256 Prof. Barton and Miss Browning on Coupled 



The ratios o£ the amplitudes of the quick vibrations to 

 those of the slow ones are seen to be 



1: ±4/(1+0) (44) 



in the y and z vibrations respectively. 



(ii.) Single Displacement. — Now let one bob be pulled 

 horizontally aside while the other is held in the zero position, 

 both constraints ceasing at the same instant. We may thus 

 write 



y = 0, z=b, |=0, g=0, for t=0. . . (45) 



These conditions inserted in (36)-(39) yield equations 

 satisfied by 



E— 2,F= g , e= f , #= f . • . . (46) 



And these values in (36) and (37) give, for this mode of 

 starting, the special solution 



b , b mt ,,-v 



y=-2 cosm * + ^ 00 V(i+£) ' • • • (47) 



b b mt /ao . 



z= - oos mt + -cos y^-^y ... (48) 



Thus the ratios of amplitudes of quick and slow vibrations 

 in the y and z traces are respectively 



+ 1 (49) 



That is, the amplitudes of the superposed vibrations are 

 numerically equal for any values of the coupling. 



The symmetry of the equations shows that the motions 

 will interchange simply if the other pendulums be struck or 

 displaced. 



(iii.) Double Displacement. — Let one bob be drawn hori- 

 zontally aside, the other hanging motionless in its equilibrium 

 but slightly-displaced position. Thus if 



z — b it follows that y= - — ^, 

 the other conditions being y . , (50) 



it=° and di=°- 



