to superinduce the partial actions; as appears very distinctly in an experiment contrived by Mr Hawkins of London, by stretching between two bridges a long and spirally coiled brass wire, the spirals being about the diameter of a quill, and extended considerably more than those of a cork-screw. The tension was such as hardly to emit any sound, but to leave the vibrations when touched quite visible to the eye. If, when the whole string is vibrating, we oppose a slight obstacle, say at the middle, the string then instantly divides itself into two, and along with its principal vibration performs two others between the centre and each extremity. If the obstacle—and even a puff of wind from the mouth is sufficient—be placed at the distance of one-fourth the length of the string from either extremity, then it divides itself into four, and along Acoustics, with its principal vibration performs four others; and so on, according to whatever aliquot part of the string the obstacle be presented; the latter always subdividing itself into the same number of parts, and performing a series of subordinate vibrations between each, which can often be observed and heard for several minutes accompanying the principal one. The same effects are beautifully observed, as was first shown by Professor Robison, by means of a monochord sounded by an ivory wheel. When the string was vibrating simply, if its middle point was then touched slightly with a quill, this point instantly stopped, but the string continued to vibrate in two parts, sounding the octave; and the same thing happened if it was touched at one-third,—it then divided itself into three parts, with two nodal points, and sounded the twelfth; and any thing soft, such as a lock of cotton, put in the way of the wide vibrations of the string, was sufficient to produce the effect.

From these facts and observations, therefore, no doubt can remain that the acute harmonics accompanying the fundamental note arise entirely from these partial vibrations, of which every string appears to be so susceptible. But one important question still remains, and has never yet received any satisfactory solution; namely, what are the circumstances which determine these partial vibrations so generally in every string? Very slight obstacles are no doubt sufficient, but still there must be some particular causes to determine every string to divide itself so regularly into two, three, four, or more parts; executing in such regular succession the vibration peculiar to each. Inequalities in the densities or thicknesses of the different parts of the strings have been suggested as a probable reason; but this appears quite inadequate to produce effects so very constant in their operation, and which besides occur, let the string or wire be ever so uniform in its texture. This subject requires further examination, and the trial of several experiments. The only probable circumstance to which we can ascribe these effects consists in the re-action of the supports to which the extremities of the strings are attached. It is well known, that when two strings in the same instrument are tuned to the same note, if one of them be struck, the other begins to vibrate, and to sound at the same time. Strings differing by an octave have the same effect; and even those differing more in tone, yet appear to act and to re-act on each other. These effects have usually been ascribed to the action of the air, communicating the vibrations from one string to another. This, however, is not the case, as is proved by a simple experiment. If we stretch two similar strings on two different boards, and tune them to the same note; then if we detach the one wholly from the other, the vibrations of the one do not in the least affect those of the other, let them be even brought quite close together. But the moment we place the boards in contact with each other, though the strings themselves be thereby farther separated than before, yet the one vibrates uniformly in sympathy with the other; and the moment the boards are again detached, the effect ceases. It is clear, therefore, that the vibrations of the one string are communicated to those of the other, not through the medium of the air, but through that of the wood; and this can only take place by means of the vibrations of the one string communicating through the supports to the wood. This agitates the supports of the other string, and these giving at first a very slight vibrating motion to the extremities, this gradually increases as it is continued, until the whole string is made to vibrate with the other. Instead of the effect, therefore, proceeding from the middle of the string to either extremity, as would be the case with aerial pulses, it is just the reverse. When once the motion is begun, the aerial undulations may certainly aid and augment the effect; but this, without doubt, originates at the extremities of the strings, and from these is communicated to the centre. That strings in unison with each other should vibrate in this manner more readily than any other, is quite easily accounted for; because in that case the vibrations of the one, however they may be produced, are quite isochronous with those of the other: they are more readily excited, therefore, because when they are once begun, however minute they may be at first, they yet harmonize exactly with the other, and every succeeding impulse from it just arrives in the proper time to augment the effect of the preceding, until the two motions become exactly alike. Were the string not isochronous, the succeeding impulses might tend sometimes to augment and sometimes to diminish the effect of the preceding ones; so that no sensible vibration would take place, as really happens with all discordant strings. With octaves, however, and other concords, the vibrations are still produced, because the vibrations of the fundamental note always concur with those at certain intervals, and at others do not oppose them. All these effects are very similar to what occurs in the motion of pendulums; a very slight impulse is sufficient to put a large pendulum into very wide vibrations, if it be often repeated, and always at the proper time to augment and not oppose the motion already communicated. Here it is that pendulums swinging in the same frame affect wonderfully the vibrations of each other, exactly like musical strings in the same instrument. If when two isochronous pendulums are at rest one of them be set in motion, in a short time the other commences to vibrate, in a very slight degree at first, but always increasing and increasing, until at last its oscillations become quite as wide as the other. This effect arises entirely from the vibrations of the one pendulum communicating through the wood very slight impulses to the axis or centre on which the other moves. This sets it in motion, and the impulses being constantly repeated and properly timed, excite at last vibrations as wide as the original. In a similar manner it is that one string causes another in the same instrument to vibrate, by means of the supports on which it is stretched; and if this be the case so remarkably in different strings, may not the vibrations of any single string re-act on itself through its own supports, and thus excite all those secondary vibrations which produce the acute harmonics? The string being first struck in the centre, the fundamental note must first of all be emitted. But the vibration of this agitating the supports at each extremity, these must necessarily re-act upon the string itself, must modify the original central impulse, and excite all those subordinate vibrations which are observed. That this is the true cause of these partial actions, and consequently of all those varied tones which they excite, appears extremely probable. But experiments, as we observed, are wanting to demonstrate these effects satisfactorily.

