Because little quantitative verification of proposed headland-bay shapes could be
found, one objective of the present study is to investigate the generality of these
equilibrium shapes through analysis of a large data set of beach response to headlands
ranging from regional scale to local engineering projects. The second objective is to
develop guidance for applying the idealized shapes in engineering and morphology
studies. In the process of investigating the shapes and developing engineering guidelines,
a new shape was developed which we call the hyperbolic-tangent shape.
Previous researchers proposed two functions for describing the equilibrium shoreline
of a headland-bay beach, the logarithmic spiral (op. cit.) and the parabolic shape (e.g.,
Hsu et al. 1987; Hsu and Evans 1989; Silvester and Hsu 1993). None of these shapes,
including the hyperbolic-tangent shape proposed in this study, is derived directly from
the acting physical processes that developed the shape; rather, they are observational.
The empirical approach appears strong qualitatively, because of the existence of the
many headland beaches found on all coasts of the world. The weakness of the approach
lies in the quantification process of fitting a shape to data and for design, where data will
not exist, requiring the engineer to exercise judgement in the fitting procedure. Many
more references were consulted and reviewed than can be given here. We plan to discuss
these, the assembled database, and calculation algorithms in another publication.
As discussed in the original sources, headland-bay equilibrium shoreline shapes arise
through wave sheltering by diffraction at the object serving as the headland, combined
with refraction, which will dominate with distance along the beach away from the
headland. This explanation tends to require the condition of a strongly predominant
wave direction. LeBlond (1972) studied the existence and properties of a planimetric
shape towards which a headland beach asymptotically approached. He also attempted
a numerical simulation of the erosion of a linear beach in the presence of a headland, but
was not fully successful. Komar and Rea (1975) and Walton (1977) investigated cause
and effect in formation of headland beaches. The problem has yet to be resolved fully,
although modern modeling technology appears capable of doing so. Wind (1994)
presented an analytical model of crenulate-shaped beach development, for which the
beach shape remains constant with time, but with the entire form expanding at a rate
according to a time function.
A logarithmic-spiral (hereafter abbreviated as log-spiral) shape eventually turns
around the headland and, at some ambiguously determined point whose location depends
on the site, it loses meaning for describing shoreline position. The question as to where
this cutoff should be is problematic because of limitations of a static (equilibrium) form.
Site-specific constraints such as presence of other headlands or sediment-impounding
features, trend of bathymetric and topographic contours, and underlying geologic
structure exert controls that cause deviations of the shoreline from a simple and smooth
form. Sometimes, the turning of the log spiral fits the shape of the shoreline produced
by impoundment at a headland-type feature located down drift. In summary, the log-
spiral shape might best be viewed as applicable to the beach located between two
headlands for a coast with predominant wave direction.
Moreno & Kraus