S.A. Hughes / Coastal Engineering 51 (2004) 10671084
1079
Dividing both sides by qgh2 yields the nondimen-
and the total pressure, which is given at first order
as hydrostatic, is expressed as a function of zs at
sional expression for the wave momentum flux
parameter
the crest as
!
2
MF
1 H
H
2
PTzs qggs zs qgH h zsŠ
24
qgh2 max 2
h
h
&
!
N2 H
M H
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 tan
1
where C--solitary wave celerity gH hŠ;
2M h
2
h
!'
gs--sea surface elevation measured from the sea
1 3 M H
floor; z s--vertical coordinate-directed positive
tan
1
28
3
2
h
upward with origin at the sea floor; h--water
depth from the bottom to still water level; M, N--
The first bracketed term in Eq. (28) arises from the
coefficients that are functions of H/h.
dynamic pressure, and interestingly, this term is
Because the reference coordinate system has its
exactly twice the value of the long wave approxima-
origin on the sea floor, the depth-integrated momen-
tion of the wave momentum flux parameter derived
tum flux definition equation changes slightly to
from linear wave theory and shown in Eq. (15). The
second term represents the contribution of horizontal
Z
gx
velocity to the wave momentum flux parameter.
pd qu2dzs
MFx
25
The coefficients M and N are typically presented in
0
graphical form (e.g., Wiegel, 1964; Shore Protection
Manual, 1984). To accommodate calculations, a
nonlinear curve fit was applied to the plotted curves
and the maximum depth-integrated wave momentum
to produce the following simple equations that give
flux for a solitary wave is found as
reasonable values for M and N
&
!'0:44
Z
Z
Hh
h
H
M 0:98 tanh 2:24
29
MFmax
qgH h zsŠdzs
qgzsdzs
h
0
0
Z
Hh
C2N 2
!
26
q
!2 dzs
H
N 0:69 tanh 2:38
30
M zs
0
h
1 cos
h
The empirically fit equations (solid lines) are plotted
along with the data points taken from the Shore
The first integral in Eq. (26) is the total pressure, and
the second integral is the hydrostatic pressure between
underprediction and overprediction errors for Eq.
the bottom and the still water level. Subtracting the
(29) are 0.018 and 0.023, respectively. Overall, root-
second integral from the first results in depth-
mean-squared error is 0.010. Eq. (30) has maximum
integrated wave dynamic pressure. Performing the
under- and over-prediction errors of 0.010 and 0.006,
integration and substituting for wave celerity, C,
respectively, with overall root-mean-squared error of
results in the expression
0.0056.
The variation of the wave momentum flux param-
eter for solitary waves as a function of H/h is shown
qg 2
MFmax
H 2H h
in Fig. 8. These values represent the upper limit of the
2
!
&
nonlinear (Fourier) wave case when h /(gT 2)
qg H hN 2h
M H
1
tan
approaches zero (see Fig. 4). At a value of H/h=0.1,
2
M
2
h
!'
the velocity term contributes only about 7% of the
1 3 M H
calculated momentum flux, whereas as at H/h=0.8,
tan
1
27
3
2
h
the percentage increases to around 38% of the total. It
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