This section has the aim of validating the damage characterization methods, which include the damage concepts (Section 3.1), damage parameters (Section 3.2), and measuring techniques (Section 3.3). For that, the physical modelling tests presented in the previous section are used. As a first step, the damage parameters considered in this study are described in more detail, including the procedure for their calculation in Figure 6, for the S and E2D parameters (6a) and for the E3D,m parameters (6b).

The first parameter considered is S, as defined by Broderick [4] (see Equation (1)). This damage parameter is widely used and describes the damage to the structure as the number of units eroded in the width-averaged profile.

$$S(number\ of\ units) = \frac{\langle Ae\rangle\_{\text{w}}}{D\_{\text{w}50}^2} \tag{1}$$

where *Ae w* (m2) is the eroded area from the averaged profile obtained over a given characterization width w and Dn50 (m) is the nominal median diameter.

The parameters E2D [5] and E3D,m [6,7] are shown in Equations (2) and (3), respectively. These two parameters estimate the damage to the structure considering the maximum erosion depth perpendicular to the slope. The distinction between them is that for E2D, this erosion depth is measured in the width-averaged profile, while for E3D,m, this erosion depth is obtained as the maximum erosion depth recorded at any point of the structure or defined characterization width. In addition, for E3D,m,

the initial profile and the profile after the test are averaged with a circular spatial moving average with mDn50 diameter.

$$E\_{2D}\text{ (erosion depth in units)} = \frac{\max(\langle \varepsilon \rangle\_w)}{D\_{n50}}\tag{2}$$

$$E\_{3D, \text{m}}\text{ (erosion depth in units)} = \frac{\max\left(\langle \varepsilon \rangle\_{mDn\mathbb{S}0}\right)\_{\text{yr}}}{D\_{n\mathbb{S}0}}\tag{3}$$

where *max*(*e w*) (m) is the maximum erosion depth from the averaged profile obtained over a given characterization width w, *max*(*e mDn*50)*w* (m) is the maximum erosion depth averaged over an area of mDn50 diameter obtained over a given characterization width w, and Dn50 (m) is the nominal median diameter.

**Figure 6.** Damage parameters methodology. (**a**) S and E2D parameters obtained from a width-averaged profile, (**b**) E3D,m parameters obtained from the maximum erosion depth observed within a given characterization width.

#### *3.1. Damage Concepts*

Two main damage concepts will be discussed here; namely characterization width (i.e., the width over which the damage is evaluated in a physical model) and the damage limits (i.e., damage initiation, intermediate damage, and failure).

#### 3.1.1. Characterization Width

This is addressed given that using different characterizations widths in the assessment of coastal structures would lead to different damage results for all damage parameters and would influence the definitions of damage limits. Among others, the authors of [11] recommend the use of characterization widths larger than 15–20 rock diameters in order to achieve representative results, but do not discuss the influence it has on the measured damage.

#### Mean Damage Evolution

The influence of the characterization width on the mean measured damage can be observed in Figure 7, obtained from the survey of a 54Dn50 wide slope in the Deltares tests Series 1 (see Table 1), carried out five times in five identical repetitions. The values for each characterization width are obtained as the average of all the values for that given width, being between 135 values for the 2 Dn50 width and 5 values for the 54 Dn50 width. This average damage is then normalized by the damage obtained for the 54 Dn50 characterization width.

Regarding the parameters obtained from width-averaged profiles (S and E2D in Figure 7a), it can be observed that the measured mean damage reduces with increased characterization width because the damage to certain areas is hidden by the accretion in other areas. Regarding the parameters obtained as the maximum erosion depth observed within a given characterization width (E3D,1 and E3D,5 in Figure 7b), it can be observed that the measured damage increases with increased characterization width.

The main observation for the damage parameters E3D,1 and E3D,5 (also referred in general form as E3D,m) is that the measured damage continues to increase with increasing characterization widths. This suggests that there is no upper limit for the damage to the structure and that when considering wider structures, the probability of observing a larger extreme damage will continue to increase. Thus, besides considering the characterization width, this length effect for the design and characterization of coastal structures should then be taken into account (see Section 3.2.2 and [9]).

**Figure 7.** Characterization width. Mean damage variation (bias), calculated as the mean damage from each characterization width (μ) divided by the mean damage from the 54Dn50 characterization width (μ54Dn50 ). (**a**) S and E2D parameters, (**b**) E3D,1 and E3D,5 parameters.

#### Damage Variability Evolution

The variability of measured damage is evaluated for different characterization widths. The five identical realizations of Deltares tests Series 1 are considered again, which consists of four test runs each (see Table 1). In Figure 8, the standard deviation normalized by the mean for the four parameters can be observed.

Regarding the parameters obtained from width-averaged profiles (S and E2D in Figure 8a), it can be observed that the increase in the characterization width leads to a general reduction of the variability in the measurements. Regarding the parameters obtained as the maximum erosion depth observed within a given characterization width (E3D,1 and E3D,5 in Figure 8b), they also show a reduction in the variability (mainly because of an increase in the mean values) with increasing characterization width, with significantly less variability than for parameters S and E2D. For additional details on damage variability, consider the work of [9].

**Figure 8.** Characterization width. Standard deviation of the damage (σ) divided by the mean of the damage (μ), for each characterization width (random error). (**a**) S and E2D parameters, (**b**) E3D,1 and E3D,5 parameters.

Conclusions Regarding Characterization Width

Following the previous results, it can be stated that the characterization width is an important factor in the definitions of damage to coastal structures. According to the present results, a characterization width of approximately 25Dn50 is recommended, but the following aspects should be taken into account:

#### 3.1.2. Damage Limits

This study aims to establish unified damage limits for the characterization of coastal structures such as damage initiation, intermediate damage, and failure. On the basis of the contributions and limitations found in literature (see Section 1.2) and the need for establishing unified concepts for damage characterization, the following damage limits are proposed for a nDn50 thick rock armored slope:

In order to characterize a given rock armored slope according to these damage limits, the parameter E3D,1 is used as the calibration method. This E3D,1 parameter can be described as "the erosion depth measured in Dn50 perpendicular to the slope averaged over a circular area of 1Dn50", which allows capturing the damage limits described above.

Traditional damage assessment methods were based on slope surveys with profilers with a limited number of profiles and the exposure of the filter layer for the failure limit; D50/2 for Thompson and Shuttler (1975), 1Dn50 for Melby and Kobayashi (1998), and to a less clear extent for other authors (see Section 1.2). According to such methods, damage initiation and intermediate damage are based on defined parametric limits, while failure is determined by the visual observation of images or using a cylindrical gauge with a hemispherical foot. In contrast, the use of high resolution surveys such as the ones provided by DSP (see Figure 2) allows a more precise visualization of the state of the structure for all damage levels. In this case, all the damage limits can be quantified according to the depth and extension of the damage area.

Thus, considering all the measurements carried out with a high resolution technique (DSP) and the damage parameter E3D,1, the damage limits can be calibrated based on the Deltares tests (characterization width of 27Dn50), see Figure 9 and Table 2.

**Table 2.** Damage limits (for 27Dn50 characterization width).

**Figure 9.** *Cont*.

**Figure 9.** Comparison of all damage parameters (for 27Dn50 characterization width). (**a**) E3D,1 vs. S parameters, (**b**) E3D,1 vs. E2D parameters, (**c**) E3D,1 vs. E3D,5 parameters, (**d**) E3D,5 vs. S parameters, (**e**) E3D,5 vs. E2D parameters, (**f**) E2D vs. S parameters.

An important remark from the previously defined calibrated damage limits is the fact that, according to the available data, they are constant for the slopes applied (1:3 in Deltares tests). Nevertheless, this should be further investigated considering a larger number of tests and different slope configurations in order to validate these conclusions.

#### *3.2. Damage Parameters*

This section addresses the need for establishing and validating a robust damage parameter able to accurately characterize the damage and remaining strength of rubble mound structures. This will be done by analyzing the Deltares shallow water test results in more detail and using a characterization width of 27 Dn50 and the previously established damage limits. Only parameters S, E2D, and E3D,5 will be evaluated (E3D,1 calibration parameter will not be considered).

The preference for E3D,5 (E3D,m with m = 5, which describes the erosion depth in Dn50 over an averaged circular area of 5Dn50 diameter) over other values of m is justified given the fact that in the erosion profile from the test results, the damage cavity repeatedly showed a cross-profile extension of approximately 5Dn50.

#### 3.2.1. Damage Parameters Analysis

The analysis of damage parameters is done considering the following four criteria: bias, random error, distinction of damage range, and its values for different structures.

#### Bias

The first element to be considered, the bias in damage characterization of coastal structures, is related to the ability of the damage parameters to provide the most precise description of the state of the structure. This can be achieved with damage parameters that do not lead to hidden erosion, defined as the conditions in which the damage in one location of the structure is hidden by the accretion in other location when considering width-averaged profiles.

Damage parameters based on width-averaged profiles (S and E2D) are associated with a larger bias because a larger characterization width will lead to an averaged profile, which reduces (averages out) the magnitude of the maximum erosion areas, especially for smaller damage levels (damage initiation). On the contrary, damage parameters that capture the maximum erosion depth observed within a given characterization width (E3D,5) have limited bias because they provide an assessment of the complete structure surface and identify all damaged areas.

#### Random Error

This second aspect, the random error in the measured damage to coastal structures, is related to the ability of damage parameters to describe the state of the structure with the smallest variability and larger confidence in the measured result. This will be analyzed further in order to estimate the random error associated with each of the damage parameters.

The damage data for Series 1 (composed of four test runs of 60%, 80%, 100%, and 120%) and Series 2 (composed of one test run of 100%) is described in the table below. For each test run, 10 observations are considered, obtained from five realizations in a model with a width of 54Dn50 (twice the considered characterization width of 27 Dn50). For these 10 observations and the 3 damage parameters, Table 3 and Figure 10 show the 80% confidence prediction interval, applying a Student's *t*-distribution with a critical t-value of 1.383 (δ = ±1.383·σ· √1 + 1/10) expressed as a percentage of the mean.

According to these results, it can be observed that damage parameters E3D,5 present a lower variability (random error), especially for the lower conditions associated with a lower amount of damage (60% and 80%). Damage parameters S and E2D present low variability in higher damage levels (120% in Series 1 run 4), but much higher variability (random error) in the other damage levels.

**Figure 10.** Eighty percent prediction interval with respect to the mean (for 27Dn50 characterization width). (**a**) S parameter, (**b**) E2D parameter, E3D,5 parameter.

#### Distinction of Damage Range

The third element to be considered is the ability of damage parameters to distinguish the range of damage levels between initial damage, intermediate damage, and failure. According to the experimental data from the tests, the S, E2D, and E3D,5 parameters were able to distinguish the different damage levels with a similar degree of scatter (for parameter E3D,1, a larger scatter was observed). For additional details, consider the work of [12].

#### Values for Different Structures

The fourth aspect to be considered in this analysis is the suitability of damage parameters to be used in different structures. In this case, different structures do not only include variations in the characteristics of the rock slope, but also 3D features such as roundheads or slopes with a berm. Of the three parameters considered here, only the parameter that captures the maximum erosion depth observed within a given characterization width (E3D,5) is suitable for describing damage to coastal structures with any geometry or 3D feature.

#### Conclusions on Damage Parameters Analysis

According to the four criteria previously discussed, the damage parameter E3D,5 is considered as the most suitable damage parameter for the characterization of damage to coastal structures based on the following four reasons:

#### 3.2.2. Extreme Damage Distribution for E3D,5

According to the results from this study, the measured damage with parameter E3D,5 can be adjusted to an extreme value distribution. This illustrates that for increasingly long structures, a larger damage is expected, in the same way that for an increasingly long time period, a larger wave height is expected, or that for increasingly long dikes, a larger probability exists that one section can fail [13].

In order to define an extreme value distribution for the measured damage with E3D,5, the results shown in Table 4 are considered. Each test condition (four for Series 1 and one for Series 2) had five realizations each with a width of 54Dn50, which leads to 10 damage results for a characterization width 27Dn50. For each test condition, the measured damage was normalized by the mean value, so all 50 measurements are combined in one single distribution. These values are adjusted to a Gumbel distribution, with the distribution parameters determined by the least square method.

The exceedance probability for the damage is translated to a return width in a similar form to that used for the study of extreme wave climate (see Equation (4)). The combined extreme distribution for the normalized damage parameter EN3D,5 is shown in Figure 11.

$$R\_w = \frac{\lambda}{1 - F\_X(x)}\tag{4}$$

where *Rw* (m) is the return width, λ (m) is the characterization width (27Dn50 in this study), and 1 − *FX*(*x*) (-) is the exceedance probability.

**Table 4.** Parameters for Gumbel distribution for EN3D,5 parameter (for 27Dn50 characterization width).

**Figure 11.** Gumbel extreme distribution for EN3D,5 parameter (for 27Dn50 characterization width).

Figure 11 shows that a single realization of a test run for a characterization width of 27Dn50 is expected to present damage of approximately 0.8 as the mean damage (calculated from 10 realizations), while for a structure 10 times wider (approximately 270Dn50 wide), damage of 1.2 as the mean damage is expected. Thus, according to this study, the damage to a structure 270Dn50 wide is expected to present damage up to 50% (1.2/0.8 = 1.5) larger than that obtained in a single realization in a 27Dn50 characterization width. Detailed guides on how to account for the length effect on rock armored structures are presented in the work of [9].

