55
Equation (4.31) contains four simultaneous equations with six unknowns. Two bound
ary conditions should be specified. At each boundary a functional relationship is established
between the grid inside the domain and the grid outside of the domain. For the wave prop
agation model there are two types of boundary conditions that are readily available for use
in a computer model. These are a totally reflective boundary condition which would be
used to model a laboratory experiment in a wave basin, and non-reflecting non-interfering
boundary which will allow waves to freely pass through the boundary. For the reflective
boundary, the mathematical statement of the boundary condition is ^ = 0. At the lower
boundary this is finite differenced as = 0 which gives the functional relationship
A1 = Ao. (4.32)
At the upper boundary the functional relationship for a reflective boundary is
An+i = An.
(4.33)
For the non-reflective boundary the mathematical statement is = imA where m is
the longshore component of the wavenumber. Since the lateral boundaries are sufficiently
far from any diffraction effects, only refraction and shoaling effects are present. Thus
using Snells Law m = k sin# = k0 sin0o. The non- reflective boundary condition is finite
differenced at the lower boundary as
Ai Aq im , . ,
£y~~ =Y[Al + A] (434)
which gives
Aq =
fj.nAY)
1 +
tmAYA
2 )
A\
(4.35)
Similarly at the upper boundary the functional relationship used to eliminate Ajv+i is
(l+taAX)
Aw+1 = '(l imAY\AN (4-36)
The use of either pair of boundary conditions, (4.32) and (4.33) or (4.35) and (4.36) to
eliminate Aq and A;v+i will then yield an N by N tridiagonal matrix equation comprising N