# Do Rock Design Formulas Based on Wave Flume Experiments Reliably Model Their Performance at Sea?

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## Abstract

**:**

## 1. Introduction

_{sI}, peak wave period, T

_{p}, wave direction and persistence). The function and behavior of these maritime structures against random sequences of storms determines their deterioration and ageing and, eventually, their destruction. This knowledge is essential to estimate the total lifetime costs of the structure, including the costs of maintenance, when, how much and how, the occasional costs of repair or reconstruction. In the 21st century this way of working is applied in most branches of engineering including civil engineering. Exceptions to this working trend are few; one of them is in the design of slope breakwaters to protect harbor and coastal areas.

- Asset management, make risk-informed decisions about investments based upon understanding the conditions and consequences of failure of assets;
- Life cycle portfolio management, look at servicing the portfolio in terms of systems and the interrelationships and interoperability across those systems;
- Alternative financing, focus on investments that will achieve an outcome over time.

_{d,}under a sea state; (2) the interplay of h/L, and wave steepness H

_{sI}/L determined at the toe of the structure, on damage evolution; and (3) the dependence of damage evolution, S

_{d}, including the initiation of damage and destruction on the experimental design and technique. On the other hand, experimental data show that the slope angle is differentially involved in the damage evolution. Finally, Losada [16] showed that the Van der Meer formula does not adequately replicate the experimental damage level, especially when it grows, i.e., when it is necessary to decide whether to repair or not the structure.

_{μ}, is the main independent parameter, defining the wave steepness as the quotient between the significant wave height at the toe of the breakwater and the wavelength at deep water, L

_{0}. Lastly, he proposed a global stability parameter to facilitate the graphic application of this global parameter and the breaker parameter (ξ

_{m}). Van der Meer [17] called for caution not to apply the formula outside its area of validity. Despite these precisions and the desire to maintain a single formula (with two expressions), the elaboration of the formula relies on some hypotheses and assumptions that are not met by the original experimental data itself.

_{sI}/L (both at the toe of the structure) in the progression of damage. The conclusions of this work are set out in Section 4.

## 2. Background, Proposal, and Validation of Van der Meer Stability Formula (1988, 2021)

_{sI}/L

_{0}, for the Reynolds number is usually larger than some minimum value above which variations in its actual value do not significantly affect the resultant motion, while for waves breaking on the slope, the value of the relative depth in front of the slope is not important either”. Then, Van der Meer [15], and many others as well, assumed that only the combination $\mathrm{tan}\left(\alpha \right)/\sqrt{{H}_{sI}/{L}_{0}}$ is the sole parameter to describe the interaction and evolution of the wave train with the structure; being tan(α)$,$ the slope of the structure, H

_{sI}/L

_{0}the wave steepness and L

_{0}the wavelength in deep water, and H

_{sI}the significant wave height of the incident wave train at the toe of the slope. Since Battjes [27] this quotient is known as Iribarren parameter (Ir), or surf similarity parameter ξ

_{m}.

#### 2.1. Selection of Main Governing and Non-Dimensional Variables

_{sI}, T

_{m}, N

_{w}, h, D

_{n}

_{50}, Δ

_{s}, α), and neglected some of variables that “had no or minor influence in the considered process”. In this selection, H

_{sI}, T

_{m}, and N

_{w}are the significant wave height, the mean wave period, and the number of waves, respectively, in the experimental Test Series (TS); D

_{n}

_{50}and Δ

_{s}are the nominal diameter and the relative mass density of the armor rock, respectively; α is the slope angle, and h is the water depth at the toe of the slope.

_{s}; the second one couples the damage level, S

_{d}, and the number of waves of TS. The damage level, S

_{d}, is determined by the quotient of the cross-sectional eroded area, A

_{e}, and the square of D

_{n}

_{50}. Equation (1) also includes the so-called notional permeability factor, P, that is established for each typology structure. Depending on whether the value ξ

_{m}is less than and equal to, or greater than, a certain critical value ξ

_{mc}, the formula has two slightly different expressions, where ξ

_{mc}= f(tan(α),P). It is considered that in the domain ξ

_{m}≤ ξ

_{mc}the waves break in plunging (Equation (24) in [15] or Equation (1) in [17]) and, otherwise, the waves break in surging (Equation (25) in [15] or Equation (2) in [17]).

