**2. Background**

Since the beginning of nineteenth century, many coastal geologists, geographers and engineers have been trying to predict the shoreline plan-shape of headland-bay beaches. Despite the complexity involved in coastal processes, simple empirical expressions have been derived since the 1940s in order to fit part or whole of the bay periphery. Among these approaches, the logarithmic spiral model [12], hyperbolic tangent model [13,14] and parabolic model [15–17] have turned out to be the most acceptable expressions for practical applications to headland bay beaches in static equilibrium. However, none of these models are derived directly from the acting physical processes that developed the shape; rather, they are based on the observation of the shoreline plan-shape, and so are lacking in a correlation between shoreline response and wave forcings (refraction and diffraction).

The logarithmic-spiral model was introduced by [12] observing an early 1940 imagery of Half-Moon Bay in California, USA. With field observations, the author realized that the bay adopted an equilibrium shape that is similar to a log-spiral. A definition sketch is given in Figure 1. The log-spiral equation, in polar coordinates, is given by:

$$R\_1 = R\_2 \exp(\theta \cot \alpha) \tag{1}$$

where:


**Figure 1.** Definition sketch for the log spiral.

Despite of being the very first model proposed, representing a milestone in the field of crenulated bays research, the log-spiral model has long been criticized [3,23] because of several drawbacks which make it tricky to use. First, due to its constant curvature, the equation does not fit the relatively straight section of the headland bay beach; secondly, the pole of the log-spiral curve must be found by trial and error, because it does not coincide with the diffraction point, and could even deviate from it by a distance ranging from centimetres to kilometres (see [24]); therefore, it is not possible to predict the effect of relocating the headland (e.g., designing a headland bay beach introducing a coastal structure); and finally it does not take into account wave direction. In light of these disadvantages, two other empirical equations were developed: the hyperbolic-tangent equation and the parabolic equation.

The parabolic bay shape equation is a second-order polynomial equation developed by [15] and [16,17] in two separate works, from fitting the plan-shape of 27 mixed cases of prototype and model bays believed to be in static equilibrium:

$$\frac{R}{R\_{\notin}} = \mathbb{C}\_0 + \mathbb{C}\_1 \left(\frac{\beta}{\theta}\right) + \mathbb{C}\_2 \left(\frac{\beta}{\theta}\right)^2 \tag{2}$$

where:


The two basic parameters are the reference wave obliquity *β* and control line length *R<sup>β</sup>* (Figure 2). The variable *β* is a reference angle of wave obliquity, or the angle between the incident wave crest (assumed linear) and the control line, joining the upcoast diffraction point to a point on the near straight beach, namely the downcoast control point. The radius *R* to any point on the bay periphery in static equilibrium is angled *θ* from the same wave crest line radiating from the point of wave diffraction upcoast. The three C constants, generated by regression analysis to fit the peripheries of the 27 prototype and model bays, differ with reference angle *β*. Their analytical expressions have been provided by [14]:

$$C\_0 = 0.0000000479 \beta^4 - 0.00000879 \beta^3 + 0.000352 \beta^2 - 0.00479 \beta + 0.0715 \tag{3}$$

$$\mathcal{C}\_1 = -0.000000128 \beta^4 + 0.0000182 \beta^3 - 0.000487 \beta^2 + 0.00771 \beta + 0.955 \tag{4}$$

**Figure 2.** Definition sketch for the parabolic equations.

$$C\_2 = 0.000000944 \beta^4 - 0.000012 \beta^3 + 0.000316 \beta^2 - 0.00828 \beta + 0.0265 \tag{5}$$

In contrast to the log-spiral, the parabolic equation origin coincides with the diffraction point and therefore the equation is directly related to wave direction. However, it is affected by a drawback: the uncertainty of locating the downdrift control point. Despite that there are several interpretations of the downcoast control point of the parabolic bay shape equation [14], it is a considerable limitation which inhibits the application of the parabolic model in designing new beaches.

