Miles 



solution corresponding to i]j = const, by n = 0: 



ko = 0, ^\>^=A^^^, (2.10) 



where A is the area of H. We also note that nnore explicit results 

 may require the use of two indices to count off the individual raodes. 



The exact determination of the assumed velocity u(0,y) 

 requires u and ^ to be matched across M to the corresponding 

 solution of the exterior boundary- value problem (see §3 below). 

 This matching condition yields an integral equation for u(0,y), the 

 exact solution of which in finite terms does not appear to be possible; 

 however, simiple approximations to u(0,y) are capable of yielding 

 excellent approximations to Z|^ and Zj^ by virtue of the associated 

 variational principle (cf. Miles [1946, 1948, 1967] and Miles and 

 Munk [ 1961] ). We proceed directly to such approxlnnatlons by Intro- 

 ducing the normalized trial function f(y), such that 



u(0,y) = (l/h)f(y), \ f(y) dy = 1 , (2.11a,b) 



In the subsequent development, we neglect the dependence of f(y) on 

 k and assume that It depends only on the geometry of M. The 

 validity of this approximation, which also Implies that f(y) Is real, 

 depends essentially on the antecedent approximation ka « 1. 



Substituting (2.11) Into (2.4) and (2.7), combining the results 

 In (2.5), and Invoking (2.8), we obtain 



■I 



Cf*dy (2.12) 



and 



Z„=^ Z„, (2.13) 



n 



where 



Is the modal Impedance , and |jLn is a dlmensionless measure of the 

 excitation of the n'th mode through M (note that ixq - ^ ^^^ 

 Zq = 1/jcoiA). The Z^ in the equivalent circuit appear In series, Z^ 

 as a capacitor, and each of the remaining 7,^ as a pareillel combina- 



102 



