88 Dr. C. V. Burton on the Scattering and 



Thus exp i/irj, containing the real exponential factor 

 exTpfi 2 y, becomes infinite with rj, and at the same time 

 exp( — i/jLT)) vanishes. Accordingly, when the laminar 

 region filled with secondary vibrators is infinitely thick, 

 (25), (26) both lead to 



C 1= =0, C 2 =-2 % A/(/*+l); 



and hence (with the accents dropped) 



E = - % _1 B = 2A(> + 1) _1 exp (— i pyx) 



= 2A{ (ft + 1) 2 + fi 2 2 } _1 (^i + l + ^ 2 ) exp ( — \x 2 vx) exp ( — i fi lV x) 



= 2A{(/A 1 + l) 2 + ^ 2 2 }"*exp(— fjb 2 vx) expi ( — /*itw+f); . (30) 



where cos £ sin ?eee ^-B+^^— (31) 



30. Analytically E exp ipt represents a train of waves 

 propagated exclusively in the direction of ^-increasing, the 

 amplitude falling off in the proportion <? _1 for a distance 

 traversed equal to \/27r/-6 2 , while the wave-velocity is /a, -1 

 times what it would be if the secondary vibrators were 

 absent. The refractive index in the ordinary optical sense 

 is fii, though the changed velocity and gradual extinction 

 of the waves are both algebraically represented by the com- 

 plex refractive index fi. 



31. The velocity-potential in the regularly reflected beam 

 will evidently be of the form 



<^'" = A"'expz [pt + vx), 



where A'" is a complex constant, determined by the con- 

 dition that A + A'" must be equal to the value assumed by 

 E when x = ; that is 



A'"=-A4-2A{( / . 1 + l) 2 +/i 2 9 }- 1 ^i + l + ^ 2 ). • (32) 



In order to trace more easily the strengthening of the 

 reflected beam as the density of the swarm of vibrators is 

 increased, consider the case where % is real, and therefore 

 identical with ^ 1? while % 2 vanishes. Then {28), (29) become 



^Vii+Wti+W"- 2 ]}, ■ ■ ■ (33) 

 (x real ) 



^=V / {-i + V[l + 4% 2 »- 2 ]}- • • • (34) 



First, when %t> -1 (or \%/27r) is small, we have as far as the 





