Magazine.

{ONE PENNY.

pulse is communicated from particle to particle in the same way as if a stretched cord were employed.

As the atmosphere consists of particles of air readily acted upon by each other, the impulse travels onward with great rapidity; but the precise measure of this rapidity has been a very difficult inquiry in the hands of philosophers. There are two circumstances which regulate the velocity of this propagation, viz., the specific gravity of the air, or the weight of matter included within a given bulk, and the elasticity or resistance to compression. It was long ago determined theoretically, that as these two conditions or properties differ in different substances, the propagation of a sonorous impulse should be effected with different velocities in different cases; and this has been found true in experiment.

Sir Isaac Newton, from calculations in which the specific gravity and the elasticity of the air were taken as elements, stated the velocity of sound to be 946 feet per second; by which is to be understood that the disturbance of particles spreads so rapidly as to extend to a distance of 946 feet in one second of time. This assumed velocity, however, was found to be smaller than that deduced from carefully conducted experiments. From investigations made by Derham, Pictet, and other philosophers, at different hours of the day, and in different latitudes, it was found that sound travels at the rate of about 1130 feet per second. This discrepancy between theory and observation threw much obscurity over the subject, until Laplace brought his powerful mind to the consideration of its probable cause. It has been demonstrated that a change in the pressure of the air would not produce any change in the velocity of sound, because the more the density of the air is increased, the greater does the elasticity become; so that in rarefied air and in condensed air, provided temperature could remain the same, the velo641

The presence of aqueous particles in the air has a tendency to increase the velocity of sound, as an impulse is communicated more rapidly in a watery than in an aërial medium. It has been found that in the moist sultry plains of Cayenne the velocity was 1175 feet per second, the augmentation being due conjointly to the greater heat and greater moisture of the climate. On the other hand, some experiments made on the dry and elevated plains of Quito gave a lower velocity of vibration than that obtained in Europe.

Such was the principle adopted in the present inquiry, Time-keepers were furnished on such a construction as to work only so long as the finger pressed upon a certain spring, so that when one of the observers saw the distant flash, he instantly pressed the spring and allowed the index hand to move; and when the report was heard, the finger was removed from the spring, and the index stopped; the number of revolutions of the index indicated the time that had elapsed between the flash and the report, which at once measured the velocity of the sound.

At each station were placed, besides the clocks here spoken of, a barometer to determine the density of the air, a hygrometer to measure its humidity, a thermometer for the determination of temperature, a wind-gauge to show the direction of the wind, and a telescope to view the distant station. At each station were also planted a twelve-pounder, and a six-pounder; and tents were provided for the experimerters and their assistants.

In order that the guns at both stations should be fired at the same instant of time, the following preparatory arrange ments were made. At 755 P. M. by the chronometer of Zevenboompjes, (one of the stations,) a rocket was fired at that station, which, on being observed at the other station, was immediately answered by a second rocket. This was the signal that on both stations everthing was ready. At 8h0 by the chronometer of Zevenboompjes a cannon was fired at that station, whilst the observers at Kooltjesberg noticed with great exactness the time on their chronometer when the flash was seen. This was repeated at 8h 5', so as to enable the observers to determine exactly the difference of their chronometers, due to difference of longitude. By this determination they were able afterwards to fire their guns at precisely the same instant of time; the object being to have a double series of observation going on when the air was in the same condition as to temperature, &c. The exact correspondence in the firing of the guns was obtained in the following manner. At each station, an officer had the chronometer placed before him on a small table very near the gun; a non-commissioned officer or gentlemancadet stood ready with the port-fire, near the touch-hole, and at the instant required, the officer holding the chronometer pressed the arm of the person who was to fire the gun, which went off at the very moment. With a little practice they were certain to fire the gun at any given second. For several successive nights the experimenters were deprived of some of the desired results, in consequence

That sound travels with different velocity through diffe rent media has been abundantly ascertained. In some experiments performed by Biot on the water-pipes of Paris, when a sound was communicated at one end of a long tube filled with air, two sounds were heard at the other end, one having travelled through the mass of metal forming the tube, and the other through the contained air; the former having travelled more rapidly than the latter. If a blow be struck at one end of a brick wall, and a person apply his ear close to the wall at the other end, he will hear a duplication of the blow, first through the substance of the wall, and then through the surrounding air. It has been observed that when an earthquake has occurred near the sea-shore, the waves on the surface of the ocean, caused by the disruption, have been observed at a distant station in less than one-fourth of the time that a sound (such as a volcanic eruption) would have taken to travel that distance through the air. Sound travels through ice with about the same velocity as through water; and it is usual among the savage hordes who inhabit the sterile steppes of Tartary, where the intervention of lakes and the frequently frozen state of the ground give an icy character to the region, to place the ear to the ground on the expected approach of an enemy thereby enabling them to hear more quickly, and to esti mate more correctly the distance and the direction of any sound that might be sensible.

