

Sound- Waves of Finite Amplitude. 32 1 



along the bar. If one of the spheres were moved backwards 

 and forwards through a small range, a disturbance would 

 travel through the whole system, but owing to the weakness 

 of the connecting springs it would travel very slowly. Sup- 

 pose, now, that the last sphere on the left hand is connected to 

 a movable piston by a spring half the length of the others, 

 but otherwise similar to them ; and let this piston be suddenly 

 moved to the right with a considerable velocity which is kept 

 constant, and which we may call unity. The weak connecting 

 spring between the piston and the first sphere produces no 

 sensible effect until the two are almost in contact, when the 

 sphere rebounds with velocity 2. This first sphere then 

 strikes the second, imparting to it the velocity 2, and at the 

 same time coming to rest. The positions of the spheres after 

 successive equal intervals of time are represented in fig. 3, 

 where the number written on any sphere represents its velo- 

 city just after the impact which it is suffering. No number 

 is written on those spheres which have not so far been affected 

 by the motion. From this it will be evident that when the 

 piston moves to the right with a constant velocity which is 

 very great compared with the propagation -velocity of infini- 

 tesimal vibrations of the system, the disturbance advances to 

 the right with twice the velocity of the piston, provided that 

 the diameters of the spheres are excluded from the reckoning. 



Now suppose that the spheres are too small and too close 

 together to be individually distinguished; then, at any instant, 

 the system will appear to be divisible into two parts, in one 

 of which the velocity is unity, while in the other it is zero ; 

 and in the moving part the spheres will appear to be twice as 

 thickly condensed as in the still part. That the constant 

 velocity of the piston is very great compared with the propa- 

 gation-velocity of small vibrations is of course only a sup- 

 position introduced for the sake of simplicity. If, on the 

 other hand, these two velocities are comparable, two adjacent 

 spheres will always remain finitely separated from one another, 

 and the velocity of any individual sphere within the disturbed 

 stretch will never be as small as zero, or as great as twice the 

 velocity of the piston ; the mean velocity within the disturbed 

 stretch being equal to that of the piston. When the spheres 

 are very small and very close together, we shall still have 

 apparently an abrupt transition from finite velocity and greater 

 density to zero velocity and smaller density; and the energy, 

 which is apparently lost as the spheres pass from the la t ten- 

 condition to the former, exists as energy of relative motion 

 and unequal relative displacement amongst the spheres in the 

 disturbed stretch. 



5. Let us now compare the case just considered with the 