#### 2.2. Design, Experimental Technique and Analysed Data

_{sI}. The output of a Test Series (TS) generally consisted of two sets of data (N

_{w}= 1000 and 3000) of five pairs of values of damage level, S

_{d}, corresponding to a given h/L and five increasing values of H

_{sI}/L. Hence, for each complete test, increasing the wave height is the same as to increase ξ

_{m}, (breaker parameter), and N

_{s}.

_{n}

_{50}= 0.0360 m, relative mass densities Δ

_{s}= 1.615, 1.630, and grading, D

_{85}/D

_{15}= 1.25 and 2.25. Furthermore, five large scale (LS) runs were carried on in the Delta flume, scaled up according to Froude’s law by a linear factor 6.25. The nominal diameter of the stones was D

_{n}

_{50,LS}= 0.214 m and the grading (D

_{85}/D

_{15})

_{LS}= 1.38. The average wave period in the large scale runs was constant, T

_{m,}

_{LS}= 4.4 s corresponding to T

_{m}= 1.77 s, and h/L ≈ 0.18.

#### 2.3. The Revisited Formula by Van der Meer (2021)

_{s}, to define the so-called parameter group for stability or relative stability number, $\frac{{N}_{s}}{{({S}_{d}^{\ast})}^{0.2}}$. Next, it is assumed that it should depend on ξ

_{m}, once the slope angle α, and the notional permeability parameter, P, of the structure are chosen; that is:

_{m}, (x-axis). The Figures 1–3 of [17], show the experimental values obtained in an impermeable slope breakwater (P = 0.1), slope 1:3 and includes two spectral widths and three characteristic periods: mean, T

_{m}; peak, T

_{p}; and spectral, T

_{m-}

_{1.0}. Figures 4–6 of [17] show the graphs for impermeable core, permeable core, and homogeneous breakwater typology.

#### 2.4. Aplication and Reliability of the Van der Meer Formula

_{d,in}= 2 or 3, and the highest damage levels S

_{d,max}= 8, 12, or 17, depending, in both cases, on the slope angle. Although, based on the experimentation of Thompson and Schulter [29] the validity of the function $\sqrt{{N}_{w}}$ can be extended to the interval 1000 ≤ N

_{w}≈ 7000, the tests carried out by Van der Meer were performed with N

_{w}= 1000 and 3000 waves. Based on these data, the parameter ${S}_{d}^{\ast}$ varies in the range (see Table A1—Appendix A): 0.5 <${S}_{d}^{\ast}$ < 0.6; 0.64 < ${S}_{d}^{\ast}$ < 0.70; 0.74 < ${S}_{d}^{\ast}$ < 0.89 for the onset damage levels 1 < S

_{d}< 4; 6 < S

_{d}< 9; and 12 < S

_{d}< 30, respectively.

_{m}< 1.5 the experimental values of the relative stability number follow the proposed function for plunging breakers; for 1.5 < ξ

_{m}< ξ

_{mc}, its dispersion grows depending on h/L

_{0}and the type of spectrum; and for ξ

_{mc}< ξ

_{m}, the dispersion of the experimental data grows significantly as the breaker parameter increases and h/L

_{0}decreases. That is, the uncertainty of the formula grows in the critical design zone and keep growing with ξ

_{m}.

_{m}is used to calculate the breaker parameter, ξ

_{m}, and its critical value, ξ

_{mc}. According to Van der Meer [15], the values of the wave steepness,

_{,}and the relative water depth are calculated with L

_{0}, the wave length in deep water conditions. These points correspond to a sea state with H

_{sI}and T

_{m}, and constant N

_{w}= 3000 (see Table 1). The star-points 1 and 3 have a h/L

_{0}≈ 0.05 and star-point 5, h/L

_{0}≈ 0.15. In all cases, the water depth at the toe of the breakwater is constant h = 6 m.

