data (measurement) points for a specified bay-shaped beach. The program outputs the
following parameters: root-mean square (rms) error of the fit taken in the direction of
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,
for θ ≥ β
= C 0 + C1 + C 2
for θ ≤β
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