1072
S.A. Hughes / Coastal Engineering 51 (2004) 10671084
The instantaneous flux of horizontal momentum
that occurs during passage of a wave, i.e., the
(mf) across a unit area of a vertical plane oriented
maximum of
parallel to the wave crests is given by
Z gx
pd qu2dz
MFx; t
7
mf x; z; t pd qu2
4
h
that occurs at the wave crest when g(x)=a. Note that
where pd--instantaneous wave dynamic pressure at a
on the surface of a perfectly reflecting, impermeable
specified position; u--instantaneous horizontal water
vertical wall, the horizontal velocity u is zero, and
velocity at the same specified position; q--water
Eq. (7) becomes simply the integral over the water
density.
depth of the dynamic pressure exerted by the wave
Longuet-Higgins and Stewart (1964) defined the
on the wall, or the total instantaneous wave force
component of bradiation stressQ perpendicular to the
on the wall (excluding the hydrostatic pressure
wave crest as the wave momentum flux integrated
component).
over the water depth and averaged over the wave,
Maximum depth-integrated wave momentum flux,
i.e.,
as defined by Eq. (7) with g(z)=a, can be determined
Z
Z
for any surface wave form provided the velocity and
gx
L
1
pd qu2dzdx
5
Sxx
pressure field under the crest can be specified. In
L
h
0
theory, this means that a wave parameter based on
momentum flux has the potential of applying to both
They substituted linear (Airy) theory expressions for
periodic and transient wave types, which may be a
pressure and horizontal velocity and completed the
useful property. More importantly, the physical
integration by applying first-order wave kinematics
relevance of wave momentum flux to force loading
above the still water level, which is not strictly first-
on structures seems logical, thus fulfilling an impor-
order theory. This bextended linear theoryQ resulted
tant criterion for the proposed wave parameter.
in the expression
4.1. Linear (Airy) wave theory
1
1
2kh
qga2
6
Sxx
2
2 sinh2kh
In linear wave theory, dynamic pressure and
horizontal wave velocity are in phase with the sea
surface elevation, and the maximum wave momentum
where L--local wave length; h--water depth from
flux occurs at the wave crest. The first-order
bottom to the still water level; g--instantaneous sea
approximation of depth-integrated wave momentum
surface elevation relative to still water level; z--
flux is found by substituting the dynamic pressure and
vertical coordinate directed positive upward with
horizontal wave velocity at the wave crest from Airy
origin at the SWL; x--horizontal coordinate positive
wave theory into Eq. (7) and integrating from the
in the direction of wave propagation; g--gravita-
bottom only up to the still water level because
tional acceleration; a--wave amplitude; k--wave
kinematics are not specified above still water level
number [=2p/L]. Note that Sxx has units of force
in first-order theory. From linear wave theory with no
per unit length of wave crest.
unidirectional current,
There is significant variation of depth-integrated
wave momentum flux over a wave length from large
coshkh z
positive values at the crest to large negative values
8
pdzcrest qga
in the trough. So instead of adopting a wave-
coshkh
averaged value (i.e., Sxx) which is quite small
and
compared to the range of variation, it is logical
when considering some coastal processes, such as
coshkh z
the wave force loading on structures, to focus on the
9
uzcrest ax
sinhkh
maximum, depth-integrated wave momentum flux
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