points thus found is drawn the curve having the line c d for its axis. This curve may be regarded as the trace of the composite vibration of a molecule of air, or of a point of the tympanic membrane, on a surface which moves near these points. Hence if we slide this curve along, in the direction of its axis, under a slit in a screen which allows only one point of the curve to appear at once, we shall reproduce in this slit the vibratory motion of the aerial molecule and of the point on the tympanic membrane. The writer has exhibited this motion in a continuous, or rather recurring manner, as follows: On a piece of Bristol board he drew a circle, and in one quadrant of this circle he drew 500 equidistant radii. On these radii, as ordinates, he transferred the corresponding values of the same ordinates of the resultant curve of fig. 11, diminished to one fourth of their lengths. He thus deflected the axis of the curve of fig. 11 into one fourth of a circle curve; and this, repeated four times on rotating the disk one can readily follow the compound vibratory motion of the spot of light; but on a rapid revolution of the disk, persistence of visual impressions causes the vibrating spot to appear elongated into a band. This band is not equally illuminated; it has six distinct bright spots in it, beautifully re vealing the six inflections in the curve. By sticking a pin in the centre of fig. 12, as an axis about which revolves a piece of paper having a fine slit, the reader can gain some ides of the complex motion we have described. 0: course it is understood that in the above experiment the amplitudes of the vibrations are enormously magnified when compared with the wave lengths, and that it is really only when the amplitudes of the elementary pendulum vibrations are infinitely small that the resultant curves we have given can be rigorously taken as representing what they purport to; for the law of "the superposition of displacements depends on the condition that the force with which a molecule returns to its position of equilibrium is directly proportional to the amount of displacement, and this condition only exists in the case of infinitely small displacements; yet the law holds good for the majority of the phenomena of sound. It is alse to be remarked that in order to simplify the FIG. 12. the Bristol board, rendered the curve continuous and four times recurring, as shown in fig. 12. He now cut this figure out of the board and used it as a template. He placed the latter centred on a glass disk 20 in. in diameter. This disk was coated on one side with opaque black varnish, and with the template and the separated points of a pair of spring dividers he removed from the glass disk a sinuous band, as shown in fig. 12. The glass disk was now mounted on a horizontal axis and placed in front of a lantern, the diameter of whose condensing lens was somewhat greater than the amplitude of the curve. The image of that portion of the curve which was in front of the condenser was now projected on a screen, and then a piece of cardboard having a narrow slit cut in it was placed close to the disk, in the direction of one of its radii. On revolving the disk he reproduced on the screen the vibratory motion of a molecule of air, or of a point on the tympanic membrane, when these are acted on by the joint impulses of the first six harmonic or pendulum vibrations, forming a musical sound. On slowly 15, we have drawn the resultant curves formed by combining the curves of musical sounds corresponding to the various consonant intervals indicated below the figures. As these fraction of an inch, and that it occupied of a second in making this forward motion; then the length of air compressed at the instant the piston has come to rest will be equal to 1980, or 109 ft. If the piston makes its movement in and in Too of a second, the length of air compressed in the tube will be respectively 10.9 and 1.09 ft. But such a compressed portion of air cannot remain at rest, by reason of its elasticity. It immediately expands, and in so doing presses forward on the undisturbed air in front of it and on the interior wall of the tube. The column of compressed air in thus regaining its natural density has com FIG. 15.-Resultant Curve formed by combining the curve of pressed an air column of equal depth in front on. a musical note with that of its major third. A:A:: 1:. of it; this in its turn reacts on the back colcurves are the resultants formed by the com- umn and prevents it from rarefying, while at bination of the curves of composite musical the same time it has compressed another colsounds, it follows that the components of these umn of equal depth in front of it, and so curves are not simple sinusoidal curves, as in Thus the sonorous pulse, as it is called, is the case of fig. 11, but are derived from the transmitted through the whole length of the resultant of fig. 11 by reducing to one fourth tube. A beautiful illustration of the manner the amplitude of that curve, and by taking in which a sound pulse is propagated is affordwave lengths corresponding to intervals indi- ed by attaching to a sounding board a long, cated below the figures. From the curves of elastic spiral spring of brass, while the other figs. 13, 14, and 15 can be reproduced their end is held in the hand. On separating two generating motions in the same manner as we of the coils of the spring with a finger nail, have done in the case of the curve of fig. 11. and then allowing them suddenly to come toAs a periodic or recurring vibration can alone gether, a pulse or compression will be thrown produce in the ear the sensation of sound, and through the