Resonant Response of Harbors (The Harbor Paradox Revisited) 



is the power- spectrum- amplification factor for the n'th mode. Sub- 

 stltuting (4.9), (4.10) and (4.12) into (4.1 6b) , we obtain 



/2 



p„ = i«r. 



(4.17) 



from which we infer that narrowing the harbor mouth does not affect 

 the mean response to a random input except in the Helmholtz mode, 

 but that it does increase significantly the response in that mode [this 

 conclusion ignores the increase in viscous dissipation that would be 

 associated with narrowing the mouth] . 



V. EQUIVALENT CIRCUIT FOR CANAL 



We now interpose a canal of breadth b and length i between 

 the harbor and the coast, as shown in Fig. 3, and obtain the equivalent 

 circuit on the assumption that only plane waves need be considered 

 in the canal. This approximation is strictly valid only for kb « 1, 

 but a more complete analysis shows that the effects of the cross-waves 

 (y- dependent modes) are not likely to be significant for kb < ir. 



Invoking the plane-wave approximation, u = u(x) and t, = 4(x), 

 in (2,3) and (2.4), we obtain 



I(x) = bhu(x) and V(x) = t,(x). 



(5.1 a, b) 



Assuming 1(0) = I, and I(i) = I2, we obtain the transmission-line 

 solution 



I(x) = CSC ki [ I, sin k(i -x) + I2 sin kx] 



(5.2a) 



and 



V(x) = (jbc sin ki)"' [ I, cos k(i-x) - Ig cos kx] . (5. 2b) 



Setting V(0) = V, and V(i ) = Vg in (5.2b), we obtain the matrix 

 equation 



Zj \\ Zj \ 



1-^12^22 J 



2 J 



(5.3) 



t 



We use canal in the same sense as Lamb [ 1932, §l69ff] • Some might 

 regard the synonym channel as more appropriate in the present context. 



107 