_{sI}, T

_{m}, and N

_{w}; that is, with the intrinsic uncertainty. In other words, it is assumed that the probability of failure of the structure is mainly associated with the selection of the principal sea state parameters: H

_{sI}, T

_{m}and N

_{w}. Once the slope, typology and damage level are chosen, and the relative mass density of the rock (Δ

_{s}) is known, the nominal design diameter D

_{n}

_{50}is inversely proportional to the relative stability number, $\frac{{N}_{s}}{{({S}_{d}^{\ast})}^{0.2}}$. The experimental data with approximately the same values of ξ

_{m}as those of the worked example, scatter from proposed curve (Figure 1), depending on the value of the breaker parameter. The interval grows as ξ

_{m}increases for, point (3), 2.6 ≤ $\frac{{N}_{s}}{{({S}_{d}^{\ast})}^{0.2}}\le 3.1$; point (1), 2.2 ≤ $\frac{{N}_{s}}{{({S}_{d}^{\ast})}^{0.2}}\le 2.9$; and point (2), 1.70 ≤ $\frac{{N}_{s}}{{({S}_{d}^{\ast})}^{0.2}}\le 2.9$. Then, for a constant value of ${S}_{d}^{\ast}$, the experimental scatter determines (linear impact) an interval of the design value of D

_{n}

_{50}, obtained with the formula, impinging (nonlinear impact) to the investment or total costs of the maritime infrastructure.

_{n}

_{50}) and relative density (Δ

_{s}) slope angle, damage level, (S

_{d}) and number of waves (N

_{w}), the design graph proposed by Van der Meer (its Figures 1–7 in [17]) simply represents the relationship, H

_{sI}versus $1/{H}_{sI}^{c},$ each of them affected by different constants: c = 1/2 for plunging breakers and c = P for surging breakers; P = 0.1 for impermeable core. The relative water depth, h/L

_{0}calculated with the depth at the toe of the structure does not intervene in the design value of D

_{n}

_{50}. Indeed, if the water depth of the worked example is increased or decreased, say h = 8 m or h = 4 m, and all other design parameters are not changed, the formula returns the same nominal diameter as for h = 6 m in Table 1. In the next section, following Losada [16] these strange results are further analyzed.

## 3. Reanalysis of the Applicability of the Formula and Sources of Uncertainties

#### 3.1. Experimental Space, Design and Limits of Wave Validity

_{sI}/L, depend on the slope angle. The isoline of γ constant is the geometric locus of the points that reach the toe of the slope with the same relative wave height. The parameter γ does not include explicitly the wavelength. However, each point of the isoline γ has a wavelength, that is determined, unambiguously, considering the depth, h, and the wave height, H

_{sI}

_{,}at the toe of the slope. Therefore, the maximum possible value of γ in a wave flume is imposed by the wave generation system (H

_{sI}/L) and depends on the relative water depth, h/L. Indeed, for both regular and irregular waves, the maximum value of γ in a wave flume is significantly lower than the maximum values observed in nature and is considered as a nonlinear parameter.

_{r}. Notice the correspondence between Figure 2 and the one proposed by Le Méhauté [32], which shows the wave theory that best describes the profile of the wave at the toe of the slope and its limits of validity [30].

_{sI}/L

_{0}, and the relative water depth, h/L

_{0}, are not interchangeable with the values of H

_{sI}/L and h/L at the toe of the structure.

_{sI}/L and h/L at the toe of the structure. Such condition is reasonably satisfied when $\mathrm{tan}\beta \le 1/30$ and $\gamma \le 0.35$, being $\beta $ the angle of the foreshore.

