**4. Results**

At first sight, comparisons suggest that wave front does not represent a static equilibrium profile: within the shadow zone, the wave front is located backward from the static planform position, according to the wave propagation direction. In fact, as seen in Figure 7, the hyperbolic tangent profile (red line) crosses various fronts until it tends to the asymptote; the farther the asymptote from the breakwater tip, the more the profile is placed on additional wave fronts. On the other hand, wave fronts distant less than a wave length from the diffraction tip represent an exception, since they coincide with their corresponding hyperbolic tangent profile (Figure 8).

**Figure 7.** Comparison between wave front (obtained with flat bottom configuration) and hyperbolic tangent profile (red line), images from (**a**–**c**) show profiles sketched out, starting with fronts gradually further from the headland.

**Figure 8.** Hyperbolic tangent profile (red line) corresponds to relative wave front; both pictures have been obtained with the gentle slope configuration, (**a**) with wave direction −35◦ and wave period of 10 s and (**b**) with wave direction 25◦ and wave period of 10 s.

The behaviour outlined has been detected for each wave period and wave direction investigated, as shown in Figure 9. Moreover, this behaviour has been observed in both bottom configurations (gentle slope and flat bottom); consequently, it seems that the correlation between hyperbolic tangent profile and wave fronts is not influenced by the effect of refraction phenomenon. At the same time, although as a first approximation results indicate that wave front does not represent a static equilibrium profile, a more accurate inspection revealed that equilibrium beach profiles of single headland bays correspond to a simple translation of a wave front normal to the propagation line at the headland head: once shifted, it completely superimposes on the equilibrium profile (Figure 10). Specifically, this issue is elaborated and discussed in the next section.

**Figure 9.** Comparisons between wave fronts (black line) and hyperbolic tangent profiles (red line): (**a**) wave direction 0◦, wave period 6 s, gentle slope configuration; (**b**) wave direction −35◦, wave period 10s, gentle slope configuration; (**c**) wave direction −35◦, wave period 6s, gentle slope configuration; (**d**) wave direction −35◦, wave period 15 s, gentle slope configuration; (**e**) wave direction 25◦, wave period 6 s, gentle slope configuration; (**f**) wave direction 25◦, wave period 10 s, gentle slope configuration.

**Figure 10.** Definition sketch for parameters c (distance between the diffraction point and the asymptote of the profile) and s (distance needed to overlap the front on the profile).
