To fit the hyperbolic tangent shape to a given shoreline, we must solve for six

unknowns; the location of the relative origin of coordinates, the coefficients *a*, *b*, and *m*,

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.

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

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

obtained for reconnaissance-level guidance:

(8)

(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

for reconnaissance studies prior to detailed analysis.

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:

(10)

with a correlation coefficient *R*2of 0.8696.

Moreno & Kraus

11

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