_{sI}(that is constant H

_{sI}/L or N

_{s}). The shallower data are slightly above $\gamma \approx $1/12, and the maximum relative wave height at the toe of the slope is $\gamma \approx 0.25$, but most of the tests were carried out with $\gamma \approx 0.21$. Furthermore, based on [31], the types of wave breaker are strong plunging, strong bore, weak bore, and surging, with $0.0065<\chi <0.014,0.004\chi 0.0065,0.002\chi 0.004,\chi 0.002$, respectively.

#### 3.2. Progression of Damage under N_{w} Equal to 1000 and 3000 Waves

_{d}against an increasing significant wave height (or stability number, or wave steepness) under N

_{w}= 1000 and 3000 waves is shown in Figure 3a,b, respectively.

_{d}≤ 2 or 3, and N

_{w}= 1000 the curves converge, although the random performance of the rock layer at these initial stages, possibly, is affecting the experimental data. Under the incidence of N

_{w}= 3000 waves, the dependence of damage progression on h/L, in the interval 3 ≤ S

_{d}≤ 15 is notorious. For the same level of damage, for example S

_{d}≈ 17, the stability number varies in the interval 1.5 < N

_{s}< 3.0 and the relative water depth increases in the interval 0.10 < h/L < 0.30. In summary, S

_{d}= f(N

_{s}; h/L) or f(H

_{sI}; h/L).

_{d,Nw=}

_{3000}/S

_{d,Nw=}

_{1000}and the stability number. Only for large relative water depths h/L ≥ 0.18, the damage progression ratio is approximately constant. For shallower water depths, at first sight, there is no well-defined relationship between the two sets of experimental pairs of values.

_{sI}/L. This behavior was obtained provided that the persistence (i.e., the number of waves) of each test (CTs) was not limited. Then, the progression of the slope form goes through successive equilibrium profiles that modify the type of wave breaking, the reflected energy flux and the rate of energy dissipation of the incident wave.

#### 3.3. Reanalysis of the Breaker Parameter—Relative Stability Number Graph

#### 3.4. Scaling and Relative Water Depth

_{m}serves as the sole determining factor for the suitable normalized parameters of the surf” [27].

_{n}

_{50}≈ 0.036 m. These tests were usually classified as small-scale. Their application to prototype is based on the Froude similarity between model and prototype. The large-scale tests performed on the Delta flume were run with a scaling ratio for the horizontal lengths of λ ≈ 6.25. The ratio between the wavelength at the toe of the breakwater in the Delta flume (Large Scale, L

_{LS}) and in the small scale test (L

_{SS}) is L

_{LS}/L

_{SS}= 6.25. The same scale was applied for the wave height and depth at the toe of the breakwater, although the latter was not provided by Van der Meer. Consequently, the ratio of mean periods is T

_{m}

_{LS}/T

_{m}

_{SS}= $\sqrt{\lambda}$.

_{sI}= 2 m and T

_{m}= 8.3 s, the scale ratio for horizontal length and rock size is λ ≈ 1/20. Then, the model mean wave period is T

_{m}= 1.86 s; ξ

_{m}is conserved (notice that the wavelength is determined in deep water). Applying the Van der Meer formula, the prototype rock diameter is D

_{n}

_{50}= 0.93 m for S

_{d}= 2 and D

_{n}

_{50}= 0.63 m for S

_{d}= 12.

_{0}, is conserved if, and only if, the prototype water depth at the toe of the structure, h

_{p}= 6 m, as other vertical quantities, is properly scaled, λ≈ 1/20. However, most of Van der Meer’s experimentation was carried out with constant water depth at the toe of the structure, h

_{m}= 0.80 m. In other words, the model is vertically distorted for the water depth but not for the wave height. As noted by Battjes [27], a distorted wave-model with the same value of ξ

_{m}as the prototype is similar if the pressure distribution in both is hydrostatic. Possibly, except for surging breakers, weak and strong plunging breakers, and strong and weak bore breakers, do not fulfil this requirement. Thus, a stability formula that should be applied to any relative water depth should include the relative water depth at the toe of the structure h/L in the functional relationship.

