*5.2. LDR and Equivalent Wave*

For the purposes of the present study, an understanding of longshore sediment transport is essential to sound headland control design practice, and for these scopes the concept of Littoral Drift Rose (LDR), [36,37] seems to represent a powerful resource. In fact, as stated previously, the long term sculpturing of headland bay beaches is related to the most persistent waves in their incidence and directions, since the straight section of the bay tends to lie normally to the wave rays. With minimizing the angle between dominant wave rays and the beach normal, the littoral drift progressively diminishes, so that the coast reaches its final equilibrium position. Therefore, such bays present their straight section normal to a dominant (or equivalent) sea state, embodied by the predominant swell on oceanic margins, and by the resultant energy vector in enclosed seas, where locally generated waves assume importance [5]. The LDR formulation is derived just from the energy vector concept [38], and, particularly, it holds the powerful property of estimating the climate equivalent (dominant) sea state, responsible for the sculpting of a stretch of coast; it is clear that it has a strong impact on the scopes of the article. Additionally, it is important to point out that, even though the equivalent wave concept is widely used in the field of practical coastal engineering, it lacks a firm theoretical basis. However, recent studies carried out by [39], demonstrated, through qualitative and quantitative analysis, how the equivalent wave concept may be reliable in explaining the long-term evolution of a stretch of coast. In light of this, the LDR for the case study area has been derived and the equivalent wave has been estimated and used as the wave attack that governs the plan-shape of the bay.

Now, as specified by [37], given a water climate represented by a series of N wave components, it is possible to determine the LDR, the compact polar representation of littoral transport potential for various shoreline orientations. For a segment of shoreline with outward normal azimuth *β*, it can be shown the net potential littoral drift rate, *Q*(*β*), can be calculated as:

$$Q(\beta) = \sum\_{\alpha\_{0i}=\beta-\frac{\pi}{2}}^{\alpha\_{0i}=\beta+\frac{\pi}{2}} p\_i \cdot \frac{\mathbf{K} \cdot (H\_{s0,i})^{2.4} \cdot (T\_{p0,i})^{0.2} \cdot \mathbf{g}^{0.6}}{16 \cdot (s-1) \cdot (1-n) \cdot \pi^{0.2}} \sin[2(\beta-n\_{0i})] \tag{16}$$

In the equation above, *Q* is intended as the in-place volumetric transport of sediment past a hypothetical plane perpendicular to the beach; additionally:


As known, the LDR graph is a useful tool for interpreting littoral drift trends along a section of shoreline, and, more significantly, it is able to sum up the effects of the entire wave climate into a single equivalent wave component, of parameters *Hs*0,*eq*, *Tp*0,*eq*, *α*0,*eq*. This is to say:

$$Q(\beta) = \sum\_{\mathbf{a}:\mathbf{a}\_{0i}=\beta+\frac{\pi}{2}}^{\mathbf{a}\_{0i}=\beta+\frac{\pi}{2}} p\_i \cdot \frac{K \cdot (H\_{\mathbf{a}0,i})^{2.4} \cdot \left(T\_{p0,i}\right)^{0.2} \cdot \mathbf{g}^{0.6}}{16 \cdot (\mathbf{s}-1) \cdot (1-n) \cdot \pi^{0.2}} \sin[2(\beta-\mathbf{a}\_{0i})] \stackrel{\approx}{\simeq} \mathbf{G}\_{\mathbf{q}\uparrow} \cdot \sin\left[2\left(\beta-\mathbf{a}\_{0\angle\mathbf{q}}\right)\right] \tag{17}$$

where:

$$G\_{eq} = \frac{K \cdot \left(H\_{s0,eq}\right)^{2.4} \cdot \left(T\_{p0,eq}\right)^{0.2} \cdot \mathcal{g}^{0.6}}{16 \cdot (s-1) \cdot (1-n) \cdot \pi^{0.2}}\tag{18}$$

The equivalent wave angle *α*0,*eq*, corresponds to the LDR node (null-point), as seen in Figure 15, and the magnitude (Equation (18)) can be easily inferred by using (for example) common harmonic regression techniques. The drift rose of Bagnoli bay has been derived using Equation (17) and the available wave data inferred through the MEDA-A Buoy. The range of shoreline orientations that exists at the site of interest has been also considered, and the net littoral drifts for each possible shoreline orientation have been calculated. LDR for the Bagnoli climate is shown in Figure 15: positive (transport to the right), and negative (transport to the left) lobes can be distinguished; moreover, the graph shows the null point; that is, the shoreline orientation at which no sediment transport is taking place.

**Figure 15.** Comparison between Climate-LDR and Equivalent-LDR of the Bagnoli bay; blue solid line represents drift to right when looking offshore, while red solid line represents drift to left when looking offshore; dashed lobes represent the equivalent drifts. Littoral drift in cubic meters/s.

From the LDR, the equivalent wave component parameters have been estimated according to Equation (17). The equivalent direction corresponds to the null-point of the real LDR (205◦ N for the present case), while *Hs,eq* and *Tp,eq* are fitted to have the same littoral transport magnitude, obtaining *Hs,eq* = 0.8 m and *Tp,eq* =5.89 s (Equation (18)). Additionally, it is worth noticing that, as explained by [39], in [37] the authors do not define explicitly how to derive the equivalent wave height and period. While the equivalent wave direction is much easier to interpret, as it corresponds to the null point of the LDR, the same is not true for *Hs*,*eq*, and *Teq*. However, the equivalent wave height can be easily inferred from Equation (18), considering that a relationship between wave height and period is established. As such, the equivalent period can be considered as the spectral peak period *Tp*; in fact, assuming that the energy distribution is that of a mean-JONSWAP spectrum (typical spectrum shape of Mediterranean Sea), wave height and period are related by:

$$T\_p = 8.5\pi \sqrt{\frac{H\_{\rm m0}}{4\pi}}\tag{19}$$

On the other hand, the equivalent wave period can be considered as the "longer period" representative of swell-wave conditions. The authors of [40], in fact, argued the annual littoral transport to be driven by swells that follow the most intense storms and recognized that swells "arrive on a coast from persistent direction". This implicitly supports the idea that a dominant swell-wave attack for shoreline evolution may exist. However, [11] claimed that the swell-wave predominance is more common for ocean margins rather than for enclosed or semi-enclosed seas, where, on the contrary, the swell-effect is minimum. In light of this, only the peak period *Tp* has been taken into account in the present study. However, further investigation on the effects of swell wave could be carried out and verified in future research works. In Figure 15, the Climate-LDR and Equivalent-LDR are compared.

It is surprising to observe that the equivalent wave direction (205◦ N) is extremely close to the orientation of the downcoast section of the bay (illuminated zone, Figure 13), confirming the results obtained in [37] that the LDR equivalent wave is responsible of the sculpturing of the bay, in the long run, bringing it to its static equilibrium plan-form. Moreover, this is confirmed also by the wave climate mode shown in Figure 14b: waves with the largest percentage of occurrence are those comprised within the south west wave sector, in which the 205◦ equivalent direction can be detected. In fact, it can be said that the LDR equivalent wave approximately represents the average climate; in other words, it is the wave component that usually affects a given region from a certain direction. Therefore, when in narrow wave sector, as the present case of Bagnoli bay, we are used to observe mono-modal wave climate, so with a single high-frequency direction. In such conditions, the directional mode corresponds to the average one, which corresponds, in turn, to the LDR equivalent wave direction [39].
