where 



ag cosh [k(z + d) ] 



Z = - — l — i± (306) 



cosh(kd) 



and f(x,y) is a wave function to be determined. Equation (305) is the 

 Helmholtz equation. The following boundary conditions are assumed: 



(a) f(x,y)/3n = along all fixed boundaries where n is 

 in the normal direction to the boundary. 



(b) The harbor does not affect the wave system where 

 (x 2 + y 2 ) 1/2 ■* °°; i.e., at large distances from the harbor 

 entrance. 



Lee determines the value of the unknown wave function f(x,y) by 

 determining the function f-^fxjy) in the open sea and the function f 2 (x,y) 

 in the harbor, then matching the functions at the harbor entrance; i.e., 

 the wave amplitude and the slope of the water surface must be the same 

 for fj(x,y) and f 2 (x,y) at the entrance. 



The function f2(x,y) at some position (x,y) within the harbor is 

 defined by a line integral /„ taken around the harbor boundary in a 

 counterclockwise direction giving 



-WMk-ihwAX t307 ' 



where H^ 1 -* is a zero-order Hankel function of the first kind, fz^o'^o^ 

 the function at a boundary point (x ,y ), and r the distance between 

 the boundary point (x ,y ) and the interior point (x,y). 



The wave function in the open sea is represented by the sum of three 

 functions 



f x (x,y) = f^Cx,y) + f r (x,y) + f pa ( x »y) (308) 



where f^(x,y) is the known incident wave function, f r (x,y) the reflected 

 wave function, and f^ a (x,y) the wave function for the wave radiating 

 seaward from the harbor entrance. The reflected wave function is deter- 

 mined from the incident wave function for total reflection. The radiated 

 wave function is determined as 



f ra Co,y) - - | I B HO) (kr) |- [f 2 (o,y o )] d g (309) 



145 