#### *3.3. Measuring Technique*

Significant differences can be observed in the damage characterization outcome when using high resolution and high accuracy measurements techniques compared with low resolution measurements used in traditional surveys. In contrast to low resolution techniques, high resolution techniques (e.g., DSP) are able to measure the state of the whole structure surface and capture all the erosion areas.

These high resolution measuring techniques, such as DSP, present a number of advantages for the damage characterization of coastal structures. First, they allow the introduction of more precise damage parameters, such as E3D,5, and allow the improvement in the damage limits and definitions. Thus, these high definition measuring techniques, together with suitable parameters and definitions, can describe the state of the full structure surface in order to identify the presence of weak areas,

which threatens the stability of the structure. Second, for parameters obtained from width-averaged profiles such as S and E2D, high resolution surveys that capture the state of the whole structure surface are able to reduce the variability and uncertainty in the damage characterization results, especially for situations with a relatively low amount of damage.

In summary, high resolution measurement techniques, in combination with the use of other damage parameters, are able to reduce the bias related to hidden erosion and random errors related to the position of the measured profiles, which are present in the damage characterization methods carried out with low resolution techniques. Thus, it can be concluded that innovative measuring techniques such as DSP are able to significantly improve the damage characterization of coastal structures.

#### **4. Conclusions**

Damage characterization for rock armored slopes was addressed in this paper, with a focus on three key elements: damage concepts, damage parameters, and measuring techniques. It is highlighted that these three elements should be taken into account for a reliable damage assessment. For this research, a physical modelling campaign was carried out within the EU Hydralab+ framework. The Deltares shallow water tests were performed in a wave flume with a foreshore (depth-limited waves), a 1:3 slope, and an impermeable core.

On damage concepts, the characterization width was discussed, because this influences the measured damage and the associated assessment description. The main effect is that for parameters obtained from width-averaged profiles (S and E2D), the measured damage reduces with increased characterization width, because the damage to certain areas will be hidden by the accretion in other areas, while for parameters obtained as the maximum erosion depth observed within a given characterization width (E3D,1 and E3D,5), the measured damage increases with increased characterization width. Considering this and also the damage variability, it is recommended to carry out laboratory experiments with a characterization width of around 25Dn50.

The calibrated damage limits and damage parameters for rock armored slopes with a characterization width of about 25Dn50 and an armor thickness of 2Dn50 are as follows:

Regarding damage characterization parameters, different parameters were considered. It was found that E3D,5 is the parameter that can better describe the damage and remaining strength of conventional and non-conventional structures. The key reasons for defining this parameter as the most suitable damage characterization parameter are the following:

In addition, the results from this study show that the damage measured with the damage parameter E3D,5 presents an extreme value distribution. This illustrates that for increasingly long structures, a larger damage is expected.

Regarding damage characterization measuring techniques, the importance of high resolution measuring techniques for delivering accurate damage characterization of coastal structures was demonstrated. These techniques (such as DSP) allow the use of the most precise damage parameter, that is, E3D,5.

**Author Contributions:** Conceptualization, Methodology, Validation and Investigation: E.d.A., M.R.A.v.G. and B.H.; Formal Analysis, Data Curation, Writing-Original Draft Preparation and Visualization: E.d.A.; Resources, Supervision and Funding Acquisition: M.R.A.v.G. and B.H.; Writing-Review & Editing: E.d.A., M.R.A.v.G. and B.H.

**Funding:** This project received co-funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 654110, HYDRALAB+.

**Acknowledgments:** We would like to thank our colleagues from Deltares and Hydralab+ for their contribution to the present work. Most of the research described here has first been presented and published [14] at the Coastlab 2018 conference (22–26 May 2018, Santander, Spain); the conference is acknowledged for providing the platform to present the described research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Applicability of Nonlinear Wavemaker Theory**

#### **Mads Røge Eldrup \* and Thomas Lykke Andersen**

Department of Civil Engineering, Aalborg University, 9220 Aalborg, Denmark<EMAIL_ADDRESS>**\*** Correspondence<EMAIL_ADDRESS>
Received: 22 November 2018; Accepted: 8 January 2019; Published: 14 January 2019

**Abstract:** Generation of high-quality waves is essential when making numerical or physically model tests. When using a wavemaker theory outside the validity area, spurious waves are generated. In order to investigate the validity of different wave generation methods, new model test results are presented where linear and nonlinear wave generation theories are tested on regular and irregular waves. A simple modification to the second-order wavemaker theory is presented, which significantly reduces the generation of spurious waves when used outside its range of applicability. For highly nonlinear regular waves, only the ad-hoc unified wave generation based on stream function wave theory was found acceptable. For irregular waves, similar conclusions are drawn, but the modified second-order wavemaker method is more relevant. This is because the ad-hoc unified generation method for irregular waves requires the wave kinematics to be calculated by a numerical model, which might be quite time-consuming. Finally, a table is presented with the range of applicability for each wavemaker method for regular and irregular waves.

**Keywords:** linear waves; nonlinear waves; wavemaker theory; wavemaker applicability

#### **1. Introduction**

When performing tests in laboratories or numerical models, high-quality waves representing conditions in prototype as close as possible is of highest priority. In the early 20th century, linear wavemaker theory was developed by Havelock [1], which was later extended to a fully second-order irregular wavemaker theory by Schäffer [2,3]. This extension made it possible to generate mildly nonlinear waves without spurious free waves. The spurious free waves contaminate the wave field and can easily be seen for regular waves for cases with low wave reflection, as the wave shape is not constant in space. For example, Orszaghova et al. [4] showed that using first-order wavemaker theory could lead to erroneously wave run-up and wave overtopping results compared to generating the waves with second-order wavemaker theory. Furthermore, Sriram et al. [5] showed that the breaking point of focused waves is different when using first-order and second-order wavemaker theory. This was expected to be caused by the influence from the free spurious long-wave components generated with first-order theory.

Recently, ad-hoc unified wavemaker theories were proposed by Zhang and Schäffer [6] for regular waves and by Zhang et al. [7] for irregular waves. These ad-hoc unified wave generation methods make it possible to generate highly nonlinear waves of high quality in intermediate and shallow water. These methods require a depth-averaged velocity as input to control the motion of the piston wavemaker. For regular waves, a fully nonlinear wave theory is available in the form of the stream function wave theory by Fenton and Rienecker [8], and from this, the depth-averaged velocity can be calculated. At the moment, there exists no analytical model to calculate the kinematics for highly nonlinear irregular waves, but the kinematics can be obtained by numerical models, for example, Boussinesq type wave models. The propagation of waves from deep to shallow water with numerical models can be time-consuming. Therefore, it is more efficient to use the first or second-order wavemaker theory for irregular waves when they are valid.

Unwanted free waves are generated when using a wavemaker theory outside its validity area. Schäffer [3] specified that the second-order wavemaker theory is not valid for regular waves when a secondary crest is produced in the wave trough. This happens when the second-order amplitude is larger than 1 4 of the first-order amplitude. To describe this, he introduced the nonlinearity parameter, *S* which must not exceed unity for the second-order wavemaker theory to be valid. For regular waves, Schäffer [3] defined *S* as four times the ratio between the amplitudes of the second-order and the first-order components in regular waves and is given by:

$$S = \mathcal{Q}|HG\_{\text{nm}}^{\pm}|\tag{1}$$

where *G*± nm is the second-order surface elevation transfer function given for example in Schäffer [3], *H* is the wave height. For the application of Equation (1) on irregular waves Schäffer [3] proposed to use a characteristic wave height (*H* = *H*1/3) and *f* n = *f* m = *f* p to calculate *G*± nm.

A more well-known approach to check the applicability of wave maker theories is the diagram by Le Méhauté [9], which described what wave theory was valid depending on the relative water depth and the wave steepness. Figure 1 shows an example of the Le Méhauté diagram and colored areas that illustrates different *S* ranges calculated with Equation (1). From the figure, it is seen that using the applicability criteria for second-order waves given by Schäffer [3] corresponds to fourth order Stokes waves according to the diagram by Le Méhauté [9]. This clearly shows the need of testing the applicability range of the existing methods and to provide some recommendations that can easily be followed.

**Figure 1.** Tested sea states, where black dots are for the regular waves and red dots are for irregular waves. The colored areas show different ranges for the nonlinearity parameter *S* calculated with Equation (1). (Adapted from Le Méhauté [9]).

The present paper presents new physical model test results to study the applicability of the different wavemaker methods for regular and irregular waves. A simple modification to the second-order wavemaker theory by Schäffer [3] is proposed to extend its range of applicability and to significantly reduce errors for highly nonlinear cases. Furthermore, recommendations are given for the validity of each of the tested wavemaker theories.

#### **2. Present Study**

New model tests covering linear to highly nonlinear waves have been performed in the new wave flume at Aalborg University for regular (5 tests) and irregular (3 tests) waves, see Figure 1.

The purpose of the tests is to study the applicability of each wavemaker theory and to provide a clear definition for the applicability range of each wavemaker theory. Furthermore, is it investigated if the applicability range of the second-order wavemaker theory can be extended with a simple modification. The second-order wavemaker theory is modified in the present paper by introducing a maximum allowed relation between the second-order amplitude and the first-order amplitude. This limitation is controlled by a *S*max value, so for example, *S*max = 1 means that the second-order amplitude will be reduced to 1 4 of the first-order amplitude if it was predicted higher. Without this upper limit, the second-order amplitude might be even larger than the first-order amplitude which is physically incorrect. Instead the second-order component should saturate at a level smaller than the primary component and third and higher order components should increase instead.

The second-order components generated by two interacting components were scaled by introducing a more generalized form of *S.* This is done by rewriting Equation (1) to consider the two interacting components:

$$S(f\_{\text{n}}, f\_{\text{m}}) = H G\_{\text{nm}}^{\pm} / \delta\_{\text{nm}} \tag{2}$$

where *f* n and *f* m are the frequencies of the two interacting components. δnm is 0.5 for *f* n = *f* m and 1 for *f* n = *f* m. In the present paper the characteristic wave height is on the safe side taken as the maximum wave height (*H* = 2*H*m0 ≈ *H*max).

The modification to the second-order wavemaker theory is performed by the scaling factor, λ, given by Equation (3).

$$\lambda\_{\rm nm} = \min\left(1, \frac{S\_{\rm max}}{S(f\_{\rm n}, f\_{\rm m})}\right) \tag{3}$$

The scaling factor should be calculated for all interacting frequencies in the first-order wave spectra and be multiplied to both second-order transfer functions *G*± nm and *F*± nm. *G*± nm is the transfer function for the second-order surface elevation and *F*± nm is the transfer function of the second-order paddle movement. This reduces the amount of second-order energy so no secondary crest is calculated for the interaction of the individual frequencies. The optimal value of *S*max is expected in this interval (1 ≤ *S*max ≤ 4) which corresponds to a second-order amplitude of 25–100% of the first-order amplitude. Although the criteria of *S* ≤ 1 given by Schäffer [3] is for superharmonics in regular waves the scaling is applied also to irregular waves. The present paper focus only on superharmonics (*G*+ nm and *F*+ nm), but the scaling is expected necessary also to subharmonics (*G*− nm and *F*− nm). Correct generation of the subharmonics is very important for response of many structures, but the present paper focus only on the superharmonics for which free energy is much easier to observe in the measured time series.

#### **3. Theoretical Optimal** *S***max**

The nonlinear wave theory by Fenton and Rienecker [8] can estimate the correct amount of second-order energy that exists for a given regular wave over a horizontal sea bed. With the use of nonlinear wave theory, it is thus possible to estimate what artificial limit (*S*max in Equation (3)) should be used in the second-order wave theory to obtain the best results. In Figure 2, the calculated amplitude of the second-order component by second-order wavemaker theory is compared with the calculated by stream function wave theory by Fenton and Rienecker [8]. This is done for different values of *S*max, where *S*max = ∞ corresponds to no limit on the second-order energy (unmodified Schäffer [3] method).

The given applicability range by Schäffer [3] (*S* ≤ 1) shows errors smaller than 10% when using the second-order wavemaker theory by Schäffer [3], see Figure 2. The figure shows that the recommendations given by Le Méhauté [9] based on this analysis might be on the safe side. However, this comparison though only considers the amplitude of the second-order component, but higher order components might be relevant.

By limiting the second-order energy to *S*max = 1 the modified second-order wave generation gives errors smaller than 10% for wave conditions up to *S* ≈ 1.5 as illustrated by the dotted line in the figure. Using values of *S*max = 1.5–1.75 gives a larger area where the error is below 10%. Even though *S*max = 1.5–1.75 gives a larger area where the second-order amplitude is calculated with a small error, the second-order wavemaker theory is not necessarily valid as the third, and higher order energy might have a significant contribution. Therefore, the relation between the second-order and third order components must additionally be small. In Figure 3, the third order amplitude is compared to the first-order amplitude calculated by stream function wave theory. The figure shows that for sea states with *S* ≤ 1.5 the amplitude of the third order component is below 10% of the first-order component. From these results, an optimum of *S*max = 1 is found, and the expected range of applicability is extended from *S* = 1 to *S* = 1.5 with the modified second-order wavemaker theory.

**Figure 2.** Comparison between the second-order amplitude calculated with second-order and stream function wave theory. *S* = 1.5 for the dotted line and *S* = 3 for the dashed-dotted line, which can also be seen in Figure 1 by the colored areas.