#### 3.5. Number of Waves, Sea State Persistence, and Storm Evolution

_{w}, that impinges the structures depends on its persistence, t

_{d}. If the mean wave period of the sea state is T

_{m}, the average number of waves is N

_{w}= t

_{d}/T

_{m}.

_{sI}. Then each CT is representative of a sea state. Instead to specify the persistence of each CT, he fixed the number of waves for all the CTs, independently of their persistence. The output of a CT consisted of two sets of data of values of damage level, S

_{d}, for N

_{w}= 1000 and 3000, corresponding to a given T

_{m}(or h/L), and a fixed H

_{sI}(or H

_{sI}/L). After running a CT, the slope was rebuilt. Then, the experimental results can report only information about the damage level in a sea state.

#### 3.6. The Inappropriateness of Using the Notional Permeability Factor, P

_{n}

_{50}). The same methodology can also be applied to analyze the performance of permeable core and homogeneous structures. The presence of a porous core is relevant to the hydrodynamic performance of the breakwater because it determines, among others, the phase lag between the incident and reflected wave trains and its impact on breaker type. This process is described by the relative water depth, h/L, width B/L and the relative diameter of the core D

_{50,p}/L, Vilchez et al. [37,38], Díaz-Carrasco et al. [30], and the ratio B/h, Dalrymple et al. [39].

#### 3.7. Conservation Laws and Model-Prototype Similarities

- (1)
- Surf similarity, normalized parameters of the wave progression on the slope: H
_{sI}/L and h/L must be used to compute χ and γ [30]; - (2)
- Porous flow similarity, normalized parameters of the wave performance inside the structure, h/L, B/L and D
_{50,p}/L to compute B/h and flow regime Re and KC [37]; - (3)
- Mechanical similarity, normalized parameters of the extraction and displacement and deposition of the rock armor units, h/L, H
_{sI}/D_{n}_{50}, angle of repose to compute the dimensionless eroded area Ae/(D_{n}_{50})^{2}, and the reshaped profile [40].

## 4. Conclusions

- The experimental space shows that, although the experimentation carried on by Van der Meer [28] is vast, unfortunately it does not cover many common field conditions. It is uncertain how this and other stability formulas derived from lab experiments perform for what may be the design condition;
- The surf similarity parameter is not the sole factor for the normalized stability of the slope. The relative water depth at the toe of the breakwater slope, h/L, cannot be neglected. Both are necessary to predict the spatial and temporal interaction of wave-structure;
- The formula of Van der Meer does not include h/L, then, under the same wave input, the design nominal diameter provided by the formula is independent of the water depth at the toe of the structure. An undesirable consequence is the non-compliance of Froudian model, prototype similarity considering that Van der Meer carried on most of the experiments with constant water depth;
- In the formula the parameter ${S}_{d}^{\ast}=\frac{{S}_{d}}{\sqrt{{N}_{w}}}$ scales the stability number N
_{s}_{,}called the relative stability number. Except for h/L ≥ 0.18, there is no well-defined relationship between the two sets of experimental pairs of values, S_{d,Nw=}_{3000}/S_{d,Nw=}_{1000}and N_{S}. To improve the epistemic uncertainty ${S}_{d}^{\ast}$ is one of the parameters whose presence should be reconsidered; - The formula cannot predict the progression of damage under a storm, f(H
_{sI}, T_{m}, t_{d}), of arbitrary shape. The inclusion of ${S}_{d}^{\ast}$ in the formula is one of the problems, but not the unique.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Roman letters | |

