The hyperbolic tangent shape was developed by the authors to simplify the fitting

procedure and reduce ambiguity in arriving at an equilibrium shoreline shape as

controlled by a single headland. As demonstrated above, it can be difficult to specify the

location of the pole or focus, and the characteristic angle (angle between predominant

wave crests and the control line) for developing a log-spiral shape or a parabolic shape.

In addition, the log-spiral shape does not describe an exposed (straight) beach located

far downdrift from the headland, so that another shape must be applied.

The hyperbolic tangent functional shape is defined in a relative Cartesian coordinate

system as

(5)

where *y *= distance across shore; *x *= distance alongshore; and *a *(units of length), *b *(units

of 1/length), and *m *(dimensionless) are empirically-determined coefficients.

This shape has three useful engineering properties. First, the curve is symmetric with

respect to the *x*-axis. Second, the values *y *= *a *define two asymptotes; in particular of

interest here is the value *y *= *a *giving the position of the down-drift shoreline beyond the

influence of the headland. Third, the slope *dy*/*dx *at *x *= 0 is determined by the parameter

range of *m *< 1.

According to these three properties, the relative coordinate system should be

established such that the *x*-axis is parallel to the general trend of the shoreline with the

point where the local tangent to the beach is perpendicular to the general trend of the

shoreline. These intuitive properties make fitting of the hyperbolic-tangent shape

relatively straightforward as compared to the log-spiral and parabolic shapes, making it

convenient in design applications.

Sensitivity testing of the hyperbolic tangent shape was performed to characterize its

functional behavior and assign physical significance to its three empirical coefficients.

The parameter *a *controls the magnitude of the asymptote (distance between the relative

origin of coordinates and the location of the straight shoreline), and its functioning is self-

evident. Fig 8 shows the action of *b *as a scaling factor controlling the approach to the

asymptotic limit. Fig. 9 indicates that *m *controls the curvature of the shape, which can

vary between a square and an S curve. Larger values of *m *(*m *≥1) produce a more

rectangular and somewhat unrealistic shape, whereas smaller values produce more

rounded, natural shapes.

For small values of *x*, the defining Eq. 5 is approximated as:

(6)

Moreno & Kraus

9

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