Oblique Wave Attack on Rubble Mound Breakwater Crest Walls of Finite Length

: Rubble mound breakwaters usually present a crest wall to increase the crest freeboards without a large increase of the consumption of material. Methods in the literature to design crest walls are based on estimates of the wave loads. These methods are focused on the maximum loading that attacks a single position of the crest wall. In practice, crest walls have a ﬁnite length. Since the maximum loading does not occur at the same instant over the entire length of the crest wall for oblique waves, these methods overestimate the loading in the situation of oblique waves. Wave loads under oblique wave attack have been measured in physical model tests. A method to account for the e ﬀ ect of the ﬁnite length of crest walls has been developed, and design guidelines have been derived. The results of this study in combination with the existing methods in the literature to estimate the wave forces enable a more advanced design of crest walls. of the crest wall leads to an overestimation of the actual forces on crest walls under oblique wave attack. This study describes a method to account for the effect of the finite length of crest walls on the forces caused by oblique wave attack. The physical model tests conducted by [5] in a wave basin were used. A triangular shape is applied to model the temporal shape of the (horizontal and vertical) force events (exceeded by 0.1% of the incoming waves) with 0.912 ≤ r ≤ 0.924. It was found that the duration of the force events ( duration 0.1% ) (base of the triangle) is the same for both horizontal and vertical forces. Equation (7) is proposed to estimate the duration of the force events ( rMSE = 0.472). Equation (6) is derived to estimate the position of the peak within the force event; the rising time of the horizontal forces is faster than for the vertical forces. The method proposed in [5] was suggested to calculate the maximum loading ( F 0.1% ) . The agreement between the calculated and measured force register was good (0.874 ≤ r ≤ 0.879). The temporal shape of the force events was transformed into the space domain by means of the velocity at which the force event travels along the structure ( v f ). To this end, the travel time between two measurement points was determined, and v f was obtained by dividing the distance between both points between the travel time. The same travel time and v f were obtained for horizontal and vertical forces. The travel time obviously is 0 for perpendicular wave attack, since the whole crest wall is impacted by the wave in the same instant. Equation (10) was proposed to calculate v f for oblique wave attack with a rather high accuracy


Introduction
Crest walls are usually built on top of rubble mound breakwaters to achieve higher crest freeboards without a severe increase in the amount of granular material needed. They also protect the crest, improve the accessibility, and provide space for equipment and infrastructure. Crest walls, also called crown walls, are built with concrete and are located on top of the core. During storms, crest walls are impacted by waves, experiencing both (horizontal) forces at the front of the crest wall and (vertical uplift) forces underneath the crest wall. Such wave loads on crest walls determine their size, since crest walls are designed to allow minimal or no displacements under extreme wave conditions. Thus, an accurate prediction of wave loads on crest walls is essential for their design.
Guidelines for crest walls design [1][2][3] exist, but they are only valid within their range of tested cross sections. In Jacobsen et al. [4], a numerical model is presented to estimate wave loads on the crest walls of rubble mound breakwaters. This model provides valuable estimates of the wave loads for cross-sections that have not been tested before (under perpendicular wave attack). Oblique wave attack has been proven to affect the wave loads on crest walls in [3,5]. Van Gent et al. conducted a systematic study on the reduction of the wave loads on crest walls due to the obliqueness of waves and derived a method to account for such an effect [5].
All the studies related to wave loads on crest walls on rubble mound breakwaters are focused on the (horizontal or vertical) forces that attack the breakwater cross-section (maximum loading). In practice, a crest wall on a rubble mound breakwater has a finite length (e.g., with expansion joints between two parts of the crest wall). If the maximum loading on a single position is used for the entire length of the crest wall, the loading will be overestimated for situations with oblique waves, because the maximum loading does not occur at the same time over the entire length of the crest wall. In other words, methods in the literature assume a rectangle-shaped force diagram all along the crest wall and T m−1,0 = m −1 /m 0 as the spectral wave period). Van Gent et al. [5] derived new estimators for the horizontal and vertical forces exceeded by 0.1% of the incoming waves (F H,0.1% and F V,0.1% ) based on Ru 2% (see Equations (1) and (2)).
where K e,H = 1.6, K e,v and c F,v are empirical coefficients, ρ is the density, g is the gravity acceleration, H wall is the height of the crest wall including the key (if any), A C is the crest level of the armor in front of the crest wall, B wall is the width of the crest wall including the key (if any), and F b is the level of the bottom of the base plate of the crest wall above the still water level. K e,v was calibrated as a function of the s m−1,0 ; K e,v = 2.4 for s m−1,0 = 0.018; and K e,v = 1.6 for s m−1,0 = 0.048 (F v,0.1% = (2.88-32s op ) F v,2% , see [3]). c F,v includes the reduction of the vertical forces due to the presence of a key; c F,v = 0.4 includes those for the crest wall configuration without key, and c F,v = 0.3 includes those for the crest wall configuration with a key. Van Gent et al. [5] included the effect of oblique waves through Ru 2% . Equation (3) given in [13,14] was proposed to estimate Ru 2% .
γ β = 0.5 cos 2 β + 0.5 (4) where β is the wave direction at the toe of the structure, where β = 0 • corresponds to perpendicular wave attack. The mentioned methods propose expressions to calculate the maximum wave loading of the crest wall in a single position (chainage) along its length. No methods that incorporate the effect of finite length in the estimation of the wave loading of the crest walls on mound breakwaters are known.

Test Set-Up
In this study, the tests by [5] are used. Those tests were conducted in the Delta Basin (50 m × 50 m) at Deltares, Delft. Waves were generated using a multi-directional wave board composed of 100 paddles and equipped with both active absorption and second-order wave steering. This means that the motion of the paddles compensates for the wave reflected by the structure in order to prevent them from re-reflecting on the wave paddles and that the second-order effects of the first lower and higher harmonics of the wave field are considered, ensuring that the generated waves resemble waves that occur in nature.
The experimental set-up in the wave basin is illustrated in Figure 1. A structure with a width of 18.3 m was built with an angle of 37.5 • between the model and the wave generator. Gravel beaches were built on both sides of the model to ensure wave damping in the basin.
where γ = γf γβ is the reduction factor to take into account the effect of both the roughness and the oblique wave attack, respectively, ξm-1,0 = tanα/(2πHs/gTp 2 ) 0.5 is the surf-similarity parameter or Iribarren number, c0 = 1.45, c1 = 5.0, c2 = 0.25 c1 2 /c0, and p = 0.5c1/c0. Van Gent et al. [5] used γf = 0.45 for the double-layer rock armor and proposed a new expression for γβ. = 0.5 0.5 (4) where β is the wave direction at the toe of the structure, where β = 0° corresponds to perpendicular wave attack. The mentioned methods propose expressions to calculate the maximum wave loading of the crest wall in a single position (chainage) along its length. No methods that incorporate the effect of finite length in the estimation of the wave loading of the crest walls on mound breakwaters are known.

