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difference could
be used to estimate γL .
Figure 5.14 presents the relationship between initial (θ0) and final (θF) swash
angles for varying values of non-dimensional linear friction.
Figure 5.13 Swash trajectories, effect of linear fric-
tion.
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5.6 SWASH ZONE DYNAMICS 119
Figure 5.14 Relationship between initial and final swash angles as a function of β (for linear
friction).
5.6.4 NORMALLY INCIDENT WAVES WITH NONLINEAR FRICTION
We commence with Eqs. (5.65) and note for normal incident waves, V ′y = 0. Because
of the absolute value sign in the friction term, it is necessary to consider uprush and
backrush separately.
Uprush – During uprush, V ′ > 0, and the first of Eqs. (5.65) becomes
dV ′x
dt ′
= −2− γ V ′2, (5.77)
a solution of which is
V ′(t ′) =
√
2
γ
tan
[√
2γ (t ′max − t ′)
]
(5.78)
in which t ′max is the time at which the maximum particle excursion occurs and is
t ′max =
1√
2γ
tan−1
(√
γ
2
)
(5.79)
The associated uprush displacement is determined by integrating Eq. (5.78) to
yield
x ′(t ′) = 1
γ
�n
{
cos(
√
2γ t ′max)
cos[
√
2γ (t ′max − t ′)]
}
, (5.80)
and the maximum displacement occurs at t ′ = t ′max and is
x ′(t ′max) = x ′max =
1
γ
�n{cos
√
2γ t ′max} (5.81)
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120 WAVES AND WAVE-INDUCED HYDRODYNAMICS
Figure 5.15 Effect of nonlinear friction parameter γ on swash characteristics.
Backrush – On the basis of a development similar to that used for the uprush, for
backrush V ′< 0 and Eq. (5.65) becomes
dV ′
dt ′
= −2+ γ V ′2, (5.82)
the solution of which can be shown to be
V ′(t ′) = −
√
2
γ
tanh[
√
2γ (t ′ − t ′max)] (5.83)
Equation (5.83) can be integrated to yield
x ′(t ′) = x ′max −
1
γ
�n{cosh
√
2γ (t ′ − t ′max)} (5.84)
The natural period is altered by nonlinear friction and can be determined from
Eqs. (5.79) and (5.84) as
T ′n =
1√
2γ
{
tan−1
(√
γ
2
)
+ cosh−1[eγ x ′max ]
}
(5.85)
Figure 5.15 shows the effect of varying nonlinear friction on natural period T ′n,
maximum downrush velocity (V ′0), and maximum uprush displacement x ′max.
5.6.5 FIELD AND THEORETICAL STUDIES
Hughes (1992) carried out a comparison of theoretical andmeasured swashmotions.
He applied the long-wave frictionless equations to predict the swash motions on a
planar beach face. It was found from the theory that the trajectory of the leading
edge of the swash motion was described qualitatively by Eq. (5.60) and that the
thickness of the swash decreased as the swash traversed up the beach face. The field
experiments were conducted in southeast Australia on several beaches ranging in
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REFERENCES 121
slope from 0.093 to 0.15. The beach faces were considerably steeper than the profile
immediately seaward, leading to bore collapse on the beach face. The swash mo-
tions were documented by capacitance gauges and stakes driven into the beach face.
Documentation from the capacitance gauges was complemented by photographing
the swash motion in the vicinity of the stakes. It was found that long wave theory
provided a good representation for the form of the uprush trajectories and the swash
thicknesses; however, the magnitudes of both were overpredicted. The measured
uprush trajectories were found to be approximately 0.65 of the theoretical, and the
swash thicknesses were overpredicted by a factor of 2 to 3. It was concluded that a
major reason for the differences between measurements and theory was the lack of
friction and infiltration in the theoretical formulation.
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