# Headland-Bay Beach

**DOI:**https://doi.org/10.1007/978-3-319-48657-4_165-2

A headland is defined in common language as: (1) a point of usually high land jutting out into a body of water: promontory; (2) high point of land or rock projecting into a body of water. Therefore, a headland-bay beach is a beach whose shape is mainly conformed by the fact that it is located between such headlands, or at least adjacent to one. Some of the synonymous terms that can be found in literature to describe a headland-bay beach are: bay-shaped beach, pocket beach, zeta bay, bow-shaped bay, and half-heart bay. This type of feature shows a gradually changing curvature which Krumbein (1944) noted resembled that of a logarithmic spiral curve.

Johnson (1919) gave an incisive description of wave refraction caused by headlands along an embayed coastline, and Krumbein (1944) showed a simplified diagram of wave refraction into a bay lying to the lee of a headland. According to Yasso (1965), a headland was considered to be any natural or artificial obstruction that extended seaward from the coastline and caused a change in some element of the coastal wave pattern because of its presence.

## Historical Development

Observations of headland beach morphology in nature Krumbein (1944) was the first author to describe a beach planshape— Halfmoon Bay, California (USA)—as being similar to the increasing radius of curvature found in the logarithmic spiral, although the restricted classification was removed some years later and the paper was reprinted in 1947. Yasso (1965) selected four US beaches for testing goodness of fit to the logarithmic spiral approximation. Poles for three of the best- fitting logarithmic spirals were located in close proximity to the seaward end of each headland. The spiral angle *α* was found to range between 41.26° and 85.64°. Berenguer and Enríquez (1988) reviewed data from 24 beaches around Spain and derived empirical correlations between geometrical characteristics of the layout of the offshore breakwaters and beach planshape. Moreno and Kraus (1999) performed fittings of analytical planshapes to a data set of 46 beaches in Spain and the United States and derived preliminary engineering design guidance including the proposal of a new functional shape.

Observations of headland beach morphology in the laboratory Silvester (1960) tracked the time evolution of a beach in a physical model and observed that the beach between headlands tended to reach an equilibrium shape—logarithmic spiral shape—in response to persistent swell directions under a certain wave angle of attack. The model coastline was allowed to erode without replenishment of sand at the updrift end. Yasso (1965) pointed out that the conditions of Silvester’s physical-model test (lack of continuous sediment supply to the updrift end of the model and close spacing of the headlands) suggested that the equilibrium form achieved in the model may not be identical to that achieved under natural conditions of sediment supply and wider separation of headlands.

Silvester (1970) performed additional model tests in which three different wave conditions were generated at three different angles of incidence to the alignment of a headland on an initially straight sandy beach. It was observed that the coastline developed three distinct curvature zones: first, a near circular section in the lee of the upcoast headland; second, a logarithmic spiral; and finally, a segment tangential to the downcoast headland. The time evolution of the spiral constant angle was plotted for the three incident wave angles tested, but only one wave condition was run long-enough as to see an asymptotic trend. A graphical linear relationship between the logarithmic spiral constant angle *α* and the wave angle β was provided based on the three angles tested.

Spătaru (1990) studied Romanian Black Sea beaches subjected to normal wave incidence by means of physical models. Equilibrium beach planshapes were considered to be described by arcs of circumferences, and provided design guidance based on geometry.

## Numerical Model Approaches

Mashima (1961) constructed wave-energy diagrams from wind-rose diagrams and studied the configuration of stable coastlines based on energy considerations. When the wave energy is a semi-ellipse, the configuration of the stable coastline is approximately parabolic on which the tangent direction at the apex of the parabola coincides with the direction of the major axis of the wave-energy diagram.

LeBlond (1972) attempted to study how wave-induced longshore currents in the presence of a headland could erode a linear beach by developing a numerical model. LeBlond (1972) stated that if there existed a planimetric shape which the headland beach asymptotically approached, it must have the following properties: (1) it should be concave outwards, near the headland, and then convex outwards. (2) the sand transport should increase monotonically along it. (3) erosion, by causing the beach to be displaced normal to itself, should not qualitatively change the shape of the beach. LeBlond also pointed out that the logarithmic spiral did not satisfy the first condition because it is always concave outwards, and that there may be other curves satisfying all of the above three conditions, and one could not decide *“*a priori*”* which one will be the equilibrium one.

The main modification implemented by Rea and Komar (1975) with respect to previous numerical modeling efforts, was the combination of two orthogonal one-dimensional grids to simulate beach configurations, so that beach erosion could proceed in two directions without the necessity of a full two-dimensional array. Testing of the model in a hooked beach coastline configuration indicated that the coastline would always attempt to achieve an equilibrium configuration governed by the pattern of offshore wave refraction and diffraction and the distribution of wave-energy flux.

