March 17, 1899.] 



SCIENGE. 



395 



ing out the front tube (§12). Additional 

 change may be obtained by allowing colored 

 balls to ride on the levers. In case of equal 

 periods the result is chiefly interesting when 

 the amplitude varies from particle to parti- 

 cle. A. linear variation is well represented 

 by a plane wave oblique to the direction of 

 the axles, and in action is very striking. 



The more important wave with an ex- 

 ponentially varying amplitude is only given 

 when the axis of motion is along the corre- 

 sponding exponential curve horizontally, 

 but the effect to an observer at a little dis- 

 tance in front is none the less good. 



II. Preceding Case (I) with Additional Ve- 

 locity Superimposed on Either Wave Train. IC. 

 Beats. — If the component waves are trans- 

 mitted in like periods or velocities* and 

 amplitudes, the compound wave is trans- 

 mitted unchanged in form ; but if any of 

 these quantities vary, the compound wave 

 continually changes form. With the ap- 

 paratus as here adjusted the last case is 

 readily realized by sending on one wave 

 faster than the other. For instance, if the 

 component wave velocities be as 3 : 4 (rear 

 wave of greater speed) , then in 4 complete 

 turns at the crank the original wave will 

 be reproduced, while all intermediate phase 

 differences between corresponding particles 

 are passed continuously in turn. All pairs 

 of cams are undergoing like continuous 

 change of phase. 



The shadow picture of this case (sun- 

 light) shows a line elongating to maximum 

 displacement and then contracting to a 

 point in S. H. M. The slow change at 

 maximum elongation is in strong contrast 

 to the rapid change of length on passing 

 through the position of equilibrium. Sim- 

 ilarly in §14 the speed ratio must be care- 

 fully adjusted if the linear compound wave 

 is to persist. 



* In the present ^special case variation of one im- 

 plies the other;, in the sequel, period and velocity 

 mnst be carefully distinguished. 



The wave corresponding to this present 

 experiment is an excellent example of an 

 infinite beating wave train, two wave- 

 lengths of which are accessible at a given 

 place. The beats are due to a difference 

 of wave velocity and frequency together. 

 Though the two cases are usually gener- 

 ically different, the gross effect is here coin- 

 cident. As a luxury a cam axle containing 

 a small fraction of a wave-length more than 

 two complete wave-lengths might be sup- 

 plied. This would then show beats due to 

 difference of wave velocity for the same 

 period or (with the proper pulley) beats 

 due to difference of period for the same wave 

 velocity. The specific difference is this, 

 that, whereas in one case (equal compo- 

 nent wave-lengths) the compound harmonic 

 is at every instant (for all pulley ratios) 

 sinusoidal, in the other case (slightly dif- 

 ferent component wave-lengths) it is at no 

 instant strictly so. The latter adjustment 

 thus admits of beats either when the com- 

 ponent periods alone or the component 

 wave velocities alone are not the same. In 

 the former both necessarily change together. 



17. Doppler^s Principle. — If the beats are 

 obtained by a diiierence of wave velocity 

 the faster wave may be treated as having 

 an additional linear velocity virtually im- 

 pressed upon it in the direction of motion 

 from without. Its interference with the 

 wave not so affected is then an illustration 

 of Doppler's principle. 



III. Preceding Cases {I and II') with the Ve- 

 locity of Either Wave Train Reversed. 18. Equal 

 Velocities. Stationary Waves. — If one of the 

 component waves be passed along the axis 

 positively and the other in a negative di- 

 rection, i. e., if one axle be rotated clock- 

 wise and the other counter clockwise by 

 cross-belting equal pulleys, the compound 

 wave is of the stationary type, since ampli- 

 tudes were made effectively equal and 

 periods are necessarily equal. The effect on 

 the machine is striking, since the nodes are 