Test Set-Up
In this study, the tests by [5] are used. Those tests were conducted in the Delta Basin (50m × 50m) at Deltares, Delft. Waves were generated using a multi-directional wave board composed of 100 paddles and equipped with both active absorption and second-order wave steering. This means that the motion of the paddles compensates for the wave reflected by the structure in order to prevent them from re-reflecting on the wave paddles and that the second-order effects of the first lower and higher harmonics of the wave field are considered, ensuring that the generated waves resemble waves that occur in nature.
The experimental set-up in the wave basin is illustrated in Figure 1. A structure with a width of 18.3 m was built with an angle of 37.5° between the model and the wave generator. Gravel beaches were built on both sides of the model to ensure wave damping in the basin.   The tested cross-section corresponds to a rubble mound breakwater with 1:2 slope and two crest walls configurations. The model cross-section is depicted in Figure 2. A double-layer rock armor with high density stones (∆ = 2.69) is used in order to limit the armor damage during the tests while keeping a realistic size of the stones. A 1:1.5 slope was used in the rear side of the structure.
Water 2020, 12, 353 4 of 21 The tested cross-section corresponds to a rubble mound breakwater with 1:2 slope and two crest walls configurations. The model cross-section is depicted in Figure 2. A double-layer rock armor with high density stones (Δ = 2.69) is used in order to limit the armor damage during the tests while keeping a realistic size of the stones. A 1:1.5 slope was used in the rear side of the structure.

Figure 2.
Tested cross-section [5]. Figure 3 presents the two crest wall configurations used in the tests, made of aluminum, as well as the location of the pressure transducers. The two crest walls configurations are identical except for the key placed at the intersection between the front wall and the bottom of the crest wall. The presence of a key is desirable from the geotechnical perspective, since the passive earth pressure is increased. Crest wall A does not present the key, while Crest wall B does. Pressures were measured at the front side of the crest wall and underneath it for both crest wall configurations using 18 pressure transducers. These transducers have a frequency response of 5 kHz, an accuracy of 0.06% of the full scale (Best Straight Line), and a range of 350 mBar. In Crest wall A, five pressure transducers were located in the front wall, and three pressure transducers were placed in the baseplate. In Crest wall B, two additional pressure transducers were located: one in the front wall and one in the baseplate. No movement of crest walls was ensured by fixing them to steel frames, as shown in Figure 4.   Figure 3 presents the two crest wall configurations used in the tests, made of aluminum, as well as the location of the pressure transducers. The two crest walls configurations are identical except for the key placed at the intersection between the front wall and the bottom of the crest wall. The presence of a key is desirable from the geotechnical perspective, since the passive earth pressure is increased. Crest wall A does not present the key, while Crest wall B does. Pressures were measured at the front side of the crest wall and underneath it for both crest wall configurations using 18 pressure transducers. These transducers have a frequency response of 5 kHz, an accuracy of 0.06% of the full scale (Best Straight Line), and a range of 350 mBar. In Crest wall A, five pressure transducers were located in the front wall, and three pressure transducers were placed in the baseplate. In Crest wall B, two additional pressure transducers were located: one in the front wall and one in the baseplate. No movement of crest walls was ensured by fixing them to steel frames, as shown in Figure 4. The tested cross-section corresponds to a rubble mound breakwater with 1:2 slope and two crest walls configurations. The model cross-section is depicted in Figure 2. A double-layer rock armor with high density stones (Δ = 2.69) is used in order to limit the armor damage during the tests while keeping a realistic size of the stones. A 1:1.5 slope was used in the rear side of the structure.  Figure 3 presents the two crest wall configurations used in the tests, made of aluminum, as well as the location of the pressure transducers. The two crest walls configurations are identical except for the key placed at the intersection between the front wall and the bottom of the crest wall. The presence of a key is desirable from the geotechnical perspective, since the passive earth pressure is increased. Crest wall A does not present the key, while Crest wall B does. Pressures were measured at the front side of the crest wall and underneath it for both crest wall configurations using 18 pressure transducers. These transducers have a frequency response of 5 kHz, an accuracy of 0.06% of the full scale (Best Straight Line), and a range of 350 mBar. In Crest wall A, five pressure transducers were located in the front wall, and three pressure transducers were placed in the baseplate. In Crest wall B, two additional pressure transducers were located: one in the front wall and one in the baseplate. No movement of crest walls was ensured by fixing them to steel frames, as shown in Figure 4.    Waves were measured using directional wave gauges (GRSM) at two locations in front of the model so the incident and reflected waves could be separated. The directional wave gauges were located to be in the same line as the measurement cross-sections for a wave angle of 37.5°, which is the average of all the tested angles and perpendicular to the wave paddle (see Figure 1). Small variations observed between both measurement points were accounted for in the following analysis; measurements of GRSM-A were used for Test section A, and measurements of GRSM-B were used for Test section B. In following analysis, incident waves at these locations at the toe were used. The spectral significant wave height (Hs = Hm0 = 4√m0) and the spectral mean wave period (Tm-1,0 = m-1/m0) were obtained from the measured wave energy spectra. Tm-1,0 was first found to better describe the influence of wave energy spectra on wave run-up and overtopping in [13,14]. Later, Tm-1,0 was applied as the best wave period to describe other interaction processes between waves and coastal structures, see for instance [15]. Thus, this spectral wave period was also used in this study. The mean overtopping discharge was also measured in [5], using two overtopping chutes and boxes (see Figure  5).
Runs of 1000 random waves were generated following a JONSWAP wave spectrum (peak enhancement factor of 3.3). Each configuration was tested with a constant value of sm-1,0 = 0.018 or 0.048. Six wave directions were considered β = 0°, 15°, 30°, 60, 45°, and 75°, where β = 0° corresponds to the perpendicular wave attack. Three water depths, hs = 0.70, 0.75, and 0.80 m, and crest freeboards, RC = 0.15, 0.20 and 0.25 m, were used. For the lowest hs, only the perpendicular waves were tested, since most of the conditions with oblique waves would not have caused forces on the crest walls. In total, 30 tests were used in this study. Table 1 summarizes the main experimental ranges in these tests.  Waves were measured using directional wave gauges (GRSM) at two locations in front of the model so the incident and reflected waves could be separated. The directional wave gauges were located to be in the same line as the measurement cross-sections for a wave angle of 37.5 • , which is the average of all the tested angles and perpendicular to the wave paddle (see Figure 1). Small variations observed between both measurement points were accounted for in the following analysis; measurements of GRSM-A were used for Test section A, and measurements of GRSM-B were used for Test section B. In following analysis, incident waves at these locations at the toe were used. The spectral significant wave height (H s = H m0 = 4 √ m 0 ) and the spectral mean wave period (T m−1,0 = m −1 /m 0 ) were obtained from the measured wave energy spectra. T m−1,0 was first found to better describe the influence of wave energy spectra on wave run-up and overtopping in [13,14]. Later, T m−1,0 was applied as the best wave period to describe other interaction processes between waves and coastal structures, see for instance [15]. Thus, this spectral wave period was also used in this study. The mean overtopping discharge was also measured in [5], using two overtopping chutes and boxes (see Figure 5).
Runs of 1000 random waves were generated following a JONSWAP wave spectrum (peak enhancement factor of 3.3). Each configuration was tested with a constant value of s m−1,0 = 0.018 or 0.048. Six wave directions were considered β = 0 • , 15 • , 30 • , 60, 45 • , and 75 • , where β = 0 • corresponds to the perpendicular wave attack. Three water depths, h s = 0.70, 0.75, and 0.80 m, and crest freeboards, R C = 0.15, 0.20 and 0.25 m, were used. For the lowest h s , only the perpendicular waves were tested, since most of the conditions with oblique waves would not have caused forces on the crest walls. In total, 30 tests were used in this study. Table 1 summarizes the main experimental ranges in these tests.

