*3.2. Methodology*

The method followed involves comparing the diffracted wave fronts generated by the BW simulations with static equilibrium profiles sketched out by the hyperbolic-tangent model [13,14]. The procedure obeys to the following steps:


The third step is directly related to one of the major findings in the field of static equilibrium bays, namely that equilibrium crenulated beaches tend to align transverse to the direction of dominant waves. The main assumptions in applying the methodology here proposed are that (1) the equilibrium headland-bay planform follows the wave front; and (2) that the predominant wave direction is perpendicular to the straight area of the bay. In this way, in order to obtain the hyperbolic tangent plan-form of every model configuration, we supposed that the asymptote of the hyperbolic tangent matches the illuminated zone of the wave front, out of the shadow zone where the influence of diffraction is negligible. Therefore, once the distance between asymptote and diffraction point is measured (distance *c* in Figure 5), the origin of the hyperbolic tangent is automatically obtained through Equations (8) and (9), and the x,y coordinates of hyperbolic tangent profile are achieved through Equation (7) (Figure 5). It is worth pointing out that, in order to compare equilibrium static profile and wave front, the latter must be able to expand without any kind of physical interference (e.g., presence of additional obstacles).

Additionally, in order to investigate the influence of wave characteristics on the relationship between wave fronts and equilibrium profiles, we performed different numerical simulation scenarios by varying wave direction, wave period and refraction conditions. For each scenario, more wave fronts were extrapolated from the BW module, each progressively further away from the headland tip, in order to examine the influence of dimensionless distance (*c/L*) on the researched correlation.

Finally, before describing the model set up, a clarification regarding the choice of the headland bay shape model is necessary. The two possible models to be implemented to sketch out the static equilibrium profile were the hyperbolic-tangent model [13,14] and the parabolic model [15–18] (the logarithmic spiral model [12] has been rejected a priori given its difficulty in practical application). Thus, the same methodology has been implemented to both the hyperbolic model and parabolic model. The results proved that there was no difference: there is no change, regardless of the model adopted. Nevertheless, given the uncertainty due to the determination of the downdrift control point of the parabolic profile, the hyperbolic tangent model [13,14] turned out to be the best static equilibrium model to be compared to the BW simulations wave fronts.