**Figure 3.** Comparison between the third order and first-order amplitude calculated with stream function wave theory. *S* = 1.5 for the dotted line and *S* = 3 for the dashed-dotted line.

*J. Mar. Sci. Eng.* **2019**, *7*, 14

#### **4. Model Test Setup and Methodology**

To verify the results from Section 3, physically experiments have been performed where the surface elevation was measured in different locations in the wave flume, see Figure 4.

**Figure 4.** Experimental setup of the wave flume. Measures in meters.

The flume is equipped with a piston-type wavemaker, cross-mode dampers and a passive absorption system with perforated sheets in the end of the flume. The cross-mode dampers are a permanent installation in the flume and not installed due to specific cross-mode problems with the generated waves in the present study. The wavemaker has two sets of flush mounted resistance wave gauge on the wave board, which is used to avoid re-reflected waves from the wave board with use of the active wave absorption method by Lykke Andersen et al. [10]. The active absorption system has proven also to be effective for nonlinear irregular waves, cf. Lykke Andersen et al. [11]. The control signals are thus the wavemaker position and the surface elevation in the nearfield.

To evaluate the validity of each wavemaker theory, the measured surface elevation is compared to the theoretical surface elevation. The regular waves are compared to the predicted surface elevation by Fenton and Rienecker [8] and the irregular waves are compared to the COULWAVE Boussinesq wave model by Lynett and Liu [12]. The COULWAVE model is an accurate model when it comes to shoal waves from deep to shallow water including nonlinear interactions, see [13,14]. The results from the COULWAVE model was also used to generate the input for the ad-hoc unified irregular wave generation method by Zhang et al. [7].

The COULWAVE model was used to propagate irregular waves from deep water to the water depth used in the laboratory (0.5 m). The water depth at the wavemaker in COULWAVE is a compromise between the wave regime where the model is valid, *kh* < 6, and that regime where first-order wavemaker theory can be used. The waves were generated in the COULWAVE model on a horizontal part using a narrow banded JONSWAP spectrum (γ = 10) and truncated at 0.5 to 1.5 times the peak frequency. Thus, the primary spectrum does not significantly overlap with the bound harmonics. Then followed a 1:100 slope to the depth of 0.5 m and finally a horizontal part with a water depth of 0.5 m. The point where the 1:100 slope end, the surface elevation time series was extracted and used as a target for the different wave generation methods. The last horizontal part is where the target values are extracted to be compared with the measured surface elevation in the laboratory. Table 1 shows the numerical model parameters and the sea state conditions at the generation point and at *h* = 0.5 m.

**Table 1.** Wave conditions of the irregular waves at the generation point in COULWAVE.

*J. Mar. Sci. Eng.* **2019**, *7*, 14

The number of cells pr. wavelength is based on the wavelength of the peak wave period at the generation point. Thus, the discretisation of the model is, Δ*x* = (cells per wavelength)/wavelength. The courant number is calculated by *C* = *c*0Δ*t*/Δ*x*, where *c*0 is the shallow water celerity at the generation point and Δ*t* is the timestep.

The method by Zhang et al. [7] uses the averaged velocity in the wave direction, but since only the surface elevation is provided from the COULWAVE model, a conversion to depth-averaged velocity is performed by assuming shallow water wave theory being valid. That is not entirely correct in intermediate water, but a fair approximation.

Because the generated primary spectrum in COULWAVE is narrow banded and truncated (0.5 to 1.5 times the peak frequency), the primary components can be separated from the bound harmonics by bandpass filtering at 0.5 *f* P < *f* < 1.5 *f* P. This truncated spectrum was used as primary spectrum for the second-order wave generation method. The signal without bandpass was used as input for the method by Zhang et al. [7]. The measured waves in the laboratory were then compared with the unfiltered signals from the numerical model in the wave gauge locations.

The theoretical surface elevations calculated with COULWAVE and Fenton and Rienecker [8] might differ from the measured surface elevation due to the cross-mode dampers. The cross-mode dampers might slightly dissipate some of the energy in the physical flume. This is likely to be seen as a reduction in wave height when comparing the measured surface elevation at *x* = 2.15 m and *x* = 6.90 m. Furthermore, the theoretical surface elevation is without reflected waves which is difficult to entirely avoid in the physical model. Therefore, reflection in the experimental tests should be reduced to a minimum. The amplitude reflection coefficient in the physical tests was in the range of 11% to 16%, which was calculated with the nonlinear irregular wave separation method by Eldrup and Lykke Andersen [15]. The amount of reflection in the physical experiments is found acceptable for comparing the total measured surface elevation with the theoretical surface elevation.

#### **5. Regular Wave Results**

For regular waves, the different wavemaker theories are evaluated against the theoretical stream function wave theory. This evaluation is performed at different distances from the wavemaker. Linear to highly nonlinear waves were generated and compared to the theoretical wave profiles. The measured surface elevations are shown after the ramp-up of the wavemaker is completed and if possible before reflection is present in the signals.

Figure 5 shows the results for Sea State A. The measured profiles are almost identical for the different wave generation methods. However, a reduction in the amplitude is observed for *x* ≥ 6.90 m. The reduction in amplitude might be due to the cross-mode dampers. The shape of the measured surface elevations for all the wavemaker theories are similar to the theoretical. Therefore, it can be concluded that all the tested wavemaker methods are valid for Sea State A, and thus the first and second-order wave generation methods lead to acceptable waves, in a more extensive area than given by the Le Méhauté diagram. However, Sea State A is within the validity range given by Schäffer [3] (*S* < 1).

The results for Sea State B are shown in Figure 6 from which it appears that the first-order wavemaker method lead to some minor deviations in the wave shape, indicating some free higher harmonics energy exist. The surface profile for the second-order methods and the ad-hoc unified generation method are similar, and these are close to the theoretical profile. The wave height of the generated waves is slightly lower than the theoretical, but the wave shape is a close match to the target. From the section with the theoretical analysis of an optimum of *S*max, a difference was expected to be seen between the original second-order and the modified second-order method when *S* > 1.0, but this is not observed from Figure 6 for a case with *S* = 1.46.

Results for Sea State C shows that the wave profile generated with first and second-order wavemaker theory is not constant in the various gauge positions, cf. Figure 7. This is due to free unwanted waves being generated by these theories when the nonlinearity is too high. The second-order wavemaker theory is slightly better than first-order theory, but the ad-hoc unified wave generation is a close match to the theoretical profile. For *x* = 2.15 m, the wave crest of the generated waves by the second-order wavemaker theory by Schäffer [3] is much larger than the theoretical wave crest, and the proposed correction to the second-order method leads to a wave profile closer to the theoretical profile. These results shows that Schäffer [3] overestimates the second-order amplitude when *S* > 1.

**Figure 5.** Theoretically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State A.

**Figure 6.** Theoretically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State B.

Sea State D is shown in Figure 8. Again, large waves are observed in *x* = 2.15 m for the second-order wavemaker theory, and a small wave crest is seen in the wave trough. The first-order wavemaker theory is actually better than the second-order wavemaker theory for *x* = 2.15 m. This indicates the second-order method actually increases the amplitude of the free waves compared to first-order wavemaker theory. The large waves generated by second-order wavemaker theory were causing breaking waves, which did not occur for the other generation methods. The modified second-order wavemaker theory is performing better than first-order and second-order wavemaker theory, but free waves are also observed for that method. The ad-hoc unified wave generation is a close match to the theoretical wave profile except for the observed reduction in amplitude for *x* ≥ 6.90 m, which is likely due to the cross-mode dampers.

**Figure 7.** Theoretically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State C.

**Figure 8.** Theoretically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State D.

Results for Sea State E is shown in Figure 9. For that case second-order wavemaker theory generates too large waves in *x* = 2.15 m and a secondary wave is also seen in the wave trough of the primary wave. The first-order wavemaker theory is significantly better than the second-order wavemaker theory as no secondary waves are seen in *x* = 2.15 m. As in Figure 8, the waves generated by second-order wavemaker theory are breaking during the tests, which is the reason for the large wave crest are not observed for the two other locations in the figure. The modified second-order wavemaker theory is slightly better than the first-order wavemaker theory, but they are both far from the theoretical profile. The ad-hoc unified wave generation is a close match to the theoretical profile except for small deviations in the trough, but this is most likely due to reflections from the passive absorber in the wave flume.

**Figure 9.** Theoretically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker method for Sea State E.

A more detailed analysis of the measured waves for Sea State E is shown in Figure 10 for *x* = 2.15 m. The time series of the primary and the first three superharmonics is calculated by Fourier transformation on the discrete frequencies *nω*. Thus, the measured components can be compared with the theoretical stream function wave components. The figure shows that the original second-order wavemaker theory generates higher harmonics that are significantly higher than the theoretical stream function wave. It also shows that only the ad-hoc method generates the correct amplitudes of the different contributions. Furthermore, a phase shift between the superharmonics and the first-order component is observed especially for the first-order generation method and the original second-order method. This phase shift demonstrates significant free energy to be present. Only the ad-hoc method has correct amplitudes and phases for all components.

It has been shown that first-order wavemaker theory can be used for *S* ≤ 0.8 and that the modified second-order wavemaker theory gives identical results as the second-order theory by Schäffer [3] when the second-order theory is valid. However, the modified second-order method performs significantly better when used outside the validity area. This is due to a more realistic value of the amplitude for the second-order harmonic, which for the method by Schäffer [3] has no upper limit and for the present tests is calculated to approximately two times the first-order amplitude in the worst case (Sea State E). In that case, the higher order harmonics have a significant influence, and therefore second-order theory is not valid for this case. This is shown in Figure 10 by comparing the contributions of the first

four harmonics with the theoretical amplitudes. The second-order wavemaker theory can be used for *S* ≤ 1.5 with reasonable results. The ad-hoc unified generation can be used for all the tested conditions (tested up to *S* = 7.7).

**Figure 10.** Contribution of the first four harmonics to the surface elevation for Sea State E at *x* = 2.15 m. First-order, second-order, modified second-order and the ad-hoc unified wavemaker method are compared with the theoretical harmonics in stream function waves.

#### **6. Irregular Wave Results**

The performance of the different wavemaker methods for the irregular waves are compared with the results from the COULWAVE model at different distances from the wavemaker. The measured surface elevations are shown for a time window including the highest wave.

Results for Sea State F are given in Figure 11. The first and second-order wavemaker theories are according to the Le Méhauté diagram not expected to be valid for this sea state. Furthermore, the test is also outside the applicability range given by Schäffer [3]. However, the figure shows that all the tested methods give similar results and that they are a close match to the COULWAVE model with only minor deviations.

**Figure 11.** Numerically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State F.

Figure 12 presents the results for Sea State G. For this case is the wave crest of the largest wave with the first-order wavemaker theory significantly smaller than the theoretical wave profile at *x* = 2.15 m. The original second-order wavemaker theory is closer to the numerical profile except for some deviations in the wave trough for *x* = 2.15 m. For the three wave gauges, it can be seen that the wave crest is too small for the largest waves when the waves are generated with the modified second-order wavemaker method, but except for that, it is a close match. The shape of the waves generated by the ad-hoc unified generation method is a close match but with a deeper wave trough at *x* = 2.15 m, but this is likely due to long reflected waves as it is not seen for the two other gauges.

**Figure 12.** Numerically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State G.

For Sea State H, the first-order wavemaker theory is deviating significantly from the numerical profile as the wave trough is not flat and long and the wave crest is not high and narrow, cf. Figure 13. The wave profile generated by the first-order wavemaker theory has a step front followed by a gentler rear as the waves propagate away from the wavemaker—this is due to the free and bound waves having different celerity. For *x* = 11.37, the free and bound waves are phase shifted to such a degree that the crest of both the free and the bound waves are visible. The second-order wavemaker method has a secondary crest in the wave trough, which is reduced significantly with the modified second-order method. The wave crest with the modified second-order wavemaker method is though significantly smaller than the theoretical but is still closer to the theoretical compared to the first-order wavemaker method. The modified method is better than the original second-order method as less free energy is generated, but for this case the amount of secondary energy is also much smaller than the target. The ad-hoc unified wavemaker method provides a close match to the numerical profile with only minor deviations, and for this sea state, only this method leads to acceptable results.

**Figure 13.** Numerically surface elevation compared with the first-order, second-order, modified second-order and the ad-hoc unified wavemaker methods for Sea State H.

For irregular waves, the present tests showed that first-order wavemaker theory can be used for *S* ≤ 1.2, while the second-order wavemaker theory can be used for *S* ≤ 2 with reasonable results. The ad-hoc unified generation showed good performance and can be used for all the tested conditions (tested up to *S* = 7.0).

#### **7. Conclusions**

Model tests have been performed with the purpose to generate waves of various nonlinearity and evaluate different wavemaker methods. First-order, second-order and ad-hoc unified generation methods have been tested, and the validity range of each method has been found by physical experiments. Furthermore, a modification to the second-order wavemaker method was proposed to increase the validity to more nonlinear waves. The ad-hoc unified generation method for the regular waves was the most accurate wavemaker method, and if possible, it should be used, otherwise the modified second-order is preferable. For the irregular waves, the ad-hoc unified generation method is also the most accurate one, but it is also much more time-consuming in synthesize the signals. This is because a time series has to be prepared by a numerical model before waves can be generated in the physical model. Therefore, it is recommended to use the modified second-order wavemaker theory when it is valid.

