358 BELL SYSTEM TECHNICAL JOURNAL 



As a first step in building up a stationary pattern, in which there is no 

 steady propagation of energy in any direction, we combine two progressive 

 wave components (5) which are identical, except that their directions of 

 phase propagation along, say, the z axis are opposite. The signs of the last^ 

 terms are then opposite and the sum can be written 



s = lA cos (co/ ± kjcX ± kyj) cos kaZ 



Proceeding in the same way for x and y, we arrive at the standing wave 

 pattern, 



s = ^A cos co/ cos kxX cos kyV cos keZ. (6) 



Components of this sort, each with its own amplitude and phase, may be 

 combined to build up possible stationary patterns. However, we shall not 

 attempt here to build such patterns, but rather to deduce what information 

 we can from a study of a single component. 



Moving Wave Patterns 



In order to represent approximately a particle in uniform linear motion, 

 we are to look for a solution of (3) which represents a moving wave pattern 

 For this we make use of two functions which may readily be shown to be 

 such solutions, 



s = g+ U{oi + VK)t - ^(h + ^ j X ± kyy ± Kzj , 



s = g_ (b{c^ - Vkjt + /3 U, - ^j X ± k,y ± k^zj , 

 where co, kx , ky and k^ are real and satisfy (4), F is a real constant, and 



C2 



g+ represents a plane progressive wave the propagation of which along tht 

 X axis is in the positive direction. g_ represents one of lower frequencyJ 

 propagating in the negative x direction. Their wave numbers in the .v direc-j 

 tion differ in such a way that those in the y and z direction are the same foi 

 the two. In the plane wave case, where ky — kg = and co = ckx , they re- 

 duce to 



The two waves then travel in the .v direction with velocities c and 



c -\- V 



their frequencies are in the ratio — . 



^ f - V 