Walton (1977) presented an analytical model to describe the equilibrium shape of a coastline sheltered by a headland using a continuous wave-energy diagram consisting of representative offshore ship wave height and direction observations. It was found to produce coastline shapes similar to the logarithmic spiral shape for sheltered beaches in Florida. The model worked by establishing that the coastline orientation at a certain point was normal to the average direction of the so-called energy of normalized wave attack—that is, the energy which is allowed by the headland to reach the shore.

Yamashita and Tsuchiya (1992) constructed a numerical model for three-dimensional beach change prediction to simulate a pocket beach formation. The model consisted of three modules to calculate waves, currents, and sediment transport and beach change. The wave transformation module was based on the mild-slope equation of hyperbolic type; the current module was horizontally two-dimensional with direct interaction with sea-bottom change, which was evaluated by the sediment transport model formulated by Bailard (1982) in the third module.

*r*, the angle δ, and time

*t*, the evolution of the coastline with a constant shape implies that the coastline might be described as

*r*

_{0}is the constant,

*f*(δ) is the shape function, and

*e*(

*t*) the evolution function of the coastline in time. With respect to the time function, it was shown that in the diffraction zone it should follow a

*t*

^{1/3}law, whereas for the refraction zone a

*t*

^{1/2}law was found. This implied that the evolution of a headland-bay beach in the diffraction zone should initially be faster and on the long term, slower than the evolution in the refraction zones. With respect to the shape, the function shape

*f*(δ) is expressed in terms of functions representing the diffracted wave field. The logarithmic spiral is obtained by taking the functions for the group velocity and the geometrical part for the driving force as constants.

## Equilibrium Planshapes of Headland-Bay Beaches

Three functional shapes have been proposed to describe the equilibrium planshape of headland-bay beaches, namely the logarithmic spiral shape, the parabolic shape, and the hyperbolic tangent shape.

*logarithmic spiral*(also named equiangular, or logistic spiral), first described by Descartes, was described as the curve that cuts radii vectors from a fixed point O under a constant angle

*α*(Fig. 1). The equation of the logarithmic spiral can be written in polar coordinates as

*R*is the length of the radius vector for a point P measured from the pole O, θ is the angle from an arbitrary origin of angle measurement to the radius vector of the point P,

*R*

_{0}is the length of radius to arbitrary origin of angle measurement, and

*α*the characteristic constant angle between the tangent to the curve and radius at any point along the spiral. The pole of the spiral is identified as the diffraction point (Silvester 1960; Yasso 1965), and the characteristic angle of the spiral is a function of the incident wave angle with respect to a reference line. For headlands of irregular shape and for those with submerged sections, the diffraction point cannot be specified unambiguously, a problem entering specification of all equilibrium shapes. The reference line extends from the approximate location of the diffraction point to a downdrift headland. This shape is extremely sensitive to variations in the characteristic angle

*α*because the angle enters the argument of an exponential function (Moreno and Kraus 1999), being the practical consequence that

*α*has to be accurately defined. For engineering application four unknowns must be found: location of pole (two coordinates), characteristic angle

*α*, and scale parameter

*R*

_{0}. The shape of the log spiral is controlled only by

*α*, with the parameter

*R*

_{0}determining the scale of the shape. In fact, the functioning of

*R*

_{0}is equivalent to setting a different origin of measurement of the angle θ. In other words, graphically the log spiral may be scaled up or down by turning the shape around its pole.

Values of *α* for headland-bay beaches reported in the literature range from about 45° to 75°. As α becomes smaller, the log spiral becomes wider or more open. There are two singular values for a: if α = 90°, the log spiral becomes a circle, and if α = 0°, the log spiral becomes a straight line.

Various authors have noted that fitting of the log-spiral shape is difficult in the downdrift section of the beach. It is a particular concern in attempting to fit to long beaches or to beaches with one headland. However, even in these situations, a good fit could be achieved for the stretch near the headland (Moreno and Kraus 1999).

*parabolic shape*of a headland-bay beach was proposed by Hsu et al. (1987) and is expressed mathematically in polar coordinates by Eq. 3 for the curved section of the beach and by Eq. 4 for the straight downdrift section of the beach (Moreno and Kraus 1999),

*R*is the radius to a point P along the curve at an angle θ,

*R*

_{0}is the radius to the control point at angle β to the predominant wave-front direction, β is the angle defining the parabolic shape, θ is the angle between line from the focus to a point P along the curve and predominant wave-front direction; and

*C*

_{0},

*C*

_{1}, and

*C*

_{2}are the coefficients determined as functions of β—the coefficients

*C*

_{0},

*C*

_{1}, and

*C*

_{2}are listed in Silvester and Hsu (1993) form from β = 20° to 80° at a 2°-interval (Fig. 2).