Table 2 summarizes the applicability ranges of each wavemaker method. For the regular waves, acceptable results are found with first-order wavemaker when *S* < 0.8, while second-order methods extends the validity to *S* < 1.5. The ad-hoc method is applicable for all the tested conditions (*S* < 7.7), but is expected to be reliable for all *S* values. For the irregular waves, acceptable results are found with first-order theory when *S* < 1.2. For both second-order methods acceptable results for irregular waves were found when *S* < 2.0. The ad-hoc method showed also for irregular waves good results for all the tested conditions. As it has not been possible to test all possible sea states these results should be taken as preliminary, but can be used until more tests have been performed.

**Table 2.** Applicability of each tested wavemaker method.

**Author Contributions:** M.R.E. wrote the original draft and performed all the simulations, experiments and analysis. Furthermore, he established the modified second-order wavemaker theory; T.L.A. reviewed the paper and contributed to the performed analysis and in establishing the modified second-order wavemaker theory.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Cubipod® Armor Design in Depth-Limited Regular Wave-Breaking Conditions**

#### **M. Esther Gómez-Martín 1,\*, María P. Herrera 2, Jose A. Gonzalez-Escriva 1 and Josep R. Medina 1**

Received: 17 October 2018; Accepted: 4 December 2018; Published: 6 December 2018

**Abstract:** Armor stability formulas for mound breakwaters are commonly based on 2D small-scale physical tests conducted in non-overtopping and non-breaking conditions. However, most of the breakwaters built around the world are located in breaking or partially-breaking wave conditions, where they must withstand design storms having some percentage of large waves breaking before they reach the structure. In these cases, the design formulas for non-breaking wave conditions are not fully valid. This paper describes the specific 2D physical model tests carried out to analyze the trunk hydraulic stability of single- and double-layer Cubipod® armors in depth-limited regular wave breaking and non-overtopping conditions with horizontal foreshore (*m* = 0) and armor slope (*α*) with cot*α* = 1.5. An experimental methodology was established to ensure that 100 waves attacked the armor layer with the most damaging combination of wave height (*H*) and wave period (*T*) for the given water depth (*hs*). Finally, for a given water depth, empirical formulas were obtained to estimate the Cubipod® size which made the armor stable regardless of the deep-water wave storm.

**Keywords:** mound breakwater; armor stability; Cubipod®; breaking waves; non-overtopping; horizontal foreshore; regular waves

#### **1. Introduction**

Breakwaters are usually tested using small-scale models in non-breaking wave conditions. Since the failure of large breakwaters in the late 1970s and early 1980s (e.g., Sines West Breakwater, Portugal), most physical tests have been carried out using irregular waves to better reproduce the highest waves in prototype conditions. However, most mound breakwaters are placed in depth-limited wave breaking conditions, where the highest waves break on the foreshore before reaching the structure.

Since the pioneering work of Iribarren (1938) [1], several formulas have been developed to characterize the hydraulic stability of rock armors, such as those provided by Hudson (1959) [2], Van der Meer (1988) [3], Van Gent et al. (2003) [4] and other authors. An in-depth literature review was undertaken by Herrera et al. (2017) [5]. Most of these formulas are based on 2D physical tests with models in non-breaking wave conditions. Spectral significant wave height, *Hs* = *Hm*0 = 4(*m*0) 0.5, at the toe of the structure and wave height with a 2% exceedance probability, *H*2%, are usually considered to describe the incident wave characteristics in non-breaking wave conditions. *H*2% is strongly correlated to *Hm*0 in deep water when wave heights are Rayleigh-distributed, but this is not the case in depth-limited breaking wave conditions (see Battjes and Groenendijk 2000) [6].

For shallow water and horizontal foreshore, the water depth (*hs*) is the key variable for designing mound breakwaters because the maximum wave height (*Hmax*) attacking the structure mostly depends on *hs*. In these conditions, irregular waves may cause less armor damage than regular waves. In a given wave storm, the highest waves may break on the foreshore while smaller waves cause irrelevant armor damage; only a very low proportion of waves will have a wave height (*H*) close to the maximum wave height (*H* ≈ *Hmax*) to be relevant when analyzing the hydraulic stability of the armor layer. By contrast, runs of regular waves with the appropriate wave height (*H*) and wave period (*T*) can be generated in 2D physical experiments to attack the structure with many waves, say 100 waves, close to the maximum wave height, which would never be obtained with runs of a few thousand irregular waves propagating on a horizontal foreshore.

Given a water depth (*hs*), a test matrix of regular wave runs covering the full range of wave height (*H*) and wave period (*T*), guarantees that the structure will be attacked by many of the most damaging waves. In these conditions, armor damage under the full test matrix of regular waves is higher than damage caused by a conventional test matrix of irregular wave runs (1,000 waves). Design rules based on the results using regular waves are on the safe side because the structure in shallow water and horizontal foreshore will never be attacked by such a large number of high waves, regardless of the design storm.

This paper describes the specific 2D physical model tests carried out to analyze the hydraulic stability of single- and double-layer Cubipod® armored breakwaters on a horizontal foreshore, in depth-limited regular wave breaking and non-overtopping conditions. New empirical formulas depending on the design water depth (*h*) are proposed to estimate the size of Cubipod® units needed to design stable armors with cot*α* = 1.5 on a horizontal bottom (*m* = 0), for any deep-water wave climate. The paper is structured as follows: Section 2 describes the background of the research. Then, Sections 3 and 4 describe the experimental methodology and present the experimental results, respectively. Section 5 summarizes and details the proposed design criterion.

#### **2. Background**

The armor layer of mound breakwaters is commonly designed using empirical formulas, such as those proposed by Hudson (1959) [2] or Van der Meer (1988) [3], based on 2D small-scale physical tests conducted in non-breaking and non-overtopping conditions. However, most of the breakwaters built around the world are located in breaking wave conditions, where they must withstand design storms having some percentage of large waves which break before reaching the structure. In these cases, it is not clear whether the design formulas for non-breaking wave conditions should be applied. The Hudson formula given below (Equation (1)), based on the pioneering work of Iribarren (1938) [1], was popularized by USACE (1975 and 1984) [7,8]:

$$N\_s = \frac{H\_s}{\Delta D\_n} = \left(K\_D \cot a\right)^{1/3} \tag{1}$$

where *Dn* is the nominal diameter of the units, Δ = (ρr − ρw)/ρw is the relative submerged mass density, ρr is the mass density of the units, ρw is the mass density of the sea water, *KD* is the stability coefficient, α is the armor slope, *Hs* is the significant wave height (*Hs* = *H*1/3 = average of one-third highest waves or *Hs* = *H*m0 = spectral significant wave height), and *Ns* is the stability number.

Equation (1) was based on regular wave tests in non-breaking and non-overtopping conditions. USACE (1975 and 1984) [7,8] recommended a change in the stability coefficient (*KD*) to use Equation (1) in breaking wave conditions, reducing *KD* values for breaking waves. *KD* takes into account the geometry of the armor unit, number of layers, breakwater section (trunk or head), armor slope (*α*), and also an implicit safety factor for design (see Medina and Gómez-Martín 2012 [9]). Equation (1) does not explicitly consider the duration of the wave storm, the wave period or wave steepness, the permeability of core and filter layers, or other relevant factors affecting the hydraulic stability of the armor layer. Nevertheless, Equation (1) is still widely used by practitioners for preliminary designs and feasibility studies to compare construction costs of different breakwater designs using different armor units.

In addition, the generalized Hudson formula is extensively used in practice for single- and double-layer armors; however, this formula is not reasonable for interlocking units because interlocking increases with the armor slope. Furthermore, armor design to Initiation of Damage (IDa) popularized by USACE (1975) [7] for double-layer randomly-placed armors must be adapted to single-layer armors with brittle failure functions when IDa and Initiation of Destruction (IDe) are very close. In order to maintain a reasonable safety factor to IDe, the recommended *KD* for single-layer armors corresponds to an armor damage level much lower than IDa. The values of *KD* for single- and double-layer armors are based on small-scale physical tests and may change slightly over time, depending on experience and implicit or explicit safety factors.

For double-layer rock armors in breaking wave conditions, Van der Meer (1988) [3] proposed replacing *Hs* with *H*2%/1.4 in Equations (2) and (3), after conducting several physical tests in breaking wave conditions with an m = 1/30 bottom slope and a permeable structure. The relationship *H*2%/1.4 = *Hs* is valid if wave heights are Rayleigh distributed (non-breaking wave conditions) but it is a conservative criterion if the highest waves break before reaching the structure (breaking wave conditions).

$$N\_s = \frac{H\_s}{\Delta D\_{n50}} = 6.2 S^{0.2} P^{0.18} N\_Z^{-0.1} \xi\_m^{-0.5} \text{for } \xi\_m < \xi\_{\text{mc}} \text{ (plunging waves)}\tag{2}$$

$$N\_{\rm s} = \frac{H\_{\rm s}}{\Delta D\_{\rm n50}} = 1.0 S^{0.2} P^{-0.13} N\_{\rm Z}^{-0.1} (\cot a)^{0.5} \zeta\_{\rm m}^{P} \text{for } \xi\_{\rm m} > \xi\_{\rm mc} \text{ (surging waves)}\tag{3}$$

in which *Ns* is the stability number, *Dn*50 is the equivalent cube size or nominal diameter of the units, *S* is the dimensionless armor damage, ξmc = 6.2*P*0.31(tan*α*) 0.5) 1/(P+0.5) is the critical breaker parameter, 0.1 ≤ *P* ≤ 0.6 is a parameter which considers the permeability of the structure, *Nz* is the number of waves, and ξm = tan*α*/(2π*Hs*/(g*Tm*2))0.5 is the surf similarity parameter based on the mean period, *Tm*.

Although several hydraulic stability formulas for rock armors are used in breaking wave conditions, considering different characteristic wave heights, most experimental validations are carried out in non-breaking wave conditions.

Melby and Kobayashi (1998) [10] and Van Gent et al. (2003) [4] also proposed hydraulic stability formulas for double-layer randomly-placed rock armors based on specific laboratory tests in breaking wave conditions (0.64 < *Hs*/*hs* < 1.11 and 0.15 < *Hs*/*hs* < 0.78, respectively) with different bottom and armor slopes. Furthermore, available formulas require knowing *Hs* or *H*2% at the toe of the structure; however, these formulas are calibrated with waves measured at a certain distance from the structure.

Van Gent (2013) [11] established that breakwaters with a berm can significantly reduce overtopping and reduce the required rock size, especially for the slope above the berm, compared to straight slopes without a berm. Van Gent (2013) [11] provided empirical relationships to quantify the required armor rock size for non-overtopped rock slopes with a horizontal berm, with a 1:2 slope above and below the berm, or a 1:4 slope above and below the berm; this formula is only valid with a horizontal foreshore on which no wave breaking occurs. Nevertheless, the experimental range investigated by Van Gent (2013) [11] is limited to rather high berms, i.e., characterized by small values of water depth over the berm compared with the water depth at the toe.

Herrera et al. (2017) [5] proposed Equation (4) for double-layer rock armors in breaking wave conditions (0.20 < *Hs*/*hs* < 0.90), considering the observed potential 6-power relationship between the equivalent dimensionless armor damage (*Se*) and the spectral significant wave height (*Hm*0) at a seaward distance of 3*hs* from the breakwater toe, where *hs* is the water depth at the toe.

$$S\_t = 0.066 \left(\frac{H\_{\rm m0}}{\Delta D\_{\rm n50}}\right)^{6.0} \tag{4}$$

Celli et al. (2018) [12] provided a design criterion for the armor layer elements of conventional breakwaters with submerged berm marked by a minimal thickness compared to water depth, extending the validity range of the method suggested by Van Gent (2013) [11].

CLI (2018) [13] specified stability coefficients for single-layer armors with AccropodeTM (*KD* = 15) and other interlocking armor units (*KD* = 16), valid for armor slopes with cot*α* = 1.33 to 1.5 in

non-breaking conditions. CLI (2018) [13] as well as USACE (1975, 1984) [7,8] recommended using lower values of *KD* for wave breaking conditions. On the contrary, Xbloc® (2014) [14] recommended *KD* = 16 and 14.2 for armor slopes with cot*α* = 1.33 and 1.5, respectively, in breaking or partially breaking wave conditions (*Hs*/*hs* > 0.4), and *KD* = 10.7 and 8.0 for 0.29 < *Hs*/*hs* < 0.4 and *Hs*/*hs* < 0.29, respectively, when cot*α* = 1.33.

To rationalize the use of *KD* when characterizing the hydraulic stability of the armor layers, Medina and Gómez-Martín (2012) [9] proposed including explicit safety factors (SF) for IDa and IDe associated to each recommended *KD*. Based on the experimental results for each armor unit, the safety factors SF(IDa) and SF(IDe) are the ratio between the stability number to IDa and IDe and the design stability number (*Nsd*), SF(IDa) = *Ns*(IDa)/*Nsd* and SF(IDe) = *Ns*(IDe)/*Nsd*, where *Nsd* is given by Equation (1) when *Hs* = *Hsd* and *Hsd* is the design significant wave height. Medina and Gómez-Martin (2016) [15] recommended *KD* values for single- and double-layer Cubipod® armors in non-overtopping and non-breaking conditions, with the corresponding explicit safety factors.

