ACOUSTICS. 



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two different waves, the type* of which are drawn, and the broad 

 line is the type of their united effect.. We know that any two 

 lengths are either in the proportion of two whole numbers, or if 

 not, two whole numbers can be found, which are as nearly propor- 

 tional to them as we please. We have, to take a simple case, drawn 

 the lengths of the waves in the proportion of 5 to 4. (t'iy. 7.> The 



types of the waves are different portions of straight lines, one whole 

 condensation and rarefaction taking place, as indicated by A a B 4 c in 

 the first, and by A;> Poq in the second. We suppose the waves to 

 commence together. Thin Kiippocition, of the condensation and rare- 

 faction proceeding in such a way that their types shall be partH of 

 straight lines, is not to be obtained in [wactice, since, as we have seen, 



uch motion as that of a spring, and (we may add) of n string or of a 

 drum, would produce regular curves. But it in as allowable in illus- 

 trating the effect* of combined undulations as any other ; and if, inure- 

 over, we round the comers of the types of the single wares, thus making 

 them present an appearance similar to that in the preceding figures, a 

 slight rounding of the comers of the broad line will show sufficiently 

 well what the combined wave would have been, if the preceding figures 

 had been rounded. And the supposition of rectilinear types facilitates 

 the drawing of such figures (which we would recommend to our 

 readers), since, as they will observe, the type of the combined wave 

 consists also of portions of straight lilies which break off only when the 

 type of one of the single waves changes from one line to another. The 

 general rule for forming the broad line, derived from a preceding 

 observation, is let the perpendicular or ordiimte [ABSCISSA] be the 

 MM of the perpendiculars of the types of the waves, when they fall on 

 the tame side of A p, and the difference when they fall on different 

 sides ; observing, in the latter case, to let the broad line fall on the side 

 of that wave which has the greatest perpendicular. Thus at the first 

 M, 11 T is the sum of M u and at v, and particles at M are in a 

 greater state of compression than the first wave would give them, 

 which arises from the second ; similarly at the second M there is an 

 increase of rarefaction. At X, the air is compressed by one wave, and 

 rarefied by the other, but more compressed than rarefied. At p, B, 

 Q, c, Ac., where one of the waves causes neither compression nor rare- 

 faction, the brood line coincides with the other wave. 

 On looking at the figure thus produced, we see 



1. That it is composed of a cycle of successive compressions and 

 rarefactions, in which, however, the rarefactions differ in kind from the 

 preceding compressions; so that we must not give the term inu-e to 

 each set of compressions or rarefactions, as we reserve this word to 

 denote cycles of changes, which are followed by similar cycles of con- 

 trary changes. 



2. That when the lengths of two waves are as five and four, four of 

 the first will be as long as five of the second ; HO that the waves recom- 

 mence together at w, but not exactly as before, the wave of condensa- 

 tion from the first being accompanied by the wave of rarefaction from 

 the second. This difference, however, is not found at the end of the 

 second similar cycle of four and five ; so that after eight of the first 

 waves, corresponding to ten of the second, the combined wave begins 

 again to have the same form as at first. 



3. The complete cycle denoted by the broad line may be divided into 

 two, joining at w; in the second of which a series of rarefaction* \,~ 

 found similar to every series of compressions in the first, and rife rend. 

 We may, therefore, give the name of wave to the part of the broad 

 line intercepted between A ami w, consistently with our definition of 

 thin word. 



4. If the waves had not begun together, a wave would have resulted 

 of the same length as the preceding, if we began at any point where 

 the compression from one was exactly compensated by the rar< : 

 from the other. 



5. If Iw.tli waves had been of the same length, the resulting wave 

 would have had that length ; or if the first wave luul been contained 

 an exact number of times in the second, the reuniting wave would li i\ . 

 been of the length of the second. \Vc subjoin a cut (fy. 8) repre- 

 senting a wave contained three times in anotln-r wave, and the resulting 

 wnvr. 



Klg. 8. 



