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COASTAL ENGINEERING 2004
3.
Analytic Solution of Shoreline Change in a Groin Compartment
For a groin compartment or for a beach enclosed by headlands, where no
transport occurs across the boundaries, an analytical solution may be derived
that displays some of the broad features of the response of a shoreline to
seasonality in the wave climate (Dean 1983). The groins are represented by the
boundary condition Q = 0 at each groin location. Mathematically, this boundary
condition can be expressed as (cf., Eq. 5):
∂y
= tan αo
(9)
∂x
This equation states that the shoreline at the respective groin is at every
instant parallel to the breaking wave crests. In this case, the boundary condition
in Eq. (9) should be employed both at x = 0 and x = B, where B is the length of
the groin compartment or enclosed beach. The breaking wave angle is assumed
to vary according to the following expression,
αo (t) = αao sin ωt
(10)
direction. The solution at steady-state conditions may be written (c.f. Larson et
al. 1997),
⎡ ζx
⎛x ⎞
αao ε / ω
⎞ ⎞ ζ ⎜ B -1⎟
π
π
⎛
⎛x
⎛
x⎞
⎢e B sin ⎜ ωt - + ζ
- 1⎟ ⎟ + e ⎝ ⎠ sin ⎜ ωt - + ζ ⎟
y(x, t) =
⎜B
2 ( cosh ζ + cos ζ ) ⎢
B⎠
4
4
⎝
⎠⎠
⎝
⎝
⎣
⎤
⎛x ⎞
x
-ζ ⎜ -1⎟
π
π
⎛
⎛ x ⎞ ⎞⎥
⎛
x ⎞ -ζ B
sin ⎜ ωt - - ζ ⎟ - e
sin ⎜ ωt - - ζ ⎜ - 1⎟ ⎟
-e ⎝ B ⎠
(11)
⎝ B ⎠ ⎠⎥
4
4
B⎠
⎝
⎝
⎦
where:
ω B2
ζ=
(12)
2ε
This non-dimensional parameter ζ is called the morphodynamic response
factor as it is an indicator of the response time of the shoreline to the variation in
input wave conditions. As seen from Eq. (11), the (steady-state) solution is
uniquely determined by the parameter ζ for a fixed αao; that is, cases with the
same ζwill have identical dimensionless shoreline evolution (y/B) expressed in
x/B and tε/B2. Figures 2 and 3 display the solution (Eq. 11) at different phase
values for ζ= 1.0 and ζ = 6.0, respectively, where the dimensionless shoreline
position was also normalized with αao. A small value of ζ implies rapid
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