#### **3. Experimental Methodology**

2D physical model tests using regular waves in breaking wave conditions were conducted in the wind and wave test facility (30 × 1.2 × 1.2 m) of the Laboratory of Ports and Coasts at the Universitat Politècnica de València (LPC-UPV). The piston-type wavemaker was equipped with an active wave absorption system (AWACS), which generates regular and irregular waves. Water surface elevation was measured using capacitance wave gauges at ten points along the wave flume. One group of four wave gauges (S1, S2, S3 and S4) was placed near the wavemaker and other group of four wave gauges (S5, S6, S7 and S8) near the model. Wave gauges of each group were distanced according to the criterion given by Mansard and Funke (1980) [16], and incident and reflected waves where separated sing the LASA-V method developed by Figueres and Medina (2004) [17]. The water depth ranged between *hs*(cm) = 30 and *hs*(cm) = 42 at the model area, and between *h*(cm) = 55 and *h*(cm) = 67 at the wave generation zone, with a 4% slope bottom transition between the two horizontal wave flume bottoms at different levels. Figure 1 shows a longitudinal cross-section of the LPC-UPV wave flume with the location of the wave gauges used in these tests.

**Figure 1.** Longitudinal cross-section of the Laboratory of Ports and Coasts at the Universitat Politècnica de València (LPC-UPV) wave flume (dimensions in meters).

Three models with different armoring protection (single- and double-layer Cubipod® armors with toe berm and double-layer Cubipod® armor without toe berm) were tested in breaking wave conditions, using the same core and filter layers (see Vanhoutte and De Rouck 2009) [18]. The breakwater crest elevation was high enough for non-overtopping conditions, and the armor slope was α with cot*α* = 1.5. The armor porosities were 39% and 40% for single-and double-layer armors. The characteristics of the core, filter and armor layer are specified in Table 1, where *Dn*50 is the equivalent cube size or nominal diameter of the units, *D*85 and *D*15 is the equivalent cube size corresponding to the 85% and 15% percentile, respectively, ρr is the mass density, and M is the unit mass.

**Table 1.** Characteristics of the tested models.

The reference scale was 1/50, and the objective was to determine, for a given water depth, a unit size which made the armor safe regardless of the deep-water wave storm conditions. In wave breaking conditions, the maximum wave height attacking the breakwater depends on the water depth at the toe (*hs*), the wave period (*T*) and the bottom slope (*m* = 0 in this study). The worst test conditions were those generating runs of regular waves with a variety of wave periods and with series of increasing wave heights (*H*), for each water depth (*hs*). For a given water depth (*hs*) and a wave period (*T*), the energy of the wave attack on the breakwater model increased with increasing H until waves broke in the horizontal foreshore before reaching the structure.

For each water level, *hs*(cm) = 30, 35, 38, 40 and 42 (*hs*(m) = 15.0, 17.5, 19.0, etc. at prototype scale), trains of 100 regular waves with different wave periods *T*(s) = 0.85, 1.28, 1.70, 2.13 and 2.55 (6 < *T*(s) < 18 at prototype scale) were generated increasing wave height (*H*) from no damage up to the armor being severely damaged (IDe) or wave breaking on the foreshore. *H* was increased in steps of 1 cm within the range 9 ≤ *H*(cm) ≤ 26 (4.5 < *H*(m) < 13 at prototype scale). For each water depth, around 3000 waves attacked the structure, which covers the range of periods for Spanish coasts in deep water. For a given water depth (*hs*) and wave period (*T*), if *H* is too high, the waves break before reaching the structure and there is no damage; if *H* is too small, the waves will not cause the maximum damage to the structure. The methodology described above ensured that the most damaging waves attacked the structure for a given water level.

Figure 2 shows the double-layer Cubipod® armored breakwater model without toe berm, and Figure 3 shows the cross-sections of single- and double-layer Cubipod® armors with toe berm. A detail of the toe berm used in these experiments is shown in Figure 4. The toe berm was built with filter material (G1). Equivalent dimensionless armor damage (*Se*) was measured with the virtual net method described by Gómez-Martín and Medina (2014) [19], considering armor unit extractions, armor layer sliding as a whole, and heterogeneous packing (HeP) failure modes simultaneously. The Cubipods in the bottom armor layer were painted white to enhance color contrast, while those in the upper armor layer were painted with stripes of different colors to facilitate the visual counting. The armor was photographed before and after each run of 100 regular waves, allowing armor damage to be measured with the virtual net method. The virtual net method was applied to the photographs taken perpendicular to the armor to calculate the corresponding equivalent dimensionless damage parameter, *Se*. After each series of tests with constant water depth, the armor damage was obtained, considering the previous armor damage obtained for each water depth. The accumulated armor damage was measured to obtain the maximum armor damage associated to each water depth, increasing water levels from *hs*(cm) = 30 to 42, for each model. Different armor damage was observed for each water depth (*hs*); the higher the water depth, the greater armor damage.

Some long irregular wave trains were generated in this study, but they caused little damage compared to the armor damage caused by the runs of regular waves corresponding to the experimental methodology described above. Most of the higher waves of the irregular trains broke on the horizontal foreshore and only a few relevant waves attacked the structure. The methodology used in this study was defined to ensure that 100 waves attacked the armor with the most damaging combination of wave height and period for the given water depth. Therefore, the design based on the results of these experiments with regular waves in breaking conditions is on the safe side if the breakwater is placed on a horizontal sea bottom.

**Figure 2.** View of a double-layer Cubipod® armored breakwater model without toe berm tested at the LPC-UPV.

**Figure 3.** Cross-sections of (**a**) single- and (**b**) double-layer Cubipod® models with the toe berm.

**Figure 4.** Toe berm used in the experiments: (**a**) Cross-section and (**b**) zenithal view.

#### **4. Experimental Results**

According to Medina et al. (1994) [20], rough quarrystone armor damage and wave height given by USACE (1984) [8] and Van der Meer (1988) [3] follow the one-fifth power relationship, and according to Gómez-Martín and Medina (2014) [19], cube and Cubipod® armor damage observations in non-breaking conditions also follow this relationship. Therefore, the linearized equivalent dimensionless armor damage *Se* 1/5 is used in this study to show the stability of Cubipod® armors in depth-limited wave breaking conditions.

Herrera et al. (2017) [5] showed that the rough quarrystone armor damage data presented in the USACE (1975) [7] fit with the empirical Equation (5), considering *KD* = 3.5. Where *H* is the wave height corresponding to damage *S*, and *HD*=0 is the design wave height corresponding to dimensionless armor damage parameter, *S* = 0.7. Based on Equation (5), a general damage function (Equation (6)) for any given armor unit can be calculated taking into account the corresponding *KD* value.

$$\frac{H}{H\_{D=0}} = \left(\frac{S}{1.4}\right)^{1/3} \tag{5}$$

$$S^{\frac{1}{5}} = 1.4^{0.2} \left(\frac{3.5}{K\_D}\right)^{1/3} \frac{H}{H\_{D=0[K\_D=3.5]}}\tag{6}$$

The accumulated linearized equivalent dimensionless armor damage, *Se* 1/5, measured in singleand double-layer Cubipod® armor experiments with and without toe berm is represented in Figure 5, as a function of the dimensionless wave height (considering the wave height, *HD*=0, that produces IDa in rough quarrystone (*KD* = 3.5) with the Hudson formula). It was observed that the *KD* for singleand double-layer Cubipod® armors is approximately 10 times higher than that for rough quarrystone armors in breaking-wave conditions (USACE 1975 [7]). For damage levels below IDa, the toe berm slightly increases the hydraulic stability of the double-layer Cubipod® armor; however, for armor damage above IDa, the toe berm has no significant influence on armor damage. The stability coefficient for single- and double-layer Cubipod® armors with breaking waves is *KD* ≈ 35 (10 times higher than *KD* ≈ 3.5 for rough quarrystone). The double-layer Cubipod® armors with a toe berm were tested up to *hs*(cm) = 42 (corresponding to 21 m at prototype scale). The other two armors were tested up to *hs*(cm) = 40 (20 m at prototype scale).

**Figure 5.** Accumulated linearized equivalent dimensionless armor damage (*Se* 1/5) as a function of dimensionless wave height in wave breaking conditions for single- and double-layer Cubipod® armors with and without toe berm.

Figure 6 shows the accumulated dimensionless armor damage as a function of the measured water depths at the toe of the structure. Given the observed relationship (Figure 6) between the dimensionless water depth (*hs*/Δ*Dn*) and the linearized equivalent dimensionless armor damage (*Se* 1/5), Equation (7) is proposed as the dimensionless failure function to describe the hydraulic stability of Cubipod® armored breakwaters on horizontal bottoms in depth-limited wave breaking conditions.

$$s\_c = 10^{-4} \left[ \frac{h\_s}{\Delta D\_{\text{fl}}} \right]^5 \tag{7}$$

where *Se* is the equivalent dimensionless armor damage, *hs* is the water depth at the toe of the structure, Δ is the relative submerged mass density, and *Dn* is the nominal diameter or equivalent cube side length.

Considering that IDa corresponds to an equivalent dimensionless damage 1.0 < *Se* < 2.0, say *Se* = 1.6, then *hs* < 7.0(Δ*Dn*) is the water depth which guarantees that both single- and double-layer armors will not exceed IDa under any deep water wave climate conditions. Furthermore, the tests carried out indicate a relevant safety margin to IDe. Taking a safety factor SF(IDe) = 1.15 for double–layer Cubipod® armors and SF(IDe) = 1.30 for single–layer Cubipod® armors, the following design values (Equations (8) and (9)) are obtained for safe Cubipod® armored breakwaters placed on horizontal seafloors (*m* = 0):

$$h\_s < 7.0(\Delta D\_n) \quad \text{for double-layer Cabibpod}^{\otimes} \text{ arms} \tag{8}$$

$$h\_{\mathbb{B}} < 6.2(\Delta D\_{\mathbb{B}}) \qquad \text{for single-layer Cabibpod}^{\otimes} \text{ arms} \tag{9}$$

**Figure 6.** Linearized equivalent dimensionless armor damage *Se*1/5 as a function of the dimensionless water depth.

Figure 7 shows an example of the incident wave height (*H*) measured at the model area as a function of the incident wave height measured at the wave generation zone, corresponding to a water depth *hs*(cm) = 30 and double-layer Cubipod® armored model without toe berm.

**Figure 7.** *H* measured at the model area as a function of *H* measured at the wave generation zone (*m* = 0).

The highest wave heights measured in the model area were approximately 60% of the water depth, *H*max/*hs*≈0.60 with a horizontal seafloor (*m* = 0). For gentle bottom slopes (*m* > 0), a criterion similar to that proposed by Goda (2010) [21] and Herrera et al. (2017) [5] is reasonable. The design water depth (*h*) is calculated at a distance three times the water depth at the toe of the structure (*hs*), *h* = *hs*(1 + 3m). This approximation is reasonable for gentle bottom slopes (*m* ≤ 0.02); for steep bottom slopes (e.g., *m* = 0.10), the toe berm design is critical, and armor layer and toe berm must be studied simultaneously (see Herrera and Medina 2015 [22] or Herrera et al. 2016 [23]). Small-scale models are highly recommended when breakwaters are placed on steep sea bottoms in depth-limited breaking wave conditions.

Once the design water depth is calculated, *h* = *hs*(1 + 3m), the size of Cubipod® units can be estimated to design safe Cubipod® armored breakwaters (cotα = 1.5 and perpendicular wave attack) for any deep water wave climate, regardless of the wave height at the toe of the structure, considering the following equations:

$$D\_{\rm ll} > h/7.0\Delta = h\_5(1 + 3\text{m})/(7.0\Delta) \qquad \text{for double-layer Cabipod}^{\text{ $\rm O}$  armors} \tag{10}$$

$$D\_n > h/6.2\Delta = h\_i(1+3\text{m})/(6.2\Delta)\qquad\text{for single-layer Cabibpod}^{\otimes}\text{ arms}\tag{11}$$

#### **5. Summary and Conclusions**

The armors of conventional mound breakwaters are usually pre-designed using simple empirical equations such as the Hudson formula, based on results from small-scale physical tests with Froude similarity. Most of the tests used to characterize the hydraulic stability of different armor units are conducted in non-overtopping and non-breaking conditions. However, most of the world's breakwaters are built to withstand design storms having some percentage of large waves breaking before reaching the structure and in depth-limited breaking wave conditions. Nevertheless, there are few studies in the literature, and often with contradictory recommendations, to design breakwaters in breaking wave conditions based on formulas validated with experiments in non-breaking wave conditions.

A specific experimental methodology is described to analyze the hydraulic stability of Cubipod® armored breakwaters on horizontal seafloors in depth-limited breaking wave and non-overtopping conditions. This methodology ensures that the main armor is attacked by runs of 100 regular waves with the most damaging combination of wave height (*H*) and period (*T*) for the given water depth (*hs*). In wave breaking conditions, the maximum wave height attacking the breakwater (*Hmax*) depends on the water depth at the toe (*hs*), the wave period (*T*) and the bottom slope (*m* = 0 in this study). This research verifies that, for a given water depth, any irregular wave train caused less armor damage compared to that caused by the series of regular waves corresponding to the experimental methodology used in this study. The worst test conditions were obtained generating regular waves for a range of wave height and periods, which can be observed in nature; for each water depth (*hs*) and wave period (*T*), a series of runs of 100 waves with increasing wave heights (*H*) was generated until waves broke on the foreshore before reaching the structure. The accumulated equivalent dimensionless armor damage (*Se*) was measured to obtain the maximum armor damage associated with each water depth (*hs*) and multiple combinations of wave height (*H*) and wave period (*T*).