Test Results
As previously mentioned, 18 pressure transducers were used to measure the pressure signals on the crest walls. Their sampling frequency was 1000 Hz. Before a test, the transducers were set to zero, so the measurements were relative to the pressures caused by the still water level and hydrostatic forces were not included in the measurements. This is especially relevant for Crest wall B, since the key is submerged during the tests with the higher water depth.
Each pressure transducer provided a point of pressure (kN/m 2 ). The pressure distribution along the front and base of the crest wall was obtained by assembling the points of pressure (see Figure 5). The pressure distribution was extrapolated toward the edges of the crest wall. Pressure transducers were located as close as possible to the edges of the crest wall to minimize extrapolation, and the extrapolated pressures were set to a lower limit of 0. In Figure 6, the extrapolation zone is indicated with blue dashed lines. The pressure distribution was integrated along the front and base of the crown wall to obtain the force on the crest wall. It resulted in a horizontal force on the front of the crest wall (F H ) and a vertical force on its base (F V ).

Test Results
As previously mentioned, 18 pressure transducers were used to measure the pressure signals on the crest walls. Their sampling frequency was 1000 Hz. Before a test, the transducers were set to zero, so the measurements were relative to the pressures caused by the still water level and hydrostatic forces were not included in the measurements. This is especially relevant for Crest wall B, since the key is submerged during the tests with the higher water depth.
Each pressure transducer provided a point of pressure (kN/m 2 ). The pressure distribution along the front and base of the crest wall was obtained by assembling the points of pressure (see Figure 5). The pressure distribution was extrapolated toward the edges of the crest wall. Pressure transducers were located as close as possible to the edges of the crest wall to minimize extrapolation, and the extrapolated pressures were set to a lower limit of 0. In Figure 6, the extrapolation zone is indicated with blue dashed lines. The pressure distribution was integrated along the front and base of the crown wall to obtain the force on the crest wall. It resulted in a horizontal force on the front of the crest wall (FH) and a vertical force on its base (FV). In this study, the temporal shape of the (horizontal and vertical) force events was studied. Thus, the duration of the force events and the time of the peak (tpeak) was determined. The start, tpeak, and end of the force events were determined for both horizontal and vertical forces. They were extracted in three steps: (1) peaks were identified applying the Peaks-Over-Threshold method; in case of two peaks closer than 0.5 s, only the maximum was considered; (2) zero-up and zero-down crossing points were determined using a threshold of 2 to account the points slightly above zero and related to each peak as the start and end points of the force event; and (3) correction was applied in case the start and end points were located even more above zero. If two peaks presented the same start or end point, it was replaced by the minimum value between both peaks. The duration is calculated as the difference between the start and end of the force event. The exceedance values used in this study are not based on the total number of force peaks but rather on the number of waves within a test. Note that some vertical force registers were discarded, since the events were too low to be distinguished. Therefore, 59 duration values (29 for Crest wall A + 30 for Crest wall B) were obtained for the In this study, the temporal shape of the (horizontal and vertical) force events was studied. Thus, the duration of the force events and the time of the peak (t peak ) was determined. The start, t peak , and end of the force events were determined for both horizontal and vertical forces. They were extracted in three steps: (1) peaks were identified applying the Peaks-Over-Threshold method; in case of two peaks closer than 0.5 s, only the maximum was considered; (2) zero-up and zero-down crossing points were determined using a threshold of 2 to account the points slightly above zero and related to each peak as the start and end points of the force event; and (3) correction was applied in case the start and end points were located even more above zero. If two peaks presented the same start or end point, it was replaced by the minimum value between both peaks. The duration is calculated as the difference between the start and end of the force event. The exceedance values used in this study are not based on the total number of force peaks but rather on the number of waves within a test. Note that some vertical force registers were discarded, since the events were too low to be distinguished. Therefore, 59 duration values (29 for Crest wall A + 30 for Crest wall B) were obtained for the horizontal forces, while 47 (22 for Crest wall A + 25 for Crest wall B) were identified for vertical forces. An example of the determination of beginning and ending points is displayed in Figure 6 for vertical forces of Test #10 on Crest wall A.
Water 2020, 12, 353 7 of 21 An example of the determination of beginning and ending points is displayed in Figure 6 for vertical forces of Test #10 on Crest wall A. Figure 6. Example of determination of beginning, peak, and ending point for vertical forces of Test #10 on Crest wall A.
As [5] pointed out, it is common practice to assume that the maximum value registered in tests is the maximum (horizontal or vertical) force, which lasts 1000 waves (force exceeded by 0.1% of the incoming waves) [1,2,11]. However, the smaller the exceedance probability, the more hampered the value is by the coincidence within a test. Thus, Van Gent et al. [5] compared the maximum forces (peak of the force event) within a test (forces exceeded by 0.1% of the incoming waves) with the forces exceeded by 2% of the incoming waves. An almost constant ratio was observed for the horizontal forces and for many vertical forces. Here, a similar comparison is conducted for the duration and the position of the peak (tpeak) to determine whether the force event exceeded by 2% of the incoming waves is a valid approach to describe the shape of the maximum force events (see Figure 7).
In order to assess the correlation between the variables, the correlation coefficient (−1 ≤ r ≤ 1) was used. Here, the higher the absolute value of r, the higher the correlation.
where No is the number of observations, ti are the first variable observations, ei are the corresponding observations of the second variable, and ̅ and ē are the average values of both variables. As shown in Figure 7, reasonable correlation (r = 0.790) was found for the duration of the horizontal force events exceeded by 0.1% and 2% of the incoming waves (durationH,0.1% and durationH,2%). Poor correlation (0.148 ≤ r ≤ 0.392) was observed for the duration of the vertical force events exceeded by 0.1% and 2% of the incoming waves (durationV,0.1% and durationV,2%), as well as for the position of the peak (tpeak) of both horizontal and vertical force events exceeded by 2% and 0.1% of the incoming waves. As a result, it was decided to conduct the following analysis using the force events exceeded by 0.1% of the incoming waves. As [5] pointed out, it is common practice to assume that the maximum value registered in tests is the maximum (horizontal or vertical) force, which lasts 1000 waves (force exceeded by 0.1% of the incoming waves) [1,2,11]. However, the smaller the exceedance probability, the more hampered the value is by the coincidence within a test. Thus, Van Gent et al. [5] compared the maximum forces (peak of the force event) within a test (forces exceeded by 0.1% of the incoming waves) with the forces exceeded by 2% of the incoming waves. An almost constant ratio was observed for the horizontal forces and for many vertical forces. Here, a similar comparison is conducted for the duration and the position of the peak (t peak ) to determine whether the force event exceeded by 2% of the incoming waves is a valid approach to describe the shape of the maximum force events (see Figure 7).
In order to assess the correlation between the variables, the correlation coefficient (−1 ≤ r ≤ 1) was used. Here, the higher the absolute value of r, the higher the correlation.
where N o is the number of observations, t i are the first variable observations, e i are the corresponding observations of the second variable, and t and e are the average values of both variables.
As shown in Figure 7, reasonable correlation (r = 0.790) was found for the duration of the horizontal force events exceeded by 0.1% and 2% of the incoming waves (duration H,0.1% and duration H,2% ). Poor correlation (0.148 ≤ r ≤ 0.392) was observed for the duration of the vertical force events exceeded by 0.1% and 2% of the incoming waves (duration V,0.1% and duration V,2% ), as well as for the position of the peak (tpeak) of both horizontal and vertical force events exceeded by 2% and 0.1% of the incoming waves. As a result, it was decided to conduct the following analysis using the force events exceeded by 0.1% of the incoming waves.