It is desirable now to offer a brief explanation respecting the state of the particles of air during the propagation of sound. When we say that sound travels 1125 feet in second, it is not to be understood that any of the particles actually move to that distance; their individual movement is confined within extremely narrow limits, but the effect of this movement is propagated from particle to particle with the rapidity here indicated. As soon as the particles first disturbed have moved to such a distance as their elasticity will allow them, they return to their former position, and acquire in so doing a momentum sufficient to carry them to a short distance in the opposite direction; and thus they acquire an oscillatory or vibrating movement. Each particle receives its disturbing force later than the one preceding it; and thus the particles are in different stages of movement, some moving onwards, while others are moving backwards, the two sets being separated by particles occupying their original positions. This has given rise to the term, ware of

sound, a wave being understood to include particles in all the various stages of vibration, each wave being exactly similar to all the others. If we know the number of times that each particle vibrates to and fro in a second, we can ascertain the length of the wave of sound; for by dividing 1125 feet by the former we obtain the latter. The successive waves of sound have been beautifully illustrated by the analogy of a waving field of corn. Supposing the stems of corn to be vertically situated and to be equidistant, if a blast of wind were to blow over the corn-field, it would first influence those stalks nearest to the point from whence the wind comes, and they would be bent in the contrary direction. But by the time the impulse had been communicated to other stalks a few feet distant, those first disturbed would rear their heads again; and not only so, but they would acquire a momentum which would carry them to the other side of the vertical line. The resulting effect may be illustrated by the annexed cut, in which the points from a to b represent the ears of corn congregated more closely than in their quiescent position, while at other points they would be separated more widely asunder. If we suppose the elasticity of the stalks to be equable and uniform throughout, the distance from a to b, or a wave of undulation, would depend on the length of the stalk, and not on the violence of the wind; the principal circumstance which determines it being the rapidity with which the stalk would return to its quiescent position after disturbance.

When speaking of the distance to which a sonorous impulse is propagated in a second of time, we confined our remarks to a row of particles situated in a straight line; but in practice this limitation cannot be maintained. Any impulse which tends to urge one particle of an elastic fluid in a certain direction, cannot do so without exerting a lateral pressure on the surrounding particles; for the equalising tendency of elasticity renders impossible the forward motion of particles without at the same time disturbing the equilibrium of those situated laterally.

We shall attain greater accuracy if we suppose sound to be propagated spherically round the vibrating body. When an explosion takes place, the waves form a series of concentric hemispheres, since the surface of the earth may be considered as bisecting the spheres into which the waves would otherwise be formed; and in the absence of wind, we have every reason to believe that the sound is equally loud and travels equally far in every direction. But if we vibrate a string or a flat surface, we have no reason to expect the sound to be equally loud in every direction. If, for instance, a tuningfark be vibrated, and held so that the plane of vibration shall pass through the ear, the percussion on the tympanum, or the loudness of the sound, (which are analogous expressions,) is greater than when the fork is held at right angles to that position. We may illustrate this by the annexed figure. Let ab represent the two prongs or arins of the tuning-fork, with their ends presented towards the eye; and let the concentric segments represent the circles or rather spheres of

undula which are in constant process of generation from the central point. The employment of segments of circles indicates that the velocity is the same in every direction; but the thickening of the segments from a to c and from & to d may be taken to indicate the greater loudness or intensity of the impulse at those points from a b to e, or from a b to f, on account of the vibrations of the parallel bars being performed in the direction a 6 toc, and ab to d. If the plane of vibration of a string were constant in its position, it is probable that we should find a different intensity of the sound elicited from it, according to the position in which the ear

were placed with respect to that plane; but it is almost impracticable so to regulate the impulse given to the string as to confine its vibrations to one plane.