Considering that IDa corresponds to an equivalent dimensionless damage *Se* = 1.6 and a safety factor SF(IDe) = 1.15 and 1.30 for double- and single-layer Cubipod® armors, respectively, the water depths at the toe of the structure which guarantee that both double- and single-layer armors will not exceed IDa are *hs* < 7.0(Δ*Dn*) and *hs* < 6.2(Δ*Dn*), respectively. Armor unit size can be estimated with Equations (10) and (11) to design safe Cubipod® armored breakwaters on horizontal seafloors with cotα = 1.5 for any deep-water wave climate. Taking a design water depth (*h*) at a distance three times the water depth at the toe of the structure, *h* = *hs*(1 + 3m), Equations (10) and (11) can be safely be used to design Cubipod® armored breakwaters on gentle bottom slopes (0 ≤ *m* ≤ 0.02) regardless of the wave height at the toe of the structure and deep-water wave climate.

**Author Contributions:** M.E.G.-M. wrote the original draft and was responsible for the conceptualization, experimental methodology, and analysis. M.P.H., J.A.G.-E. and J.R.M. supervise the investigation, review and approved the manuscript.

**Funding:** This research was funded by Conselleria d'Educació, Investigació, Cultura i Esport (Generalitat Valenciana) under grant GV/2017/031.

**Acknowledgments:** The authors thank data provided by SATO-OHL Group (Cubipod Project 2007–2009). The authors thank Debra Westall for revising the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Wave Impact Pressures on Stepped Revetments**

#### **Nils B. Kerpen \*, Talia Schoonees and Torsten Schlurmann**

Ludwig-Franzius-Institute for Hydraulic, Estuarine and Coastal Engineering, Leibniz University Hannover, 30167 Hannover, Germany<EMAIL_ADDRESS>(Ta.S<EMAIL_ADDRESS>(To.S.)

**\*** Correspondence<EMAIL_ADDRESS>Tel.: +49-511-762-3740

Received: 19 October 2018; Accepted: 5 December 2018; Published: 13 December 2018

**Abstract:** The wave impacts on horizontal and vertical step fronts of stepped revetments is investigated by means of hydraulic model tests conducted with wave spectra in a wave flume. Wave impacts on revetments with relative step heights of 0.3 < *Hm*0/*Sh* < 3.5 and a constant slope of 1:2 are analyzed with respect to (1) the probability distribution of the impacts, (2) the time evolution of impacts including a classification of load cases, and (3) a special distribution of the position of the maximum impact. The validity of the approved log-normal probability distribution for the largest wave impacts is experimentally verified for stepped revetments. The wave impact properties for stepped revetments are compared with those of vertical seawalls, showing that their impact rising times are within the same range. The impact duration for stepped revetments is shorter and decreases with increasing step height. Maximum horizontal wave impact loads are about two times larger than the corresponding maximum vertical wave impact loads. Horizontal and vertical impact loads increase with a decreasing step height. Data are compared with findings from literature for stepped revetments and vertical walls. A prediction formula is provided to calculate the maximum horizontal wave impact at stepped revetments along its vertical axis.

**Keywords:** Stepped revetment; wave impact; physical model test

#### **1. Introduction**

The increasing population living in coastal areas poses new demands in terms of the environmental and touristic compatibility of coastal protection structures. As aesthetically pleasing coastal protection structures gain increasing importance, and accessible revetments, e.g., stepped revetments become more attractive. Recent installations can be found in Margate, UK (finished 2013), Blackpool, UK (finished 2017), or Chicago shoreline project, US (finished 2018). Furthermore, the stepped surface of a stepped revetment induces additional turbulence in the flow, which leads to increased energy dissipation compared to smooth impermeable structures, however, traditional permeable structures provide more dissipation due to filtration and percolation effects [1,2]. Consequently, wave energy available as kinetic energy for the wave run-up process is reduced. Presently however, practical design guidance is limited for stepped revetments. A comprehensive overview of existing research focusing on the general wave-interaction with stepped revetments is presented in Reference [3]. References [2,4] highlight the reduction capabilities of stepped revetments for wave run-up and overtopping compared to smooth impermeable structures for regular and irregular waves. Reference [5] discusses the energy dissipation within the wave run-up at stepped revetments and detected similarities to steady flow conditions over stepped surfaces like spillways. Thus far, wave impacts on stepped revetments have not been analyzed systematically and are discussed in this study. The paper is structured as follows: First, the current understanding of wave impacts on stepped revetments is presented in Section 2 and key knowledge gaps outlined. Section 3 describes the geometrical and hydraulic boundary conditions of the conducted model tests. The analysis and discussion of the results for wave impact pressures with special focus on (1) the probability distribution,

(2) the time evolution including load case definitions, and (3) the specific evolution of wave-induced pressure is given in Section 4. Section 5 focuses on the underlying laboratory and scale effects in the physical model. Finally, the results are summed-up and contrasted in context of the present state-of-the-art and the advancement of knowledge outlined through this new study.

#### **2. Previous Studies on Wave Impacts on Stepped Revetments**

Knowledge of loads on a coastal protection system is essential to determine its final design and performance over its lifetime. Hydraulic loads can be classified as either hydrostatic or hydrodynamic. Only the hydrodynamic load cases are addressed in this paper.

A general classification of breaking wave loads on vertical structures for different breaker types is given by Reference [6]. Dynamic impact loads most often occur due to plunging waves impacts on a marine structure. The progress of a single wave impact is dependent on the hydraulicand geometry-related boundary conditions (described in this section). A stepped revetment has horizontally and vertically aligned fronts in relation to the direction of gravity. The loads on these two fronts are different due to the asymmetry and phase-shift of orbital velocities in shallow waters and the asymmetric structure of the wave-form represented by the wave steepness (wave height *H* « wave length *L*). Furthermore, the effect of wave loads is dependent on the overall slope (*n*) of a structure. A unique loading case is represented by a plain vertical wall without horizontal planes where an overturning, i.e., breaking wave crest violently slams against the wall without any damping or dissipation due to preceding waves. The other extreme case is a very gentle slope (nearly horizontal) where plunging and spilling breaking waves run-up on an inclined, often impermeable plane (dependent on the Iribarren number ξ = tan α/ √*H*/*L*). In the case of spilling breaking waves the exposed load on the structure is regularly damped by previous waves. On a structure with horizontal and vertical fronts, tongues, and jets of water can be created from the breaking wave and be directed upwards like an up-rushing jet of water [1]. This jet of water instantaneously collapses and falls downwards, and in turn, induces an additional impact on the horizontal fronts. It is evident that this loading on a horizontally aligned front (tread of the step) is smaller than on a vertical front (riser of the step).

Recent literature provides only little insight on wave impacts on stepped revetments. According to Reference [7], the forces on stepped structures should be calculated for design purposes with the same method as for vertical walls since the dynamic pressures are in the same range. Reference [8] conducted hydraulic model tests with irregular waves for a stepped seawall in a 76 m long wave flume following a Froude scale of 1:19. These tests focused on measuring the reduction of wave overtopping and the wave-induced impacts on stepped seawalls. Two sloping structures (*n* = 1.5, *n* = 2) with step heights of *Sh* = 0.026 m were analyzed. A recurved seawall was incorporated at the crest of both sloping structures. Reference [8] analyzed the wave loads on stepped structures in a specific range of Iribarren numbers of 2.8 < ξ < 6.3 and step ratios in a range of 9.0 < *Hm*0/*Sh* < 11.0 with *Hm*0 as zeroth moment wave height. Similarly, Reference [8] remarks the importance of the short duration shock pressures (impacts) resulting from the rapid compression of an air pocket trapped between the front of a breaking wave and the wall. For vertical walls the authors in Reference [6] note that 'the shock pressure exerted by a breaking wave is due to the violent simultaneous retardation of a certain limited mass of water that is brought to rest by the action of a thin cushion of air, which in the process becomes compressed by the advancing wave front' [9]. The position of the highest measured impacts was dependent on the initial still water level (SWL). According to the analysis of impact distributions, the maximum impacts at different wall elevations rarely occur simultaneously. This finding is particularly valid in the case of a non-vertical wall, such as the stepped wall studied here, since some wave energy is dissipated through turbulence. In some cases, a negative impact duration was measured, which is interpreted as a characteristic of turbulence and air entrainment occurring at the base of each seawall step. Finally, Reference [8] summarizes a discussion about the importance of shock pressures for the actual design of a stepped seawall. According to the discussion, pressures of such short duration should not be

used for establishing the design load case. Rather, it is recommended to consider the smaller surge pressures with a longer duration to determine the critical dynamic load.

In addition, Reference [10] conducted hydraulic model tests in a Froude scale of 1:20 for stepped revetments (*Sh* = 0.015 m), focusing on wave run-up and wave overtopping. The vertical wave impact was measured on a single step with a sampling rate of only 100 Hz, which is considered as rather inappropriate for measuring rapid, i.e., almost instantaneous, wave impact pressures. As such, analyzed data represent merely averaged maximum impact pressures *Pmax* of six test repetitions (with a standard deviation of *STD*~0.1 *Pmax*) to depict the inherent loading bias within each experiment.

Both studies lack a comprehensive discussion on the instantaneous pressure impact events with corresponding wave conditions. Hence, the systematic analysis of the wave impacts on stepped revetments conducted in the experiments in the present study includes a comparison with the data and evaluations provided by References [8,10].

#### **3. Experimental Set-Up, Test Conditions and Procedures**

Hydraulic model tests focusing on the wave interaction with stepped revetments were conducted in a wave flume, which has a length of 110 m, a width of 2.2 m and an overall depth of 2.0 m. For these tests, wave spectra were calculated with second order wave theory routines. The irregular wave profiles (*H*1/3*,max* = 0.42 m with *Tmax* = 2.0 s) were generated with a piston-type wave maker. Two model set-ups, constructed from plywood, with varying step heights (large steps: *Sh* = 0.3 m, small steps: *Sh* = 0.05 m) are placed in a 0.7 m wide sub section over a horizontal flume bottom at a distance of *LF* = 81.6 m from the wave paddle (Figure 1). The relative flume length with respect to the tested wave length *Lp* is 10 < *LF*/*Lp* < 36.

**Figure 1.** Model set-up of pressure sensors and position of the *SWL* for two 1:2 inclined stepped revetments. (**a**) Cross-section of the flume set-up, (**b**) detail of the large step configuration (*Sh* = 0.3 m), and (**c**) detail of the small step configuration (*Sh* = 0.05 m).

The surface elevation is measured by five ultrasonic sensors with a measuring range of 200 to 1200 mm, a superior resolution of 0.36 mm and a sampling rate *fsample.* = 50 Hz. Three of the sensors are positioned at a distance larger than two wave lengths *L* from the toe of the revetment, to determine incident wave conditions, as calculated by a reflection analysis. One sensor is placed at the toe of the stepped revetment and another in the shallow water region of the still water level. Pressure impacts on the stepped revetment are recorded by seven pressure transducers, which are placed along the horizontally (*fsample.* = 2.4 kHz) and vertically (*fsample.* = 19.2 kHz) orientated step fronts (Figure 1). An impression of the set-ups is given in Figure 2 for the analyzed step heights of 0.05 m (a) and 0.3 m (b). In order to capture a profile of wave induced loads on an 1:2 inclined stepped revetment, sensor locations are varied in relative water depths —6.0 < *z*/*Hm*0 < 2.0, relative to the still water level. The pressure sensors (ATM.1ST/N fabricated by sts-sensors) have a range from zero to 150 mbar and a non-conformity of ±0.1% from full scale. The sensors are connected by a serial interface connection (RS232) to the data acquisition and provide an output signal from zero to 10 V. The configuration of the probes allows a local (over a single step) and a more global interpretation (for the whole revetment).

**Figure 2.** Model set-up with instrumentation for (**a**) step height 0.05 m and (**b**) step height 0.3 m.

The hydraulic boundary conditions of the wave impact experiments are listed in Table 1. The parameter choice covers a wide range of dimensionless variables with a wave steepness of 0.015 < *Hm*0/*Lp* < 0.04 (with *Hm*0: spectral wave height, *Lp*: wave length calculated from the peak period *Tp* measured at the gauge array in a distance of *x* = 8.6 m (1 *Lp* < *x* < 3.8 *Lp*) from the toe of the revetment) and Iribarren numbers of 2.5 < ξ < 4.9. Three different water levels *hs* with intermediate water depths (0.13 < *hs*/*Lp* < 0.49) are tested. Additionally, the corresponding freeboard height *Rc* and the number of waves *N* in each test are given. A total of 13 tests are conducted for steps with a height of *Sh* = 0.05 m (1.1 < *Hm*0/*Sh* < 2.8) and 10 tests with a step height of *Sh* = 0.3 m (0.2 < *Hm*0/*Sh* < 0.6).

**Table 1.** Hydraulic boundary conditions for the wave impact tests.

Raw data from the pressure sensors are offset-corrected by means of the first five seconds of the data. The time and amplitude of the peaks are calculated for minimum peak heights of 10 mbar and a minimum distance between two peaks of 0.8 times the wave period *Tp*.

