**1. Introduction**

Crenulate-shaped or headland bays are quite common on exposed sedimentary coasts containing headlands, and represent about 50% of the world's coastline [1,2]. The term headland-bay beach has been used to define a shoreline bounded by rocky outcrops or headlands, either natural or man-made, which lead to diffraction of incoming waves. Particularly, the predominant waves are diffracted in such a way as to break simultaneously around the periphery of the bay once an equilibrium plan-shape has been established (static equilibrium condition, [3]). The fame of the headland bay beaches, in fact, lies in their equilibrium condition, static or dynamic, which ensures that they are considered a way to achieve coastline stabilization [4]. Static equilibrium is a condition characterized by the absence of littoral drift, without the need for sediment supply to preserve its long-term stability; on the other hand, a dynamic equilibrium condition requires sediment supply, from updrift and/or from another kind of source, to maintain its stability and not retreat towards the limit defined by static equilibrium position [3].

Typically, the plan-shape of a single-headland bay is characterized by an upcoast curved zone (diffraction zone), a gentle transition zone and a relatively straight tangential segment on the downdrift end of the bay (illuminated zone), which is largely orthogonal to the dominant wave direction; this equilibrium plan-form is that assumed by the bay at a relatively long-term scale (e.g., annual to decadal) as a response to the predominant

**Citation:** Buccino, M.; Tuozzo, S.; Ciccaglione, M.C.; Calabrese, M. Predicting Crenulate Bay Profiles from Wave Fronts: Numerical Experiments and Empirical Formulae. *Geosciences* **2021**, *11*, 208. https:// doi.org/10.3390/geosciences11050208

Academic Editors: Jesus Martinez-Frias, Gianluigi Di Paola, Germán Rodríguez and Carmen M. Rosskopf

Received: 19 February 2021 Accepted: 6 May 2021 Published: 10 May 2021

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wave direction. Short-term fluctuations arising from beach-storm interactions, which could cause severe beach berm retreat, can be neglected due to their reversibility.

Devising a headland bay beach system in static equilibrium in such a way that the reorientation of the shoreline cancels the longshore sediment transport is called "*headland control*", and can be considered as a valuable option for coastal stabilization, which is frequently addressed via traditional detached low-crested breakwaters [5,6] or artificial reefs [7,8].

The headland control concept has been suggested for engineering use by [9,10] as a naturally functioning and preferable means of shore protection. In this regard, the formation of a crenulate-shaped bay on a sedimentary coastline, under oblique attack of persistent swell, is the most stable beach generated by nature [11]. The headland control approach appears to be of the utmost importance in the field of coastal stabilization against beach erosion, which increasingly affects many parts of the world's coastline. Hence, the employment of a crenulate-shaped bay to stabilization of a shoreline may be a powerful tool for engineering purposes.

Several studies and researches have been carried out in order to develop functional models describing static equilibrium bays' shapes: the logarithmic-spiral model [12], hyperbolic-tangent model [13,14] and parabolic model [15–17]. However, they are merely of an empirical nature, lacking further insight on relationships between incident wave characteristics and beach shape. As a matter of fact, none of these shapes are derived directly from the acting physical processes that developed the shoreline; rather, they are observational. Consequently, despite the strength in the assessment of stability of existing beaches, they are affected by uncertainties in the design of new artificial beaches through the headland control approach, which instead requires a deep knowledge of the dominant physical processes that govern the plan-shape of the bay. Without that, the project design of a new beach could result extremely challenging.

With the aim of overcoming such drawbacks and, broadly, to establish a relationship between wave forcings (diffraction and refraction) and bay shape response, the main task of this paper is founding a possible correspondence between static equilibrium profiles and wave characteristics, accounting for the relationship between equilibrium shape profiles and wave fronts. In fact, it is commonly believed that equilibrium beach profiles follow the wave front trend, but this has not been proved in literature so far, and no research has clarified in depth how wave characteristics, and particularly wave fronts, could shape a crenulated stretch of coast. For these reasons, this research represents a first step towards a development of a guidance which could help in engineering and morphological practice.

The analysis has been carried out via numerical modeling, which has long been proved to be a powerful tool, suited to even complex hydrodynamic phenomena [18–20].

Numerical experiments have been performed using the MIKE 21 Boussinesq Wave Module (BW) [21], where wave fronts have been compared to the hyperbolic-tangent equilibrium profile, analysing the influence of wave direction, wave period and refraction phenomenon. A correspondence function, called the "wave-front-bay-shape equation" has been established, offering an easy application to engineering uses due to the simple geometric interpretation of its controlling parameters. It is noteworthy that, being based on wave front analysis, the employed approach is essentially linear (e.g., [22]), although Boussinesq wave models account for weak non-linearity. As such, the parametric study discussed in Section 3 treats the wave height as an invariant of the problem.

The simulation's outcomes seem to indicate that equilibrium beach profiles of a single headland bay correspond to a simple translation of wave fronts normal to the propagation line at the headland tip. Moreover, the application of the "wave-front-bay-shape equation" to the case-study bay of the Bagnoli coast (south-west of Italy) is described in the paper.