Results
In this section, the temporal shape of the (horizontal and vertical) force events is discussed. A triangle-shaped model is proposed. Therefore, the base width and the height of that triangle are needed to characterize the temporal shape. The formulas given in [5] are used to estimate the maximum loading (height) for both the horizontal and the vertical forces. Formulas to calculate the duration of the force events and the velocity at which the force events travel along the structure are proposed. This way, the duration is transformed into the space domain (base width); a force diagram along the longitudinal dimension of the crest wall is obtained. This study assumes that the force received by a point of the crest wall in a certain instant is at some point of the spatial force diagram, and the rest of the points receive forces according to the shape of that diagram. The force diagram is integrated to estimate the actual force that attacks the crest wall, including the finite length effect. Such force is compared to the force obtained using the existing methods and a coefficient to account for the effect of finite length on crest walls under oblique wave attack is given.

Temporal Shape of Force Events
In this section, the temporal shape of the force events is discussed and parameterized. In Figure  8, typical registers for horizontal and vertical forces are presented.

Results
In this section, the temporal shape of the (horizontal and vertical) force events is discussed. A triangle-shaped model is proposed. Therefore, the base width and the height of that triangle are needed to characterize the temporal shape. The formulas given in [5] are used to estimate the maximum loading (height) for both the horizontal and the vertical forces. Formulas to calculate the duration of the force events and the velocity at which the force events travel along the structure are proposed. This way, the duration is transformed into the space domain (base width); a force diagram along the longitudinal dimension of the crest wall is obtained. This study assumes that the force received by a point of the crest wall in a certain instant is at some point of the spatial force diagram, and the rest of the points receive forces according to the shape of that diagram. The force diagram is integrated to estimate the actual force that attacks the crest wall, including the finite length effect. Such force is compared to the force obtained using the existing methods and a coefficient to account for the effect of finite length on crest walls under oblique wave attack is given.

Temporal Shape of Force Events
In this section, the temporal shape of the force events is discussed and parameterized. In Figure 8, typical registers for horizontal and vertical forces are presented.
In Section 3.2, it was decided to analyze the maximum force event in each test (force events exceeded by 0.1% of the incoming waves). The maximum force event is extracted for each test, and a new register is created grouping them (Extracted register, from now). A five-second spacing was left Water 2020, 12, 353 9 of 21 between the events in order to allow for the afterwards analysis. Figure 9 shows a fragment of such an Extracted register. In Section 3.2, it was decided to analyze the maximum force event in each test (force events exceeded by 0.1% of the incoming waves). The maximum force event is extracted for each test, and a new register is created grouping them (Extracted register, from now). A five-second spacing was left between the events in order to allow for the afterwards analysis. Figure 9 shows a fragment of such an Extracted register. A triangle-shaped model is proposed for the temporal shape of the force events. In order to assess the goodness of fit of such a model, the correlation between the Extracted register and a new triangle-shaped register is evaluated. The new triangle-shaped register is created using the measured duration and the measured peak values of the forces. The location of the peak value of the force (tpeak0.1%) is calculated as function of the duration of the horizontal or vertical force event (duration0.1%), as shown in Equation (6).
where Kpeak is an experimental coefficient. Kpeak is calibrated for both horizontal and vertical forces in order to maximize the correlation with the Extracted register. Kpeak = 0.1 and Kpeak = 0.35 are obtained  In Section 3.2, it was decided to analyze the maximum force event in each test (force events exceeded by 0.1% of the incoming waves). The maximum force event is extracted for each test, and a new register is created grouping them (Extracted register, from now). A five-second spacing was left between the events in order to allow for the afterwards analysis. Figure 9 shows a fragment of such an Extracted register. A triangle-shaped model is proposed for the temporal shape of the force events. In order to assess the goodness of fit of such a model, the correlation between the Extracted register and a new triangle-shaped register is evaluated. The new triangle-shaped register is created using the measured duration and the measured peak values of the forces. The location of the peak value of the force (tpeak0.1%) is calculated as function of the duration of the horizontal or vertical force event (duration0.1%), as shown in Equation (6).
where Kpeak is an experimental coefficient. Kpeak is calibrated for both horizontal and vertical forces in order to maximize the correlation with the Extracted register. Kpeak = 0.1 and Kpeak = 0.35 are obtained A triangle-shaped model is proposed for the temporal shape of the force events. In order to assess the goodness of fit of such a model, the correlation between the Extracted register and a new triangle-shaped register is evaluated. The new triangle-shaped register is created using the measured duration and the measured peak values of the forces. The location of the peak value of the force (tpeak 0.1% ) is calculated as function of the duration of the horizontal or vertical force event (duration 0.1% ), as shown in Equation (6).
tpeak 0.1% = K peak duration 0.1% (6) where K peak is an experimental coefficient. K peak is calibrated for both horizontal and vertical forces in order to maximize the correlation with the Extracted register. K peak = 0.1 and K peak = 0.35 are obtained for horizontal and vertical forces, respectively; 0.912 ≤ r ≤ 0.924. Figure 10 shows the superposition of the Extracted register (the actual maximum force events) and the triangle-shaped model.
for horizontal and vertical forces, respectively; 0.912 ≤ r ≤ 0.924. Figure 10 shows the superposition of the Extracted register (the actual maximum force events) and the triangle-shaped model.

