into Uniform Undulations of Flat Wavelets. 215 



reveals to us when we attempt to pry into her operations at 

 close quarters. 



18. The other of the two substitutions spoken of in § 4, 

 may be effected in almost exactly the same way. When two 

 u f w's of wave-frequencies fa and <j> 2 (or of the corresponding 

 wave-lengths X x and \ 2 ) travel in the same direction, they 

 may, if similarly polarized and of equal intensity, be repre- 

 sented by equations which, by a suitable selection of origin 

 and coordinates, simplify into 



^ 1 = Asin[27r(o?-i;0^i] \ ^ 



y 2 — A sin [ftir^x — vt)<f> 2 ~\ i 



When both of these undulations are present they produce a 

 resultant effect represented by 



y,+jfc=2Aain [2v(*-rt)^±&] . cos [fa (x- vt) &=*i] , (4) 



which tells us that at each station in space beats occur with a 

 frequency (fa — </>i)/2, and that the loops between the nodes 

 consist of waves of which the frequency is (<fi 2 + fa)/'2, the " 

 phases of the corresponding waves in any two consecutive 

 loops being the reverse of one another. 



19. A convenient way to deal with them is to imagine the 

 waves to become stationary and rigid at a given instant of 

 time *, and then to imagine the whole of this rigid system to 

 be borne forward, in a direction perpendicular to the wave- 

 fronts, with the speed v, i. e. with the speed of light. It will 

 then sufficiently represent what occurs in nature. 



Let us select £ = as the time when the system becomes 

 rigid. The components will thenceforth be represented by 

 the equations 



#i = A sin (2irxfa) 



X .... (5) 

 y 2 = A sin (2tt xfa) J 

 and the resultant by 



yi -y 2 =2A sin (W^±*i) . cos hwx *•=*•), . (6) 



which is a rigid undulation f with loops and nodes, the middles 



* Another convenient device is to make .r=0 in eqn. (4). and in this 

 way to ascertain what happens at a given station in space as time Hows 

 on. It is quite immaterial which method we employ,: of the two. that 

 in the text is perhaps the more easily handled. 



t In speaking of the rigid system it is convenient to continue to use 

 the terms wave, undulation, &c, although there is no motion. Thev 

 are used in the same sense as when one speaks of an undulating land- 

 scape. 



T 2 



