Forced Vibrations of Electromagnetic Systems. 381 



wave would run to and from between the origin and boundary 

 unceasingly. This is to a great extent true ; and therefore 

 there is no truly permanent state (the electric flux, namely, 

 alone) ; but examination shows that the reflexion is not clean, 

 on account of the electrification of the boundary, so that there 

 is a spreading of the magnetic field all over the region within 

 the screen. 



48. Alternating f with reflecting barriers. Forced vibrations. 

 — Let the medium be nonconducting between the boundaries 

 a and a x . Equation (2SS) then becomes 



H= J^_ (tf«-yoM«)(tt-yittQ , (297) 



p 1 y\-y Q ■ • v ) 



giving H outside the surface of/. We see that y =0 and 

 u a = make H = 0. That is, the forced vibrations are confined 

 to the inside of the surface of/ only, at the frequencies given 

 by u a =0, provided there is no internal screen to disturb, but 

 independently of the structure of the external medium (since 

 y 1 is undetermined so far), with possible exceptions due to the 

 vanishing of y x simultaneously. But (297), sinusoidally 

 realized by p 2 — —n 2 , does not represent the full final solution, 

 unless the nature of y and y x is such as to allow the initial 

 departure from this solution to be dissipated in space or killed 

 by resistance. Ignoring the free vibrations, let ?/o = 0, and 

 yi = iii/wi, meaning no internal, and an infinitely conducting 

 external screen. Then 



(out) ~H.= (va/fjLvi')ii a ^uw 1 / /u i / — ivlf, -j 

 (in) R=(va/fivr)u {u a Wi[ juj -w a \f. J ' ' (298) 

 If 10/= 0, or in full, 



(v/na^ tan (najv) = 1 — (v/naj) 2 , 

 we obtain a simplification, viz. 



H (in0 rout) = — (va/fivr)(uw a or u a w)f ; . . (299) 



and the corresponding tangential components of electric force 

 are 



E (in or out) = (va/fj,vr)(u / iv a or u a w')(cp)- l f. . (300) 



But if w/ = 0, the result is infinite. This condition indi- 

 cates that the frequency coincides with that of one of the free 

 vibrations possible w T ithin the sphere r = a x without impressed 

 force. But, considering that w r e may confine our impressed 

 force to as small a space as we please round the origin, the 



