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



These begin to operate at r—a when vt=2a ; and later, the 

 range is from r = a to rssvt— a. 



This completes the mathematical work. As a check upon 

 the accuracy, we may test satisfaction of differential equa- 

 tions, and of the initial condition, and that the four solutions 

 join together with the proper discontinuities. 



73. The following is a general description of the manner 

 of establishing the steady flux. We put on e in the cylinder 

 when £ = 0. The first effect inside is E x = ^ at the surface 

 and H 1 =E 1 //av. This primary disturbance runs in to the 

 axis at speed v, varying at its front inversely as the square 

 root of the distance from the axis, thus producing a momen- 

 tary infinity there. At this moment t = a/v, Ej is also very 

 great near the axis. In the meantime E x has been increasing 

 generally all over the cylinder, so that, from being \e initially 

 at the boundary, it has risen to 'lie, whilst the simultaneous 

 value at r = ^a is about '956. 



Now consider E 3 within the cylinder, it being the natural 

 continuation of E x . The large values of E T near the axis 

 subside with immense rapidity. But near the boundary 

 E x still goes on increasing. The result is that when vt = 2a, 

 and the front of the return wave reaches the boundary, E 3 

 has fallen from qo to 1*154^ at the axis ; at r—\a the value 

 is 1*183<? ; at r=ja it is l'2Ble; and at the boundary the 

 value has risen to l'lle, which is made up thus, l*21g+ \e ; 

 the first of these being the value just before the front of the 

 return wave arrives, the second part the sudden increase due 

 to the wave-front. E 3 is now a minimum at the axis and 

 rises towards the wave-front, the greater part of the rise being 

 near the wave-front. 



Thirdly, go back to £ = and consider the outward wave. 

 First, E 2 = —\e at r=a. This runs out at speed v, varying 

 at the front inversely as ri As it does so, the E 2 that 

 succeeds rises, that is, is less negative. Thus when vt = a, 

 and the front has got to r=2a, the values of E 2 are —-"2326 

 at r = a and — '353^ at r=2a. Still later, as this wave forms 

 fully, its hinder part becomes positive. Thus, when fully 

 formed, with front at r=3«, we have E 2 =. — *288<? at r = Sa ; 

 — •1450 at r = 2a ; and '2le at r=a. This is at the moment 

 when the return wave reaches the boundary, as already 

 described. 



The subsequent history is that the wave E 2 moves out to 

 infinity, being negative at its front and positive at its back, 

 where there is a sudden rise due to the return wave E 4 , behind 

 which there is a rapid fall in E 4 , not a discontinuity, but the 

 continuation of the before-mentioned rapid fall in E 3 near its 