The circumstance which determines the distance to which sound can be propagated, is the intensity of the first impulse; and whatever that intensity may be, the effect upon the ear decreases as the square of the distance increases. But this rule remains permanent only when the sound is capable of a spherical propagation. If it be interrupted by any solid obstacle, those waves which would otherwise proceed in a straight line from the sounding body, are diverted from their original direction: they are reflected or carried along the surface of the obstacle, and extend to a considerable distance with very little loss of intensity.

A series of valuable and curious experiments on the effect of tubes in aiding the audibility of sound at a great distance were performed by M. Biot some years ago. He availed himself of the opportunity afforded by a combination of cast-iron pipes which had been newly laid down for the purpose of supplying the city of Paris with water. These pipes formed a continuous canal of equal internal diameter throughout, and had two flexures about the middle of its length. M. Biot, after stating that the pipes extended to a distance of 951 metres (about 3120 English feet), proceeds thus:-"Sounds uttered in the lowest tone of voice were heard at this distance, so as not only to enable me to distinguish the words, but to keep up a lengthened conversation. I wished to determine the lowest tone at which the voice ceased to be heard, but I was unable to do so: words spoken as low as when we whisper in another's ear were heard and appreciated; so that in order not to be heard there was positively but one means, which was not to speak at all. Between the asking and the answering of a question in this manner there elapsed 58 seconds; this then was the space of time that the sonorous impulse occupied in travelling twice the length of the column of air, that is, 951 × 2=1902 metres (6240 feet, about a mile and a quarter). In order to ascertain whether grave and acute sounds, or strong and weak sounds, were propagated with equal rapidity, or if there were under such circumstances any difference between them, I caused some airs to be played upon a flute at one of the extreme ends of the pipes. It is known that a musical piece is generally subject to a certain 'measure' which regulates very exactly the interval between the succeeding sounds. Consequently if some of the notes were propagated either more rapidly or more slowly than others, they would have arrived at my ear in a confused manner, the various tones mingling with those which preceded or followed them in the order of the tune, which, thus heard, would have been entirely changed in character. But instead of that, it was per fectly regulated in strict conformity with its natural time. Whence it results that all sounds are propagated with equal velocity." Biot states that in order to perform this experiment successfully, he found it necessary to choose the calmest period of the night, to avoid the interruption which the busy hum of day would have occasioned. He also fired a pistol at one end of the tube, and says that the report "occasioned a considerable explosion at the other end, when the shock arrived there. The air was driven from the last pipe, with sufficient force to produce upon the hand, the sensation of an impetuous wind, to hurl light bodies placed in its path, to a distance of more than half a metre, and to extinguish lighted tapers, although we were at the distance of 951 metres from the spot at which the pistol had been fired two seconds and a half before."

2. MUSICAL SOUNDS.

The difference between a noise and a musical sound, and

the diversities of the latter, are the points to which we next direct our attention.

A noise is the impression produced on the ear when the vibrations of the sounding body are irregular, whereas a musical sound owes its beauty to the regular and equaltimed vibrations of the body which yields it. If a body be struck and it produces a dull or a grating sound, we may generally conclude that it possesses little elasticity, or at least that it is not fitted for the performance of vibrations. Such is the case with a beam of wood, a clod of earth, a mass of stone, &c. But when, either from the nature of 641-2

the substance itself, or from the form into which it is worked, a body is capable of vibrating regularly, then the emitted sounds partake of that character which we are accustomed to call "musical."

Moreover it is found that when two bodies vibrate regularly, but one more rapidly than the other, the former yields a higher tone than the latter. An experiment_made by Savart, a learned Parisian philosopher recently lost to science, will enable us to illustrate these points. He caused a metallic wheel to be made with 360 equidistant teeth cut in its edge; and held a piece of card in such a position as to be struck by every tooth in succession as the wheel revolved. Now the slip of card, not being of elastic material, was not calculated to vibrate either continuously or equably after percussion, and therefore a musical sound could not result from any one single blow. But if the wheel revolved so fast as to cause a large number of blows per second to be given to the card, then a musical sound was heard, not due to the vibrations of the card itself, but to those of the air moved by it at each blow. When the wheel was turned so that six or eight teeth struck the card in a second of time, nothing but a succession of slight blows was heard; but when the rapidity increased to sixteen, a low humming sound was distinguishable: the individual sounds lost their separate identity (so far as the sense of hearing was concerned), and became blended in a continual hum or low musical note. When the velocity of rotation was increased and the rapidity of contact between the teeth and the card thereby augmented, the sound, still continuing musical, was heightened in pitch, becoming more and more acute as the rapidity increased. Savart was thus enabled to attain a rapidity of 4000 strokes of the card in a second; and by a different form of apparatus he reached so high a number of 24,000 vibrations, which yielded a note of extraordinary

acuteness.