Every single wave in a wave spectrum causes an individual impact *p* on the structure. The magnitude of the single impact depends on its individual wave kinematics and the influence of the previous wave (remaining water layer over a pressure sensor and amount of aeration in the wave). The analysis of wave loads includes a number of parameters such as the maximum impact pressure *Pmax*. The quantity of a finite number *N* of waves that cause *N* individual impacts *p* within a single test is defined as *P*. If the quantity *P* is sorted in a descending order, the maximum recorded impact *Pmax* is defined as max{*P*} or *P*(*i*=1), as given in Equations (1) and (2).

$$P = \{p\_1, p\_2, p\_3, \dots, p\_N\} \quad \text{with} \;:\; p\_1 > p\_2 >, p\_3 > \dots > p\_N \tag{1}$$

$$P\_{\max} = \max\{P\} = P\_{(i=1)} = p\_1 \tag{2}$$

While the maximum induced impact *Pmax* is very important for the design of a structure, it presents significant scatter when formulating predictions for practical design purposes. Therefore, the collected pressure magnitudes will rather be described with a probability of exceedance (e.g., 2% of all incident waves reveals the probability of exceedance impact pressure *P*98%, Equation (3)), which offers more representative and reliable predictions.

$$P\_{98\%} = \frac{\sum\_{i=1}^{n} p\_i}{n} \text{ with: } n = N(1 - 0.98) \in \mathbb{N} \tag{3}$$

In order to compare the measured data to data from previous investigations (e.g., Reference [11]), the impact magnitudes exceeded by the four highest waves out of a 1000-wave test should be calculated. Hence, *P*99.6% is calculated on the basis of the number of waves (*N*) given in Table 1. The impact with a certain probability of exceedance (*P*99.6% in this case) is calculated finally by the mean of the *n* highest impact events according to Equation (4).

$$P\_{\\$9,6\%} = \frac{\sum\_{i=1}^{n} p\_i}{n} \text{ with: } n = N(1 - 0.996) \in \mathbb{N} \tag{4}$$

#### **4. Results and Discussion**

The general differences in the wave impact and wave run-up for a plain slope and stepped revetments with two different step ratios are given in Figure 3 for a better comparison and understanding of the results presented in this section. The color-scheme of the initial black and white images recorded with a frame rate of 50 Hz is inverted for the sake of visual clarity. Small tracers (tiny black dots in the images) that follow the main flow field induced by the up-rushing wave are visible on each image. Intense dark areas represent a high amount of aeration due to wave breaking and wave-structure interaction, as such, a corresponding high flow velocity. Figure 3 shows the wave-induced processes on a 1:2 sloped revetment with smooth impermeable surface in chronological order, i.e., the wave impact (a), the wave run-up (b,c), and finally, the wave run-down (d) of a breaking wave. The time step *t* of each frame is given in relation to the wave period *T*. In analogy, Figure 3e–h shows the wave impact and wave run-up of a wave with the same boundary conditions over a revetment with small steps (*H*/*Sh* = 1.7) and Figure 3i–l over a revetment with large steps (*H*/*Sh* = 0.3). For the two stepped configurations, the still water level *SLW* is at the level of a step edge.

**Figure 3.** Demonstration of flow processes for relative time-steps in the run-up process on revetments with smooth surface (**a**–**d**), small step heights (**e**–**h**), and large step heights (**i**–**l**).

On a smooth plain slope, a breaking wave induces air in the water body. The wave run-up (b) contains water and air. At the maximum point of wave run-up excursion (c) the entire fraction of air is leaked from the water body, and in consequence, the wave run-down (d) contains only water. During the wave run-down the preceding wave then interacts with the next incoming wave and the described run-up and aeration process is repeated. For revetments with small steps (*H*/*Sh* = 1.7), the general flow processes of wave breaking, wave run-up and wave run down sequence is almost in analogy with the processes on a smooth slope. The main differences are: (1) The impact of the breaking wave causes a splash-up at the vertical step front (f) during the wave run-up process, (2) the aeration during the wave run-up is more intense during the run-up process caused by larger friction induced by the steps while interacting with the incoming waves, and (3) higher turbulence and continuous air intrusion (g). Yet, in contrast, the wave run-down induces air into the water body and is retarded compared to the smooth revetment (h). For large step heights (*H*/*Sh* = 0.3), the flow processes differ significantly from a normal wave run-up as shown in (a–h). The incident breaking wave interacts with the reflected backflow of a previous wave at the first step edge (i). The interaction leads to an impact on this first step edge, an upwards-directed flow and a wave breaking in very shallow waters over the step edge (j). Then, the wave is propagating as spilling breaker over the horizontal step front (k) until it impacts and resonates with the second vertical step front (l).

The previous described processes during the wave run-up over a plain and stepped revetment resemble the major dissimilarities, namely (1) the higher volume of aeration, (2) the stronger interaction with and the influence of previous waves, and (3) the higher breaking wave impact characteristic (impact direct on the slope/step front, impact on a water cushion induced by the previous wave). In summary, these previously outlined effects feature the structure and functioning of stepped revetment by means of reflecting and dissipating incoming wave energy on a sloping beach leading to less wave run-up and, in consequence, to lower wave overtopping rates. In order to quantify the influence of the formerly described processes, the induced wave impact pressures on stepped revetments are analyzed. This examination focuses on (1) the probability distribution of impact pressures, (2) the temporal evolution and typical load cases of impact pressures, and (3) the spatial distribution of impact pressures. Attention is drawn to these three foci in reference to the step ratio. Recent literature on wave impacts on impermeable plain slopes, stepped revetments and vertical seawalls are presented to complement the findings of this research and ensure a comprehensive analysis of processes.

#### *4.1. Probability Distribution of Impact Pressures*

According to References [12,13], the probability distributions of wave impact pressures can be described in agreement with mathematical fitting approaches based log-normal functions. Additionally, it has been argued that these log-normal functions are valid in model scale and prototype.

Hence, Figure 4 gives the log-normal probability distribution over the maximum impact pressure for two exemplary tests (test number 103 and 209 according to Table 1). The tests were conducted with a spectral wave height of *Hm*0 = 0.084 m and *Hm*0 = 0.089 m (for test 103 and 209, respectively) and a corresponding Iribarren number of ξ = 2.9 (2.8 for test 209). Figure 4a provides measurements of wave impacts on slopes with a small step height (*Hm*0/*Sh* = 1.68, *hs* = 1.1 m, sensor *P1*). A total of 876 individual wave impacts have been recorded with pressure sensor *P1* with a maximum recorded pressure impact of *Pmax* = 1.26 kPa. The wave impacts are presented with a certain probability of exceedance. The figure presents an idealized normal-distribution, shown as a dashed line, as reference. As is evident from the presented figure, the wave impacts on this stepped revetment follow a log-normal distribution. Over and under predictions from the idealized normal-distribution given with the dashed line appear for impacts larger than *P*95% and smaller than *P*10%.

**Figure 4.** Recorded pressure impacts and log-normal probability distributions of the maximum impact pressures for (**a**) small step heights (test 103, pressure sensor *P1*, *Hm*0/*Sh* = 1.7) and (**b**) large step heights (test 209, pressure sensor *P2*, *Hm*0/*Sh =* 0.3). (**c**) Gives a detail of the log-normal probability distribution of the 18 largest impacts in test 209.

Figure 4b presents the log-normal distribution of wave impact pressures on a revetment with large steps (*Hm*0/*Sh* = 0.3, *hs* = 0.921 m, sensor *P2*) for the same hydraulic boundary conditions, as in Figure 4a. The maximum pressure impact *Pmax* of 1114 individual impacts is 1.68 kPa, which is about 35% larger when compared to the small steps. As for Figure 3c, slight deviations from the ideal log-normal distribution are detected for impacts smaller than *P*10%, while impacts larger than *P*90% deviate significantly from this trend. These deviations indicate that the influence of the wave-interaction with large steps differs for high and low impact scenarios. The apparent difference in the wave breaking and wave run-up at these large steps compared to slopes with and without smaller steps (Figure 3) is a significant transformation of the incident wave which breaks over the very shallow horizontal step front. Hence, the log-normal distribution of the induced pressures of incident wave heights is disturbed and transformed over large steps directly at the structure, and in analogy, the wave impact distribution is affected. The increase in the wave impacts for large steps can be explained by the fact that the vertical step fronts fall completely dry during the run-down process, whereas for small

step heights, there is always a thin water flow remaining from the previous run-down. This thin water layer (often aerated) reduces the wave impact of the next breaking wave.

In parallel, the geometry of a wave front has a significant influence on its resulting wave impact. Generally, the wave height plays a dominant role. It is observed, that small wave heights in the spectrum lead to pulsating extreme loads, whereas the largest waves lead to impacting loads. Distribution of wave impacts deviate strongly from the log-normal distribution for less frequent waves indicating occasional extreme loads on the structure, which is most relevant for practical design attempts. Reference [14] indicated that for vertical walls impacts smaller than 0.4 *Pmax* are based on pulsating loads and larger impacts on impacting loads. Reference [15] found that often only a part of all impacts show a good agreement with the log-normal distribution according to the stochastic pattern of the pressures in the impact area. This observation is confirmed for the present experimental configuration. In contrast, when re-considering only the highest impacts during one wave impact event on the area, it turns out that measured data points follow an almost linear relation in a log-normal plot. Hence, Figure 4c gives a detailed log-normal probability distribution of only the 18 largest impacts (>*P*98%) of test 209, confirming evaluations stemming from Reference [15] to be also valid for stepped revetments.

It can be seen from the left-hand side of this diagram that only a part of them show good agreement with the log-normal function according to the stochastic pattern of the pressures in the impact area. However, if the highest pressure on the area during one wave impact event is selected and plotted in log-normal scale the measured points nearly fall on a straight line which verifies the log-normal distribution.

#### *4.2. Temporal Evolution and Load Cases of Impact Pressures*

The temporal evolution of wave impact pressures on vertical structures was initially categorized by Reference [6] and analyzed in depth by Reference [16]. According to Reference [16], impacting loads on vertical structures are defined as impact peaks that are minimum 2.5 times larger than the subsequent quasi-static peak *Pq*. Partially breaking waves induce peaks in a range of 1.0 < *Pmax*/*Pq* < 2.5. Pulsating impacts due to standing waves occur when the impact peak and its subsequent quasi-static peak *Pq* are in the same order of magnitude. Figure 5 sketches a parameter definition describing both a pressure impact event (a), and an exemplary time series of the maximum impact event during test 204 for horizontal oriented pressure sensors *P1*, *P2,* and *P3* (b). A typical time evolution with an impact and subsequent quasi-static peak is observed. The highest impact is recorded by sensor *P2*, which is located near the still water level *SWL*. The oscillation of the signal and negative pressures, as also observed by Reference [8], are interpreted as characteristically mimicking physical processes of turbulence and air entrainment occurring at the base of each revetment step. With increasing distance to the *SWL,* the impact amplitude decreases (*P1* and *P3*). While sensor *P2* recorded a violently impacting load, *P1* was impacted by a slightly breaking wave and *P3* experienced a pulsating impact.

**Figure 5.** (**a**) Parameter definition describing a pressure impact event and (**b**) exemplary time series of the maximum impact event during test 204 for pressure sensors *P1*, *P2* and *P3.*

Figure 6 presents the normalized time evolution of impact events with a certain probability of exceedance for a 1:2 stepped slope with large (test 103) and small step heights (test 209). These tests were conducted with a similar spectral wave height and wave steepness (Table 1). Raw data (black line) are given to indicate the largest impact. A filtered time series (red line, 6th order low pass filter with 38.4 Hz cut-off frequency) is superimposed in order to highlight a clearer temporal progress. Subfigures (a–f) correspond to step ratios of *Hm*0/*Sh* = 1.7 (small steps) and subfigures (g–l) to step ratios of *Hm*0/*Sh* = 0.3 (large steps). Each subfigure displays the measured impact event of a certain probability of exceedance (*Pmax* to *P*10%) over the relative time. The time *t* is normalized by the wave period *Tm–*1,0. Time step *t/T* = 0 is set at the time of maximum impact. Each peak impact (based on the raw data) is given proportionally to the maximum measured peak *Pmax* of the entire test duration.

**Figure 6.** Comparison of pressure events with a certain probability of exceedance between small steps ((**a**–**f**), *Hm*0/*Sh* = 1.7, test 103) and large steps ((**g**–**l**), *Hm*0/*Sh* = 0.3, test 209).

The maximum measured impact for the small steps (a) was 1.26 kPa, which is about 30% lower than the maximum impact for large steps (g). For the large steps the maximum peak amplitude decays more rapidly with increasing probability of exceedance than for the small steps. In the case with smaller steps, a peak amplitude of about 35% of the maximum impact *Pmax* is reached by 5% of all impact events (*P*95%) (d), whereas in the case of the large steps (h), it is only 0.4% (*P*99.6%) of all impact events.