A_{e} | Cross-sectional eroded area |

B | Breakwater width |

D_{n}_{50} | Nominal armor rock diameter |

D_{50,p} | Nominal diameter of the core |

D_{85} | Percentile 85 of the armor rock diameter |

D_{15} | Percentile 15 of the armor rock diameter |

h | Water depth |

H_{sI} | Significant incident wave height |

Ir | Iribarren number |

KC | Keulegan-Carpenter number |

L | Wave length |

L_{0} | Wave length at deep water |

N_{s} | Stability number |

N_{w} | Number of waves |

P | Permeability factor |

Re | Reynolds number |

S_{d} | Damage level |

S_{d, in} | Damage level start of damage |

S_{d, max} | Highest damage level |

S_{d}^{*} | Non-dimensional damage level |

t_{d} | Sea State persistence |

T_{m} | Mean wave period |

T_{m-}_{1.0} | Spectral wave period |

T_{p} | Peak period |

U_{r} | Ursell number |

Greek letters | |

α | Slope angle |

β | Foreshore angle |

γ | Relative wave height |

λ | Scaling ratio |

Δ_{s} | Relative mass density of the armor rock |

χ | Alternate similarity parameter |

ξ_{m}^{*} | Surf similarity parameter with H_{sI}/L |

ξ_{m} | Surf similarity or breaker parameter with H_{sI}/L_{0} |

Acronyms | |

CT | Complete tests |

LS | Large scale tests |

SS | Small scale tests |

NA | Narrow spectra |

PM | Pierson Moskowitz spectra |

TS | Test Series |

WI | Wide spectra |

## Appendix A

**Table A1.**Experimental data conditions for some of the Pierson Moskowitz data tested by Van der Meer [28]. The damage level and the error calculated by the formula of Van der Meer and the sigmoid function proposed by Losada [16] are included. The wave breaker types according to Van der Meer and following the works of Moragues and Losada [31] are also compared. By colors the values of relative water depth in deep water are included following Figure 1.

N° Test | S_{0m} | ${\mathit{\xi}}_{\mathit{m}}$ [17] | ${\mathit{\xi}}_{\mathit{m}.\mathit{c}}$ [17] | Breaker [15] | N_{s} | H_{sI}/L | h/L | χ | Breaker [31] | γ | S_{d, fixed} | N_{d}^{*} | S_{d} lab | S_{d} [17] | S_{d} [16] | % Error [17] | % Error [16] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

27 | 0.0366 | 1.74 | 2.55 | plunging | 1.69 | 0.0381 | 0.3108 | 0.0118 | strong plunging | 0.1226 | 1–4 | 0.449–0.5925 | 2.15 | 2.62 | 1.4 | 21.86 | −34.88 |

40 | 0.0164 | 2.6 | 2.55 | surging | 1.49 | 0.02 | 0.1844 | 0.0037 | weak bore | 0.1085 | 2.93 | 3.58 | 2.88 | 22.18 | −1.71 | ||

3 | 0.0115 | 3.11 | 2.55 | surging | 1.46 | 0.0158 | 0.1469 | 0.0023 | weak bore | 0.1073 | 3.09 | 2.96 | 3.2 | −4.21 | 3.56 | ||

5 | 0.0096 | 3.4 | 2.55 | surging | 1.20 | 0.0131 | 0.1486 | 0.0019 | surging | 0.0881 | 1.88 | 1.06 | 1.96 | −43.62 | 4.26 | ||

18 | 0.0114 | 3.21 | 2.55 | surging | 1.48 | 0.0157 | 0.1461 | 0.0023 | weak bore | 0.1075 | 3.66 | 3.16 | 3.52 | −13.66 | −3.83 | ||

8 | 0.0076 | 3.83 | 2.55 | surging | 1.44 | 0.0122 | 0.1157 | 0.0014 | surging | 0.1055 | 1.47 | 2.49 | 1.26 | 69.39 | −14.29 | ||

9 | 0.0076 | 3.64 | 2.55 | surging | 1.57 | 0.0134 | 0.1167 | 0.0016 | surging | 0.115 | 2.35 | 3.93 | 2.22 | 67.23 | −5.53 | ||

13 | 0.006 | 4.3 | 2.55 | surging | 1.62 | 0.0113 | 0.0949 | 0.011 | surging | 0.1188 | 2.66 | 4.23 | 2.26 | 59.02 | −15.04 | ||

28 | 0.0495 | 1.5 | 2.55 | plunging | 2.32 | 0.0516 | 0.3068 | 0.0158 | weak plunging | 0.1682 | 6–9 | 0.6426–0.6968 | 7.45 | 8.8 | 7.82 | 18.12 | 4.97 |

