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COASTAL ENGINEERING 2004

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 ω*t*

(10)

direction. The solution at steady-state conditions may be written (*c.f. *Larson *et*

⎡ ζ*x*

⎛*x * ⎞

αao ε / ω

⎞ ⎞ ζ ⎜ *B *-1⎟

π

π

⎛

⎛*x*

⎛

⎢*e *B sin ⎜ ω*t *- + ζ

- 1⎟ ⎟ + *e *⎝ ⎠ sin ⎜ ω*t *- + ζ ⎟

⎜*B*

2 ( cosh ζ + cos ζ ) ⎢

4

4

⎝

⎠⎠

⎝

⎝

⎣

⎤

⎛*x * ⎞

-ζ ⎜ -1⎟

π

π

⎛

⎛ *x * ⎞ ⎞⎥

⎛

sin ⎜ ω*t *- - ζ ⎟ - *e*

sin ⎜ ω*t *- - ζ ⎜ - 1⎟ ⎟

-*e * ⎝ *B * ⎠

(11)

⎝ *B * ⎠ ⎠⎥

4

4

⎝

⎝

⎦

where:

ω *B*2

ζ=

(12)

2ε

This non-dimensional parameter ζ is called the *morphodynamic response*

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

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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