1092
S.A. Hughes / Coastal Engineering 51 (2004) 10851104
where
with the coefficients M and N approximated by the
2:0256
empirically determined functions
H
A0 0:6392
29
h
!'0:44
&
H
0:391
32
M 0:98 tanh 2:24
H
h
A1 0:1804
30
h
Even though the empirical coefficients and exponents
!
H
in Eqs. (29) and (30) are expressed to four decimal
N 0:69 tanh 2:38
33
h
places, corresponding accuracy is not implied.
Rounding to two decimal places should be reasonably
adequate for practical application of these empirical
Note that maximum depth-integrated wave momen-
equations.
tum flux for solitary waves is a function of only
The empirical equation represented by Eq. (28),
relative wave height, H/h. The first bracketed term in
along with Eqs. (29) and (30), provides an easy
Eq. (31) arises from the dynamic pressure, and the
method for estimating maximum wave momentum
second term represents the contribution of horizontal
flux for finite amplitude, steady, regular waves. This
velocity to the maximum wave momentum flux.
formulation gives more accurate estimates of the true
The solitary wave estimates of the wave momen-
maximum depth-integrated wave momentum flux
tum flux parameter represent the upper limit of the
nonlinear (Fourier) wave case when h/(gT2) ap-
than linear and extended linear theory because it
better represents the momentum flux in the wave
proaches zero (see Fig. 1). At a value of H/h=0.1,
crest, which is expected to be critical for most
the velocity term contributes only about 7% of the
applications to coastal structures.
calculated momentum flux, whereas as at H/h=0.8 the
For irregular wave trains, Hughes recommended
percentage increases to around 38% of the total.
that the wave momentum flux parameter be repre-
sented by substituting frequency-domain irregular
wave parameters Hmo (zeroth-moment wave height)
3. Wave run-up as a function of wave momentum
and T p (peak spectral period) directly into the
flux parameter
empirical Eqs. (28)(30). While this might not be
the best set of irregular wave parameters to use, these
In the following sections the maximum, depth-
frequency-domain parameters are commonly reported
integrated wave momentum flux parameter is corre-
for laboratory and field measurements, and numerical
lated to existing available data of normally incident,
irregular wave hindcast and forecast models output
breaking and nonbreaking wave run-up on smooth,
frequency-domain parameters.
impermeable plane slopes. Included are data for
regular waves, irregular waves and solitary waves.
2.2. Estimates for solitary waves
3.1. Wave run-up deviation
nondimensional wave momentum flux parameter
using first-order solitary wave theory given as
correlation between the time series of wave run-up on
!
2
a beach and the time series of depth-integrated mass
MF
1
H
H
flux within the swash zone. They also noted that the
2
qgh2 max 2
h
h
local depth-integrated momentum flux was balanced
&
!
mainly by the weight of water in the swash zone,
N2
H
M
H
which was approximated as a triangular wedge. Their
1
1
tan
h
2
h
2M
observation suggests that maximum wave run-up on
!'
an impermeable slope might be directly proportional
1
M
H
to the maximum depth-integrated wave momentum
tan3
1
31
3
2
h
flux contained in the wave before it reaches the toe of
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