Mites 



change In the mean value) of the velocity In M and stationary with 

 respect to first-order variations of this velocity about the true solu- 

 tion to the integral equation (cf. Miles and Munk [ 1961] and Miles 

 [ 1946, 1948, 1967] j we omit the explicit formulation of the integral 

 equation and further discussion of the variational principle in the 

 present development). The resulting representation of Z.^ is rela- 

 tively insensitive to the geometry of H and yields a simple, ex- 

 plicit approximation that depends essentially only on ka. The cor- 

 responding representation of Z^ requires Green's function (subject 

 to a Neumaxin boundaiy condition) for the closed harbor, the explicit, 

 analytical construction of which is possible only for those boundaries 

 (rectangular, circular or circular-sector, and elliptic or elliptic- 

 hyperbolic sector) that permit separation of variables; however, we 

 may infer the matrix representation of this Green's function for a 

 polygonal approximation to an arbitrarily shaped harbor fronn Lee's 

 [ 197l] collocation solution of the general problem. We give explicit 

 results for a circular harbor in §6 with special emphasis on the 

 Helmholtz mode. It appears from these results that a large harbor 

 with a short entrance or a small harbor with an entiy canal of length 

 comparable with R may resonate in the Helmholtz mode under 

 tsunami excitation. 



II. HARBOR IMPEDANCE 



Let X and y be the Cartesian coordinates in the free sur- 

 face, t the time, cx> the angular frequency, h the depth, 



c = (gh) and k = w/c (2.1a,b) 



the wave speed and wave number, t, the free-surface displacement, 

 u the X- component of the particle velocity, t, and u the corre- 

 sponding complex amplitudes, such that 



{C(x,y,t),u(x,y,t)} = B[{C(x,y),u(x,y)}eJ''^] , (2.2) 



where ^ implies the real part of and j = y-i » 



I = \ u dS (dS = h dy) (2.3) 



the flow through M, 



V =(r u* dy) r ;u*dy (2.4) 



M M 



* 



a weighted measure of the displacement in M, where u is the 



i 



100 



