380 Mr. 0. Heaviside on Electromagnetic Waves, and the 



47. Solution with outer screen ; K^co ; f constant. — Let 

 there be no inner screen and let the outer he perfectly con- 

 ducting. As J. J. Thomson has considered these screens*, I 

 will he very brief, regarding them here only in relation to the 

 sheet of/and to former solutions. The determinantal equation is 



< = 0, or tan*=<L— a*)- 1 , . . . (292) 



if x=ipa/v. Roots nearly tt, 27r, 37t, &c; except the first, 

 which is considerably less. The solution due to starting/ 

 constant, by (289) is therefore 



Il = (fva/ f ivr)t(uu a w 1 '/a 1 u 1 ")€P t ; . . (293) 



which, developed by pairing terms, leads to 



„ fva v a? 4 — <« 2 + l . vtx( a x . \xr ( a x . \xa /rxn , 



H = ' / — 2, o, 9 — ^ 2 sin — ( cos sin J — ( cos L sm ) — , (294 



fivr or (or — 2) a^ \ xr 7 a 1 \ xa Ja 1 ' v ' 



which of course includes the effects of the infinite series of 

 reflexions at the barrier. By making a 1 =co, however, the 

 result should be the same as if the screen were non-existent, 

 because an infinite time must elapse before the first reflexion 

 can begin, and we are concerned only with finite intervals. 

 The result is 



fivr 



- I dx l l — I cos sin jxM cos sin )x } a, (295) 



7rJ x x \ x x r ) 1 \ x,a ) l ' v ' 



which must be the equivalent of the simple solution (142) of 

 § 21, showing the origin and progress of the wave. 



Now reduce it to a plane wave. We must make a infinite, 

 and r— a—z finite. Also take fv=e, constant. We then 

 have 



H= JL I f^ 1 ( s i n M^) + sin ^+*) ), . (2 96) 



QfAV TTjo \ X\ X\ J 



showing the H at x due to a plane sheet of vorticity of e 

 situated at 2=0. This is the equivalent of the solution (12) 

 of § 2, indicating the continuous uniform emission of 

 H = e/2fJLV both ways from the plane z=0. 



Returning to (294), it is clear that from t=0tot= (a 1 — a)/v, 

 the solution is the same as if there were no screen. Also if a 

 is a very small fraction of a 1? the electromagnetic wave of 

 depth 2a will, when it strikes the screen, be reflected nearly as 

 from a plane boundary. It would therefore seem that this 



* In the paper before referred to. 



