ERDC/CHL CHETN-III-67
September 2003
Hlimit
= 0.142 tanh (kh)
(4)
L
which overestimates limiting steepness for long waves and underestimates limiting steepness for
short waves.
An empirical equation for estimating the wave momentum flux parameter for finite amplitude steady
waves was established using the calculated curves of constant H/h shown in Figure 1. A nonlinear
best-fit of a two parameter power curve was performed for each calculated H/h-curve, and the
resulting coefficients and exponents for each power curve were also approximated as power curves.
The resulting, purely empirical, equation representing the curves of constant H/h shown on Figure 1
is given as
-A1
⎛ MF ⎞
⎛ h ⎞
= A0 ⎜ 2 ⎟
(5)
⎜
⎟
ρ gh2 ⎠max
⎝ gT ⎠
⎝
where
2.026
⎛H⎞
A0 = 0.639 ⎜ ⎟
(6)
⎝h⎠
-0.391
⎛H⎞
A1 = 0.180 ⎜ ⎟
(7)
⎝h⎠
Goodness-of-fit of Equation 5 compared to the computed values given in Figure 1 is shown in
Figure 2. For smaller values of nondimensional (MF)max, there is reasonable correspondence except
for the left-most points of each curve (shown below the line of equivalence). This divergence was
caused by the power curve tending toward positive infinity as h/gT 2→ 0. Greater deviation begins to
occur for dimensionless (MF)max > 0.6. Nevertheless, the only poorly fitted curve is for H/h = 0.8
which is at or slightly above the limiting steepness for waves on a horizontal seabed. This poor
correspondence was likely the result of forcing the numerical computation beyond appropriate limits.
The empirical equation represented by Equation 5, along with Equations 6 and 7, provides an easy
method for estimating the nondimensional wave momentum flux parameter for finite amplitude,
regular waves. However, most coastal structure design guidance developed in the past 2025 years
has used wave parameters representative of unidirectional irregular wave trains. It is recommended
that the wave momentum flux parameter for irregular wave trains be estimated by substituting
irregular wave parameters Hmo (zeroth-moment wave height) and Tp (peak spectral period) directly
into the empirical Equations 5, 6, and 7. Application is simple, and estimates of maximum wave
momentum flux should be reasonably representative of the irregular wave train. However, other
irregular wave parameters might be used depending on the application. Therefore, before applying
any design guidance using the wave momentum flux parameter, it is important to ascertain which
wave parameters were used to establish that particular empirical relationship.
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