*Relationship between Wave Fronts and Equilibrium Profiles*

As described in the above paragraph, numerical experiments demonstrated that, in contrast to what is usually supposed, wave fronts do not represent a static equilibrium profile. However, it is possible to establish a correlation between them. Regarding the fact that an equilibrium profile actually is a wave front translated perpendicularly to the wave direction at the headland head (Figure 10), and recognizing that wave period, wave direction and wave refraction do not influence this behaviour, it is possible to calculate a direct relationship between wave fronts and equilibrium profiles, called the "*wave-frontbay-shape equation*".

Therefore, in order to derive this relationship, for each comparison between wave front and hyperbolic tangent profile, we measured the minimum distance between the diffraction point and the asymptote of the profile (distance *c* of Figure 10), and, also, the distance needed to overlap the front on the profile (distance *s* of Figure 10). These two distances have been standardised to the local wave length, *L* (the wave length at the diffraction point water depth), so obtaining *c/L* and *s/L*.

Moreover, it is necessary to take into account that wave fronts far from the diffraction point by less than one wave length coincide with the corresponding hyperbolic tangent profile. Therefore, it has been assumed that for a value of *c*/*L* ≤ 0.7, it is not necessary to shift the wave front to overlap it on the profile; they are already superimposed (Figure 8).

Hence, for each configuration, results have been plotted on a graph where on the x-axis there is "*c/L* − 0.7" and on the y-axis there is "*s/L*". It can be noted that all results analysed tend to follow an increasing trend, which suggests that an equation to describe the correlation between diffractive wave fronts and static equilibrium profiles can be derived (Figure 11). The relationship is:

$$\frac{s}{L} = 0.0484 \left( \frac{c}{L} - 0.7 \right)^2 + 0.4694 \left( \frac{c}{L} - 0.7 \right) + 0.3567 \tag{15}$$

with a correlation coefficient R<sup>2</sup> of 0.9548.

**Figure 11.** numerical simulations values of *s*/*L* in function of *c*/*L*, together with the best line.
