SOUND 181 15, we have drawn the resultant curves formed by combining the curves of musical sounds corresponding to the various consonant in- tervals indicated below the figures. As these FIG. 15. Resultant Curve formed by combining the curve of a musical note with that of its major third. A. : A : : 1 : . curves are the resultants formed by the com- bination of the curves of composite musical sounds, it follows that the components of these curves are not simple sinusoidal curves, as in the case of fig. 11, but are derived from the resultant of fig. 11 by reducing to one fourth the amplitude of that curve, and by taking wave lengths corresponding to intervals indi- cated below the figures. From the curves of figs. 13, 14, and 15 can be reproduced their generating motions in the same manner as we have done in the case of the curve of fig. 11. As a periodic or recurring vibration can alone produce in the ear the sensation of sound, and as the duration of the period is always equal to the least common multiple of the periods of the pendulum vibrations of the components, it follows that in the case of a sound formed of a harmonic series the period equals the time of one vibration of the fundamental ; but in the cases of other combinations the duration of the period increases with the complexity of the ratio of the times of vibration of the compo- nents ; thus, the durations of the periods of the following combinations are placed after them in fraction sofa second: G 3 + C*=?fa; 3 + G 3 = T ; C 3 +E 3 =^V; Cs+Es + Ga^^; C 3 + E 3 + Gr 3 + C4= 1 1 7 - of a second. (0 3 stands for the treble ; C 4 is the of the octave above it.) Transmission of Sound. If air were in- compressible, a motion produced at any point of its mass would instantaneously be trans- mitted to every other point of the atmosphere. Thus, if we imagine a long tube, open at one end and closed at the other by a piston which moves in the tube without friction, it is evident that if this piston were pushed into the tube a certain distance, the air would at the same time move out of the tube at the open end. But air is compressible and elastic, and after the piston has been pushed into the cylinder, a measurable interval of time will have elapsed before the air moves out of the open end of the tube. This interval is the time taken by sound to traverse the length of the tube. The velocity of sound is 1,090 ft. in a second at 32 F., and it increases almost exactly one foot in velocity for each degree of elevation of tem- perature above 32. Now imagine the piston to move forward into the tube over a minute fraction of an inch, and that it occupied -fa of a second in making this for ward motion; then the length of air compressed at the instant the piston has come to rest will be equal to J ^^ a , or 109 ft. If the piston makes its movement in y^g- and in y^ of a second, the length of air compressed in the tube will be respectively 10-9 and T09 ft. But such a compressed por- tion of air cannot remain at rest, by reason of its elasticity. It immediately expands, and in so doing presses forward on the undisturbed air in front of it and on the interior wall of the tube. The column of compressed air in thus regaining its natural density has com- pressed an air column of equal depth in front of it ; this in its turn reacts on the back col- umn and prevents it from rarefying, while at the same time it has compressed another col- umn of equal depth in front of it, and so on. Thus the sonorous pulse, as it is called, is transmitted through the whole length of the tube. A beautiful illustration of the manner in which a sound pulse is propagated is afford- ed by attaching to a sounding board a long, elastic spiral spring of brass, while the other end is held in the hand. On separating two of the coils of the spring with a finger nail, and then allowing them suddenly to come to- gether, a pulse or compression will be thrown through the whole length of the spring to its further end, where striking on the sounding board it will cause a sharp rap. This action against the board will be reflected from the board to the hand, and again from the hand to the board, and so on several times in succes- sion. When the piston above spoken of makes a backward movement, it creates a vacant space in the tube, into which the air rushes by virtue of its elasticity, and thus a certain depth of air is rarefied; this first cylinder of rarefied air in retracting to its natural dimen- sions causes rarefaction in an equal depth of air in front of it ; this second rarefied cylinder of air now reacts on the first, brings it to rest, and causes rarefaction in a third equal column of air, and so on. Thus the rarefaction, like the compression, is transmitted through the whole length of the tube. When a compression trav- erses the tube it successively brings the mole- cules of air nearer together, while a rarefac- tion in its progress separates the aerial mole- cules ; hence, if we imagine the piston to move backward and forward with a regular vibratory motion we have rarefaction succeeding com- pression in regular order, and the effect on any one molecule of air is to give it a like regular motion backward and forward. In the above discussion we have, for simplicity, supposed the piston to have a uniform velocity during its motions ; but this, as we have already seen, is not the case with freely vibrating elastic bodies, for they have the same character of reciprocating motion as that of a freely swing- ing pendulum. To explain what will be the effect on the air of such a motion, we will suppose that the piston vibrates through a
