**7. Conclusions**

The main purpose of present paper is to establish a relationship between static equilibrium profiles of crenulated bays and wave front shapes and, more generally, understanding the correlation between bay shoreline response and wave forcings. The peculiar equilibrium plan-form (static or dynamic) of headland bays is assumed at a relatively long-term scale (e.g., annual to decadal) as a response to predominant wave direction; the downdrift segment of the bay, in fact, tends to align to the dominant direction of incoming waves, while, the curved up-coast segment is modelled by the diffracted wave fronts. In this regard, it is commonly believed by engineers that equilibrium beach profiles follow the wave front trend; however, this has been never fully proved so far. The most important works concerning headland bay beaches provided by the literature are empirical models, which simply describe the geometry of static equilibrium bays shape, neglecting the acting physical processes that sculpture the coast. In fact, they are merely of an empirical kind, lacking in a further insight on relationships between incident wave characteristics and beach shape. Therefore, in the lack of a model in which plan-shape is strongly correlated with wave characteristics, the project design of a new headland bay beach (necessary if the *headland control* practice has to be implemented) could, as a result, be extremely challenging.

To assess a possible correlation, numerical experiments have been carried out using the MIKE 21 Boussinesq Wave Module (BW), where wave fronts have been compared to the hyperbolic-tangent equilibrium profile, analysing the influence of wave direction, wave period and refraction phenomenon. Results proved that equilibrium profiles are located seaward compared to their relative wave fronts. Hence, designing a new beach following a wave front trends ensures a shoreline accretion in the sheltered zone (i.e., leeward to the headland). Additionally, 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. The function seems to indicate that equilibrium beach profiles of a single headland bay correspond to a simple translation of wave front 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) has been performed. The numerical model has been set up, and the LDR equivalent wave concept has been used to embody the dominant wave attack that rules the long-term evolution of the little bay. Results confirm the behaviour observed from numerical experiments outcomes; the "*wave-front-bay-shape equation*" has been successfully verified, thus confirming that a correlation between equilibrium plan form and wave fronts can be found.

Nevertheless, the research still stands at a primary stage, and requires improvement and accurate verification in future research works in order to develop an enhanced guidance which could help in engineering and morphological practice.

**Author Contributions:** Conceptualization, M.B. and M.C.; Methodology, M.B. and S.T.; Software, S.T.; Supervision, M.C.; Writing—original draft, M.C.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Experimental data are available from the corresponding author upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.