If we inquire why the noise produced by the avocation of the itinerant knife-grinder is an unmusical one, it will not be difficult to show that it must result from that inequality of vibration before alluded to. The small elevafions which cause the roughness of his grindstone are situated at irregular distances one from another; and the sounds resulting from the friction between those elevations and the blade of steel which he applies to the wheel, do not follow each other at exactly equal intervals. It is probable that if his wheel were cut across its edge with regular groves like a file, the sound would be somewhat musical, so long as the grooves retained their figure. We may similarly account for the grating, unpleasant noise of a saw, by considering the very unequal rapidity with which the arm moves. This unequal motion gives an irregularity to the rapidity of the concussions between the teeth of the saw and the object against which they strike.

We thus see that equability in repetition is the agent to which we are to look as the regulator of musical sounds, and we choose the materials of our musical instruments with especial reference to their vibrating quality. A glass goblet and a porcelain vessel of the same shape yield sounds different in musical character, not because one is harder than the other, but because one is more elastic and performs its vibrations more equably. The cords of stringed instruments are formed either of metallic wires, or of an animal substance which regains its primitive position after disturbance, with great regularity and intensity. Cymbals and bells are formed of a metal whose vibrations are both extensive in range and regular in recurrence. Wind instruments, which depend for their action on the longitudinal vibration of a column of air, derive their musical sounds from the circumstance that air is perfectly elastic, and thus performs all its vibrations in equal times. We thus arrive, by various paths, at the conclusion before stated, that all our musical instruments are merely contrivances by which the circumambient air may be set into equable and rapid vibration.

We have stated that if two musical notes be due to different velocities of vibration, the higher velocity produces the more acute tone. We have now to speak of a third sound resulting from the simultaneous production of the other two. When we put a string into vibration, every time it passes the axis or position of rest it gives one impulse to the surrounding air and to the ear. Suppose this to occur 100 times in a second, and that we take another string whose vibrations amount to 200 in a second; then if we vibrate both together, we cause two separate systems of undulations to be propagated, the one with a rapidity of 200, and the other of 100 in a second. Every second vibra

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tion of the former will coincide with one of the latter, and an augmented impulse will be the consequence. This augmentation of intensity, occurring at intervals of 1 of a second, furnishes a new element, as if a third note were actually produced, equal in pitch to the lower of the two original notes.

But this will be rendered more clear if one of the strings vibrates 100 and the other 150. In such case we shall find every second vibration of the one coinciding in direction and in ultimate effect with every third vibration of the other; these coincidences would thus occur 50 times in a second, and would thus form a third series, springing from, but independent of, the other two.

It is from this source that many writers on Harmony trace the pleasure which results from musical sounds, that is, from the production of a third sound from two others heard together. We are aware that other theories have been offered in relation to this matter; but without entering on the controversies which the subject has excited, we shall offer such details as will serve to convey a few general notions on the matter.

If the performers on two musical instruments produce two dissimilar tones, or if a performer places his fingers on two keys of the pianoforte at the same time, and the united sound be pleasant to the ear, we shall always find that the rapidity of vibration producing those tones has such a ratio, that a third series, resulting from the coinciding intensities of the other two, will have a considerable rapidity of repetition. If the two series are equal in rapidity, the musical effect is what is called unison, which is the most perfect agreement of tone. If the velocity be as one to two, the latter is the "octave" to the former, the next approximation to unison. If they are as two to three, the one is (in musical language) a "fifth" to the other. If, on the contrary, we take any more complex ratio, such as eleven to seven, we should find that no musical interval is represented thereby. In what way this bears on the principles of harmony we shall endeavour to explain.

Every one is familiar with the expression "a musical tone" or "semitone," and there are but few individuals so entirely destitute of musical feeling as to be unable to produce a few notes of an air or song with an approximation to correctness. Now it is found that the notes sung or played or hummed by any number of persons, bear, within a comparatively small range, a close analogy to each, as regards the intervals between the notes; and this tacit agreement among both learned and unlearned well deserves our consideration.

