The measured average wave overtopping has been compared with the predicted values using the existing formulas. A reduction coefficient for each direction has been assessed using the FHR tests results; both mean value and standard deviation of the reduction coefficient were calculated. The distribution of overtopping along the overtopping boxes was analyzed and correlated to the total wave height measured at the toe of the structure. For the analyses of the overtopping reduction due to the obliqueness, only tests without crest berm or with very short crest berm (5 m in prototype) were considered. The influence of long crest berms has been analyzed afterwards.

To cope with the spatial variation, the calculation of the mean overtopping discharge starting from the collected overtopping volume has been done based on a geometrical rule, summarized as follows:

For incident wave height to be used in the formulas, the one coming from the star array after reflection analysis is employed since the information of the wave gauges placed along the dike only corresponds to total wave height (incident + reflected).

#### 4.1.1. Vertical Quay Wall

The results of the tests indicate a clear decrease in the overtopping volumes with the increase of the wave angle. An increase of the overtopping volumes along the structure was observed for all cases, except for the perpendicular waves. In

Figure 8, an example is shown, and the horizontal axis represent the quay extension, from 0 to 8.0 m, where the 0 is taken in the corner of the structure closest to the wave paddle. Each line plotted in every figure represents the results from one model test. The distribution of the wave overtopping along the vertical quay is generally consistent with the distribution of the total wave height at the toe.

However, stem wave formation can play a role in increasing wave height and consequently increasing overtopping along the structure. Research on stem waves along vertical wall with different researchers, ref. [

16,

17] reveal that the normalized significant stem wave height becomes large as the incident angle of wave become large. It was found also that the wave breaking suppresses the growth of the stem waves. These studies were based on wave tank experiments and on various numerical wave models with regular and irregular waves, but the predictions did not match very well the observations. In the present study the effect of the stem waves was not investigated since just one value of mean discharge along the whole quay was considered in the final analysis, a value measured by the instruments array.

Figure 9 shows the results of the FHR tests in a graph with the measured discharges plotted against the predicted ones, expressed in l/s/m (prototype scale). The plotted data include cases without a crest berm (distance of the wall from the edge of the quay, d

_{w}, equal to 0 m) and with a crest berm (d

_{w} larger than 0 m). The dash–dot lines indicate a prediction of 10 times larger and smaller with respect to the central line (ratio predicted/measured equal to 1:1). The formula overestimates the overtopping discharge for the 70° and 80° directions, while for the 0°, 45°, and 60° directions results are in reasonable agreement or within the above mentioned range.

The effects of the obliqueness on the overtopping discharge were evaluated calculating the reduction coefficient of each case, starting from Equation (14), as follows:

The calculation has been performed both for the FHR data and for the selected CLASH data.

Figure 10 shows the variation of the reduction coefficient with the wave angle. The existing formulations were analyzed to calculate the reduction coefficient as function of the wave angle. Despite the scattering of the results (similar scatter can also be noticed in Goda, 2009) a certain trend can be identified.

The tests clearly show that the overtopping discharge is inversely proportional to the wave angle: the larger the wave angle, the smaller the wave overtopping. Different formulas propose constant values for the overtopping volumes for waves larger than 37° (long crested waves, [

18], or 45° [

1]). Franco and Franco formula [

18] for short-crested waves seems to be the closest to FHR results, although the FHR tests were conducted using just long-crested waves. However, the differences due to the “short-crestedness” lie within the scattering of the formula, similar to previous studies [

19]. Franco and Franco [

18] stated that the directional spreading might allow reducing the freeboard with 30% in respect to cases with only long-crested waves.

The results of the experiments indicate that no formula, among those previously proposed predicts accurately the overtopping reduction. However, it is preferable to use the formula proposed by Goda [

3] for large angles due to two main reasons:

- (a)
the correction coefficient represents an upper limit (safe approach) for the present cases with very oblique waves, although not excessively high as EurOtop [

1]; and

- (b)
the expression for γ

_{β} is applicable up to 80°, meanwhile EurOtop [

1] indicates a constant value for wave angles larger than 45°.

The mean overtopping discharge is generally expressed by means of an exponential function as follows:

where

A = 0.040 and B = 2.6 in EurOtop [

1],

A = 0.033 and B = 2.3 in Goda [

3], and

A = 0.116 and B = 3.0 in Franco and Franco [

18].

Note that the reduction coefficient γ

_{β} is a function of the A and B coefficients. The differences between Goda [

3] and EurOtop [

1] can be considered negligible because the values of A and B coefficients are rather similar.

New values for the reduction coefficient are presented here based on the FHR data and it is proposed to be used for similar conditions (

Table 3). The resulting values, based on the FHR measurements, including the standard deviation, can be summarized as follows:

γ_{β} = 0.76 (σ = 0.23), for β = 45°;

γ_{β} = 0.75 (σ = 0.17), for β = 60°;

γ_{β} = 0.44 (σ = 0.21), for β = 70°; and

γ_{β} = 0.28 (σ = 0.04), for β = 80°.

The calculated gamma value is the mean value for each wave angle. The mean values and standard deviation values were calculated for each wave angle starting from the results of γ

_{β} estimated for each single test. The confidence interval represented in

Figure 10 is calculated as ±σ with respect to the mean value. As general approach, the mean value of γ

_{β} has to be used for design purposes. It can be noticed that the difference in the reduction coefficient between 0.72 (calculated value using EurOtop [

1]) and 0.28 might cause a difference in the calculated discharge of at least 1 order of magnitude (10 times) in the selected data range.

