chap, xxxn.] THE PRODUCTION OF BEATS. 431 



slightly shorter to counterbalance their increased 

 rapidity. The oscillation, so to speak, of his foot will 

 be faster, but the wave lengths will be shorter. The 

 result will be that at the second step his foot will be 

 slightly in advance of his friend's ; his friend's foot, 

 that is, will be slightly behind. With each step this 

 difference will be slightly increased, till after several 

 steps his right foot will touch the ground in front, 

 when his friend's is just being lifted from the ground 

 behind. The two feet, that is to say, will be in oppo- 

 site phases. The foot of each will be on opposite 

 sides of the middle line, the position of rest ; one 

 will be nearly as far forward as the other is behind, 

 and the mean position would be very nearly in the 

 position of rest. Suppose the walking to be pro- 

 ceeded with, with each step the difference will be 

 added to ; but now the foot that is behind will with 

 each step come farther and farther forward, till 

 after a few more steps the right foot of each man will 

 again touch the ground in front at the same instant, 

 and the two feet will be in the same phase. If the same 

 relative pace be maintained, after a few more steps 

 the feet will again be in opposite phases, and after a 

 few more again in the same phase ; they will alter- 

 nately coincide and be opposite to one another. The 

 more nearly the speed of the two is similar, the more 

 seldom will they coincide and be opposed ; and the 

 greater the difference in speed the oftener will the 

 coincidence and opposition happen. Now, for the 

 feet of the two men substitute two wave movements, 

 whose periods are very nearly the same, one being, 

 however, shorter than the other, that is, with slightly 

 quicker vibrations, and suppose them to start together 

 through the air. Because of the slight difference 

 they will alternately coincide and be opposite. When 

 they coincide the vibrating particles will be in the 

 same phase, will be moved by both in the same 