Such, then, are the singular phenomena of the acute harmonics, and of the subordinate vibrations of strings from whence they arise. Whatever be the cause of these, no doubt whatever can remain of the fact itself; and it is to this circumstance that we must perhaps ascribe much of that agreeable effect, so pleasing to the ear, which we experience in the tones of our musical instruments. Regularity of succession may determine a certain pitch in the sound; but it is from the harmonic tones mingling together, and blending with the gravity of the original note, that appears to arise all that sweetness and rich melody which so peculiarly belongs to them. Hence also we can now understand the reason of the musical effect arising from the sound of thin plates, cups, or bells; in all these, when they are examined minutely, we observe numerous vibrations among the parts; the substance of the plate dividing itself into various subdivisions, separated by nodal lines, between which all the parts are thrown into vibration. Hence arise a variety of harmonious notes; and these mingling together, produce that highly musical effect which we observe. It is to Professor Chladni that we owe a variety of curious facts regarding these vibrations of plates; which he observed by strewing them with fine sand: the nodal points and lines were then shown by the sand throwing itself from all the vibrating parts, and accumulating in little heaps along the lines which divide the vibrating surface, and where, therefore, there was no agitation at all. Squares of plate glass are well adapted for showing these effects. If we hold these by the centre in a sort of small wooden vice, as at Fig. 13, and then cause them to sound by drawing a bow along any of the edges, which are smoothed for the purpose with emery, the gravest tone is produced by applying the bow to one of the angles; and then the sand assumes the figure represented in Fig. 14. The tone next to this in graveness is produced by applying the bow to the middle of one of the sides; and then the sand assumes the figure represented in Fig. 15. By varying the points of application in this manner, and the figure of the plates, we obtain many other figures and curves for the nodal lines; such as Fig. 16, 17, 18, 19, &c. These effects are extremely curious and interesting: they are also very beautifully exhibited by plates of paper or other flexible membranes, stretched on frames like those of a drum or tambourine. But we must refer for further details on this subject to the Traité d'Acoutique of Chladni.

Besides the lateral vibrations above described, of strings, membranes, and plates, all these bodies are capable of vibrating longitudinally, and of emitting sounds which are in general much more acute than the others, and are regulated in their pitch by the length of the rods or strings. In this manner even the air itself, confined in a tube or pipe, is made to vibrate regularly, and to emit musical tones; and from this source arises another extensive and very important class of sounds, namely, those which are produced by the various kinds of wind-instruments, and of which also the tones of the human voice and of that of different animals present interesting examples. One is apt to imagine that the sound of a pipe or flute is emitted from the wood or other material of which it is composed; it really, however, arises from the cylinder of inclosed air, which is thrown into vibration by the current of air striking against the reed of the pipe, or against the sides of the hole in the plate. For a particular account of this class of sounds, see Wind-Instruments, Organ, Trumpet, &c.; and for a more particular account of the scale, and of musical harmony, see Harmonics, Music, Temperament.