PARABOLIC SHAPE

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PARABOLIC SHAPE

For the parabolic shape, the focus of the parabola is taken to be the diffraction point.

The three coefficients needed to define the shape (e.g., see Silvester and Hsu 1991,

1993) are functions of the predominant wave angle with respect to a control line. The

control line is defined similarly to the case of the log-spiral shape as the line that extends

from the diffraction point to a reference point. Down drift of the reference point, the

shoreline is assumed to be aligned parallel to the incident wave crests. This shape

pertains to that of a long straight beach with shape controlled by one headland.

The parabolic shape of a headland bay beach was proposed by Hsu et al. (1987) and

is expressed mathematically in polar coordinates by Eq. 3 for the curved section of the

beach and by Eq. 4 for the straight down-drift section of the beach,

β 

for θ ≥ β

= C 0 + C1  + C 2  

(3)

θ 

sin β

for θ ≤β

(4)

sin θ

where R = radius to a point P along the curve at an angle θ; Ro = radius to the control

point, at angle β to the predominant wave front direction; β = angle defining the

parabolic shape; θ = angle between line from the focus to a point P along the curve and

predominant wave front direction; and C0, C1, and C2 = coefficients determined as

functions of β. The variable R of the parabolic shape is expressed as a second-order

polynomial of β/θ for the curved section of the shape; otherwise, it is a straight line. For

θ = β, the condition R = R0 must be met, which forces C0+C1+C2 = 1 by Eq. 3.

Fig. 4 is a definition sketch for the parabolic shape, and the revised values of the

coefficients C0, C1, and C2 are listed in Table 4.2 of Silvester and Hsu (1993). To our

knowledge, the C-coefficients appear for the first time in the literature in Hsu and Evans

(1989). We believe that their values were obtained by fitting of data from 14 bay

beaches in Australia and from seven physical-model beaches (Ho 1971). Values of β

ranged in prototype beaches from 22.5to 72.0 whereas the variation in model beaches

was from 30to 72 Values of the C-coefficients were later revised in Silvester and

Hsu (1993) and given in tabular form from β = 20to 80at a 2-deg intervals.

As a sensitivity test, we analyzed the response of the parabolic shape to a change of

the value of the characteristic angle β and of Ro. The angle corresponds to the angle

between the control line and the predominant wave crest orientation. Ro is a scaling

parameter the length of the control line.

Moreno & Kraus