for 0 # tan αb # 1.23

(III-2-27a)

and

for 1.23 < tan (αb)

(III-2-27b)

tan (αb )

where *K*a and *K*b are constants. The growth of river deltas was studied with these equations.

(4) Bakker (1968) extended a one-line shoreline change theory to include the shoreline and an additional

offshore depth contour to describe beach planform change. Bakker hypothesized that the two-line theory

provides a better description of sand movement downdrift of a long groin since it describes representative

changes in the contours seaward of the groin head. Near structures such as groins, offshore contours may

have a different shape from the shoreline. The two lines in the model are represented by a system of two

differential equations which are coupled through a term describing cross-shore transport. According to

Bakker (1968), the cross-shore transport rate depends on the steepness of the beach profile; a steep profile

implies offshore sand transport; and a gently sloping profile implies onshore sand transport. Additional

complex solutions of cases with groins under very simplistic assumptions are discussed in Bakker, Klein-

Breteler, and Roos (1971). Le Mhaut and Soldate (1977) provide an analytical solution of the linearized

shoreline change equation for the spread of a rectangular beach fill. Walton (1994) has extended this case

to the fill case with tapered ends.

(5) Walton and Chiu (1979) present two derivations of the linearized shoreline change equation. The

difference between the two approaches, which both arrive at the same partial differential equation, is that one

uses the so-called "CERC Formula" (see Equation 2-5) for describing the longshore sand transport rate by

wave action and the other a formula derived by Dean (1973) based on the assumption that the major sand

transport occurs as suspended load. Walton and Chiu (1979) also present solutions for beach fill in a

triangular shape, a rectangular gap in a beach, and a semi-infinite rectangular fill, and present previous

analytical solutions in the literature in a nondimensionalized graphical solution form.

(6) Dean (1984) gives a brief survey of some analytical shoreline change solutions applicable to beach

nourishment calculations, especially in the form of characteristic quantities describing loss percentages. One

solution describes the shoreline change between two groins initially filled with sand. Larson, Hanson, and

Kraus (1987) provide a review of a number of analytical solutions to the one-line model as well as additional

solutions where the amplitude of the longshore sand transport rate is a discontinuous function of *x*, the

shoreline coordinate in the longshore direction.

(7) Analytical solutions presented here are in the nondimensionalized form of Walton and Chiu (1979)

and easily adaptable to solving simple scenarios. More difficult scenarios are best handled by a numerical

model. In arriving at all solutions, it is tacitly assumed that sand is always available for transport unless

explicitly restricted by boundary and/or initial conditions.

(8) The first case to be considered is that of a structure trapping sediment. This formulation can be

applied to the prediction of the shoreline updrift and downdrift of a littoral barrier extending perpendicular

to the initially straight and uniform shoreline. At the barrier, all sediment is assumed to be trapped by the

barrier (no bypassing). This boundary condition requires that the shoreline at the structure be parallel to the

incoming wave crests. Figure III-2-30 shows the resulting shoreline evolution with increasing time updrift

(accretion) and downdrift (erosion). Figures III-2-31a, III-2-31b, and III-2-31c are nondimensionalized

solution graphs (at different scales) for the condition of no sand transport at the structure location (*x *= 0),

Longshore Sediment Transport

III-2-59

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