ANALYSIS OF THE IONOSPHERE 463 



Since it is entirely the signal-speed which determines the delay of 

 the echo, the wave-speed may seem a pointless side-issue. This 

 quantity u is, however, essential in the theory from which are derived 

 first the conditions of echoing and then the dependence of v upon /o. 

 Postponing the topics of signal-speed and echo-delay in order to return 

 to them later with better preparation, we now take up the theory. 



Theory of Wave-Speed, Total Reflection and Group-Speed 



IN THE Ionosphere 



It will now be proved from Maxwell's theory, combined with the 

 concept of mobile ions, that total reflection of wireless waves must 

 occur in the ionosphere at the level where the ion-density attains a 

 certain value depending on the wave-frequency. 



The famous equations of Maxwell melt together into a wave- 

 equation. The waves which it describes consist of an oscillating 

 electric field which I will denote by -Eo sin nt, and an oscillating mag- 

 netic field which we are permitted to ignore. When of high enough 

 frequency these are the waves of light, as Maxwell knew; when 

 of the frequency-range with which we are now concerned they are 

 the waves of radio, as Maxwell was never to know because of his 

 premature death. In the wave-equation there figures of course the 

 wave-speed u. Here then is a paraphrase of the great idea of Maxwell : 

 the square of the wave-speed varies inversely as the current-density provoked 

 by unit amplitude ^ of the oscillating field. 



Now we see at once that in the ionosphere the wave-speed must be 

 affected by the presence of the free electrons, since they are set into 

 oscillation by the waves and therefore make a contribution to the 

 current-density. 



At this point those who were educated in the electronic era (an 

 ever-increasing fraction of the population) are in some danger of 

 falling into a serious error. One may in fact assume that the electrons 

 form the whole of the current, and deduce that in vacuo the oscillating 

 field provokes no current at all, and the wave-speed must therefore 

 be infinite — an absurd conclusion ! Maxwell was wiser. He under- 

 stood, and made it a part of his theory, that wherever there is an 

 electric field which is changing in time the rate-of-change of that field 

 is equivalent to a current. This he called the "displacement-current," 

 and for the case of vacuum he said that the displacement-current- 

 density is precisely equal to the rate-of-change of the field, multiplied 

 by l/4x. 



^ I introduce the words "unit amplitude" to shield the reader from drawing the 

 false inference that wave-speed depends upon wave-amplitude. 