A comparison of the occurring load cases defined by Reference [16] elucidates that large steps (*Hm*0 < *Sh*) demonstrate a behavioral function like vertical walls in reflecting incoming wave energy. Functional processes are mimicked in two distinct phases: A clear initial impact followed by a quasi-static peak *Pq*. For small steps (*Hm*0 > *Sh*) the impact peak is also clearly visible, but the subsequent quasi-static peak *Pq* is not as prominent as for large steps or vertical walls. These differences are caused due to different flow principles. At large steps, the water level in front of the step front is constantly rising after the initial impact, thus inducing the quasi-static load. At small steps, which dissipate energy from the up-rushing wave tongue, a highly aerated flow emerges after the initial impact (Figure 3f,g) and the absolute depth under the wave crest is evidently lower. This leads in general to smaller quasi-static peaks. For the specific case presented in Figure 6, impacting loads are observed in the range of about 5% of all impacts at the small steps and 2% of all impacts at the large steps.  As a result, it can thus be deduced that the overall energy dissipation at small step heights is larger, i.e.,

more effective. Furthermore, the impact-mitigating effect due to the practically permanent existence of a thin water layer on top of the individual small steps of the revetment during the wave run-down phase stemming from the preceding wave (Figure 3) plays a dominant role in impact reduction.

The characteristics of an impact pressure on stepped revetments can be described by the rising time *tr* and impact pressure duration *td*, as defined in Figure 7 and depicts the correlation between the dimensionless impact duration *td*/*tr* and the dimensionless impact rising time *tr*/*Tm*–1,0. The data are grouped in terms of the step ratios *Hm*0/*Sh*, the direction of the measured wave impact (horizontal or vertical with respect to gravity) and the position of the impact (above or below the still water level *SWL*). As a meaningful reference, Figure 7 presents the empirical projections for rising times and durations of impacts at vertical walls provided by Reference [16]. Impacting load cases (*Pmax* > 2.5 *Pq*) are characterized by a short relative rising time (0.02 < *tr*/*Tm–*10 < 0.2), while pulsating loads are distinguished by their longer peak durations (*tr*/*Tm–*10 > 0.2). The dominance of a peak is characterized by the relative impact duration *td*/*tr*. The highest impact loads (e.g., *P*max or *P*99.6%) correspond to short relative impact durations *td*/*tr* in combination with a short relative rising time *tr*/*Tm–*10. The larger the relative impact duration *td*/*tr*, the more critical the impact gets in terms of causing damage or instability of the revetment.

**Figure 7.** Correlation of the dimensionless impact duration *td*/*tr* and the dimensionless impact rising time *tr*/*Tm–*1,0 at stepped revetments compared to findings for vertical walls. Given trend lines following Equation (5) represent upper envelopes of the findings. Straight horizontal lines represent the corresponding lower envelope line.

Impacts measured on stepped revetments below the *SWL* (green and red markers) show relative rising times of *tr*/*Tm–*10 > 0.2. Horizontal impacts (filled marker) or vertical impacts (empty marker) have comparable impact rising times and impact durations. Below the *SWL*, no impacting load case was detected for either small or large steps. The recorded loads have a pure hydrostatic nature induced due to the water level changes induced by wave run-up and run-down over the pressure sensors. Stronger impacts are mitigated by means of a sufficient water layer ("cushion") effectively sheltering the submerged steps' fronts from more violent impacts. The stepped shape of the slope retards the wave run-down leading to a permanent water cover of the revetment below the *SWL*.

The relative impact rising time of horizontal impacts above the *SWL* (filled blue squares for large steps 0.2 < *Hm*0/*Sh* < 0.6 and filled black diamonds for small steps 1.0 < *Hm*0/*Sh* < 3.5) ranges from very short impacts (*tr*/*Tm–*10 < 0.05) up to long load cases (*tr*/*Tm–*10 > 0.3). The latter ones represent a full run-up and run-down phase inducing pulsating loads. While the impacts at small steps show an increase in the peak duration for decreasing relative peak rising times up to *td* = 4*tr*, the maximum duration of *td* = 2*tr* is observed for large steps. This finding can be explained with the fact that impacts at large steps are often caused by breaking waves that directly hit the step front, whereas for small steps, the step fronts are covered by thin water layers. Rising times and peak durations of vertical impacts above *SWL* (empty blue squares for large steps 0.2 < *Hm*0/*Sh* < 0.6) and empty black diamonds for small steps 1.0 < *Hm*0/*Sh* < 3.5) scatter significantly and do not follow a clear, yet visual trend. The formation and progression of an aerated water layer over the horizontal step front influences the recorded impact significantly and explains the scattering in the data. Pulsating loads can be explained by the water layer thickness during the wave run-up and run-down. Impacting conditions occur only very close to the *SWL*. The impact rising times for vertical walls and stepped revetments are in the same range. Reference [16] found a minimum peak duration of *td* = 2*tr* for vertical walls, whereas the minima for stepped revetments is *td* = *tr*. These differences may be due to the differences in the sampling frequency of the impact pressure sensors (0.6 to 1.0 kHz at [16] and 19.2 kHz in the present study). Lines of best fit for the different load cases are calculated according to Equation (5) with corresponding regression coefficients *a*, *b* and *c* given in Table 2.

$$t\_{d,max} = t\_r \cdot \left[ a + b \cdot \exp\left( c \cdot \frac{t\_r}{T\_{m-1,0}} \right) \right] \tag{5}$$

**Table 2.** Coefficients *a*, *b* and *c* for Equation (5) and corresponding minimal impact duration *td,min*.

#### *4.3. Spatial Distribution of Impact Pressures*

For sloping structures the maximum wave impact resulting from depth-induced wave breaking occurs slightly below the *SWL* [12]. This section presents the maximum wave impacts on stepped revetments at different locations relative to the *SWL*. Figure 8 gives the horizontal (a) and vertical (b) relative maximum pressure impact distribution over large (dashed, Test 209) and small (solid, Test 103) stepped revetments. For each pressure sensor *Pj*, the maximum impact *Pj,max* is normalized by the maximum pressure impact of all compared sensors (*Pmax* = max{*Pj,max*}). For this case, the absolute maximum was recorded by sensor *P2* of the revetment with the small steps (compare Figure 1).

**Figure 8.** Horizontal (**a**) and vertical (**b**) relative maximum pressure impact distribution over large (dashed, Test 209, *Hm*0 = 0.089 m, and ξ = 2.8) and small (solid, Test 103, *Hm*0 = 0.084 m, and ξ = 2.9) stepped revetments.

The vertical distribution of horizontal wave impacts (Figure 8a) shows that the maximum wave impact for all configurations occurs close to the *SWL*. The impact loads decrease with increasing distance to the *SWL* (|*z*| > 0). Below the *SWL,* the pressures are reduced by the presence of the water body, which mitigate the impact of a breaking wave. Above the *SWL* the energy of the breaking wave is less due to the decrease of the potential energy (reduced drop height). Impacts above the SLW are often caused by the secondary impact of the breaking wave during the wave run-up or due to splashes induced by the primary wave impact close the *SWL*. For Test 209 the *SWL* is directly located at the edge of the large step (*Hm*0/*Sh* = 0.3), while the maximum horizontal impact is about four times lower than at the small steps (Test 103, *Hm*0/*Sh* = 1.7). Additionally, data from Reference [8] are shown. These data represent very small step heights (*Hm*0/*Sh* = 11) and indicate a maximum horizontal wave impact of about 1.5 times larger than for the steps with a step ratio of *Hm*0/*Sh* = 1.7. The smaller the step height in relation to the wave height, the higher the recorded horizontal loads become. The reduced impact for an increasing step height can be ascribed to the delayed run-down of the previous wave impact (compare Figure 3a,e). This run-down causes a constant water layer on the revetment, which buffers the wave impact. Moreover, the plunging wave breaking at a 1:2 slope with small steps (*Hm*0/*Sh* ≥ 1.7) triggers impacting wave loads whereas the transformation of the breaking wave by large dominant step edges (*Hm*0/*Sh* = 0.3) leads to the formation of spilling wave breaking over the horizontal step front and triggers, thereby pulsating wave impacts.

The horizontal distribution of vertical wave impacts (Figure 8b) shows that the maximum vertical wave impact for the small steps is located slightly below the *SWL*. The maximum vertical impact has an amplitude of about 50% of the maximum horizontal impact (*Pmax,vertical* = 0.5 *Pmax,horizontal*). Note that, in the context of the onshore-orientated wave propagation and wave breaking process, this finding is reasonable for design and constructional purposes in any practical applications of stepped revetments. The vertical impact for the large steps is negligible as it represents only the hydrostatic pressure induced by the overflow of the incident wave. Additionally, data from Reference [10] are given. The relative step height of *Hm*0/*Sh* = 7.5 is comparable to the data set provided by Reference [8] for horizontal wave impacts. The maximum presented impact is about two times larger than for the steps with a step ratio of *Hm*0/*Sh* = 1.7. An increase in the vertical impact loads is observed for decreasing step heights and is explained analogous to the horizontal wave impact.

Figure 9 provides a non-dimensional relation between the pressure impact normalized by water density ρ, gravity *g,* and spectral wave height to *P*99.6%/(ρ*gHm*0) on the abscissa and the relative position to the *SWL* (*z*/*Hm*0) on the ordinate. The probability wave impact *P*99.6% is selected following the approach of Reference [11] to allow a comparison with the underlying data for wave impacts on vertical walls. For both examined step heights, the maximum pressure impact is close to the *SWL*. The maximum pressure decreases significantly within a range of ± *z*/*Hm*0 around the SWL. Within a range of ±2 *z*/*Hm*0, the normalized pressure impact becomes low. The *P*99.6% wave loads measured over the stepped revetment tend to be about 50% smaller than those measured at a vertical wall. On the contrary, the impacts on the steps around the *SWL* exhibit pressures peak impacts which are comparable to those measured on vertical walls by Reference [11] (data points of single impacts not given in this figure scatter significantly along the abscissa). A dual-sided envelope curve (Equation (6)) with coefficients *a*, *b* and *c* according to Table 3 representing the best fit of data (coefficient of determination: *R*2 = 0.62, *STD* = 0.189) is given to describe the correlation of the vertical distribution *z*/*Hm*0 of horizontal impacts 0.01 < *P*99.6%/(ρ*gHm*0) ≤ 3.6.

$$\frac{P}{\rho g H\_{m0}} = \min \left\{ \tan \left[ \frac{z/H\_{m0} + a}{b} \right] / c, 3.6 \right\} \tag{6}$$

**Figure 9.** Normalized pressure impact relative to the *SWL* (*z* = 0) for a 1:2 inclined stepped revetment.

**Table 3.** Coefficients *a*, *b,* and *c* for Equation (6).

Normalized horizontal pressure impacts with varying probability of exceedance and corresponding relative water depth *z*/*Hm*0 for revetments with large (0.2 < *Hm*0/*Sh* < 0.6, Figure 10a–d) and small (1.0 < *Hm*0/*Sh* < 3.5, Figure 10e–h) steps are shown. Each row represents data with a certain probability of exceedance, as described in Section 4.1. For comparison, the reference line for the horizontal impact forces [11] and the trend line according to Equation (6) are given. The pressure distribution predicted for *P*99.6% values by Equation (6) also shows a good agreement for *Pmax* values, if the step height is larger than the wave height (a). *Pmax* values for revetments with step heights smaller than the wave height (e) show higher impacts, which scatter in a range of ±2 *z*/*Hm*0. The higher and more variable distributed impacts for small steps compared to large steps can again be ascribed to the influence of higher aeration [8,9]. With increasing probability of exceedance ((c,d) for large steps and (g–h) for small steps) the peaks around the *SWL* become less prominent, although the peaks are still visible. This trend is reasonable as the statistical distribution of the individual wave heights in a wave spectrum follows a Rayleigh distribution and the mean wave height of all waves, exceeding a certain threshold, decreases significantly with increasing probability of exceedance. Additionally, the shape of the front of a breaking wave governs the subsequent wave impact as the wave's asymmetry has a secondary importance on the resulting impact. The overall distribution over the water depth becomes more homogeneous with the decreasing probability of exceedance for reasons discussed in Section 4.2.

It has to be clarified that Equation (6) is based on experimental data by means of mathematical fitting routines. Each data-point represents the average of the four maximum impact values recorded by all implemented pressure sensors during a test. Hence, these four impacts do not necessarily correspond to or stem from the same wave or wave train. Furthermore, Equation (6) is based on a fitting through all data points. Individual under-estimations larger than the given mean standard deviation (up to factor 3) are present for single measurements. Similar observations are stated by Reference [11]. Despite these constrains, the approach is applicable to reliably estimate the location, magnitude and trend of impact pressures on stepped revetments. Further, the formula enables an estimation of the wave-induced local impacts and loads on a stepped revetment, and thus provides

the first robust estimate in the practical design and dimensioning of adequate foundations and anchoring systems.

**Figure 10.** Normalized horizontal pressure impacts with varying probability of exceedance and corresponding relative water depth *z*/*Hm*0 for large (0.2 < *Hm*0/*Sh* < 0.6) and small (1.0 < *Hm*0/*Sh* < 3.5) stepped revetments including theoretical laws for stepped revetments according to Equation (6) and for vertical walls according to Reference [11].

#### **5. Laboratory and Scale Effects**

The tests are conducted in a 2D wave flume. The natural multi-directionality of prototype wave conditions is simplified. This affects the wave breaking and accordingly the measured impact forces. Waves are generated in intermediate water depth over a horizontal flume bottom. Hence, waves transform non-linearly during propagation onto and in interaction with the structure. Furthermore, reflected waves from the structure are re-reflected at the piston wave board as experiments were conducted without active reflection compensation. The influence of both boundary conditions on the calculated wave parameters (*Hm*0, *Tp*), minimized by an analysis of the incident waves very close to the structure.