*Article* **Predicting Crenulate Bay Profiles from Wave Fronts: Numerical Experiments and Empirical Formulae**

**Mariano Buccino, Sara Tuozzo \*, Margherita C. Ciccaglione and Mario Calabrese**

Department of Civil and Environmental Engineering, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy; buccino@unina.it (M.B.); margheritacarmen.ciccaglione@unina.it (M.C.C.); mario.calabrese@unina.it (M.C.)

**\*** Correspondence: sara.tuozzo@unina.it

**Abstract:** For crenulate-shaped bays, the coastal outline assumes a specific shape related to the predominant waves in the area: it generally consists of a tangential zone downcoast and a curved portion upcoast. Many coastal engineers have attempted to derive an expression of the headland bay shapes that emerge when a full equilibrium is reached (stable or dynamic). However, even though models for static equilibrium bays exist, they are merely of an empirical kind, lacking further insight on relationships between incident wave characteristics and beach shape. In addition, it is commonly believed that shoreline profiles tend to follow wave fronts, but this has been never fully verified. In this paper, we investigate a possible correlation between static equilibrium profiles and wave front shapes. Numerical experiments have been performed using the MIKE 21 Boussinesq Wave module, and the generated wave fronts have been compared to the hyperbolic-tangent equilibrium profile. A thoughtful analysis of results revealed that a single-headland equilibrium profile is merely the wave front translated perpendicularly to the wave direction at the headland tip, without any influence of wave period or in wave direction. A new function called the "*wave-front-bay-shape equation*" has been obtained, and the application and validation of this formula to the case-study bay of the Bagnoli coast (south-west of Italy) is described in the paper.

**Keywords:** crenulated-bay beach; hyperbolic-tangent shape; shoreline response; Boussinesq Wave model; coastal defences
