1897.] Mr Pocklington, Electrical Oscillations in Wires. 329 



7. Induced Vibrations in a Ring. We will now consider the 

 case of a ring upon which plane waves are incident in a direction 

 parallel to the axis of the ring. 



Let the coordinate axes be chosen as before, and let the 

 incident vibration be given by 



P = R = 0, Q = Ae^ev*. 



The tangential force at the point 6 due to this wave together 

 with a disturbance induced in the wire of the fundamental mode 

 and magnitude B, is to be equated to zero, giving 



cos f w 

 = Ae 1 ^ cos - 2Be l P t - d(f>Tl cos <f> (1 - a 2 a 2 cos 6). 



a Jo 



If we neglect e/a in comparison with unity, this becomes, as in § 6, 



= A + —- 



a? 



-f + (f-iV 



+ f )- % J, (•) + (2 - f ) J, (,■) + (f - l) \'W. (.)} 



7T 



x 2 [* , . cos (# sin ilr) — 1 

 + sj/^ - \ in ^ (1 - 2 «» 2f + cos ty) 



TT 



(x- , \ f a 7 . cos (x sin ■&) — 1 _ , " 

 + (4- 1 )j ^- sin/ ~ C ° B2f 



Unless a; =2, this gives 



2 A a 2 



B = 



(4>-x?)L: 



so that in general the induced vibration is small, and the 

 thinner the wire the smaller the induced vibration. The phase is 

 the same as or opposite to that of the electric displacement in 

 the plane of the ring due to the incident wave. If, however, 

 x = 2 = 2 + £, where £ is small, we may put x = 2 in all the terms 

 not involving L, and get 



„ Aa 2 



- 2££ + -970 - l-055t 



The maximum amplitude of the induced vibration is obtained 

 when 2X| = -970, or a = x/2a = {1 + -243/Z}/a, i.e. when the period 

 of the incident wave is the same as the free period of the ring ; 

 the amplitude then is *948 Aa 2 , and the phase is in quadrature 



