61 



ACOUSTICS. 



ACOUSTICS. 



03 



cessively repeated in quantity, but changed into their contraries ; that 

 is, each portion undergoes successive rarefactions, equal in amount to 

 the former condensation*, and the particles move toicardf c with the 

 same velocities which they formerly had from c. This continues until 

 the piston reaches A again, after which the same phenomena recom- 

 mence in the same order. Thus it appears that the absolute velocity 

 of each particle is in the direction of the propagation so long as it is 

 compressed ; but in the contrary direction, when it is rarefied, and that 

 each particle, during the progress of a double series of compressions 

 and rarefactions, moves forward in the direction of propagation, and 

 back again to its former place, where it rests, unless a third vibration 

 follow the first two. When we talk of the compression of a particle, we 

 mean that it is nearer the succeeding particle than it would have been 

 in its natural state ; and rice rersd for rarefaction. We may represent 

 these phenomena in the following table, which, to save room, is made 

 on the supposition that A c was divided into four parts, and might be 

 equally well constructed if the number of parts into which A c was 

 divided had been greater. The numbers in the top horizontal line are 

 the successive portions of the tube, those in the left vertical column 

 the successive instants of time, and under any portion of the tube, 

 opposite to any instant of time, will be found the state in which that 

 portion of the tube is at that instant of time, 1 denoting its first com- 

 pression, 1' its first rarefaction ; these latter numbers recommencing 

 when a complete cycle of changes is finished. The blanks denote that 

 the effect has not yet reached the corresponding particles. 



2 3 -1 



6 7 8 9 10 11 12 13 14 15 16 





On carting the eye down any vertical column, we see the state of the 

 same portion in successive instants of time : on looking along a hori- 

 zontal column, we see the state of all the portions of the tube, at the 

 mine instant, as far as the effect has reached them. In the latter case, 

 we see that all the successive states are continually repeated, in such a 

 way, that whatever states two portions may be in, the intermediate 

 lortions have all the intermediate states. There is also at the beginning 

 an unfinished series in process of formation. If we look down a column, 

 we see that any one particle successively undergoes the different states, 

 from the moment when the effect first reaches it. We shall now sup- 

 pose the division of A c to go on without end, and examine the final 

 result. The different states of compression or rarefaction will then 

 become more and more numerous, but the difference of quantity 

 between each and its preceding will become less and less, so that when 

 we at laet give to the piston a continuous or gradually increasing and 

 decreasing velocity, we must also suppose a continuous or gradually 

 increasing and decreasing compression or rarefaction of the air in the 

 tube. This being premised, we return to the figure, and construct the 

 type of the motion of the piston, both backwards and forwards, and also 

 i* of the state in which the particles of air actually are for two 

 "T three several positions of the spring; as in the figure below, which 

 we proceed hi explain. (Pitj. 5.) 



Fif. 



In f-j. ^ (1) the pinion ban travelled frnm A to the small perpen- 



dicular, through something more than a quarter of a vibration : 

 the first disturbance has reached D, and the curve D K is the type of the 

 state of each particle as to velocity ; that is, the perpendicular F G is 

 the rate per second at which the particle P is moving from c, and the 

 same for every other perpendicular. 



If the piston be performing its third complete vibration, or its second 

 vibration forwards, there will have been a preceding series of com- 

 pressions and rarefactions propagated onwards, as in fy. 5 (1). In 

 fig. 5 (2), a vibration forwards has been completed ; the curve on c D 

 now represents a complete undulation, as far as the compressions are 

 concerned. In fy. 5 (3), the return of the piston has commenced, and 

 the particles between c and D are rarefied, and moving towards c ; this 

 we explain by placing the type beneath the tube, and dotting the curve ; 

 F G expressing the velocity per second of the particle p towards c. The 

 length of the whole wave c D is easily calculated. If, for example, the 

 single vibrations of the piston are made in ^ of a second, the first 

 impulse will have travelled through one hundredth part of 1125 feet, 

 or 11J feet. This is the length of c D, in fy. 5 (2). The complete 

 series of compressions is called a wave of compression ; and that of 

 rarefactions a ware of rarefaction. And the same type which repre- 

 sents to the eye the velocities of the various particles, will also serve 

 to represent the degrees of compression or rarefaction. For those par- 

 ticles which are moving quickest from c are most compressed, and those 

 which move quickest towards c are most rarefied. In returning to 

 fy. 4, we see that A 1, 1 2, 2 3, &c., are spaces described in equal times, 

 and are therefore in the same proportions as the velocities, that is, as 

 1^), 2q, Sr, &c. But these spaces, in the preceding explanation, are 

 proportional to the degrees of condensation ; these latter then are 

 proportional to the velocities. If, then, we suppose the series of 

 compressions and rarefactions to have gone on for some time, and 

 an unfinished wave of compression to have been formed at the instant 

 we are considering, we may represent the whole state of the particles 

 iu the tube at that instant by the following figure (fy. 6) : B G N 

 is a line parallel to the tube, and therefore a F is of the same 

 length for all positions of F. It is to be made 1125 feet iu 

 length. Its use depends upon the following proposition : That in 

 the simple undulation which we are now considering, so long as the 

 disturbance is small, the velocity of any particle bears to the velocity 

 of propagation (two very distinct things, as we have before observed) 

 the same proportion as the change in the density bears to the 

 density of undisturbed air. This follows from the investigation at- 

 tached to fg. 4 : for, in the fourth instant for example, the column 3 4 

 of air is forced into c P, and 3 4 and c V being spaces described in equal 



Fig. C. 



times with velocities 4 and 1125 feet per second, are spaces propor- 

 tional to these velocities. And the compression will be the same if we 

 increase c P in any proportion, provided we increase the quantity of air 

 forced into it in the same proportion. A similar proposition holds for 

 rarefactions. Or, in other words, F K being the velocity with which 

 the particle at F is moving towards c, the rarefaction of the particles at 

 F is that which would be obtained by allowing the air naturally con- 

 tained in a tube Q F, 1125 feet long, to expand into the length G K. 

 Similarly, the compression at L is that which would be obtained by 

 compressing the air in a tube N L into the shorter tube N M. If we 

 wish to see the state of these particles at any succeeding instant, let 

 the curvilinear part of the figure travel uniformly forward at the rate 

 of 1125 feet per second, new curves being continually formed and 

 finished at c : we shall thus have the state of the whole tube at any 

 succeeding moment. Before proceeding to apply this explanation to 

 the phenomena of sound, we must see what will take place if the tube 

 be agitated by several different undulations at once. 



All readers, however little acquainted with Mechanics, are awre, 

 that if a body be impressed by two forces in the same direction, it will 

 proceed with the sum of the velocities produced by the two forces ; and 

 with the difference of the velocities, if the forces act in contrary direc- 

 tions, the motion in the latter case being iu the direction of the greater 

 of the forces. Hence, if there be different undulations excited in the 

 same column of air, the velocities of each particle will be made up of 

 the mm or difference of those which it would have received from each 

 undulation, had each acted alone ; the mm when it would have been 

 compressed by both, or rarefied by both, and the difference when it 

 would have been compressed by one and rarefied by the other. And 

 the compressions or rarefactions being proportional to the velocities, a 

 similar proposition will hold of them. We have represented in Jig. 7, 

 the state in which a column of air would be at n given instant from 



