21
where A is the complex amplitude, u/ is the absolute frequency, a is the intrinsic or relative
frequency following the current, and h0t h and k are as previously explained. Equations
governing $ and a parabolic approximation solution for A are presented in the next chapter.
The total velocity vector tT< is expressed as
ut =
= ^ *(§!)] ?+[v+*(!;)]
where SR indicates the real part, so that its magnitude is given by
Kl f'+ + (I) + 2V* (I?)+ [ (£)]+ [* (£)]
The x and y components of the time averaged bottom friction are thus
- 'r?£[u+*{£)\
(2.19)
(2.20)
(2.21)
(2.22)
As in Ebersole and Dalrymple, the assumption is made that only the gradients of A are
retained and the gradients of the depth and spatially varying quantities such as k and a
are ignored. This is essentially the assumption of a flat bottom and very slowly varying
currents.
Figure 2.1 plots the magnitude of the longshore current as a function of the distance
offshore for the various friction formulations discussed above. These velocity profiles were
obtained numerically for an eleven second wave with a deep water wave height of 1.028
meters (1 meter at offshore grid of the computational domain, using a AX of 10 meters
and 30 grid rows) and a deep water wave angle of 17.28 degrees (10 degrees at the model
boundary) approaching a 1:25 plane beach. A friction factor, F, of .01 was used for each case.
Using a friction formulation that ignores the contribution of the waves results in a very weak
friction and thus a large longshore current. Using the Longuet- Higgins formulation gives a
friction slightly less than the more proper time averaged formulation, while a combination of
the Longuet-Higgins formulation and the strong current model (no wave contribution) yields
a slightly stronger friction. For practical numerical considerations the latter formulation is