lim \fr 



jk (J) J = 



(7) 



where 



r - radial polar coordinate 



(ff = velocity potential of the scattered wave 



The complete boundary value problem is specified by Equations 2, 5, and 7. 

 A hybrid element method is employed to solve the boundary value problem A 

 conventional finite element grid is developed and solved in Region A. The 

 triangular elements allow detailed representation of harbor features and 

 bathymetry within Region A. An analytical solution with unknown coefficients 

 in a Hankel function series is used to describe Region B. For a given grid, short 

 wave period tests (relatively large values of k) require more terms than long 

 period tests to adequately represent the series. A variational principle with a 

 proper functional is established such that matching conditions are satisfied along 

 dA. Details are given by Chen (1986) and Lillycrop and Thompson (1996). 



Experience with the model has indicated that the element size Ax and local 

 wavelength L should be related by 



L 

 £ — 



6 



(8) 



Typically, harbor domains include some shallow areas in which many elements 

 would be needed to satisfy the constraint in Equation 8. In practice, Equation 8 

 is at least satisfied in the harbor channel and basin depths. If additional elements 

 can be accomodated, it is generally preferred to extend the semicircle further 

 seaward rather than to greatly refine shallow harbor regions. 



Input information for HARBD must be carefully assembled. In addition to 

 developing the finite element grid to suit HARBD requirements, a number of 

 parameters must be specified. Critical input parameters and ranges of typical 

 values are summarized in Table 8. 



The principal output information available from HARBD consists of 

 amplification factor and phase at each node. These are defined as 



6 = tan" 



Re {<j)} 



1*1 



(9) 



46 



Chapter 4 Numerical Model 



