238 Computational Modelling in Hydraulic and Coastal Engineering
Inside the breaker zone the wave height is linearly changed from the H b
value to the value λΔη on the coastline. In addition the offshore water
flux due to the undertow can be approximated either as (O’Connor et al.
1998)
Q
H
T
kh
H
T
=
+
π
2
2
4
1
0 9
tanh( )
.
(8.45)
or
Q
H
T
= κ
2
(8.46)
where O(κ) = 2 ÷ 3. This flux (or specific discharge) is transported offshore
in a water column of height equal to 0.8h (the space under the trough of the
breaking waves). The resulting current velocity, U c , is
U
Q
h
c = 0 8
.
(8.47)
The amplitude of the orbital water velocity, U o , can be described using the
long-waves theory as
U
H
h
gh
o = 2
(8.48)
From the pairs of the velocity values U c and U o , inside and outside the break
line, the combined bed shear due to wave motion and the wave induced
current is
τ
ρ
wc
c
h
w o
gU
C
f U
=
+
2
2
2
0 25
.
(8.49)
where C h and f w are the bed friction coefficients due to currents and waves,
respectively, defined by
C
h
h =

 

 
18
12
log ε
(8.50)
where ε is the absolute bed roughness, and
ln
.
.
.
f w
w
= −
+

 

 
−
5 99 5 12
0 914
ξ
ε
(8.51)
with ξ w being the amplitude of the horizontal wave orbital motion. A mean
value of the friction coefficient f w is approximately equal to 0.05.