Fig. 4. Definition sketch of the parabolic shape.

Fig. 5 depicts change in parabolic shape for different values of for fixed *R*0 and a

given focus position. Because the parabolic shape is defined only for ? ≥ , the

alongshore extent of the shape decreases as β increases. In addition, as β increases, the

slope of the shape at the point where θ = β departs farther from horizontal. Similarly,

Fig. 6 shows the change in parabolic shape for different values of *R*o for constant , for

a given focus position. The scaling effect of the radius *R*o is evident. In summary, the

angle controls the shape of the parabola, and *R*o controls its size.

Because the control line intersects the beach at the point where the curved section

meets the straight section of the beach, the sensitivity of the parabolic shape to errors in

the estimation of the control point was examined. This was done by jointly changing *R*o

and while keeping the distance from the headland to the straight shoreline constant.

Fig. 7 shows the resultant parabolic shapes for slightly different locations of the

control point. It is apparent that the parabolic shape is insensitive to . This observation

means that the control point is not well defined, i.e., uncertainty in selection of the

control point and hence the radius *R*o and has little influence on the final result.

To analyze the validity of the assumption that a parabolic shape can describe

headland-bay beaches, the location of the focus of the parabola and the characteristic

angle must be specified, together with the scaling parameter *R*o. Because the parabolic

shape is defined in a particular coordinate system, in design applications an extra

unknown enters the orientation of the entire parabolic shape in plan view. Therefore,

five unknowns must be solved for in practical applications. For this purpose, a computer

program was prepared that solves for the five unknowns by minimizing the composite

radial (for the curved section of the beach) and Cartesian (for the straight-line section of

the beach) rms error of fit for a given bay-shaped beach.

Moreno & Kraus

7

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