Duration of Force Events
As mentioned in Section 3.2, 59 values (29 for Crest wall A + 30 for Crest wall B) and 47 (22 for Crest wall A + 25 for Crest wall B) of the duration corresponding to the force exceeded by 0.1% of incoming waves (duration0.1%) were obtained for the horizontal (durationH,0.1%) and vertical (durationV,0.1%) forces, respectively. durationH,0.1% and durationV,0.1% corresponding to the same test were compared in order to determine if they present the same behavior. Note that 47 duration0.1% values could be compared (i.e., the minimum number of duration0.1% values, which corresponds to the vertical forces). Figure 11 compares durationH,0.1% and durationV,0.1% corresponding to the same test. Comparison of Extracted register (the actual maximum force events) and the triangle-shaped model using the measured F H,0.1% , F V,0.1% , and duration 0.1% for: (a) horizontal forces, and (b) vertical forces.

Duration of Force Events
As mentioned in Section 3.2, 59 values (29 for Crest wall A + 30 for Crest wall B) and 47 (22 for Crest wall A + 25 for Crest wall B) of the duration corresponding to the force exceeded by 0.1% of incoming waves (duration 0.1% ) were obtained for the horizontal (duration H,0.1% ) and vertical (duration V,0.1% ) forces, respectively. duration H,0.1% and duration V,0.1% corresponding to the same test were compared in order to determine if they present the same behavior. Note that 47 duration 0.1% values could be compared (i.e., the minimum number of duration 0.1% values, which corresponds to the vertical forces). Figure 11 compares duration H,0.1% and duration V,0.1% corresponding to the same test.
Water 2020, 12, 353 10 of 21 for horizontal and vertical forces, respectively; 0.912 ≤ r ≤ 0.924. Figure 10 shows the superposition of the Extracted register (the actual maximum force events) and the triangle-shaped model.

Duration of Force Events
As mentioned in Section 3.2, 59 values (29 for Crest wall A + 30 for Crest wall B) and 47 (22 for Crest wall A + 25 for Crest wall B) of the duration corresponding to the force exceeded by 0.1% of incoming waves (duration0.1%) were obtained for the horizontal (durationH,0.1%) and vertical (durationV,0.1%) forces, respectively. durationH,0.1% and durationV,0.1% corresponding to the same test were compared in order to determine if they present the same behavior. Note that 47 duration0.1% values could be compared (i.e., the minimum number of duration0.1% values, which corresponds to the vertical forces). Figure 11 compares durationH,0.1% and durationV,0.1% corresponding to the same test. Reasonable correlation was observed (r = 0.778) between duration H,0.1% and duration V,0.1% . Thus, it is reasonable to analyze them together. Equation (7) was proposed to estimate duration 0.1% , based on the difference between Ru 2% and A C .
Note that if A C ≥ Ru 2% , no waves reach the crest, which leads to duration 0.1% = 0. Equation (7) also shows that the duration of the force event increases for the larger crest walls (H wall ); larger crest walls are a bigger obstacle to dissipating phenomena (infiltration and wave overtopping), such that the water remains longer at the breakwater crest. Equation (7) is valid within the ranges 0.260 ≤ γ β (Ru 2% − A C )/H s ≤ 0.863. In order to assess the goodness of fit, r and the relative mean squared error (rMSE) were used.
where MSE is the mean squared error, var is the variance of the target values, N o is the number of observations, t i and e i are the target and estimated values, respectively, and t is the average of the target values. The variation coefficient (CV) of the data was also calculated as CV = σ/µ, where σ is the standard deviation of the observations and µ is the observations' mean. Figure 12 compares the measured and estimated dimensionless duration (duration 0.1% /T m−1,0 ) using Equation (7), as well as the 90% confidence interval. Here, r = 0.726 and rMSE = 0.472. Figure 11. Comparison between the measured dimensionless duration0.1% (duration0.1%/Tm-1,0) for horizontal and vertical forces.
Note that if AC ≥ Ru2%, no waves reach the crest, which leads to duration0.1% = 0. Equation (7) also shows that the duration of the force event increases for the larger crest walls (Hwall); larger crest walls are a bigger obstacle to dissipating phenomena (infiltration and wave overtopping), such that the water remains longer at the breakwater crest. Equation (7) is valid within the ranges 0.260 ≤ γ β (Ru2% -AC)/Hs ≤ 0.863. In order to assess the goodness of fit, r and the relative mean squared error (rMSE) were used.
where MSE is the mean squared error, var is the variance of the target values, No is the number of observations, ti and ei are the target and estimated values, respectively, and ̅ is the average of the target values. The variation coefficient (CV) of the data was also calculated as CV = σ/μ, where σ is the standard deviation of the observations and μ is the observations' mean. Figure 12 compares the measured and estimated dimensionless duration (duration0.1%/Tm-1,0) using Equation (7), as well as the 90% confidence interval. Here, r = 0.726 and rMSE = 0.472. The MSE remained stable with increasing values of duration0.1%/Tm-1,0. Thus, the 90% confidence interval was calculated based on the variance of the error (var(ε) = 0.0167). Assuming a Gaussian error distribution, the confidence interval is obtained as Figure 12. Comparison between the measured and estimated dimensionless duration 0.1% using Equation (7).
The MSE remained stable with increasing values of duration 0.1% /T m−1,0 . Thus, the 90% confidence interval was calculated based on the variance of the error (var(ε) = 0.0167). Assuming a Gaussian error distribution, the confidence interval is obtained as Similar to Section 4.1, the correlation between the Extracted register and a new triangle-shaped register was assessed. Here, the triangle-shaped register was created using the estimated F H,0.1% , F V,0.1% , and duration 0.1% using the method in [5] (Equations (1) to (4)) and Equation (7). tpeak 0.1% in Equation (6) was again calibrated by maximizing the correlation; K peak = 0.15 and K peak = 0.35 were obtained for horizontal and vertical forces, respectively; 0.874 ≤ r ≤ 0.879. Figure 13 shows the superposition of the Extracted register (the actual maximum force events) and the triangle-shaped model generated with the estimated F H,0.1% , F V,0.1% , and duration 0.1% . Note that if (Ru 2% − A C ) ≤ 0 or (Ru 2% − 0.75A C ) ≤ 0, F H,0.1% or F V,0.1% result in 0, respectively. Similar to Section 4.1, the correlation between the Extracted register and a new triangle-shaped register was assessed. Here, the triangle-shaped register was created using the estimated FH,0.1%, FV,0.1%, and duration0.1% using the method in [5] (Equations (1) to (4)) and Equation (7). tpeak0.1% in Equation (6) was again calibrated by maximizing the correlation; Kpeak = 0.15 and Kpeak = 0.35 were obtained for horizontal and vertical forces, respectively; 0.874 ≤ r ≤ 0.879. Figure 13 shows the superposition of the Extracted register (the actual maximum force events) and the triangle-shaped model generated with the estimated FH,0.1%, FV,0.1%, and duration0.1%. Note that if (Ru2% -AC) ≤ 0 or (Ru2% -0.75AC) ≤ 0, FH,0.1% or FV,0.1% result in 0, respectively.

