ERDC/CHL CHETN-III-70
July 2005
Hughes (2003a, 2004a) established an empirical equation for estimating the wave momentum flux
parameter for finite amplitude, nonlinear waves based on a numerical solution technique (Fourier
approximation). The resulting, purely empirical equation, was given as
- A1
⎛ h ⎞
⎛ MF ⎞
(6)
= A0 ⎜ 2 ⎟
⎜
2⎟
⎝ ρgh ⎠max
⎝ gT ⎠
where
2.026
⎛H⎞
A0 = 0.639 ⎜ ⎟
(7)
⎝h⎠
-0.391
⎛H⎞
A1 = 0.180 ⎜ ⎟
(8)
⎝h⎠
and H and T are the regular wave height and period, respectively. More information and a sample
calculation of wave momentum flux are given in CHETN-III-67 (Hughes 2003a).
Hughes (2003b, 2004b) derived a simple generic expression for wave runup in terms of wave
momentum flux. He argued that the weight of fluid within the hatched wedge area (ABC) of
Figure 2 is proportional to the maximum depth-integrated wave momentum flux of the wave just
before it reached the toe of the structure slope. The resulting equation was in the form
1/ 2
⎛M ⎞
R
= C ⋅ F ( α ) ⎜ F2 ⎟
(9)
⎝ ρgh ⎠
h
where R is the maximum vertical runup from swl, and C ⋅F(α) is an unknown function of structure
slope angle. For convenience the subscript "max" has been dropped from the wave momentum flux
parameter.
In the Equation 9 runup equation, relative runup (R/h) is directly proportional to the square root of
the wave momentum flux parameter. (Note that representing the runup sea surface slope as a straight
line is an approximation and may not be fully appropriate for nonbreaking waves on steep slopes
where the water surface has pronounced curvature.)
Applying Equation 9 to irregular wave runup requires that regular wave height and period (H and T)
used to estimate the wave momentum flux parameter using Equations 6, 7, and 8 be replaced with
representative irregular wave parameters (Hmo and Tp). This substitution does not imply that an
equivalence exists between values of wave momentum flux parameter calculated for regular and
irregular waves; it only provides a convenient standard for application with irregular waves when
establishing empirical relationships. For irregular waves, always check that the depth-limited wave
height does not exceed the rule-of-thumb Hmo ≤ 0.6 h.
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