Figure 11 shows the FHR data, the CLASH data and the EurOtop predictions in term of non-dimensional discharge Q = q/(g·H

_{m0}^{3})

^{^0.5}. Only the FHR cases with the wall on the edge of the quay are plotted in order to avoid misinterpretations due to the effects of the width of the crest berm. Three different plots are shown in

Figure 11:

- (a)
the values of Q are plotted against the non-dimensional freeboard R_{c}/H_{i};

- (b)
the values of Q are plotted against the non-dimensional freeboard R_{c}/H_{i}γ_{β} (EurOtop), where γβ (EurOtop) is the correction coefficient calculated using the EurOtop (2007) formula; and

- (c)
(the values of Q are plotted against the non-dimensional freeboard Rc/Hiγβ(Goda), where γβ (Goda) is the correction coefficient calculated using the Goda [

3] formula.

The use of Goda [

3] formula is improving the wave overtopping prediction in case of oblique wave attack with respect to the EurOtop [

1] formula. In most of the cases, especially for very oblique angles, the EurOtop formula seems to overestimate the overtopping, while using Goda correction factors the results are spread around the formula prediction and only few of them are still overestimated.

The analysis on the berm length effects (distance between the seaward edge of the quay and the storm wall) and on the wall height has been carried out.

Figure 12 shows the non-dimensional overtopping discharge in function of two different non-dimensional parameters: (i) the ratio between the wall height and the incident wave height, (ii) the ratio between the berm length and 1.56

T_{p}^{2} that can be assumed as the wave length in deep water conditions. The combination of obliqueness, wall height and berm length made it challenging to have a clear view of the phenomena occurring at the structure. Despite the rather wide data scatter, there are clear differences between short or no berm layouts and wide berm layouts. A dependence on the berm length can be detected, the overtopping was reduced when the ratio of the berm length over the wave length was increased and this trend was clearer for larger wave angles. The waves travelled at the dike crest before approaching the storm wall and it was expected that the waves would refract on the berm, and therefore approach the wall with less obliqueness, but still not perpendicular. The distance travelled by the waves to reach the wall was larger for larger angles, so the amount of energy dissipated on the crest might have been larger. The configurations without berm, and with short berm length, 5 m in prototype, show a similar behaviour leading to larger overtopping discharge than the configurations with wider berms (25 m and 50 m in prototype).

#### 4.1.2. Sloping Dike

The results of the tests for a sloping dike are similar with those for a vertical quay, indicating the same decrease in the overtopping volumes with the increase of the wave angle. The measured overtopping discharges for FHR data are plotted in

Figure 13 against the values predicted using Equations (3) and (4) [

3]. As noticed in the previous cases, the formula seems to overestimate the overtopping discharge for very oblique wave attacks.

The effects of the obliqueness on the overtopping discharge were evaluated calculating the reduction coefficient of each case starting from Equation (3) as follows:

The calculation has been performed both for the FHR data and for the selected CLASH data. Three different datasets were selected from CLASH (for only non-breaking wave conditions):

Dataset 030 [

20]: 1:2 slope with 1:20 foreshore;

Dataset 220 [

21]: 1:2.5 slope with 1:1000 foreshore; and

Dataset 222 [

21]: it includes data for 1:2.5 and 1:4 slope with 1:1000 foreshore.

Figure 14 shows the variation of the reduction coefficient with the wave angle. The CLASH data are labelled as red triangles whose size is proportional to the slope (e.g., 1:2 larger size than 1:4). Several proposed formulations were analyzed to calculate the reduction coefficient as function of the wave angle. The formulas predictions and the confidence interval for the FHR data are also plotted.

The results display a scattered distribution, but similar scatters can be observed in other studies performed in similar conditions [

3,

22]. However, a certain trend is visible, and the reduction of the FHR data are in agreement with the reduction of the CLASH data.

Figure 15 shows the FHR data, the CLASH data and the EurOtop predictions. Three different plots are depicted:

- (a)
the values of Q are plotted against the non-dimensional freeboard R_{c}/H_{i};

- (b)
the values of Q are plotted against the non-dimensional freeboard R

_{c}/H

_{i}γ

_{β(EurOtop)}γ

_{prom_v}, where γ

_{β(EurOtop)} is the correction coefficient calculated using the EurOtop [

1] formula and γ

_{prom_v} is the reduction coefficient calculated by means of Van Doorslaer [

9]; and

- (c)
the values of Q are plotted against the non-dimensional freeboard R

_{c}/H

_{i}γ

_{β(Goda)}γ

_{prom_v}, where γ

_{β(Goda)} is the correction coefficient calculated using the Goda [

3] formula.

Similar improvement of the wave overtopping prediction, as in the case of a vertical quay when Goda formula is used over EurOtop formula, can be observed for sloping dike cases.

The influence of the geometrical layout is not easily detected due to interreference between three involved parameters: obliqueness, wall height, and berm length. However, the existence of the wall significantly reduces the wave overtopping for all cases. The position of the storm return wall is also important, larger berms leading to a decrease in the overtopping volumes.