Travel Time and Velocity of Force Events
As discussed in Section 4.1, a triangle-shaped model is adopted for the time evolution of both horizontal and vertical force events. Since the area of a triangle depends on the base (duration0.1%) and the height (maximum loading), the method proposed in [5] (Equations (1) to (4)) and Equation (7) can be used to estimate the area of the actual force in the time domain. In order to transform the temporal shape into the space domain, the velocity (vF) at which the force event travels is needed. To assess vF, the travel time of the force events between the two measurement points was obtained. vF was calculated dividing the distance between the measurement points (6.7 m) between the travel time.
In order to extract the travel time from the experimental registers, the register measured at point B (see Figure 1) was moved forward in steps of 0.01 s from 0 s (no displacement) to 5 s. For each step, the correlation between Register A and the displaced Register B was calculated. The travel time corresponded to the displacement time, which maximizes the correlation. This procedure could not be applied to three out of the 30 analyzed vertical force registers, since they were too low. Thus, 57 travel time values (30 values for horizontal forces and 27 for vertical forces) were obtained. Figure 14 shows an example of the exposed procedure for the horizontal force registers in Test #5, whereas Figure 15 presents a comparison between Register A and the original and displaced Register B until the travel time, which maximizes the correlation for Test #5.

Travel Time and Velocity of Force Events
As discussed in Section 4.1, a triangle-shaped model is adopted for the time evolution of both horizontal and vertical force events. Since the area of a triangle depends on the base (duration 0.1% ) and the height (maximum loading), the method proposed in [5] (Equations (1) to (4)) and Equation (7) can be used to estimate the area of the actual force in the time domain. In order to transform the temporal shape into the space domain, the velocity (v F ) at which the force event travels is needed. To assess v F , the travel time of the force events between the two measurement points was obtained. v F was calculated dividing the distance between the measurement points (6.7 m) between the travel time.
In order to extract the travel time from the experimental registers, the register measured at point B (see Figure 1) was moved forward in steps of 0.01 s from 0 s (no displacement) to 5 s. For each step, the correlation between Register A and the displaced Register B was calculated. The travel time corresponded to the displacement time, which maximizes the correlation. This procedure could not be applied to three out of the 30 analyzed vertical force registers, since they were too low. Thus, 57 travel time values (30 values for horizontal forces and 27 for vertical forces) were obtained. Figure 14 shows an example of the exposed procedure for the horizontal force registers in Test #5, whereas Figure 15 presents a comparison between Register A and the original and displaced Register B until the travel time, which maximizes the correlation for Test #5.  Once the travel time was determined for both the horizontal and vertical forces, the travel time values corresponding to the same test were compared to determine whether the behavior was equal for both types of forces (see Figure 16). In total, 27 pairs of values could be compared (minimum number of travel time values, which corresponds to those available from measured vertical forces).   Once the travel time was determined for both the horizontal and vertical forces, the travel time values corresponding to the same test were compared to determine whether the behavior was equal for both types of forces (see Figure 16). In total, 27 pairs of values could be compared (minimum number of travel time values, which corresponds to those available from measured vertical forces). Once the travel time was determined for both the horizontal and vertical forces, the travel time values corresponding to the same test were compared to determine whether the behavior was equal for both types of forces (see Figure 16). In total, 27 pairs of values could be compared (minimum number of travel time values, which corresponds to those available from measured vertical forces). As shown in Figure 16, a very good agreement between the travel time of both horizontal and vertical force events corresponding to the same test was found (r = 0.974; rMSE = 0.050). Therefore, it is reasonable to analyze the travel time for horizontal and vertical forces together.
vF was determined by dividing the distance between the two measurement points (6.7 m) between the obtained travel time. The higher the angle, the lower the observed velocity. Note that the travel time for tests with perpendicular waves was zero, since the crest wall is impacted by the whole wave front at the same time. Thus, the tests with perpendicular wave attack could not be used to analyze vF; 45 vF values were obtained.
A very high correlation was observed between the dimensionless velocity of the maximum force along the structure (vF/(g hs) 0.5 , where hs is the water depth at the toe of the structure) and sinβ (r = 0.826). Therefore, Equation (10) was proposed for vF/(g hs) 0.5 , .
Equation (10) is valid for 15° ≤ β ≤ 75° and 0.018 ≤ sm-1,0 ≤ 0.048. Figure 17 presents the comparison between the measured and estimated dimensionless velocity vF/(g hs) 0.5 with Equation (10), as well as the 90% confidence interval. Here, 93.4% of the variance was explained by the model. As shown in Figure 16, a very good agreement between the travel time of both horizontal and vertical force events corresponding to the same test was found (r = 0.974; rMSE = 0.050). Therefore, it is reasonable to analyze the travel time for horizontal and vertical forces together. v F was determined by dividing the distance between the two measurement points (6.7 m) between the obtained travel time. The higher the angle, the lower the observed velocity. Note that the travel time for tests with perpendicular waves was zero, since the crest wall is impacted by the whole wave front at the same time. Thus, the tests with perpendicular wave attack could not be used to analyze v F ; 45 v F values were obtained.
A very high correlation was observed between the dimensionless velocity of the maximum force along the structure (v F /(g h s ) 0.5 , where h s is the water depth at the toe of the structure) and sinβ (r = 0.826). Therefore, Equation (10) Equation (10) is valid for 15 • ≤ β ≤ 75 • and 0.018 ≤ s m−1,0 ≤ 0.048. Figure 17 presents the comparison between the measured and estimated dimensionless velocity v F /(g h s ) 0.5 with Equation (10), as well as the 90% confidence interval. Here, 93.4% of the variance was explained by the model.
The 90% confidence interval is calculated following the methodology given in [16]. As the MSE increases with increasing v F /(g h s ) 0.5 , the variance of the error (var(ε)) is calculated as Assuming a Gaussian error distribution, the 90% confidence interval for v F /(g h s ) 0.5 estimated using Equation (10) is given in Equation (12). The 90% confidence interval is calculated following the methodology given in [16]. As the MSE increases with increasing vF/(g hs) 0.5 , the variance of the error (var(ε)) is calculated as Assuming a Gaussian error distribution, the 90% confidence interval for vF/(g hs) 0.5 estimated using Equation (10) is given in Equation (12).

