stages with the plan view dimension of the fill as a function of time given in Figure III-2-46. For this

particular scenario, the solution (at time *t*) analogous to Equation 2-37 only with the planform beach built in

staged increments (as per Figure III-2-46) is:

% *Y*1 & *Y*o erfc

% *Y*2 & *Y*1 erfc

(III-2-39)

2 ε*t*

2 ε (*t *& *t*1)

2 ε (*t *& *t*2)

which can be nondimensionalized and solved utilizing the nondimensionalized solution graph provided by

the simpler case of Equation 2-37 given in Figure III-2-34.

(25) It should be recognized that the linearized form of shoreline solution will produce a higher rate of

shoreline change by overestimating the longshore transport rate because 2αb > sin(2αb) in the linearized sand

transport solution. Thus, under properly estimated parameters for idealized conditions, a higher rate of

attenuation of beach fills will be obtained than is expected to occur in reality, thus providing a conservative

answer to project losses.

(26) As a final point, it is noted that when wave angles are very large and the difference between the

wave direction and shoreline orientation exceeds 45 deg, the true form of the shoreline diffusivity constant

based on the nonlinear sin(2α) term in the sand transport equation will be negative, which totally changes the

characteristics of the shoreline change model equation. In these cases, the shoreline evolves in an unstable

manner equivalent to running the previous stable solution forms backwards through time. In other words,

a shoreline having a perturbation placed on it (such as a beach fill) would see a growth of the perturbation

toward an elongated cuspate feature as time progresses rather than seeing the smoothing out of the

perturbation as given in solutions of the preceding paragraphs. This type of shoreline instability may possibly

explain certain shoreline features such as cuspate forelands which are ubiquitous on elongated bays where

the dominant wave action is along the major axis of the bay and at large angles to the prevailing shoreline

trends (Walton 1972).

Longshore Sediment Transport

III-2-75

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