Resonant Response of Harbors (The Harbor Paradox Revisited) 



complex conjugate of u, 



ZH=V/I = h"'|\ udyl' \ ;u*dy (2.5) 



the harbor impedance , and 



P = iR{pgh\ Cu* dy} = i pgR(VI*) (2.6) 



the rate at which enej-gy flows through M. We may regard O'V, PI, 

 (Qr/P)Z^, and apR(VI ) as the voltage, current, impedance, and 

 power in am equivalent electrical circuit, where the constants of 

 proportionality, a and P, may be chosen to obtain convenient 

 electrical units. The choice a = f> ~ i is implicit in the discussion 

 in §1, but not in what follows except as noted. 



Solving the shallow-water equations (Lamb [1932] , §189) for 

 an assumed velocity in M, subject to the boundary condition that the 

 normal derivative of ^, jq • Vt» vanish on B, the lateral boundary 

 of the free surface in H, we obtain 



;(x,y) = (jco/g) \ G(x,yjO,Ti)u(0,Ti) dTl (2.7) 



where 



G(x,y;e,tl) = 2^ (kn-kVipnU,y)^n(^,^), (2.8) 



n 



is the point- source Green's function for H, the ijj^ are the nor- 

 malized eigenfunctions for the closed harbor, and the summation is 

 over the complete set of these functions. The i|j„ are real and satisfy 



(V^ + k^n)^n = (x»y in H) , (2.9a) 



(n • V)qjn = on B, (2.9b) 



and 



^m^n dA = 6„„, (2.9c) 



i 



where kr, are the eigenvalues (resonant wave numbers), and 6mn 



is the Kronecker delta. We designate the degenerate (but non-trivial) 



101 