Integration of the Actual Force
The triangle-shaped force diagram along the longitudinal dimension of the crest wall was obtained for both the horizontal and vertical force events using the method proposed in [5] to calculate the maximum loading (Equations (1) to (4)) and Equations (7) and (10) to estimate duration0.1% and vF. The actual force (AF0.1%) that attacks the structure can be calculated using the formula for the triangle area, . % = 0.5 ( . % ) . % (13) where F0.1% is the maximum estimated loading for the horizontal or vertical force with [5]. Equation (13) is valid while the length of the crest wall (Lcrest) is higher than the width of the force event (duration0.1% vf). If duration0.1% vf > Lcrest, the considered portion of the area within the triangle needs to be the maximum possible value to obtain the dominant force (see the right panel of Figure 18). For every case, 20 values of Lcrest were considered. Lcrest = KA duration0.1% vf, where KA was varied from 0.05 to 1 in steps of 0.05. The maximum area within the triangle was numerically integrated for every case and Lcrest. Similar to the estimated AF0.1%, the measured AF0.1% was obtained as the maximum numerically integrated area from the force register in the space domain. For the numerical integration, the following steps were performed: (1) a vector with the same length as the time register is created, and all its values are set equal to the time step (0.001 s in this study), (2) the vector is multiplied by the measured velocity, (3) the cumulative sum is determined, and (4) the force register is numerically integrated over this new space vector. Figure 19 shows the comparison between the

Integration of the Actual Force
The triangle-shaped force diagram along the longitudinal dimension of the crest wall was obtained for both the horizontal and vertical force events using the method proposed in [5] to calculate the maximum loading (Equations (1) to (4)) and Equations (7) and (10) to estimate duration 0.1% and v F . The actual force (AF 0.1% ) that attacks the structure can be calculated using the formula for the triangle area, where F 0.1% is the maximum estimated loading for the horizontal or vertical force with [5]. Equation (13) is valid while the length of the crest wall (L crest ) is higher than the width of the force event (duration 0.1% v f ). If duration 0.1% v f > L crest , the considered portion of the area within the triangle needs to be the maximum possible value to obtain the dominant force (see the right panel of Figure 18). For every case, 20 values of L crest were considered. L crest = K A duration 0.1% v f , where K A was varied from 0.05 to 1 in steps of 0.05. The maximum area within the triangle was numerically integrated for every case and L crest . Similar to the estimated AF 0.1% , the measured AF 0.1% was obtained as the maximum numerically integrated area from the force register in the space domain. For the numerical integration, the following steps were performed: (1) a vector with the same length as the time register is created, and all its values are set equal to the time step (0.001 s in this study), (2) the vector is multiplied by the measured velocity, (3) the cumulative sum is determined, and (4) the force register is numerically integrated over this new space vector. Figure 19 shows the comparison between the measured and estimated AF 0.1% using the method in [5] (Equations (1) to (4)) and Equations (7) and (10). Here, r = 0.938 and rMSE = 0.945.
Water 2020, 12, 353 16 of 21 measured and estimated AF0.1% using the method in [5] (Equations (1) to (4)) and Equations (7) and (10). Here, r = 0.938 and rMSE = 0.945.    Every alignment of points in Figure 19 represents the AF0.1% variation for one test due to the different values of KA. As shown in Figure 19, the proposed method is on the conservative side. In order to obtain the best fitting approach, AF0.1%* = Kopt AF0.1% is considered where AF0.1%* is the best fit estimated AF0.1% and Kopt is a coefficient to be calibrated. Kopt was calibrated by minimizing the rMSE. The best fit was obtained for Kopt = 0.6 with rMSE = 0.133. Figure 20 presents the comparison between the measured AF0.1% and estimated AF0.1%* using the method in [5] (Equations (1)-(4)) and Equations (7) and (10) with Kopt = 0.6, as well as the 90% confidence interval. Every alignment of points in Figure 19 represents the AF 0.1% variation for one test due to the different values of K A . As shown in Figure 19, the proposed method is on the conservative side. In order to obtain the best fitting approach, AF 0.1% * = K opt AF 0.1% is considered where AF 0.1% * is the best fit estimated AF 0.1% and K opt is a coefficient to be calibrated. K opt was calibrated by minimizing the rMSE. The best fit was obtained for K opt = 0.6 with rMSE = 0.133. Figure 20 presents the comparison between the measured AF 0.1% and estimated AF 0.1% * using the method in [5] (Equations (1)-(4)) and Equations (7) and (10) with K opt = 0.6, as well as the 90% confidence interval.

