589-592] 



Non-conducting Media 



515 



Either of these solutions represents the propagation of a plane wave. 

 The direction-cosines of the direction of propagation are I, m, n, and the 

 velocity of propagation is a. Usually it will be found simplest to take the 

 value of u given by equation (547) as the solution of the equation and reject 

 imaginary terms after the analysis is completed. This procedure will of 

 course give the same result as would be obtained by taking equation (548) 

 as solution of the differential equation. 



Propagation of a Plane Wave. 



592. Let us now consider in detail the propagation of a plane wave of 

 light, the direction of propagation being taken, for simplicity, to be the axis 

 of as. The values of X, Y, Z, a, ft, y must all be solutions of the differential 

 equation, each being of the form 



u = Ae iK(x ~ at} (549). 



The six values of X, Y, Z, a, ft, 7 are not independent, being connected 

 by the six equations of 574, namely 



KdX 



C dt 



KdY_ 

 G dt 



KdZ 



G dt 



dy dz 



dx 



dy 



> (A), 



From the form of solution (equation (549)), it is clear that all the differ- 

 ential operators may be replaced by multipliers. We may put 



d d 



1=1=0 

 dy dz 



The equations now become 



Ka v 



Y = 



c 



Ka 



Z= 



(A'), 



Since X = 0, a = 0, it appears that both the electric and magnetic forces 

 are, at every instant, at right angles to the axis of x, i.e. to the direction of 

 propagation. From the last two equations of system (A') we obtain 



ftY+ryZ=0, 



shewing that the electric force and the magnetic force are also at right angles 

 to one another. 



332 



