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COASTAL ENGINEERING 2004
holding Qn fixed. In this study it will also be shown that, for the same ratio
Qn/Qg, the offshore losses and the corresponding shoreline response are
dependent on the frequency of the changing transport direction.
2. One-Line Modeling
The history and basic assumptions of one-line theory, with the line taken to
represent the shoreline, are discussed extensively by Hanson and Kraus (1989).
In the one-line model, longshore sand transport is assumed to occur uniformly
over the beach profile down to a certain critical depth D called the depth of
closure. By considering a control volume of sand and formulating a mass
balance during an infinitesimal interval of time, while neglecting the cross-shore
transport, the following differential equation is obtained,
∂Q
∂y
+D
=0
(1)
∂x
∂t
where Q = longshore sediment (sand) transport rate, x = space coordinate along
the axis parallel to the trend of the shoreline, y = the shoreline position, and t =
time. Line discharges of sediment representing cross-shore transport can be
added to Eq. (1) (Kraus and Harikai, 1983; Hanson and Kraus, 1989), but this
capability is not exploited here.
Eq. (1) states that the longshore variation in the sand transport rate is
balanced by changes in the shoreline position. In order to solve Eq. (1), it is
necessary to specify an expression for the longshore sand transport rate. A
general expression for this rate in agreement with several predictive
formulations is,
Q = Qo sin 2αb
(2)
where Qo = amplitude of longshore sand transport rate, and αb = angle between
breaking wave crests and shoreline. This angle may be expressed as,
⎛ ∂y ⎞
αb = αo - arctan ⎜ ⎟
(3)
⎝ ∂x ⎠
in which αo = angle of breaking wave crests relative to an axis set parallel to the
trend of the shoreline, and ∂y/∂x = local shoreline orientation
A wide range of expressions exists for the amplitude of the longshore sand
transport rate, mainly based on empirical results. For example, the SPM (1984)
gives the following equation,
ρ 2
K
Qo =
(4)
Hb Cgb
(ρs - ρ )(1 - λ )
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