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```ρu(x, z) dzdt, (5.21)
where the time interval between t1 and t2 is a long time (many wave periods for
irregular waves; one wave period for periodic waves). If we integrate over the depth
from the bottom only to themeanwater surface, z = 0, rather than the instantaneous
water surface η, we obtain M equal to zero, as predicted by linear theory. If we
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5.2 WATER WAVE MECHANICS 101
continue the integration up to η, the mass transport becomes
M = E
C
, (5.22)
which shows that there is a nonlinear transport of water in the wave direction due to
the larger forward transport of water under the wave crest because the total depth
is greater when compared with the backward transport under the trough. From this
formula, the mass transport is larger for more energetic waves.
Thismass transporthasmomentumassociatedwith it,whichmeans that forceswill
be generatedwhenever thismomentum changesmagnitude or direction byNewton’s
second law. To determine this momentum, we integrate themomentum ﬂux from the
bottom to the surface as follows:
M = 1
t2 − t1
∫ t2
t1
∫ η
−h
(ρu)u dzdt (5.23)
This quantity has as a ﬁrst approximationM = MCg = En, which indicates that the
ﬂux of momentum is described by the mass transport times the group velocity.
Offshore of the breaker line, there is a depression of the mean water level from
the still water level due to the waves, which is called setdown and is denoted as η.
This quantity, originally elucidated by Longuet-Higgins and Stewart (1963), is (e.g.,
Dean and Dalrymple 1991)
η = − H
2k
8 sinh 2kh
, (5.24)
where, again, H is the wave height. Because the wave height increases as waves
shoal, the setdown increases as well, reaching a maximum at the breakerline that is
approximately 5 percent of the breaking water depth.
Longuet-Higgins and Stewart (1963) introduced the concept of wave momentum
ﬂux, designating the sum of the momentum ﬂux and the mean pressure as the radia-
tion stress, based on the analog with light, which develops a radiation pressure when
shining on an object. The quantities are related in the following way:
M+ 1
2
ρgh2 = Sxx + 12ρg(h + η)
2, (5.25)
where Sxx is the radiation stress representing the ﬂux in the x direction of the x
component of momentum and η is the mean water level elevation. The formula for
Sxx is
Sxx = E
(
2n − 1
2
)
(5.26)
There is an equivalent term for y-momentum carried in the y direction and the mean
pressure, which is
Syy = E
(
n − 1
2
)
(5.27)
These expressions apply for waves traveling in the x direction. If the waves were
traveling in the θ direction, where θ is an angle the wave direction makes with the
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102 WAVES AND WAVE-INDUCED HYDRODYNAMICS
x-axis, then we would have the following radiation stresses:
Sxx = E
[
n(cos2 θ + 1)− 1
2
]
(5.28)
Syy = E
[
n(sin2 θ + 1)− 1
2
]
(5.29)
Sxy = Syx = En sin 2θ2 (5.30)
The last equation is for the ﬂux of x momentum in the y direction, or vice versa, and
it arises owing to the obliquity of the waves to the coordinate axis. This equation can
be rewritten in the following form by introducing the wave celerity in numerator and
denominator:
Sxy = EnC cos θ
(
sin θ
C
)
The ﬁrst part of this expression is recognized as the shoreward ﬂux of wave energy,
which, on a beach characterized by straight and parallel contours, is constant until
breaking begins in the surf zone, and the second term, in parentheses, is Snell’s
law, which is also constant. Therefore, for this idealized beach, Sxy is constant from
offshore to the breaker line.
5.2.6 WAVE SETUP
The wavemomentum ﬂuxes are proportional to the wave energy. If the waves break,
then themomentum ﬂux decreases. This change inmomentum ﬂuxmust be balanced
by forces; therefore, wave breaking induces forces in the surf zone that act in thewave
direction.Wewill nowexamineeachof these forces by resolving them into anonshore
and alongshore direction.
The onshore-directed momentum ﬂux is Sxx . As the wave propagates into the
surf zone, the momentum ﬂux is equal to its value at the breaker line. At the limit
of wave uprush, the value is zero. This gradient in momentum ﬂux is balanced by a
slope in water level within the surf zone ∂η/∂x . Balancing the forces, the following
differential equation results:
∂η
∂x
= − 1
ρg(h + η)
∂Sxx
∂x
(5.31)
Integrating the mean water level slope, we get the wave set up η, as noted earlier. Us-
ing the shallow water asymptote and the spilling breaker assumption in the radiation
stress terms, Sxx becomes
Sxx = 316ρgκ
2(h + η)2,
where κ is the breaking index. As noted earlier, the value of κ is about 0.8 for spilling
breakers. For a monotonic beach proﬁle, we can solve Eq. (5.31) for η as follows:
η = ηb −K(h − Hb), (5.32)
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5.3 CROSS-SHORE AND LONGSHORE CURRENTS 103
where ηb is the mean water level at the breaker line and K = (3κ2/8)/(1+ 3κ2/8).
Experiments have been carried out to verify this model in the laboratory (with good
results; see Bowen, Inman, and Simmons 1968) and in the ﬁeld. (This is the same
wave setup that was introduced in the last chapter as a component of storm surge.)
5.3 CROSS-SHORE AND LONGSHORE CURRENTS
The wave-induced mass transport M must engender a return ﬂow (which on a long-
shore uniform beach is the undertow), for there can be no net onshore ﬂow of water
because of the presence of the beach. The amount of seaward mass ﬂux is therefore
equal to M . This ﬂow is not distributed uniformly over the depth but has a distinct
proﬁle caused by the variation in wave-induced stress over the depth.
In the alongshore direction, the greatest change in radiation stress occurs owing
to the Sxy term, which is affected, of course, by breaking. To balance this change in
momentumﬂux, a longshorewater level slope is possiblewhere the shoreline is short;
bounded at the ends by headlands, inlets, or man-made structures; or when there are
rip currents. For an inﬁnitely long uniform shoreline with uniform wave conditions,
this water-level slope cannot exist, for it leads to inﬁnite or negative water depths in
the surf zone. Some other mechanism for developing a longshore balancing force is
required.Mean currents ﬂowing along the shorelinewill develop bottom stresses and
canbalance the gradients in the radiation stress terms.The resulting longshore current
is then directly engendered by the obliquely incident waves and the process of wave
breaking. By balancing the frictional forces and the gradients in the radiation stress,
Bowen (1969a), Longuet-Higgins (1970a and b), and Thornton (1970) developed
equations for generating the longshore current and, in fact, since these early models,
numerous studies have been made of the current ﬁeld.
The steady-state equation of motion in the alongshore direction is
−ρg(h + η)∂η
∂y
−
[
∂Sxy
∂x
+ ∂Syy
∂y
]
+ [τs − τb]+ ∂[(h + η)τxy]
∂x
= 0, (5.33)
in which the last term represents lateral shear stress coupling. The y derivatives can
be neglected for the case of a long straight beach because these terms would lead to
inﬁnitemagnitudes of η and Syy if they existed. The remaining terms (after neglecting
the surface shear stress for the case of no wind and lateral shear stress coupling) can
be written as
∂Sxy
∂x
= −τb (5.34)
Determining a simple expression for the bottom friction term that results from the
mean current that is coexistent with the oscillatory wave ﬁeld is difﬁcult. Longuet-
Higgins (1970) developed what Liu and Dalrymple (1978) call the small (incident
wave) angle model deﬁned by
τb = ρ f4π umV (5.35)
The f is the empirical Darcy–Weisbach friction factor, used in pipe ﬂow calculations,
which is known to be a function of the ﬂowReynolds number and the sand roughness,
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