Resonant Response of Harbors (The Harbor Paradox Revisited) 



tion of an inductor and capacitor, Ln = UnA^nA) and Cp = A/fip. The 



dominant terms in Z^ as co ■-* are Zq and the sum of the inductive 



reactances obtained by neglecting co relative to con in the remaining 

 Zn. 



III. RADIATION IMPEDANCE 



The solution of the shallow-water equations in the exterior 

 half-space (x < 0) for a prescribed incident wave, say £,j(x,y), and 

 the assumed velocity u{0,y) in the harbor mouth is given by 

 [Miles and Munk 1961] 



Ux,y) = Ci(x,y) + Ci(-x,y) + ;s(x,y), (3.1a) 



where 



;s(X'y) = -i(^/g) \ H; [k(x + |y-n| ) ]u(0,Ti)dTi (x< 0), 



{3.1b) 



(2) 

 Ho is a Hankel function, the first two terms on the right-hand side 



of (3.1a) give the solution for total reflection from the plane x = 

 (as would occur if M were closed), and t,^ is the scattered wave. 

 Substituting u into (3.1) from (2.11), setting x = 0, and then sub- 

 stituting the result into (2,12), we obtain 



V = Vi - ZJ, (3.2) 



where 



Vi = 2 \ ^if*dy (3.3a) 



= 2;j(0,0) (ka « 1) (3.3b)'^ 



is the equivalent exciting voltage of the incident wave , and 



Z =i(co/c^)r r Hf (k|y-ii|)f*(y)f(Ti) dil dy (3.4) 



The definition of Vj implicit in (1.1) corresponds to the approxi- 

 mation (3. 3b) . 



103 