We have hitherto considered combined undulations as propagated in 

 the same direction : let us now take two waves of equal lengths pro- 

 pagated in opposite directions, rising, as we may suppose, from two 

 pistons, one at each end of the tube. After a certain time, depending 

 on the length of the tube, two waves will meet, by which we mean 

 that the particles will begin to be affected by the motion of Uith 

 piston*, and the manner in which the joint effect is represented is the 

 some as before, though the phenomena are very different. In the 

 former case, having represented the resulting wave at one instant, we 

 could trace the change of state throughout every particle of the fluid, 

 by supposing the type of that resulting wave, or a succession of such 

 types, to move along the tube at the rate of 1125 feet per second ; in 

 the present case, the waves are propagated in contrary direct i 

 that any given effect from the first wave is no longer continually 

 accompanied by another given effect from the second wave. We uiu-t 

 also recollect, that the motion of the particles in each wave of eom- 

 pression is in the direction of the propagation ; so that a (nil-tide under 

 the action of two waves of compression, has opposite velocities impressed 

 upon it, and therefore moves with the difference of the velocities; 

 and so on. 



Now let A, B, c, D, Ac., be the points where the tv. i waves 



meet in the axis, and let us choose the instant of meeting for the time 

 under consideration. Let the continued Hue represent the waves 

 propagated from left to right, and the dotted line those propagated 

 from right to left, as marked by the arrows at the parts at whieh they 

 end ; tile arrows above them representing the direction.- nf the alwolute 

 velocities whieh the waves over which they are placed give to the 

 particles. (Fly 9). All the particles are now neither eorapremed nor 

 rarefied ; for it is evident that, whatever condensation or rarefaction a 



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particle experiences from the wave moving to the right, there is a 

 contrary rarefaction or condensation from that which moves to the 

 left But every particle has the velocity derived from either wave 

 doubled by the other. Again, the particular points A, B, c, D, Ac., 

 are never put in motion : fr it is plain that by the time any point p 

 comes over c, giving it the velocity of P p to the left, the point q, 

 similarly placed on the other wave, will also have come over c, giving 

 it the equal and contrary velocity q 7 ; so that, as far as velocity is 

 concerned, all the impression produced on A, B, c, D, Ac., is equivalent 

 to two equal and contrary velocities, or to no velocity at all, for we 

 are considering the case of particles, and not of rigid bodies, where 

 such opposite equal forces would form a " copit," and produce 

 rotatory motion. But when p has come over c, the compression, 

 answering to P p, is doubled by that answering to q 7. So that 

 the particles at A. B, c, Ac., undergo no change of place, but only 

 condensation or rarefaction. Also the particles at o, b, Ac., halfway 

 between A and B, B and c, Ac., never undergo compression or rare- 

 faction, but only change of velocity. For by the tune any point n, 

 from one wave, has come over a, with the condensation answering 

 to r, i will have come over it from h other, with the equal rare- 



f. u-t ion answering to 8 ; so that the effect of the combined wave- 

 upon a, is always that answering to equal condonation and rarefaction, 

 or no change at all. But the velocities answering to R r and s i are 

 equal, and in the name direction ; so that the points a, b, Ac., have the 

 velocities which one wave would have given them doubled by the. 

 other. Hence at a. b, o, Ac., the particles suffer no change of state, 

 but are only moved backwards and forwards. Now, let the time of 

 half a wave elapse, in which case the types of the undulations will 

 coincide, and those part* will be over the capitals on the axis, which 

 are now over the small letters, and rice rmd, an in ></ '" where the 

 coincidence is denoted by a continued and dotted line together, the 

 latter being, of course, a little displaced. 



Half a wave since, all compression and rarefaction had disappeared 

 throughout the tube, the velocity of every particle being double that 

 which either wave would have caused. The case is now altered ; no 

 particle has any velocity, since there are the signs of equal and con- 

 trary velocities at every point of the tube ; but every particle is either 

 doubly compressed, or doubly rarefied, except n, b, Ac., which, as we 

 proved, are never either compressed or rarefied. In one more half 

 wave, the phenomena of the first supposition will be repeated ; that is, 



