where |tÂ¡| is written as
20
|tTj| = \Ju2 + V2 + titeos2 at + 2Uum cos at cos 9 + 2Vtim cos at sin 0 (2-14)
The integrals in Eqs. (2.12) and (2.13) are then evaluated by a 16 point summation using
Simpsons Rule.
In the present study several different formulations are used and compared. The first is
the formulation given by Longuet- Higgins (1970a) and used by several previous investiga
tors. The second is a modification of the Longuet-Higgins formulation with partial inclusion
of current strength in the friction formula.
T* = pF \uorb\ U + pF\U\U
(2.15)
rbv = pF\uorb\V + pF\U\V
(2.16)
It is thought that this formulation has some value since in the lee of structures where there
is relatively weak wave activity, there may still be relatively strong currents generated by
the gradients of the set-up outside of the sheltered region. The third formulation is similar
to that of Ebersole and Dalrymple in that the full quadratic expression is used. However the
wave orbital velocities at the bottom are expressed as gradients of the complex amplitude
and the resulting integral is determined using an eight point Gaussian integration. It is
found that an eight point Gaussian integration of a cosine squared integration over an
interval of 2x is accurate to within 1 percent.
The quadratic formulation of the bottom friction is used
Tb = pF ui|tT(| (2.17)
where tT* is composed of both the mean current and the wave orbital velocities. The wave
orbital velocities are expressed as the real part of the gradient of the complex velocity
potential $ which is expressed as
. A(,y) cosh k(h0 + z) ,-Mt
9 a cosh kh
a
(2.18)