ERDC/CHL CHETN-III-67
September 2003
Integrating the instantaneous wave momentum flux over the water depth, i.e.,
η ( x)
∫-h
(pd + ρu 2 ) dz
M F ( x, t ) =
(2)
gives the total wave momentum flux per unit crest length acting through a vertical plane parallel to
the crest. If the depth-integrated MF is also integrated over the wavelength, the result becomes the
wave "radiation stress" defined by Longuet-Higgins and Steward (1964). However, there is sig-
nificant variation of depth-integrated wave momentum flux over a wavelength from large positive
values to large negative values. So instead of adopting a mean value which is quite small compared
to the range of variation, it is logical when considering the wave force loading on structures to focus
on the maximum, depth-averaged wave momentum flux that occurs during passage of a wave, i.e.,
the maximum of Equation 2, which occurs at the wave crest.
Using Equation 2, maximum depth-integrated wave momentum flux (MF)max can be determined for
any surface wave form provided the velocity and pressure field can be specified. Therefore, (MF)max
has the potential of being a unifying wave parameter applicable to both periodic and transient wave
types.
ESTIMATING MAXIMUM DEPTH-INTEGRATED WAVE MOMENTUM FLUX: Hughes (in
preparation) derived formulas for estimating the maximum depth-integrated wave momentum flux
for periodic (regular) waves and solitary waves.
Periodic Waves
Expressions derived for maximum depth-integrated wave momentum flux from linear wave theory
do not include that part of the wave above the swl where a significant portion of the wave momen-
for linear wave kinematics are assumed to be valid in the crest region. However, the wave form is
still sinusoidal rather than having peaked crests and shallow troughs typical of nonlinear shoaled
waves; and consequently, the theory underpredicts momentum flux under the crest. Hughes (in
preparation) calculated values of (MF)max for a wide range of nonlinear uniform waves using a
numerical technique (Fourier approximation) that optimized the solution to provide the best fit of the
fully nonlinear free surface boundary conditions. Results were expressed in terms of the
dimensionless parameter
⎛ MF ⎞
⎜
⎟
ρ gh2 ⎠max
⎝
This parameter represents the nondimensional maximum depth-integrated wave momentum flux, and
it is referred to as the "wave momentum flux parameter."
Figure 1 presents the dimensionless wave momentum flux parameter versus relative depth h/gT2.
The solid lines are lines of constant relative wave height H/h. For a constant water depth, wave
period increases toward the left and decreases to the right. The range of relative depths covers most
coastal applications.
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