To fit the hyperbolic tangent shape to a given shoreline, we must solve for six
and the rotation of the relative coordinate system with respect to the absolute coordinate
system. Because of the clear physical meaning of the parameters, fitting of this shape can
be readily done through trial and error. An optimization procedure that minimizes the
rms error with respect to vertical axis values was implemented that solves for the six
unknowns.
RESULTS
The three functional headland-bay shapes were fit to the database assembled in this
study, as summarized in Table A1. Various authors have noted that fitting of the log-
spiral shape is difficult in the down-drift section of the beach, also encountered here. It
is a particular concern in attempting to fit to long beaches or to beaches with one
headland. However, even in these situations, it was found that a good fit could be
achieved for the stretch near the headland.
The parabolic shape provides good fits for beaches with a single headland, because
they consist of a curved section (well describes the portion of the beach protected by the
headland) and a straight section (well describes the down-drift section). However, this
shape is insensitive to values of the determining parameters. Interpolation of the
C-coefficients over a broad multi-valued plain makes the fitting process time consuming.
The possibility of allowing the C-coefficients to be free while keeping the second-order
polynomial shape has been implemented and will be discussed elsewhere. The goodness
of the fit increases significantly in most applications, suggesting re-evaluation of the C-
values.
The hyperbolic-tangent shape was found to be a relatively stable and easy to fit,
especially for one-headland bay beaches. According the fittings shown in Table 1 in the
Appendix and the plot of best-fit values in Fig. 10, the following simple relationships are
a b ≅1.2
(8)
m ≅ 0.5
(9)
The physical meaning of Eq. 8 is interpreted that the asymptotic location of the down-
drift shoreline increases with the distance between the shoreline and the diffracting
headland. Eqs. (8) and (9) are equivalent to selecting one family of such hyperbolic
tangent functions for describing headland-bay beaches, and these values are convenient
The mean value of the product ab from the database was ab = 1.2. Least-squares
fitting for the linear function between log10 a and log10 b led to the following relationship
shown as the line drawn through the data points in Fig. 10:
a 0.9124 b = 0.6060
(10)
with a correlation coefficient R2of 0.8696.
Moreno & Kraus
11
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