whole length of the spring to its as the duration of the period is always equal further end, where striking on the sounding to the least common multiple of the periods of board it will cause a sharp rap. This action the pendulum vibrations of the components, it against the board will be reflected from the follows that in the case of a sound formed of a board to the hand, and again from the hand harmonic series the period equals the time of to the board, and so on several times in succesone vibration of the fundamental; but in the sion. When the piston above spoken of makes cases of other combinations the duration of the a backward movement, it creates a vacant period increases with the complexity of the space in the tube, into which the air rushes ratio of the times of vibration of the compo- by virtue of its elasticity, and thus a certain nents; thus, the durations of the periods of the depth of air is rarefied; this first cylinder of following combinations are placed after them rarefied air in retracting to its natural dimenin fractions of a second: Cs+C4=; Ca+Gssions causes rarefaction in an equal depth of air =1; C+E=4; Cs+E+Gs=7; Cs+ Es+G3+C1=1 of a second. (Cs stands for the treble C; C. is the C of the octave above it.)-Transmission of Sound. If air were incompressible, a motion produced at any point of its mass would instantaneously be transmitted to every other point of the atmosphere. Thus, if we imagine a long tube, open at one end and closed at the other by a piston which moves in the tube without friction, it is evident that if this piston were pushed into the tube a certain distance, the air would at the same time move out of the tube at the open end. But air is compressible and elastic, and after the piston has been pushed into the cylinder, a measurable interval of time will have elapsed before the air moves out of the open end of the tube. This interval is the time taken by sound to traverse the length of the tube. The velocity of sound is 1,090 ft. in a second at 32° F., and it increases almost exactly one foot in velocity for each degree of elevation of temperature above 32°. Now imagine the piston to move forward into the tube over a minute in front of it; this second rarefied cylinder of air now reacts on the first, brings it to rest, and causes rarefaction in a third equal column of air, and so on. Thus the rarefaction, like the compression, is transmitted through the whole length of the tube. When a compression traverses the tube it successively brings the molecules of air nearer together, while a rarefaction in its progress separates the aërial molecules; hence, if we imagine the piston to move backward and forward with a regular vibratory motion we have rarefaction succeeding compression in regular order, and the effect on any one molecule of air is to give it a like regular motion backward and forward. In the above discussion we have, for simplicity, supposed the piston to have a uniform velocity during its motions; but this, as we have already seen, is not the case with freely vibrating elastic bodies, for they have the same character of reciprocating motion as that of a freely swinging pendulum. To explain what will be the effect on the air of such a motion, we will suppose that the piston vibrates through a very small distance, a a', fig. 16, making equal excursions on one side and the other of the position of equilibrium m m'; and as the piston vibrates like a pendulum, it will increase a. n FIG. 16. M PA | tations will travel behind the condensations, and when the piston has returned to a, in which case the series of condensations will have reached the position A' a A, these dilatations will be distributed in the space a A', and the diminution of density of the layers of air can be represented by the negative ordinates of the curve a ẞ A', below the axis of the curve a A'. The state of air in the tube at the instant when the vibrating piston, departing from a, arrives at n p, m m', n'p', a', is indicated by the curves n N, m M, n' N', a' A'. If the piston makes another comin velocity as it goes from a or from a' to plete vibration from a to a' and from a' to a, m m', and diminish in velocity as it goes from a new series of condensations and of dilatam m' to a or to a'. Let T be the time taken tions, distributed in a space equal to a A, will by the piston to make a semi-vibration, that travel behind the first series already described. is to say, a motion from a to a' or from a' to The dilatation and condensation contained in Divide this time T into exceedingly small a' A, and produced by a complete vibration and equal parts t, during which the piston of the body at the origin of sound, i. e., by will also go over very small but unequal spaces, an oscillating motion from a to a' and back increasing with the velocity from a to m m', from a' to a, is called a sonorous wave. A and diminishing with the velocity as the pis- sonorous wave is always formed of two parts, ton goes from m m' to a'. The first very small one half of air in a state of condensation, the displacement of the piston, accomplished du- other half of rarefied air. The sum of all the ring the time t, will produce in a very thin condensations in the condensed half of the layer of air, which touches the piston, a very wave is represented by the area of the curve feeble degree of compression, and this com- a'a A'; and if we divide this by the interval pression will progress forward into the air of T of a half vibration of the body, we have the the tube. The very small succeeding motion mean condensation of the half wave. This of the piston during the next succeeding t mean condensation can be