HYPERBOLIC TANGENT SHAPE
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
y = a tanh m (bx)
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
m, and the slope is infinite if m < 1. This restriction on slope indicates m to be in the
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
onshore. Also, the relative origin
of coordinates should
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:
y ≈ a (b x )m
Moreno & Kraus