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For waves on a current the dispersion relation is
oj = a + k
= yjg\k\ tanh \k\h + |Jk| cos 0 + |Jb| sin 0
(2.34)
Equation (2.34) is differentiated to eliminate and from Eq. (2.33). Using a
forward difference derivative in x and a backward difference derivative in y, 0,- is obtained
as a function of cos 0, sin 0, U,V,k,h and their derivatives as well as 0 at the adjoining grids.
The sin 0 and cos 0 terms are expressed in terms of the surrounding four grids using a 2nd
order Taylor expansion. The solution is an iterative process alternating solution of the 0
equation with solution of the dispersion relationship (2.34) for k = |it| which is accomplished
by a Newton-Raphson method until acceptable convergence is obtained for k and 0.
Next an energy equation is solved for the wave height. A conservation of energy equation
is given by Phillips (1969 ed. Eq. 3.6.21) as
^+|_{m + C(.)} + s.|^ = 0 (2.35)
Assuming steady state conditions and expanding in Cartesian coordinates gives
(V + C,cos *)1|| + (V + C, sin ) i a-~ + A(U + C, cos S)
dU
dU
dV dV
+ + C'isin 0) + axx + fyx + fxv9vy
dy
dy
dx vv dy
0 (2.36)
where
xz ~E^ZX
1
VV £ Syv
ZV
*yz
n(cos2 0 + 1) i
2
n(sin2 0 + 1) -
2
n cos 0 sin 0
Defining
+ CB cos 0) + (V + C. sin
dy
(2.37)
where
dU dU dV dV
zxd'{'Tvx~d + Txv~d(Tvv~d.
(2.38)
a =