In order to give precision to our remarks we will assume that a certain note, called C (DO) for instance, is due to 240 vibrations per second. In the standard tuning-forks used at Mr. Hullah's Singing Classes at Exeter Hall, the C or DO is taken at 256, 512, &c., vibrations per second; but our assumption is taken merely for the sake of avoiding fractions, and is not to be considered in any other light. If a tone, resulting from 480 vibrations per second, be heard at the same time as our assumed fundamental tone, the agreement is so perfect and satisfactory to the ear, and the new tone bears such a close resemblance to the fundamental tone, that it has been universally agreed to designate it by the same letter of the alphabet. We have then two tones (C and c) depending respectively on 240 and 480 vibrations per second; and the pleasure derivable from them has been attributed to the production of a third tone, called the "grave harmonic," resulting from an augmentation of tone repeated 240 times in a second. If we next estimate what ratio between the vibrations will produce a harmonic of the next greatest frequency, we shall find it to be the ratio of 3 to 2. 360 bears to 240 the same ratio as 3 to 2; and if heard with the latter, produces the very agreeable interval of the fifth. This note also, viz., that resulting from 360, if heard at the same time as the upper note 480, to which it bears the ratio of 3 to 4, gives the musical interval, almost as agreeable as the former, called the "fourth." In the same way the ratio of 5 to 4, a very simple one, gives a harmony very pleasing to the ear; and we find accordingly that 300 and 240 will yield this ratio. We thus obtain four notes, which may be represented thus:

C

DO 240

E MI 300

G

с

DO

480

all of which are connected by very simple ratios, viz., C and c=1 to 2, C and G=2 to 3, Cand E-4 to 5, E and c=5 to 8, E and G=5 to 6, G and c 3 to 4. These four notes produce

a beautiful effect when sounded together, and then form what, in musical language, is called the "common chord." An attentive ear is conscious of hearing many sounds besides those properly belonging to the four notes; these are the "grave harmonics," resulting from the frequent coincidences of the vibration of different notes, when the ratios between them are simple.

But the vacancies between these four notes are too wide for the purposes of music; and it has been the custom, in almost every country, to insert three others, so as to form a gradually ascending scale of sounds. The manner in which this has been effected is the following :— It will be seen that three of the above notes, C, E, G, have vibrations in the ratio 4, 5, and 6; this relation has been called a "triad," and two other similar triads may be so introduced as to fill up the vacancies in the scale. Let us make c the highest note of such a triad, and G the lowest note of another such; these will give respectively 320, 400, 480, and 360, 450, 540; each pair of which gives numbers in the ratio 4, 5, 6. As every note receives the same name as the note resulting from exactly half the number of vibrations, on account of the similarity of sound, we may take, instead of 540, its half 270; and then we can form our scale thus, affixing letters to the notes produced:

C D

E

F

G

RE

MI

FA

DO 240

320

B

C

DO

480

A SOL LA SI 270 300 360 400 450 This forms the usual "diatonic scale" of music, of which the intervals appear so natural, that if an unpractised person be required to utter an ascending series of tones, he is found, involuntarily and in utter ignorance of the fact himself, to produce a succession of sounds the result of a series of vibrations very nearly in the ratio here given. As far as we are justified in referring this circumstance to its source, it would seem that this natural series produces a system of grave harmonics," so rich and full as to appeal powerfully to the ear.

"

It is customary to distinguish between melody and harmony, on the ground that melody is a succession of single sounds, while harmony is a combination of sounds heard at one time. But, however correct this distinction may be, it is worthy of remark that every melody is made up of sounds so chosen with regard to each other, that different combinations of twos, threes, and fours, chosen from among them, produce those beautiful effects which we designate harmony. Melody and harmony, therefore, however different, derive their charm from the same common source, viz., a certain simplicity of numbers among the vibrations producing musical notes.

There is another point of view in which the ratios between the different notes of the musical scale may be considered, which is also convenient as enabling us to speak of the comparative lengths of a string which produce the eight sounds of the octave. It was stated in a former page, that a string, in order to vibrate with twice the velocity of another string, must be one half the length; to vibrate with thrice the velocity, it must be one-third of the length, and so on. It follows from this, that whatever ratio exists between the numbers of vibration producing any two given tones, the comparative lengths of two strings which produce those tones are in the inverse ratio. If we reduce the list of numbers, 240, 270, 300, 320, 360, 400, 450, and 480 to a similar series of which the lowest = 1, then the whole will be 1, į, į, į, į, §, 13, 2; and if we invert the divisors and dividends of the fractions we obtain 1,,,, i, i, ft, i. When, therefore, we have a string which yields the note C, due, as we are here supposing, to 240 vibrations per second, if it be shortened to of its full length, it will yield the next note D; if to, the note E; and so on.