For this parabolic shape, the focus of the parabola is taken to be the diffraction point. The three coefficients needed to define the shape (see Silvester and Hsu 1991, 1993) are functions of the predominant wave angle with respect to a control line. The control line is defined similarly to the case of the log-spiral shape as the line that extends from the diffraction point to a reference point, at an angle β between the control line and the predominant wave crest orientation. Downdrift of the reference point, the coastline is assumed to be aligned parallel to the incident wave crests. This shape pertains to that of a long straight beach with shape controlled by one headland. Values of β ranged in prototype beaches from 22.5° to 72.0°, whereas the variation in model beaches was from 30° to 72°.

Sensitivity tests were performed (Moreno and Kraus 1999), where the response of the parabolic shape to a change in the value of the characteristic angle β and of *R*_{0} was analyzed. The results proved that *R*_{0} is a scaling parameter—the length of the control line, the alongshore extent of the shape decreases as β increases because the parabolic shape is defined only for θ ≥ β. In summary, the angle β controls the shape of the parabola, and *R*_{0} controls its size. Because the control line intersects the beach at the point where the curved section meets the straight section of the beach, the sensitivity of the parabolic shape to errors in the estimation of the control point was examined. This was done by jointly changing *R*_{0} and β while keeping the distance from the headland to the straight coastline constant. This observation means that the control point is not well-defined, that is uncertainty in selection of the control point and hence the corresponding joint combination the radius *R*_{0} and β has little influence on the final result.

According to Moreno and Kraus (1999), the parabolic shape provides good fits for beaches with a single headland, because they consist of a curved section (well described by the portion of the beach protected by the headland) and a straight section (well describes the down-drift section).

*y*is the offshore distance from diffraction point to coastline,

*L*the average wavelength in the lee area (between coastline and diffraction point).

For practical use of the parabolic shape, five unknowns need to be solved for: location of focus (two coordinates), characteristic angle β, scaling parameter *R*_{0}, and the orientation of the entire parabolic shape in plan view.

*hyperbolic tangent shape*was developed by Moreno (1997) and proposed for engineering design of equilibrium shapes of headland-bay beaches by Moreno and Kraus (1999) to simplify the fitting procedure and to reduce ambiguity in arriving at an equilibrium coastline shape as controlled by a single headland (Fig. 3). As mentioned 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.

*y*is the distance across shore,

*x*is the 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 downdrift coastline 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 *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 coastline with the *y*-axis pointing onshore. Also, the relative origin of coordinates should be placed at a point where the local tangent to the beach is perpendicular to the general trend of the coastline. 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 (Moreno and Kraus 1999) to characterize its functional behavior and assign physical significance to its three empirical coefficients: *a* controls the magnitude of the asymptote (distance between the relative origin of coordinates and the location of the straight coastline); *b* is a scaling factor controlling the approach to the asymptotic limit; and *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.

To fit the hyperbolic tangent shape to a given coastline, we must solve for six unknowns: the location of the relative origin of coordinates (two 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.

## Headland-Control Concept of Shore Protection

The headland-control concept of shore protection was first proposed by Silvester (1976) and further discussed by Silvester and Ho (1972), and was described as a combination of groins and offshore breakwaters at alongshore and seaward spacings such to create long lengths of equilibrium-bay beaches (Silvester and Hsu 1993). The structural dimensions in proportion to beach length are much smaller, and headlands are spaced much farther apart than offshore breakwaters. Therefore, it is intended to be a “regional” means of shore protection. Because headlands form pocket beaches, they might best be applied in a sediment-deficient area or for stabilizing an entire littoral reach of coast.

Headland beaches compartmentalize the coastline and reorient it in the local compartments to be parallel to the wavecrests of the predominant wave direction. If a coast has a substantial change in wave direction annually, the headland-bay beach might not be as stable as beaches behind traditional detached breakwaters, or shorter headland-bay beach compartments would be required.

A headland-bay beach design requires that a tombolo forms or be created (as by beach fill) behind the anchoring headland. If this connection is lost, the pocket beach is destroyed, and sediment can move alongshore, between adjacent compartments. Headland-bay beaches function and have their main attribute in creating independent pocket beaches, for which there is little or no communication of sand alongshore. Therefore, a headland beach presents a total barrier to littoral drift and can only be considered as a shore-protection alternative if such a barrier would not pose a problem to adjacent beaches.

The assumption that there is a single predominant wave direction which controls the final coastline shape is questionable where long distances are involved. If the design goal is to stabilize a regional extent by multiple pocket beaches, the headland-control concept might be appropriate.

## Cross-References

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