piece of paper, a grain of corn, or any small object fall into it; or by throwing a stone into a smooth-surfaced pond.] A number of familiar facts prove that the apparent advancing motion of the liquid on which waves are passing is only a deception. Light pieces of wood are not hurried forward on the surface of water, but merely rise up and sink down alternately as the waves pass. The true nature of the motion is such that each particle, at the surface of the undulating liquid, describes a circle in a vertical plane, and in the direction in which the wave is advancing, the movement being propagated from each to its next neighbour, and so on. And as a certain time must elapse for this transmission of motion, the different particles will be describing different points of their circular movement at the same moment. Some will be at the highest part of their vertical circle when others are in an intermediate position, and others at the lowest, giving rise to a wave, which advances a distance equal to its own length, while each particle performs one entire revolution. [The force by which the water-waves are propagated is gravity; for, if from any cause an elevation or a depression be produced on the horizontal surface of the water, the gravity of the separate particles of water will endeavour to restore the disturbed horizontal plane, by which means an oscillatory motion is produced, which, by degrees, is propagated from one particle to another.-Professor Müller's "Physics and Meteorology," Lecture XV.] Thus, in Fig. 192, let there be eight particles of water on the surface, a m, which, by some appropriate disturbance, are made to describe the vertical circles represented at a bc de fg h, moving in the direction represented by the darts, Fig. 192. and let each one of these commence its motion one-eighth of a revolution later than the one before it. Then, at any given moment, when the first one, a, is in the position marked a, the second, b, will be in the position marked 7, c at 6, d at 5, e at 4, f at 3, g at 2, h at 1; but m will not yet have begun to move. If, therefore, we connect these various points, a 7 6 5 4 3 2 1 m, together by a line, that line will be on the surface of the wave, the length of which is a m, the height or depth of which is equal to the radius of the circle of each particle's revolution, and the time of passage through the length of one wave will be equal to the time of the revolution of each particle. By a ray of undulation we mean a line drawn from the origin of a wave in the direction in which any given point of it is advancing. A wave is said to be incident when it falls on some resisting surface, and reflected when it recoils from it. Incident rays are those ray, ip a perpendicular to the point of impact of the wave, i d the reflected ray. Then i Ρ is the angle of incidence, and dip the angle of reflection. The general law for the reflection of waves is, that "all the points in a wave will be reflected from the surface of the solid under the same angle at which they struck it." If, therefore, parallel rays fall on a plane surface, they will be reflected parallel; if diverging, they will be reflected diverging; and if converging, converging. If a circular wave advances from the centre of a circular vessel, each ray falls perpendicularly on the surface of the vessel, and is reflected perpendicularly that is to say, back in the line along which it came. The waves, therefore, all return to the centre from which they originated. If undulations proceed from one focus of an ellipse, they will, after reflection, converge to the other focus. If a surface be a parabola, rays diverging from Fig. 194, its focal point, a, will, after reflection, pass in parallel lines, b d, dcd, ed. Or if the rays impinge in parallel lines, they will, after reflection, converge to the focus. When diverging Fig. 195. rays of a circular wave fall upon a plane surface, their path, after reflection, is such as it would have been had they originated from a point on the opposite side of the plane, and as far distant as the point of origin itself. Thus, let c be the origin of a circular wave, d a g, which impinges on a plane, ef, after reflection this wave will be found at e hf, as though it had originated at c', a point on the opposite side of e ƒ, as far as c, in front of it. Now, the parts of the circular wave, d a g, do not all impinge on the plane at the same time, but that at a, which falls perpendicularly, impinges first, and is first reflected; the ray at d has to go still through the distance, de, before reflection takes place; but, in this space of time, the ray at a will have returned back to h; and, in the same way, it may be shown that the intermediate rays will have returned to intermediate positions, and be found in the line e hf, symmetrically situated, with respect to the line e n f, in which they would have been had they not fallen on the plane. And it further follows that the centre, c', of the circular wave, e hf, is as far from ef as is the centre, c, of the circular wave, e nf, but on the opposite side. By interference we mean that two or more waves have encountered one another, under such circumstances as to destroy each other's effect. If on water two elevations or two depressions coincide, they conspire; but when an elevation coincides with a depression, interference takes place, and the surface of the fluid remains plane. Waves which have thus crossed one another continue their motion unimpaired. If two systems of waves of the same length encounter each other, after having come through paths of equal length, they will not interfere; nor will they interfere, even though there be a difference in the length of their paths, provided that difference be equal to one whole wave, or two, or three, &c. But if two systems of waves of equal length encounter each other after having come through paths of unequal length, they will interfere, and that interference will be complete when the difference of the paths through which they have come is half a wave, or one and a half, two and a half, three and a half, &c. When a circular wave impinges on a solid in which there is an opening, as at a b, Fig. 196, the wave passes through, and is propagated to the spaces beyond; but other waves arise from a bas centres, and are propagated as represented at c def. This is the inflection of waves, and these new waves intersecting one another and the primitive one, give rise to interferences. Fig. 196. We have now traced the chief phenomena of vibrations in solids and on the surface of liquids. It remains to do the same for elastic bodies, such as gases. When any vibratory movement takes place in atmospheric air, the impulse communicated to the particles causes them to recede a certain distance, condensing those that are before them; the impulse is finally overcome by the resistance arising from this condensation. There, therefore, arises a sphere of