calculated, and it will produce a slightly greater condensation, has been found that for the sound given by which will travel behind the former conden- 250 vibrations per second, which corresponds sation with the same velocity. The third dis- nearly with the lowest C of the violin, this placement of the piston will produce a still compression gives for the compressed half of greater condensation, and so on, until the dis- the wave an increase of to the ordinary placement which brings the piston to the po- density of the atmosphere. The length of a sition m m', which, being the greatest of all, wave is evidently the distance through which will produce the greatest condensation. If the air has been affected the moment after the the piston continues its motion to a', with a first complete vibration of the sonorous body velocity which is now gradually decreasing, a has been made. If we designate this length new series of condensations will take place, by 1, we can calculate the wave length by diless and less in degree, which will travel be-viding the velocity v of sound in a second by hind those of the first series. These two series will be symmetrically placed on one side and the other of the maximum condensation, if we suppose that the two semi-oscillations of the piston are equal, and if we neglect the very small amplitude of oscillation a a'. If a A' is the space through which the first condensation progresses in the time T, then all the condensations which have succeeded it during the movement of the piston from a to a' will be distributed in the space a' A'. If we represent by ordinates these condensations at the moment when, the piston having arrived at a', the first condensation is at A', we will form a curve a' a A', whose maximum ordinate Ma will represent the condensation produced by the piston at the moment of its passage through m m'. Let us now suppose that the vibrating piston returns on its path, it will produce by this motion a series of increasing dilatations during the time T, and then decreasing dilatations until the instant when the piston reaches a. These dila n, the number of vibrations the sounding body makes in a second; or, l=; By a sonorous wave surface is understood that surface which is at such a distance from the point or points of origin of the sound that all points in that surface have the same phase of vibration at the same instant of time. Thus, it is evident that if we have a small sphere of air which successively and rapidly increases and diminishes its volume, we shall have alternate spherical shells of compressed and of rarefied air surrounding the vibrating sphere. If we view a surface in one of these shells, in every part of which surface the particles of air are moving in the same direction with the same velocity, we shall have the sonorous wave surface. The acoustic wave lengths and wave surfaces are not mere creations of the imagination, but have a real existence. The author of this article first devised a method by which one can readily detect the phases of vibration in the air surrounding a sounding body, and thereby has succeeded in measuring directly in the vibrating air the length of sonorous waves, and has determined in the air surrounding the vibrating body the form of the wave surface. ("American Journal of Science," November, 1872.) It is evident that the ultimate effect of the passage of sonorous waves through the atmosphere will be to cause the molecules of the air to swing to and fro with the motions of pendulums. It is also apparent that all the characteristics of the periodic motion at the source of the sound will be impressed on the surrounding air and transmitted through it to a distance.-Reflection of Sound. It follows from the very nature of sound pulses that if a sonorous wave meet a hard smooth surface, or encounter the surface of separation of two media of unequal elasticity, reflection of sound will take place, and the laws of reflection will be the same as in the case of light, viz. the angle of reflection will equal the angle of incidence, and both the incident and reflected ray will lie in the same plane, which is at right angles to the reflecting surface. These laws admit of a ready experimental proof. If two concave parabolic mirrors, formed of metal backed with hard wood or plaster of Paris, be placed opposite each other at a distance of 10 or 15 ft. with the axis of the mirrors in the same line, and a watch be placed in the focus of one of the mirrors, it will be found that the sonorous pulses emanating from the watch will be reflected from the first mirror upon the surface of the second mirror, and here by a second reflection will be conveyed to the focus. This fact can be ascertained by leading to the focus a tube terminated at one end by a small funnel, while | the ear is applied to the other end of the tube. In the article OPTICS it has been shown that the action just described is a necessary consequence of the laws of reflection given above.Refraction of Sound. Sound waves are also refracted, and their refraction is due to the same cause which produces refraction of the rays of light; i. e., to the change in velocity which occurs when the sonorous beam enters a refracting medium. When the sonorous wave surface falls upon the refracting mediu so that it is parallel to the refracting surface, there will be no refraction, or change in the direction of the sound, but only a change of velocity. But when the sonorous wave surface forms an angle with the surface of the refracting medium, the change in velocity causes the refraction of the sonorous beam, so that if the velocity of the sound is less in the refracting medium