Miles 



The voltage- amplification ratio, G ^ °^ I'^n/^i I » provides a 

 measure of the resonant response in the neighborhood of to = co^. 

 The zero'th mode, in which the harbor acts like a Helmholtz reso- 

 nator, is unique in that the equivalent circuit reduces to a series 

 combination of Rj^p I-|yj+ I-u» and Cq and exhibits a simple, series- 

 resonant behavior with a resonant frequency, say coo, that is deter- 



mined bv a balance between the potential energy stored in H, 

 Munk 19d1J suggest that the sharpness of the Helmholtz resonance 



z ^ol^ol ' ^^ ^^^ kinetic energy stored in the vicinity of M, 

 2 (L,^ + ill. III;. The results for the rectangular harbor [Miles and 

 nk 19?1 



6 = {log (R/a)}"' (1.4) 



is measured by 



cuid that 



wo=0(6'''^), (10 = 0(1/6), and Qo=0(i/6) (1.5a,b,c) 



as a/R —" 0, where U^ is the peak value of G , and Q is the 

 ratio of the resonant frequency to the half -power bandwidth of the 

 resonance curve for the n'th mode. 



The resonant response of the harbor in the higher modes is 

 strikingly different than that of a simple, series -resonant circuit in 

 consequence of the proximity of the parallel- resonant frequency, 

 coq, at which Z| = oo, and the series -resonant frequency, co , at 

 which |Z| I has a minimum and G^^ = tij^ » i. We show in §4 that 



Z;^= co„ + 0(6), S^=0(l/6), and Q„ = 0(l/6^) (n ^t 0) (1 . 6a,b,c) 



It follows from (1.5) and (1.6) that narrowing the harbor 

 mouth does not affect the mean-square response to a random excitation 

 in the spectral neighborhood of oo = co_ (which response is proportional 

 to w 'vi /Q^ if the bandwidth of the random input is large compared 

 with 6(jOj^ except in the Helmholtz mode, but that the response in that 

 mode increases inversely as 6 ' . Miles and Munk [ 1961] overlooked 

 the proximity of parallel and series resonance in the higher modes and 

 arrived at the erroneous conclusion that narrowing the harbor mouth 

 would increase conUn/Qn for all modes, rather than only the Helmholtz 

 mode, and designated the phenomenon as "the harbor paradox. " In 

 fact, as pointed out by Garrett [ 1970] , this qualitative conclusion is 

 inconsistent with their quantitative results, which actually imply 

 (1.6) for the higher modes in a narrow rectangular harbor. Garrett 

 also showed that oo^^/Qp, is similarly invariant for excitation of a 

 circular harbor through an open bottom and correctly conjectured 

 that the result holds generally for the higher modes in any harbor. 

 In brief, the harbor paradox originally stated by Miles and Munk 



98 