31 | 0.0443 | 1.58 | 2.55 | plunging | 2.04 | 0.0461 | 0.3108 | 0.0143 | weak plunging | 0.1474 | 6.44 | 5.27 | 6.82 | −18.17 | 5.90 | ||

38 | 0.0186 | 2.44 | 2.55 | plunging | 1.71 | 0.0228 | 0.1817 | 0.0041 | strong bore | 0.1255 | 6.63 | 6.45 | 6.54 | −2.71 | −1.36 | ||

2 | 0.013 | 2.92 | 2.55 | surging | 1.70 | 0.0181 | 0.1452 | 0.0026 | weak bore | 0.1244 | 6.71 | 6.53 | 6.807 | 2.68 | 1.45 | ||

4 | 0.0145 | 2.77 | 2.55 | surging | 1.85 | 0.0199 | 0.1469 | 0.0029 | weak bore | 0.1356 | 9.16 | 10.24 | 8.92 | 11.79 | −2.62 | ||

16 | 0.0145 | 2.77 | 2.55 | surging | 1.89 | 0.02 | 0.1461 | 0.0029 | weak bore | 0.1372 | 8.16 | 11.4 | 8.34 | 39.71 | 2.21 | ||

7 | 0.009 | 3.51 | 2.55 | surging | 1.70 | 0.0145 | 0.1162 | 0.0017 | surging | 0.1245 | 7.46 | 5.96 | 6.48 | −20.11 | −13.14 | ||

12 | 0.0069 | 4.01 | 2.55 | surging | 1.79 | 0.0127 | 0.0969 | 0.0012 | surging | 0.1314 | 8.7 | 7.22 | 6.7 | −17.01 | −22.99 | ||

29 | 0.0544 | 1.43 | 2.55 | plunging | 2.66 | 0.0571 | 0.2953 | 0.0169 | weak plunging | 0.1935 | 12–30 | 0.7388–0.8866 | 13.11 | 15.47 | 13.15 | 18.00 | 0.31 |

30 | 0.0582 | 1.38 | 2.55 | plunging | 2.89 | 0.0613 | 0.2916 | 0.0179 | weak plunging | 0.2102 | 17.81 | 21.42 | 16.97 | 20.27 | −4.72 | ||

25 | 0.028 | 1.99 | 2.55 | plunging | 2.35 | 0.0333 | 0.1946 | 0.0065 | strong bore | 0.1711 | 19.47 | 19.01 | 19.38 | −2.36 | −0.46 | ||

37 | 0.0233 | 2.19 | 2.55 | plunging | 2.14 | 0.0285 | 0.1817 | 0.0052 | strong bore | 0.157 | 12.34 | 15.13 | 11.48 | 22.61 | −6.97 | ||

20 | 0.0157 | 2.66 | 2.55 | surging | 2.04 | 0.0217 | 0.1461 | 0.0032 | weak bore | 0.1485 | 15.15 | 17.04 | 12.48 | 12.48 | −17.62 | ||

21 | 0.017 | 2.56 | 2.55 | surging | 2.21 | 0.0235 | 0.1461 | 0.0034 | weak bore | 0.1606 | 13.98 | 25.92 | 14.17 | 85.41 | 1.36 | ||

6 | 0.0187 | 3.22 | 2.55 | surging | 2.00 | 0.0171 | 0.1167 | 0.002 | surging | 0.1466 | 29.07 | 14.03 | 28.44 | −51.74 | −2.17 | ||

32 | 0.0083 | 3.66 | 2.55 | surging | 1.97 | 0.0147 | 0.1024 | 0.0015 | surging | 0.1431 | 18.41 | 12.2 | 17.82 | −33.73 | −3.20 |

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**Figure 1.**Application of the Van der Meer formula: breaker parameter—relative stability number graph. The symbols (circle, square, triangles) represent all the laboratory data obtained by Van der Meer [28] for an impermeable core and slope 1:3. By colors the values of relative water depth in deep water are included. The yellow curve is the formula of Van der Meer in plunging and surging wave breaker for a slope 1:3, P = 0.1 and N

_{w}= 3000. The stars-symbols are the data examples calculated with the formula for h/L

_{0}= 0.05 and 0.15. The laboratory data were taken from [28].