than it was before it entered it, the sound will be refracted toward the perpendicular to the refracting surface. The refraction will be away from the perpendicular when the velocity of the sound is greater in the refracting medium than it was before it entered it. It follows from the above action, that for the same media there will be a constant ratio existing be best adapted for this purpose is one devised by Sondhaus and represented in fig. 17. He constructed a lens, L, of sheets of collodion, having the form of portions of a sphere, and united these sheets to the opposite sides of a metal ring. On inflating the envelope thus formed with carbonic acid gas, a lenticular form was given to it. A watch was placed at W, on the axis of the lens, and it was found that the sound waves were refracted to the conjugate focus of the lens at F. If at F we place a bent pipe with a funnel-shaped mouth, and replace the watch at W by a small organ pipe, the refraction is detected by seeing grains of a light powder dance on the membrane closing the upper mouth of the bent pipe at c. -Interference of Sound. Another necessary consequence of the nature of sound vibrations and of the manner of their propagation is, that if the condensed half of a sonorous wave meet the rarefied half of another sonorous wave, and these waves have the same length and the same energy of vibration, there can be no vibratory motion at their place of meeting, for the directions of the vibrations in the two half waves are opposed, and the intensities of these opposed vibratory motions are equal. These conditions are fulfilled in various well known experiments, and it is one of the best established facts in acoustics that two sound vibrations may meet and produce silence at the place of their meeting; this is known as the phenomenon of the interference of sound. Dr. Thomas Young studied this phenomenon attentively, and its contemplation led to his great discovery of the similar phenomena of the interference of light, which formed the basis of his reasoning in establishing the undulatory theory of light. To Dr. Young we owe one of the simplest known means of exhibiting and studying the phenomena of interference of sound. If a vibrating tuning fork be held in a vertical position at a short distance from the ear, and then rotated around its vertical axis, it may be observed, when the surfaces of the prongs of the fork are oppo- | cold and heated gases, such as carbonic acid site the ear, that sound will be perceived; but gas and hydrogen. Two resonators were placed when the edges of the fork formed by the as in fig. 20 with the planes of their mouths at meeting of those surfaces are opposite the ear, it will be found that no sound, but entire silence, occurs. This phenomenon is readily explained. First, it is necessary to know that the prongs of a vibrating fork alternately approach to and recede from each other, as is readily seen when we obtain on a piece of smoked glass the trace of two delicate wires attached to the ends of the prongs of the vibrating fork. A trace thus made is accurately shown in fig. 18. When the prongs recede FIG. 20. a right angle, and in this angle was firmly fixed the fork to whose note the resonator resounded. The broad face of one of its prongs faced the mouth of one resonator, while the space between the prongs faced the mouth of the other resonator. By trial the two planes of the fork are placed at such distances from the resonators that complete interference of the vibrations issuing from their mouths is obtained, and the only sound that reaches the ear is the faint one given by the action of the fork on the air outside the angle included by the mouths of the resonators. If in these circumstances we place before the mouth of one of the resonators a flat coal-gas flame, we shall find that this flame reflects part of the sound which falls upon it, and thus partially screens the resonator, so that sonorous vibrations of diminished intensity now enter this resonator, and therefore the balance of interference no longer exists, and a sound issues from the resonator which has not the gas flame opposite its mouth. But if a piece of French tracing paper be placed before the mouth of the latter resonator, the balance of interference will be restored, thus showing that the reflecting power of a gas flame is equal to that of tracing paper. In & similar manner the writer has shown and approximately measured the reflecting power of sheets of cold carbonic acid and hydrogen gases.-Change of Pitch caused by Translation of the Sounding Body. One of the most remarkable phenomena is the change in pitch caused by the motion of a sounding body to or from the ear; or, what is the same, by the motion of the ear to or from the source of sound. When the sounding body and the ear approach, we perceive a rise in the pitch; when they recede from each other, a fall in pitch occurs. This is a fact known to all who have listened to the rapid change in pitch of a locomotive whistle which of condensed air, and the dotted lines the centres of shells of rarefied air. These shells alternate, and meeting along the planes p, p, p, p, passing through the vertical edges of the fork, they neutralize each other's action. W. Weber has shown that the points of quiescence in this case must lie in hyperbolic sheets. This must be so, for the difference in the distance of every point of quiescence from two fixed points must be a constant quantity, which in this experiment will be equal to the half of the wave length given by the fork. The writer has used this experiment of Young to show the reflection of sound from flames and from sheets of |