238 



THE SOLUTION OF THE PROPAGATION EQUATION 



inside the duct, and 



Xi 



k 



a) [A + 2a 2 (z - h)] 



above the duct reduces equation (3') to 



dx' 



+ xU = , 



(3") 



whose general solution is 



U 



_ (Aihi(xi. 

 {Aihi(x 2 



) + BMxd 

 ) + BiAiCsjO 



The "h } " functions are expressible in terms of Hankel 

 functions of order }/%: 



hj(x) 



iV a- // 



3/ ~ "* V'3** 



x, U = 1,2) 



(5) 



Condition (3b) is satisfied by setting Az = 0. A t and 

 B\ are determined by the requirement of continuity 

 of U and dU/dz at z = h. 



Al = f!^ Br[l , 



{ t ] '' 



_k_ 

 2a 2 



±V 



2a, 



h'i 



5i 



"r(f)* 



4 



- ht 



B 2 



n-i 



a] 



k \ 



h\ 



2a- 



fci 



}■ 



2a! 



(6) 

 A 



2a, 



A 



A'i 



2a i 



The characteristic values A then appear as the roots 

 of the equation 



k 



U(0) = A,h x 



2a i 



(A - 2aji) 



+ BJi, 



2a i 



(A - 2aih) 



]" 







(7) 



The constant factor 5 2 appearing throughout is 

 determined by the normalization condition (3c), 

 which takes a peculiarly simple form in this case, 

 since (3") implies the identity 



<"- ![•«" + (§)"]■ 



(8) 



Owing to the present lack of adequate tables of 

 the H l l' or h t functions for complex arguments and 

 the complicated nature of equation (7) , one is forced 

 to employ asymptotic expansions to determine the 



A's so far as possible. Due care must be exercised in 

 dealing with the branches of the multiple- valued 

 approximations appearing, as well as with the 

 so-called "Stokes phenomenon." For example, in 

 deciding between the two rival asymptotic approxi- 

 mations 



ff? (W) ^ l^Y e~ HW - : 



H ( l> (W) £! (-JpY [ e -«^-w«> + gw+iwu>] ( 



(9) 



both formally valid in the common domain <^ 

 arg W < ir, one employs the first when arg W < 

 ir/2, and the second when arg W > x/2. The ambi- 

 guity on a "Stokes line" (arg W = ir/2) apparently 

 must be resolved by taking the mean of the two 

 expressions when their difference is important, as it 

 is when strongly trapped modes exist (ai < 0). 



The nature of the results obtained is illustrated 

 in the important case of complete inversion (ai < 0) . 

 For simplicity set 



2a ih 



(10) 



Then 



I (p + 1)' 



-3ai 



Oh 



at 



(-2a Ji)* 



in - 1) 



II 1 + 



te 



= 



2io-(p + 1)S 



e-^^-z)" 2 



[arg(p + 1)>|] 

 (11) 



;< arg p<7r — S 



III 1 + tB-w^ + D* + \e~ 2i < 1 



a J 



= , (arg P ^ r) . 

 (n a positive integer) 



The corresponding regions of validity are indicated 

 in Figure 1, which also shows the dependence of one 

 characteristic value on <r for a particular ratio 02/01. 

 If one numbers the characteristic values p n in order 

 of increasing imaginary part, those for which n is 

 greater than some integer are defined by equation 