Length Coefficient
The goal of this study was to propose a reduction coefficient to directly multiply to the force obtained with the methods in the literature to account for the effect of the finite length of crest walls under oblique wave attack. Methods in the literature provide the maximum force that attacks a chainage of the crest wall (F 0.1% ). They assume that the whole crest wall is attacked by this maximum force; a rectangle-shaped diagram along the whole crest wall length is assumed, as illustrated in the left panel of Figure 21. Thus, the total force AF 0.1% on a crest wall section is generally estimated by multiplying F 0.1% by the length of the crest wall (L crest ). Figure 21 illustrates the horizontal force diagrams along the crest wall as generally applied compared to the method proposed in the present study. As previously mentioned, vertical force diagrams were defined in the same way. The length coefficient (γ L ) was calculated by dividing AF 0.1% estimated using the results of the present study ("triangular shape") with the best fitting approach by the AF 0.1% without the finite length effect ("rectangular shape"). Equation (16) is derived where Equation (3) is used for Ru 2% and Equation (4) is used for γ β . Thus, Equation (16) is only valid if L crest ≥ (duration 0.1% v F ). As mentioned in Section 4.4, when duration0.1% vf > Lcrest, the considered portion of the area within the triangle needs to be the maximum (see Figure 18). Since the area calculation is not direct, a new equation is derived. Cases within the tested experimental ranges were generated. sm-1,0 = 0.02 and 0.04, β = 15°, 30°, 45°, 60°, and 75°, hs = 0.77 m, AC = 0.12 m and Hwall = 0.15 and 0.20 m were considered. Values for Hs were selected such that the generated cases remain within the experimental range of tested values for Ru2%/AC. Test conditions were used for cross-validation. Both the generated cases and the test conditions cases were applied to five lengths of the crest wall (Lcrest = 2.25 m, 5 m, 10 m, 15 m, and 20 m on the scale of the model). Thus, 187 generated cases and 235 test conditions were As mentioned in Section 4.4, when duration 0.1% v f > L crest , the considered portion of the area within the triangle needs to be the maximum (see Figure 18). Since the area calculation is not direct, a new equation is derived. Cases within the tested experimental ranges were generated. Values for H s were selected such that the generated cases remain within the experimental range of tested values for Ru 2% /A C . Test conditions were used for cross-validation. Both the generated cases and the test conditions cases were applied to five lengths of the crest wall (L crest = 2.25 m, 5 m, 10 m, 15 m, and 20 m on the scale of the model). Thus, 187 generated cases and 235 test conditions were used to illustrate the obtained method. The considered values of L crest went from approximately the smallest L m−1,0 until 2L m−1,0 of the highest L m−1,0 .
F H,0.1% and F V,0.1% were calculated using the methodology given in [5] (Equations (1) to (4)). Note that K e,v needed to be interpolated for the generated cases (K e,v = 2.35 and 1.81 for s m−1,0 = 0.02 and 0.04, respectively). AF 0.1% using the results of this study were obtained with the best fitting option (K opt = 0.6) with Equations (7) and (10). The reduction factor γ L is calculated by dividing AF 0.1% obtained with the results from this study ("triangular shape") by the AF 0.1% calculated with the generally applied method ("rectangular shape" with the maximum force acting on the entire crest wall at the same instant).
The same trend was observed in γ L for both horizontal and vertical force events. Equation (17) was proposed based on the generated cases. Thus, for relatively short crest walls with L crest < (duration 0.1% v F ), the following expression is proposed: where Equation (3) is used for Ru 2% and Equation (4) is used for γ β , which accounts for the effect of oblique wave attack. The reduction factor expressed by Equations (16) and (17) are valid within the ranges 0.541 ≤ A C /Hs ≤ 1.102; 0 ≤ (γ β Ru 2% − A C )/L crest ≤ 0.045. Since Equations (16) and (17) do not depend on F 0.1% , they are applicable independently on the method used to estimate F 0.1% . Note that the higher the wave attack angle, the lower γ L and the higher the reduction on the crest wall forces. Figure 22 compares the goal γ L and the estimated γ L using Equation (17) Assuming a Gaussian error distribution, the 90% confidence interval for γL is given by: Note that the confidence interval was calculated for the generated cases. The agreement is rather good for both the generated cases and the test conditions (0.198 ≤ rMSE ≤ 0.227).

Conclusions
To estimate the forces on a crest wall of a rubble mound structure, the predicted maximum force Assuming a Gaussian error distribution, the 90% confidence interval for γ L is given by: Note that the confidence interval was calculated for the generated cases. The agreement is rather good for both the generated cases and the test conditions (0.198 ≤ rMSE ≤ 0.227).

Conclusions
To estimate the forces on a crest wall of a rubble mound structure, the predicted maximum force at a single position (chainage) is normally assumed to act on the entire length of the crest wall. For perpendicular wave attack, this is a reasonable approach. However, for oblique wave attack, the maximum loading is not acting on the entire length of the crest wall at the same instant. Since crest walls have a finite length, assuming that the maximum wave loading acts at the same instant over the entire length of the crest wall leads to an overestimation of the actual forces on crest walls under oblique wave attack. This study describes a method to account for the effect of the finite length of crest walls on the forces caused by oblique wave attack.
The physical model tests conducted by [5] in a wave basin were used. A triangular shape is applied to model the temporal shape of the (horizontal and vertical) force events (exceeded by 0.1% of the incoming waves) with 0.912 ≤ r ≤ 0.924. It was found that the duration of the force events (duration 0.1% ) (base of the triangle) is the same for both horizontal and vertical forces. Equation (7) is proposed to estimate the duration of the force events (rMSE = 0.472). Equation (6) is derived to estimate the position of the peak within the force event; the rising time of the horizontal forces is faster than for the vertical forces. The method proposed in [5] was suggested to calculate the maximum loading (F 0.1% ). The agreement between the calculated and measured force register was good (0.874 ≤ r ≤ 0.879).
The temporal shape of the force events was transformed into the space domain by means of the velocity at which the force event travels along the structure (v f ). To this end, the travel time between two measurement points was determined, and v f was obtained by dividing the distance between both points between the travel time. The same travel time and v f were obtained for horizontal and vertical forces. The travel time obviously is 0 for perpendicular wave attack, since the whole crest wall is impacted by the wave in the same instant. Equation (10) was proposed to calculate v f for oblique wave attack with a rather high accuracy (rMSE = 0.066).
The actual force on the entire crest wall with a finite length (AF 0.1% ) was obtained using a triangular shape in the space domain. The adopted approach resulted in conservative estimates of the reduction due to the finite length of crest walls under oblique wave attack. The best fit for actual force on the entire crest wall with a finite length (AF 0.1% ) resulted in 60% of the triangular shape (K opt = 0.6) with rMSE = 0.133.
A new reduction factor (γ L ) was defined as the ratio between AF 0.1% estimated using the findings of the present study and the maximum force acting against the crest wall (F 0.1% L crest ). Equations (16) and (16) were proposed to directly estimate γ L for both horizontal and vertical forces. Equation (16) is valid if L crest ≥ (duration 0.1% v F ). Equation (17) was developed to account for relatively short crest walls L crest < (duration 0.1% v F ). To obtain Equation (17), the methodology developed in this study using the best fit approach (K opt = 0.6) was applied on two data sets: (1) generated cases within the experimental ranges with five values of the length of the crest wall (L crest ) and (2) tests conditions with five values of L crest . The forces on the entire crest wall with finite length (AF 0.1% ) were also calculated using the generally applied method (AF 0.1% = F 0.1% L crest ). Equation (17) was derived using dataset (1), while dataset (2) was used for cross-validation (0.198 ≤ rMSE ≤ 0.227). Since the derived reduction factor, expressed by Equations (16) and (17), does not depend on F 0.1% , Equations (16) and (17) are applicable independently on the estimator used for F 0.1% .
The expressions proposed in this study are valid within the ranges of the present tests (0.84 ≤ R c /H s ≤ 1.6; 1.27 ≤ R c /A c ≤ 1.55; 0 ≤ F b /H s ≤ 0.56; 0.13 ≤ H s /h s ≤ 0.27). It is encouraged to check their validity out of the experimental ranges of this study, paying special attention to the effect of wave characteristics on the shape of the force events, different crest wall geometries, and armor layers (1:2 rock armored slopes).
This study has illustrated how large the reduction in the required size of the crest wall can be if the finite length of the crest wall is taken into account in oblique wave attack conditions. Therefore, it is recommended to take into account not only the reduction in the maximum forces due to oblique waves, but also the reduction due to the fact that the maximum forces do not occur at the same instant over the entire length of crest walls on rubble mound breakwaters.