air, the superficies or shell of which has a maximum density. Reaction now sets in, the sphere contracts, and the returning particles come to their original positions. But as a disturbance on the surface of a liquid gives origin to a progressive wave, so does the same thing take place in the air. By the intensity of vibration of a wave we mean the relative disturbance of its moving particles, or the magnitude of the excursions they make on each side of their line of rest. Thus, on the surface of water we may have waves mountains high," or less than an inch high; the intensity of vibration in the former is correspondingly greater than in the latter case. In aërial waves, precisely as in the surface waves of water, interference arises under the proper conditions. Thus, let a m p h, Fig. 197, be a wave קי W Fig. 197. advancing toward c, and m n o p be the intensity of its vibration, or the maximum distances of the excursions of its vibrating particles. Then suppose a second wave, originating at b (a distance from a precisely equal to one wave length), the intensity of vibration of which is represented by q r. The motions of this second wave coinciding throughout its length with the motions of the first, the force of both systems is increased. The intensity, therefore, of the wave, arising from their conjoint action at any point, q, will be equal to the sum of their intensities, qr, q s-that is, it will be qt, and for any other point, v, it will be equal to the sum of v w and v u— that is, So the new wave will be represented by b t g x h. vx. Now let things remain as before, except that the point of impact of the second wave, instead of being one whole wave from a, is only half a wave, the effects on any particle, such as q, take place in opposite directions, the second wave moving it with the intensity and direction q r, the first with q s-the resultant of its movement in intensity and direction will, therefore, be the difference of these quantities-that is, q t. And the same reasoning continued gives, for the wave resulting from this conjoint action, btg xh c. Under the circumstances given in Fig. 198, the systems of waves increase each other's force; under those of Fig. 197 they diminish it; or, if equal to one another, counteract completely, and total interference results. Waves in the air, as they expand, have their superficies continually increasing, as the squares of their radii of distance from the original point of disturbance. Hence the effect of all such waves is to diminish as the squares of the distance increase. SECTION VII.-THE LAWS OF SOUND.-ACOUSTICS. CHAPTER XXXII. PRODUCTION OF SOUND. The Note depends on Frequency of Vibration-Distinguishing Powers of the Ear-Soniferous Media-Origin of Sounds in the Air-Elasticity required and given in the Case of Strings by Stretching-Rate of Velocity of Sounds—All Sounds transmitted with Equal Speed-Distances determined by it-High and Low Sounds-Three Directions of Vibration— Intensity of Sound-Quality of Sounds-The Diatonic Scale. WHEN a thin elastic plate is made to vibrate, one of its ends being held firm in a vice, and the other being free, as in Fig. 199, and its length limited to a few inches, it emits a clear musical note. If it be gradually lengthened, it yields notes of different characters, and finally all sound ceases, the vibrations becoming so slow that the eye can follow them without difficulty. This instructive experiment gives us a clear insight into the nature of musical sounds, and, indeed, of all sounds generally. A substance which is executing a vibratory movement, provided the vibrations follow one another with sufficient rapidity, yields a musical sound; but when those vibrations fall below a certain rate, the ear can no longer distinguish the effect of their impulsions. The number of vibrations which such a plate makes in a given time depends upon its length, being inversely as the square of the length of the I vibrating part. Thus, if we take a given plate and reduce its length, the vib rations will increase in rapidity; when it is half as long it vibrates four times as fast; when one-fourth, sixteen times, &c. All sounds arise in vibratory movements, and musical notes differ from one another in the rapidity of their vibrations-the more rapidly recurring or frequent the vibration the higher the note. There is, therefore, no difficulty in determining how many vibrations are required to produce any given note. We have merely to find the length of a plate which will yield the note in question, knowing previously what length. of it is required to make a determinate number of vibrations in a given space of time. Thus it has been found that the ear can distinguish a sound made by fifteen vibrations in a second, and can still continue to hear, though the number reaches 48,000 per second. That all sounds arise in these pulsatory movements common observations abundantly If we touch a bell, or the string of a piano, or the prong of a tuning-fork, we feel at once the vibratory action, and with the cessation of that motion the sound dies away. Fig. 199. prove. But the pulsations of such a body are not alone sufficient to produce the phenomena of sound. Media must intervene between them and the organ of hearing. In most cases the medium is atmospheric air, and when this is taken away the effect of the vibrations wholly ceases. Thus, a bell, or a musical snuff-box, under an exhausted receiver, as we have already seen (Chapter VII., Fig. 38), can no longer be heard; but on re-admitting the air the sound becomes audible. The sounding body, therefore, requires a soniferous medium to propagate its impulses to the ear. Atmospheric air is far from being the only soniferous medium. Sounds pass with facility through water; the scratching of a pin or the ticking of a watch may be heard by the ear applied at the end of a very long plank of wood. Any uniform elastic medium is capable of transmitting sound; but bodies which are imperfectly elastic, or have not an uniform density, impair its passage to a corresponding degree. The effect of a vibrating spring, or, indeed, of any vibrating body on the atmospheric air, is to establish in it a series of condensations and rarefac tions which give rise to waves. These, extending spherically from the point of disturbance, advance forward until they impinge on the ear, the structure of which is so arranged that the movement is impressed on the auditory nerves, and gives rise to the sensation which we term sound. Both the sonorous body and the soniferous medium must, therefore, be elastic; the regularity of the pulsations of the former depends upon the uniformity of its elasticity. In the case of strings, we give them the requisite degree of elastic force by stretching them to the proper degree. And, as the undulatory movements which arise in the soniferous medium are not instantaneous, but successive, it follows that the transmission of sound in any |