The parabolic shape was developed by Hsu et al. (1987, 1989a, 1989c), Hsu and
Evans (1989), and Silvester and Hsu (1991, 1993) to improve agreement, as compared
with the log spiral, along the down-drift section of a headland-bay beach. This section
is not typically strongly curved distant from the headland, unless it intersects a sediment-
impounding structure or headland. In studying single-headland beaches, the present
authors developed the hyperbolic-tangent shape as described in this paper, which appears
to have advantages over the parabolic shape in ease of application and interpretation of
empirical parameters defining it.
In the course of this investigation, automated shape fitting routines were developed
in the Graphical User Interface of the MatLab (Version 5) language. These convenient
programs are available from the authors upon request.
A database was developed comprised of 23 beaches each in Spain and in North
America. Data were sought for beaches extending from relatively small-project scale
(hundreds of meters) to regional scale (exceeding tens of kilometers for the case of the
headland bay downdrift of Cape Canaveral). The database and resultant fitting
parameters are summarized in tabular form at the end of this paper. The observed
headland-bay beaches were classified as having one, two, or "1.5" headlands (partial
headland located down drift of the main headland). The database was developed from
nautical charts, project drawings, and aerial photographs, from which the shoreline
position and the location and geometry of the headland were digitized.
The expansive data set covers a wide range of beach lengths, sediment size from fine
to medium sand, types and sizes of headland controls, and incident wave conditions. It
therefore provides a challenge to the concept of an equilibrium shape for headland-bay
beaches, by which the validity and universality of the shapes can be examined.
LOGARITHMIC SPIRAL SHAPE
Krumbein (1944) observed that a headland-bay beach adopts an equilibrium shape
that is similar to a log spiral. The pole of the spiral is identified as the diffraction point
(Silvester 1960, 1976), and the characteristic angle of the spiral is a function of the
incident wave angle with respect to a reference line. For headlands of irregular shape and
for those with submerged sections, the diffraction point cannot be specified
unambiguously, a problem entering specification of all equilibrium shapes. The reference
line extends from the approximate location of the diffraction point to a downdrift
headland. In fitting the log-spiral to data, the location for of the pole can be considered
as free parameter to be determined in a best-fit search. In design, some ambiguity exists
as to where to locate the pole.
The logarithmic, equiangular, or logistic spiral as shown in Fig. 1 was described by
Descartes as the curve that cuts radius vectors from a fixed point O under a constant
angle α. It is expressed mathematically in polar coordinates by
R = R0 e θcot α
Moreno & Kraus