ACOUSTICS. 



ACOUSTICS. 



66 



all condensation or rarefaction will be destroyed throughout, the 

 particles however being all in motiou, except A, B, &c., but in directions 

 /// to those they had at first ; while, at the end of a fourth half 

 wave, the phenomena of the second supposition will be repeated, that 

 is, all velocity will be destroyed, the particles being all condensed or 

 run ilnt, according as they were before rarefied or condensed. The 



reader may easily convince himself of these facts by drawing the cor- 

 responding figures. To put the results before the eye, suppose the 

 tube to be of a highly elastic material (thin India-rubber, for example), 

 so as to bulge outwards a little when compressed from the interior, or 

 to contract in diameter by the pressure of the outward air when the 

 inward is rarefied. Recollect, also, that A, B, c, D, &c., remain without 



Fig. 10. 



motiuu, their only change being condensation or rarefaction; while 

 a, b, c, &c., are never compressed or rarefied, their only change being 

 that of place. We exhibit side by side the successive appearances of 

 the tube, and the relative situations of the types between A and c, the 

 arrows always representing the direction of the motion of the particles. 

 A half -wave elapses between each two configurations. (Fly. 11.) 



These phenomena will recur in the same order, and this mode of 

 undulation, though it in necessary to show how it arises from the 

 combination of two waves, is nevertheless more easy to be explained by 



itself than either of these two. For if we recollect that when particles 

 of air move away on both sides from a given point, there must be a 

 condensation in the parts towards which they move, and a rarefaction 

 in those which they quit, (2) will evidently follow from (1). At this 

 second period, the elasticity of the air will have opposed and destroyed 

 the velocities of the particles ; so that there now only remains a tube 

 of particles at rest for the moment, condensed towards the ends and 

 rarefied in the middle. There will therefore immediately commence a 

 rash of air towards the rarefied parts, which will end by producing the 



rig/11. 



state represented in (3), where equilibrium ID restored, as far as com- 

 pression and rarefaction arc concerned ; but where, at the moment 

 under consideration, nothing has yet taken place to depiive the [ar- 

 ticle** of the velocity which they received from the elasticity of the air 

 before the natural state was recovered. There is now a motion of 

 l';irtirle>. in all directions, towards B, which will go on producing com- 

 pression at B, and rarefaction at A and c, until all the velocity is 

 di'stroyi-d. Thin is the state represented in (4), from which (1) follows 

 ajfiin ; and so on. The states of the column intermediate between the 

 times of (1), (2), &c., are easily imagined. Between (1) and (2) the 

 compression at the extremities will have begun ; but not yet to the 

 complete destruction of the velocities. Between (2) and (3) the motion 

 of the particles towards the middle will have begun ; but will not yet 

 have placed them in their natural positions ; and so on. The particle 

 at B, in evidently never in motion, being always equally pressed on 

 t>oth xides. The same would be seen of A, and c, if the tube were 

 I'xU-nded on both Hides. 



It is evident also, that except at the instant when compression and 

 rarefaction are all destroyed, there must be a point at which the 

 transition occurs from condensation to rarefaction ; and rire rend. 

 It Li not however so evident, in this way of viewing the subject, that 

 these points always remain in the same position at n and 6, which is 

 the result of our previous investigation. The reader must however 

 recollect, that, when we talk of the points a and b being always free 

 from condensation or rarefaction, we do not say that it is the tame air 

 which is always uncondensed or unrarefied, but only that the different 

 poitloiui of air, which pans by a and b, are in their natural state at the [ 

 instant of the passage. 



Now it must be evident, that if, in the motion of a fluid, there be 

 'i particles which remain at rent, it is indifferent whether we 

 suppose those particles to be fluid or solid ; for all that we know of a 

 *"li.|, ;is distinguished from a fluid, is, that the particles of the latter 

 yield nen*iUy to any applied force, while those of the former do not. 

 Hence, when such impulses are communicated to a fluid, that some of 

 its particles must remain at rest, the question never arises, so to speak, 

 an to whether those particles would, or would not, move with the 

 fluid, or resist, if the conditions of motion were so altered, that forces, 

 which did not counterbalance, would be applied to those particles. 

 Let us now suppose that a solid diaphragm is stretched across the tube 

 at A ; the motion will still continue exactly as before ; and we may 

 produce this species of complex undulation by a piston at one end 

 only of the tube, provided the other end be closed. For, on this 

 Kiipposition, all the successive states into which the air at the end 



ARTS ASO CI. DIV. VOL. I. 





furthest from the piston is brought, cannot be communicated to the 

 outside air, and must, therefore, be either retained, or returned back 

 again through the column of air. The latter effect results ; and the 

 returning wave, which is of the same kind as the advancing wave, 

 produces the phenomena just explained. If A and B were both closed 

 during an undulation, no piston would be necessary, if it were not 

 that there is no substance but what will vibrate in some small degree, 

 and the vibrations communicated to the tube from the internal ail- 

 gradually destroy the internal motiou, by the communication of motion 

 to the external air. 



We have hitherto considered only the motion of air in a small tube, 

 and have found that the velocity of the particles, as well as the con- 

 densation and rarefaction, may be propagated undimiuished to any 

 extent. The case is somewhat different when we consider undulations 

 propagated in all directions at once. Imagine a small sphere, which 

 is uniformly elastic in every part, and which, by some interior 

 mechanism, is suddenly diminished in its dimensions, and afterwards 

 as suddenly restored. A wave of rarefaction and condensation will be 

 propagated in every direction ; which wave, at any instant, will be 

 contained between two spheres, concentric with the sphere already 

 mentioned, the radii of which differ by the length of the double wave : 

 at least, unless there be some reason in the state of the atmosphere, 

 why the propagation should take place more quickly in one direction 

 than another. We have no reason, at first sight, to suppose that the 

 velocity of propagation would be exactly, or even nearly the same as if 

 a portion of the air through which the waves pass had been contained 

 in a tube, unconnected with the exterior air. But it is found, both by 

 mathematical analysis and experiment, that the velocity of propagation 

 remains unaltered in both cases ; and also that the abtolute velocities 

 of the particles diminish. This last is a natural consequence of a very 

 simple principle namely, that when one body, or collection of bodies, 

 strikes a larger body, or collection of bodies, in such a way that its 

 whole motion is destroyed, the velocity of the larger body will not be 

 so great as that of the communicating body, but less in the same 

 proportion as its mass is greater. The law of this diminution should 

 lie, from theory, inrerntly a the dittance ; that is, by the time the 

 wave has moved from 3 miles to 5 miles, the compressions and velo- 

 cities should be as 5 to 3 ; but we have no direct means of submitting 

 this to experiment, the absolute velocities being imperceptible. 



We now proceed to the application of these principles. We know 

 that when the air is violently or rapidly propelled in any direction, 

 undulations such as we have described are produced, and that the 

 impression called sound is produced also. When a gun is fired, the 