**Figure 2.**Experimental space of Van der Meer’s data [28] for an impermeable core and slope 1:3. The orange lines and black lines represent the isolines with constant value of $\gamma $ and χ. By color the values of relative water depth h/L, at the toe of the slope, are identified. The data were taken from [28].

**Figure 3.**Progression of damage, S

_{d}, against the stability number, N

_{s}, (or wave height, H

_{sI}

_{,}or wave steepness, H

_{sI}/L) under number of waves (

**a**) N

_{w}= 1000 and (

**b**) N

_{w}= 3000. By colors the values of relative water depth, h/L, at the toe of the slope is identified. The data were taken from [28].

**Figure 4.**Quotient of the damage progression, S

_{d}, under N

_{w}= 1000 and N

_{w}= 3000 against the stability number, N

_{s}, (or wave height, H

_{sI}, or wave steepness, H

_{sI}/L). By colors the values of relative water depth, h/L, at the toe of the slope is identified. The data were taken from [28].

**Figure 5.**Breaker parameter at the toe of the slope (${\xi}_{m}{}^{\ast})$—relative stability number $\left(\frac{{N}_{s}}{{({S}_{d}^{\ast})}^{0.2}}\right)$ graph $\left({S}_{d}^{\ast}=\frac{{S}_{d}}{\left({N}_{w}{}^{0.5}\right)}\right)$: all the experimental data of Van der Meer [28] for an impermeable core and slope 1:3 under N

_{w}= 3000, is included. By colors, the values of relative water depth, h/L, at the toe of the slope are identified. The dotted black lines and blue bands represent the constant $\chi $ isolines (range) and thus the limits of wave breaker types. The dotted yellow lines are the isolines with constant $\gamma $. The data were taken from [28].

**Table 1.**Calculation of the nominal design diameter of a breakwater for a three sea states applying the formula of Van der Meer, and the corresponding data in laboratory with the same breaker parameter, N

_{w}= 3000 and h = 6 m.

Point | H_{sI}Sea State | T_{m}Sea State | S_{d}Selected | ξ_{m} | D_{n50} | N° Test-Lab | ξ_{m}Lab | h/L_{0}Lab |
---|---|---|---|---|---|---|---|---|

1 | 2 | 8.3 | 2, 12 | 2.44 | 0.91, 0.63 | 38 PM | 2.44 | 0.15 |

2 | 1 | 8.3 | 2, 12 | 3.46 | 0.44 0.31 | 10 PM | 3.42 | 0.07 |

3 | 1 | 5 | 2, 12 | 2.08 | 0.42 0.29 | 3 LS | 2.09 | 0.15 |

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**MDPI and ACS Style**

Losada, M.A.; Díaz-Carrasco, P.; Clavero, M.
Do Rock Design Formulas Based on Wave Flume Experiments Reliably Model Their Performance at Sea? *J. Mar. Sci. Eng.* **2022**, *10*, 487.
https://doi.org/10.3390/jmse10040487

**AMA Style**

Losada MA, Díaz-Carrasco P, Clavero M.
Do Rock Design Formulas Based on Wave Flume Experiments Reliably Model Their Performance at Sea? *Journal of Marine Science and Engineering*. 2022; 10(4):487.
https://doi.org/10.3390/jmse10040487

**Chicago/Turabian Style**

Losada, Miguel A., Pilar Díaz-Carrasco, and María Clavero.
2022. "Do Rock Design Formulas Based on Wave Flume Experiments Reliably Model Their Performance at Sea?" *Journal of Marine Science and Engineering* 10, no. 4: 487.
https://doi.org/10.3390/jmse10040487