1090
S.A. Hughes / Coastal Engineering 51 (2004) 10851104
estimating nonbreaking wave run-up. For irregular
(2004) derived formulas for estimating the maximum
waves breaking on the slope, a single formula proved
depth-integrated wave momentum flux for periodic
sufficient for all slopes in the range 2/3VtanaV1/30. A
(regular) waves and solitary waves.
slightly different formula is given for nonbreaking
wave run-up. In addition, two new equations for
2.1. Estimates for periodic waves
breaking and nonbreaking solitary wave run-up are
presented.
An estimate of wave momentum flux in periodic
waves was given by Hughes for first-order wave
theory in nondimensional form as
2. Wave momentum flux parameter
MF
1 H tanhkh
qgh2 max 2 h
kh
2
!
representing the maximum depth-integrated wave
1 H
2kh
24
1
momentum flux occurring in a wave of permanent
8 h
sinh2kh
form, i.e., the maximum over the wave of the integral
The dimensionless parameter to the left of the equal
Z gx
sign represents the nondimensional maximum depth-
pd pu2 dz
MFx; t
23
integrated wave momentum flux, and it is referred to
h
as the bwave momentum flux parameterQ.
Eq. (24) expresses nondimensional maximum
where MF(x,t)--depth-integrated wave momentum
wave momentum flux as a function of relative wave
flux at x and t; pd--instantaneous wave dynamic
height (H /h ) and relative depth (kh ). However,
pressure at a specified position; u--instantaneous
integration over the water depth stopped at the still
horizontal water velocity at the same specified
water level, and Eq. (24) does not include that part of
position; q--water density; h--water depth; x--
the wave above the still water level where a
horizontal direction perpendicular to wave crests;
significant portion of the wave momentum flux is
z--vertical direction, positive upward with z=0 at
found.
still water level; g(x )--sea surface elevation at
An improved estimate of (MF)max was obtained
location x; t--time.
using extended-linear theory in which expressions for
Maximum depth-integrated wave momentum flux
linear wave kinematics are assumed to be valid in the
has units of force per unit wave crest, and Hughes
crest region so the integration could be continued up
speculated this wave parameter may prove useful in
to the free surface at the crest. This resulted in a
empirical correlations relating waves to nearshore
slightly different expression given by
coastal processes occurring on beaches and coastal
structures. He also noted that integration of Eq. (23)
MF
over a uniform periodic wave results in radiation
qgh2
max
Stewart (1964). However, values of MF vary over a
H sinhkh H =2Š
1
wave from large positive values at the wave crest to
kh coshkh
2
h
large negative values in the trough, whereas the value
2
!
of Sxx is relatively small in comparison to the
sinh2kh H =2 2kh H =2Š
1 H
maximum. When considering force loading on coastal
8 h
sinh2kh
structures, perhaps better correlations can be made
25
using a parameter representative of the maximum
force in the wave instead of one corresponding to the
integration over the entire wavelength.
However, the wave form is still sinusoidal rather
Estimates of (MF)max can be made for any wave for
than having peaked crests and shallow troughs typical
of nonlinear shoaled waves and, consequently, the
which sea surface elevation and wave kinematics are
extended-linear theory under-predicts momentum flux
known either through theory or measurement. Hughes
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