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dependency / is given by linear theory as
j cosh k(h0 + z)
cosh kh
(3.3)
where h0 is the depth of the bottom referenced to the still water line, and h is the total
water depth and k is the wave number vector. It should be noted that / is also a function
of the horizontal coordinates since k and h0 vary with horizontal position.
For incompressible, irrotational flow the governing equation for the velocity potential
is the Laplace equation
VV = 0 h0*z 0
h0 < z < i)
(3.5)
(3.6)
The governing equation is complemented by the following boundary conditions. On the free
surface the kinematic free surface boundary condition is given by
VhV = 4>t
z = r¡
The dynamic free surface boundary condition with zero atmospheric pressure
1,
t + ^(V)2 + gz = 0
z = r¡
(3.7)
(3.8)
and the bottom boundary condition
- Vh = 4>t
z = -h0
(3.9)
The derivation technique is to write Greens second identity for and / as defined by
Eqs. (3.2) and (3.3) respectively.
rf) rn
fztdz dz = [fz fz]lt
(3.10)