The hyperbolic-tangent shape model was derived by [13] through the analysis of 46 beaches around Spain and North America. Its equation is defined in a relative Cartesian coordinate system (Figure 3) as:

$$y = \pm atanh^m(b\mathbf{x})\tag{6}$$

where:


**Figure 3.** Definition sketch for the hyperbolic-tangent shape equation.

The x-axis is parallel to the general trend of the shoreline while the y-axis points shoreward; the origin of the coordinates is placed at the point where the local tangent to the beach is perpendicular to the general trend of the shoreline. The hyperbolic tangent curve is symmetric with the x-axis and produces two asymptotes, found at *y* = ±*a*. The line *y* = +*a* indicates the location on the shoreline which is no longer under the influence of the headland. The parameter *a* controls the magnitude of the asymptote (distance between the relative origin of coordinates and the location of the straight shoreline), *b* is a scaling factor controlling the approach to the asymptotic limit and *m* controls the curvature of the shape. These unknowns were found by using trial and error and an optimisation procedure that minimises rms errors. The authors of [13] found the following relationships: *ab* ∼= 1.2; *m* ∼= 0.5.

The model indirectly considers wave direction and, initially, it does not correlate the diffraction point with the origin of the hyperbolic tangent.

In 2018, [14] presented a development of the hyperbolic-tangent equation, establishing a relationship between the existing hyperbolic-tangent shape equation with the wave diffraction point. The authors used a database of case studies comprised of 46 beaches in Spain, Southern France and North-Africa, determining the coefficients *a*, *b* and *m*. The value of *m* is about 0.496, supporting the original findings of [13]. Moreover, the authors estimated a correlation between *a* and *b*, and the hyperbolic tangent shape equation became:

$$y = \pm \operatorname{atanh}^{0.496} \left( 1.794 a^{-1.097} x \right) \tag{7}$$

Overall, the research's relevance has been constituted by two relationships which link the hyperbolic tangent profile and location of diffraction point. This makes the model easy to use, both in order to verify the bay's stability and design a new beach. In the former case, starting from the distance between the location of diffraction point and asymptote, it is instantaneous to locate the origin of the equation and obtain the hyperbolic tangent profile; in the latter case, based on the shoreline advancement established in the illuminated zone, the location of the new diffraction point (i.e., new headland's tip) can be directly carried out. The relationships, which correlate parameters *c*, *d* and *a* shown in Figure 4, are:

$$\frac{c}{a} = 1.256\tag{8}$$

$$\frac{d}{a} = 0.517\tag{9}$$

**Figure 4.** Definition sketch of parameters for the modified hyperbolic-tangent equation.

It is worth noting that the hyperbolic-tangent model could be applied not only to bay in static equilibrium conditions, but also to fit a bay under non-equilibrium conditions [3]. Among the three aforementioned models, the parabolic model prevailed over the others since it was the only one that used the wave diffraction point as the origin of the co-ordinate system, ensuring that the effect of relocating the diffraction point can be assessed [14]. Despite that, in comparing the origin of the three coordinates system in studying a headland bay (using a computer program based on a trial and error approach), [25] found that the parabolic equation's origin was located in the middle of the ocean (Figure 5). This outcome weakens the strong point of parabolic model, revealing an uncertainty about its alleged robustness.

**Figure 5.** Definition sketch to obtain the hyperbolic tangent profile starting from a wave front.

On the whole, regardless of models' possible limitations, their description here presented shows how none of them model the relationship between wave characteristics and beach shape, but instead simply characterize the geometry of the headland bay profile. Nevertheless, over the last few decades, further investigations have been carried out towards the comprehension of the correlation between crenulate-shaped bays and wave characteristics. Particularly, several works focused on wave directional spreading, demonstrating that the crenulate-beach plan-form suffers under its influence. The authors of [26], searching for a new criterion to locate the downdrift control point of the parabolic shape equation, observed that the broader the directional spreading the farther the position of the downdrift control point. This behaviour indicates that, with a wider directional spreading of waves, a greater area could be influenced by wave diffraction behind the coastal structure (i.e., the transition zone increases).

During their numerical experimentation, [27] observed that a narrow directional spreading produces a more curved and asymmetric planform, confirming that the crenulated-

beach behaviour is related to the modal directional spreading of incident waves [26,28]. Furthermore, [28] evaluated how the degree of directional spreading influences diffraction's effect on the headland shape, observing that diffraction appears less important with a high directional spreading degree. The latter outcome is in accordance with [29], which demonstrated that diffraction shapes bay morphology only for wave climates with restricted approach angles.

Although important progress has been made, the effective correlation between wave forcing (i.e., diffraction and refraction) and static equilibrium profile has not been fully examined and understood. In this work we attempt to understand this issue, starting from the commonly engineering assumption that an equilibrium profile follows the wave front trend, with the aim of verifying the validity of this statement.
