54
where p = CC9 for notational convenience and the y derivative of p ^ j is written as
(rUi + pj) [(£)+, ~ (£))] (p> + p;~i) [(f) ($)'
2(AY)2
(4.28)
In order to evaluate w)+1 by the method to be explained in section 3.3 an intermediate
value of A)+1 is obtained by the explicit formulation
A,+1 A*-
3 + i(k* cos S' k'j cos 0') A'j
AX
1 [{Ct cos 0 + V)/<,])+' \(Ct cos 0 + U)/a]
1 [ic
' AX 1 [(C, cos 0 + U)/af-+1 + [(Ctf cos 0 + U)/a]
(
|}
(Cg cos0 + U)/a
j+i
2AY
+
( V
[(v;+, (W;-,!
A* *
kCg{\ cos2 0)
{2(Cecos0 + U) J .
2AY
A* 2
(Cscosi + i/)
A
n
V)
T +
}4
2 (Cg cos 0 + U)) _r 2[Cg cos 0 + U))
When Eq. (4.27) is written out using Eq. (4.28) and w*+1 obtained through use of
(4.29) there results for any particular values of i and j an equation for the three unknown
values A}**, A)+1, and A.\. The equation is put into the form
ai aYA + hi }+1 + Cj Ajt\ = dj (4.30)
where the coefficients aj, bj, and Cj are known quantities involving the properties of the
environment (depth, current, celerity, etc.) and the dj involves the environmental properties
and the known values of the amplitude on the grid row. When Eq. (4.30) is written for
all the j values, j equal to 1 through N, for a specified i value, the ensemble of N equations
takes the form of a tridiagonal matrix equation. For illustrative purposes let N be equal to
four. The resulting matrix equation is
r 4>+1i
aj ci
A+1
1 '
a2 &2 c2
A2+1
d2
as b$ Cj
A*3+1
S
a< 4 C4
4+1
. d* .
L 4+1J
(4.31)