The whole of the details which have recently engaged our attention, relate to the tones of one octave only; but we shall find this quite sufficient for our purpose, as every successive octave is formed precisely in the same way. To ascertain the number of vibrations producing the tones in the next higher octave, we have only to double the numbers attached to the corresponding letter in the lower octave; for the second higher octave we take four times the number, and

80 on.

We assumed as a convenient number, 240 vibrations per second for the note C; but we also stated that 256 is the number now fixed upon in practice. On this latter point it may be well to offer a few words of explanation. It has been deemed convenient to call the effect produced by one vibration per second (although perfectly inaudible) by the letter C, and to apply the same title to all the duplications

of that number, that is, to 2, 4, 8, 16, 32, &c., vibrations per second. It is found that, in following this series, the fifth C is, under ordinary circumstances, about the lowest sound which can be heard, and that the middle C of a pianoforte is the ninth in the series, due to 256 vibrations per second. This was the standard recommended by the late Dr. Thomas Young, and is one possessing many advantages; but in practice there has been great diversity. Experiments made at different times have shown that the middle C of the piano-forte has varied from 238 to 264 vibrations per second, in different orchestras and at different periods. To remove the obvious inconveniences resulting from this state of things, a standard has been recently determined whereby the middle C of the pianoforte shall be produced by 256 vibrations per second, the number recommended by Dr. Young. The determination was, in the first instance, made with a view to the preparation of tuning-forks for the Singing Classes conducted by Mr. Hullah, at Exeter Hall, under the sanction of the Committee of Council on Education, the C selected being an octave above that just named, and being adjusted by scientific means to 512 vibrations per second; but the standard, once attained, is calculated for general use, both in instrumental and in vocal music.

The highest note used in music is about the 14th C (five octaves above the middle C of the piano-forte), due to about 8000 vibrations per second; but much higher tones can still be heard, although their very acuteness deprives them of all value for the purposes of music.

The earlier details of this section have given a sufficient idea of the nature of the "grave harmonic," and of the probable mode in which this harmonic heightens the effect of musical combinations. But the term harmonic has likewise been applied to tones produced under very different circumstances; tones which, from being always higher than the fundamental tones whereon their production depends, are called "acute harmonics."

If we draw a bow across the string of a violoncello about midway between its two extremities, we draw the central part farther from the position of repose than any other part; the consequence of which is that the whole string continues to vibrate backwards and forwards over the line of repose. But if the bow be applied gently near one end of the string, four tones can frequently be heard at once, three of which are called "acute harmonics." The origin of this singular effect seems to be, that while the whole string vibrates in one system to produce the fundamental or proper note of the string, it is at the same time broken up into three subordinate systems, composed of aliquot parts of the string. Sometimes these lengths are,, and of the whole length, and in such case the tones resulting from them are the octave, the twelfth or octave to the fifth, and the double

octave.

Much obscurity exists as to these subdivisions of a string. A writer in the Encyclopædia Britannica attributes their formation to the reaction of the fixed termini of the string; that is, that the string by its vibration communicates a vibratory impulse to the solids to which the two ends are attached, and that those vibrations are reflected, as it were, from the solid supports back to the string, and by interfering with the continuity of the previous vibrations existing therein, generate a new series of a different rapidity from the former. These harmonics can be artificially produced by placing the finger very lightly on any of the aliquot divisions of the string, and drawing the bow across near one end of it; in this case the two divisions of the string will vibrate in different systems, and will produce tones dependent on their relative lengths.

3. THE REFLEXION of Sound.

When an obstacle opposes the motion of an elastic body in any given direction, a rebound or reflex motion occurs, whether the moving body be a solid ball, waves of water, of light, or of sound. All of these present very similar features, in respect of the direction in which the impulse is communicated; but it will be sufficient to confine our attention here to the phenomena presented by sound.

Let us take, as a simple illustration, the case of a bell sounding at a short distance from a wall. Let b c be the wall, and a the bell, from which pulses of sound proceed in all directions, of which we will here take notice only of the two ▲ 6 and ▲ c. The pulse which proceeds perpendicular to the wall, a b, is reflected back along the same line; but the pulse A c is reflected, not along the same path, not perpendicularly to the wall, but in a direction cd, which is as