228 The late Lord Ray lei gh on Resonant 



Before proceeding further it may be worth while to deal 

 with the case where 7i, and consequently k 2 , is very small, 

 but k 2 l so large that vibrations in the channels are sensibly 

 extinguished before the stopped end is reached. In this case 



cos ik 2 l— ±e k * 1 , sin ik 2 l = \ie k * 1 , 



so that in (19), tan k'l=-i. Also by (9), ///&= 1, and (15) 

 becomes 



B + l" (<r + <r')cos0' * * * * ^ Uj 



making B = when, for example, </ = 0, cos = 1. The 

 reflexion may also vanish when the obliquity of incidence is 

 such as to compensate for a finite a'. 



In examining the formula for the general case we shall 

 write for brevity 



cos0(<r + <r')/cr = S, (21) 



and drop Z, so that & 1? k 2 , k stand respectively for kj, k 2 l, kL 

 This makes no difference to the first of equations (9), while 

 the second becomes 



k 1 k 2 = inlil 2 /a 2 (9 bis) 



Thus -r, _ &S cos k' — ik' sin k' \ . . 

 k$ cos k' + ik' sink' [ } 



Separating real and imaginary parts, we find for the 

 numerator of B in (22) 



cos ki cos ik 2 kS — ; — — — k 2 tan k\ 



. ( /Q tan ik 2 k 2 tan ik 2 ") 1 

 + 1 < &>3 tan k x : % tan k x -| : v . (23) 



The denominator of (22) is obtained (with altered sign) 

 by writing — S for S in (23). 



In what follows we are concerned with the modulus of B. 

 Leaving out factors common to the numerator and denomi- 

 nator, we may take 



Mod. 2 Numerator = { jfeg- * lta "^ 2 -£ 2 tan k, ]■ ' 



' ( /j^tanikc, 7 \ 7 & 2 tan ik ) 2 _ 



+ \ r s ~i ~ k i ) tan ** + i — : } • ( u > 



