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



2 = plane, with the full magnetic field to correspond, and 

 from it immediately follows the E solution due to any initial 

 distribution of electric current in plane layers. 



Owing to H being permanently ^H u at the origin in the 

 case (49), (54), when # = 0, we may state the problem thus : 

 An infinite conducting dielectric with a plane boundary is 

 initially free from magnetic induction, and its boundary 

 suddenly receives the magnetic force H = constant. At time 

 t later (49) and (52) or (53) give the state of the conductor 

 at distance z<vt from the boundary. In a good conductor 

 the attenuation at the front of the wave is so enormous that 

 the diffusion solution (50) applies practically. It is only in 

 bad conductors that the more complete form is required. 



10. Effect of Impressed Force. — We can show that the initial 

 effect of impressed force is the same as if the dielectric were 

 nonconducting. In equations (23), (24), let p = ni, where 

 n2 it — frequency of alternations, supposing e to alternate 

 rapidly. By increasing n we can make the second terms on 

 the right sides be as great multiples of the first terms as we 

 please, so that in the limit we have results independent of 

 k and #, in this respect, that as the frequency is raised in- 

 finitely, the true solutions tend to be infinitely nearly repre- 

 sented by simplified forms, in which k and g play the part of 

 small quantities. An inspection of the sinusoidal solution for 

 plane waves shows that E and H get into the same phase, and 

 that k and g merely present themselves in the exponents of 

 factors representing attenuation of amplitude as the waves 

 pass away from the seat of vorticity of impressed force. 



Consequently, in the plane problem, the initial effect of an 

 abrupt discontinuity in e, say e = constant on the left, and zero 

 on the right side of the plane through the origin, is to pro- 

 duce 



H= — e/2fiv (58) 



all over the plane of vorticity; and 



E=+i« (59) 



on its left and right sides respectively. We may regard the 

 plane as continuously emitting these disturbances to right 

 and left at speed v so long as the impressed force is in opera- 

 tion, but their subsequent history can only be fully represented 

 by the tail formulae already given. 



Irrespective of the finite curvature of a surface, any 

 element thereof may be regarded as plane. Therefore every 

 element of a sheet of vortex lines of impressed force acts in 

 the way just described as being true of the elements of an 

 infinite plane sheet. But it is only in comparatively simple 



