(III-2-29)

4 ε tan2 (αb )

where *Y *= length of structure.

(11) Pelnard-Considre (1956) also provides a second solution for times after the structure has filled to

capacity and bypassing of sediment begins to occur naturally. The boundary conditions for his second

solution are that *y *= *Y *at *x *= 0 and *y *= 0 at *x *= 4 for all *t *> 0. The initial conditions are as in the previous

solution *y *= 0 at *t *= 0 for *x *> 0. The solution to these specific boundary conditions is as follows:

, *t *> *t*f

(III-2-30)

2 ε *t*2

which can be made dimensionless by dividing the above equation by the length of structure *Y. *The

dimensionless solution is presented graphically in Figures III-2-34a, III-2-34b, and III-2-34c (at different

scales). Pelnard-Considre (1956) used a time = *t*2 in Equation 2-30 such that areas of shoreline above the

matched solution plan areas. In this manner *t*2 was found to be *t*2 = *t *- 0.38*t*f , where *t *is the initial solution

time at which the structure begins to trap sand. Although the planforms do not match at time *t *= *t*f for the two

solutions of Pelnard-Considre, the formulations are still useful for conceptual preliminary design and

evolution of projects.

(12) The solution of Pelnard-Considre prior to bypassing (as given by Equation 2-28 and Fig-

ure III-2-31) may also be utilized for the situation in which erosion occurs on the sand-starved beach down-

drift of an impermeable coastal structure that has no bypassing (natural or man-made). In this specific

instance the solution would provide shoreline recession values as opposed to shoreline progradation values.

In this scenario the solution should only be utilized far enough downdrift of the structure (i.e., beyond the

immediate "shadow" of the structure) such that diffraction and refraction effects due to the structure do not

influence the wave field and shoreline geometry.

(13) When applied in this scenario, the solution of Pelnard-Considre suggests that the ultimate downdrift

extent of erosion caused by the structure is infinite. In practice, Equations III-2-28 and III-2-29 may be

applied to estimate the theoretical downdrift extent of erosion, prior to bypassing, in terms of the structure

length *Y*. That is, the distance downdrift of a structure at which the shoreline recession is less than or equal

to some fraction of the structure's length (i.e., *y */ *Y*) can be expressed as a multiple of the structure's length

(i.e., *x */ *Y*). Example III-2-10, below, illustrates this application. It is important to note that this solution is

idealized and assumes that the breaking wave angle *α*b can be approximated as an average, quasi-steady value.

At present, the actual downdrift extent of erosion associated with a structure or other sediment sink is not well

understood.

(14) The second case to be considered is that of a rectangular beach fill as shown in Figure III-2-35.

Figure III-2-36 is a nondimensionalized solution graph that can be utilized in estimating plan area change for

the rectangular beach nourishment fill on an initially straight reach of beach. Fill exists from -a < x < +a and

extends Y distance seaward from the original beach. The solution for this specific case is as follows:

% *erf*

1&

1%

(III-2-31)

2

2 ε*t*

2 ε*t*

Longshore Sediment Transport

III-2-63

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