33
Then Eq. (3.6) is used to substitute for zz in the first term of Eq. (3.10) and the boundary
conditions are used to substitute for z at the mean free surface and at the bottom in the
third term of Greens identity. Equation (3.2) is used and terms of order e only are retained
in order to obtain the governing equation for .
In order to obtain z at fj it is assumed that
r] fj efj.
(3.11)
The kinematic free surface boundary condition, Eq. (3.7), is then expanded in a Taylor
series about z = fj,
J*=ff
d
dn
-gj + V/,f7 4>M
= 0
J*=ff
(3.12)
solving for z |/j using Eqs. 3.2 and 3.11 while retaining only terms of order e yields
df¡
Iff
A total horizontal derivative
dt
+ U Vhfj + Vh Vfcff fjWz
*=fl
(3.13)
D 9 ft T7
Di = ai + uVh
(3.14)
is introduced so that the first two terms on the right hand side of Eq. (3.13) can be written
as the total horizontal derivative of fj. The final manipulation of Eq. (3.13) is to recognize
that ^ at the surface is the horizontal divergence of the horizontal current (= V>, ).
Thus
M |ff= 'VhT) + fjVh
(3.15)
fj is eliminated from Eq. (3.15) by expanding the dynamic free surface boundary condition,
Eq. (3.8), in a Taylor series about z = fj.
4>t + ^(V<)2 + gz
J*=rj
= 9V +
M=fl
4>t + ^4>?
, 1 3 ,
9 + 4>*t + 2^(v^)
= o
*=ff
Using Eq. 3.2 to substitute for the order t terms yield a solution for fj
(3.16)
(3.17)