106 ACOUSTICS heard by the same ear at the distances OA and OB are to each other as OB 2 to OA 2 . Influence of 30. In order to verify the above law when the atino- (Uiniiiislietl sphere forms the intervening medium, it would be necessary density of to test it at a considerable elevation above the earth s the air on sur f ace> the ear and the source of sound being separated oMul J ky a i r f constant density. As the density of the air diminishes, we should then find that the loudness of the sound at a given distance would decrease, as is the case in the air-pump experiment previously described. This arises from the decrease of the quantity of matter impinging on the ear, and the consequent diminution of its vis-viva. The decay of sound due to this cause is observable in the rarefied air of high mountainous regions. De Saussure, the celebrated Alpine traveller, mentions that the report of a pistol at a great elevation appeared no louder than would a small cracker at a lower level. But it is to be remarked that, according to Poisson, when air-strata of different densities are interposed between the source of sound and the ear placed at a given distance, the intensity depends only on the density of the air at the source itself; whence it follows that sounds proceeding from the surface of the earth may be heard at equal dis tances as distinctly by a person in a floating balloon as by one situated on the surface itself ; whereas any noise origi nating in the balloon would be heard at the surface as faintly as if the ear were placed in the rarefied air on a level with the balloon. This was exemplified during a balloon ascent by Glaishor and Coxwell, who, when at an elevation of 20,000 feet, heard with great distinctness the whistle of a locomotive passing- beneath them. PART III. Reflexion and Refraction of Sound. Laws of 31. When a wave of sound travelling through one refraction, medium meets a second medium of a different kind, the vibrations of its own particles are communicated to the particles of the new medium, so that a wave is excited in the latter, and is propagated through it with a velocity de pendent on the density and elasticity of the second medium, and therefore differing in general from the previous velocity. The direction, too, in which the new wave travels is dif ferent from the previous one. This change of direction is termed refraction, and takes place according to the same laws as does the refraction of light, viz., (1.) The new direction or refracted ray lies always in the plane of incidence, or plane which contains the incident ray (i.e., the direction of the wave in the first medium), and the normal to the surface separating the two media, at the point in which the incident ray meets it; (2.) The sine of the angle between the normal and the incident ray bears to the sine of the angle between the normal and the refracted ray, a ratio which is constant for the same pair of media. For a theoretical demonstration of these laws, we must refer to the art. OPTICS, where it will be shown that the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second ; in other words, the sines of the angles in question are directly proportional to the velocities. 32. Hence sonorous rays, in passing from one medium into another, are bent in towards the normal, or the reverse, according as the velocity of propagation in the former exceeds or falls short of that in the latter. Thus, for instance, sound is refracted towards the perpendicular when passing into air from water, or into carbonic acid gas from air; the converse is the case when the passage takes place the opposite way. Fig. 6. 33. It further follows, as in the analogous case of light, that there is a certain angle termed the limiting angl t whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but surfer total reflexion back into the first body, if the velocity in that body is less than that in the other body, and if the angle of incidence exceeds the limiting angle. The velocities in air and water being respectively 1090 and 4700 feet, the limiting angle for these media may be easily shown to be slightly above 15^. Hence, rays of sound proceeding from a distant source, and therefore nearly parallel to each other, and to PO (fig. 6), the angle POM being greater than 15i, will not pass into the water at all, but suffer total reflexion. Under such circumstances, the report of a gun, however powerful, would be inaudible by an ear placed in the water. 34. As light is concentrated into a focus by a convex Aeousl
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glass lens (for which the velocity of light is less than for lcils e s - the air), so sound ought to be made to converge by passing through a convex lens formed of carbonic acid gas. On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave. These results have been confirmed experimentally by Sondhaus and Hajech, who also succeeded in verifying the law of the equality of the index of refraction to the ratio of the velocities of sound. 35. When a wave of sound falls on a surface separating Laws < two media, in addition to the refracted wave transmitted ix-Hcxii into the new medium, which we have just been consider ing, there is also a fresh wave formed in the new medium, and travelling in it in a different direction, but, of course, with the same velocity. This reflected wave is subject to the same laws as regulate the reflexion of light, viz., (1.) the coincidence of the planes of incidence and of reflexion, and (2.) the equality of the angles of incidence and reflexion, that is, of the angles made by the incident and reflected rays with the normal. 3G. As in an ellipse (fig. 7), the normal PG at any point R, e fl e x: by a roid. bolic s faces. bisects the angle SPH (S, II being the foci), rays of sound diverging from S, and falling on the spheroidal surface formed by the revolution of the ellipse about the longest diameter AB, will be reflected to H. Also, since SP + PH is always = AB, the times in which the different rays will reach H will all be equal to each other, and hence a crash at S will be heard as a crash at H. 37. At any point P of a parabola (fig. 8) of which S is Reflex: the focus, and AX the axis, the normal PG bisects the by par angle SPX, PX being drawn parallel to AX. Hence rays of sound diverging from S, and falling on the paraboloid formed by the revolution of the parabola about its axis, will all be reflected in directions parallel to the axis. And vice versa rays of sound XP, XQ, &c., from a very distant source, and parallel to the axis of a paraboloid, will be reflected into the focus. Con sequently, if two reflecting paraboloids be placed at a considerable distance from and opposite to each other, with their axis coincident in direction (fig. 9), the tick of a watch placed at the focus S of one will be heard dis
tinctly by an ear at S , the focus of the other.
