Coastal Engineering 140 (2018) 257–271 Contents lists available at ScienceDirect Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng Empirical model for probabilistic rock stability on ﬂat beds under waves with or without currents T Shigeru Kawamataa,∗, Manabu Kobayashib, Norio Tanadac a National Research Institute of Fisheries Engineering, Japan Fisheries Research and Education Agency, Hasaki, Kamisu, Ibaraki, 314-0408, Japan Choshi Branch, International Meteorological and Oceanographic Consultants Co. Ltd, Kawaguchicho, Choshi, Chiba, 288-0001, Japan c Fisheries Research Division, Tokushima Agriculture, Forestry and Fisheries Technology Support Center, Dounoura, Seto, Naruto, Tokushima, 771-0361, Japan b A R T I C LE I N FO A B S T R A C T Keywords: Rock stability Asymmetric wave Minimum stable mass Damage Friction coeﬃcient Mobility index An empirical formula was developed for predicting the stability of isolated quarry rocks on relatively ﬂat beds under waves with or without current in terms of the damage ratio. The damage ratio was examined using 100 crushed stones (median mass M50 = 293 g and mass density ρs = 2.62 g/cm3) as scale models of quarry rocks and replicas with a lower density (ρs = 1.38 g/cm3) by placing them on beds of diﬀerent roughness in a wave ﬂume, not only under periodic waves with periods of 2–3.5 s, but also in symmetric and asymmetric oscillatory ﬂows (periods: 8–12 s), simulating waves without and with current in a circulating water channel. Comparison between hydrodynamic forces expressed in terms of three non-dimensional mobility indices: the maximum velocity, maximum semi-velocity amplitude (Ua), and maximum acceleration relative to the friction force (expressed in terms of the median coeﬃcient of friction) suggested that the damage ratio was most closely related to the Ua-based mobility index (ψ2). Nevertheless, signiﬁcant diﬀerences remained between data from the wave ﬂume and circulating water channel tests. The variation in the damage ratio, which included the eﬀects of the oscillatory-velocity asymmetry, oscillation period, superimposed steady current, mass density of stones, and bottom friction, was reasonably well described via the product of ψ2 and a function of a Keulegan–Carpenter number. The results of the ﬁeld tests on quarry rocks (with M50 = 2.04 t) placed on a thin sand layer overlaying hard substrate show that the minimum stable mass is consistent with the prediction. 1. Introduction Quarry rocks are commonly used as a cost-eﬀective material for artiﬁcial reefs in civil engineering (Deysher et al., 2002; Seaman, 2007; Bohnsack and Sutherland, 1985; Grant et al., 1982). Quarry rock reefs have been often constructed at sites having a thin layer of sand overlaying a ﬂat hard substrate, aiming to attract more ﬁsh in hard bottom habitats or to create macroalgal beds without being buried by sand (Grant et al., 1982). The proper placement of low-relief (thus smallsized) rock substrates or rocks at regular intervals on sandy bottoms can aid the development and persistence of macroalgal stands (Deysher et al., 2002; Kawamata et al., 2011; Ohno et al., 1990). The hydrodynamic stability of rocks or stones on the seabed should be treated as probabilistic in nature because of the high variabilities not only in the rock shape but also in the roughness of the substratum. The stability of rocks in coastal sites has been well studied with regard to the design of coastal structures including rubble revetments, breakwaters (Van der Meer, 1987, 1992; Kobayashi and Jacobs, 1985), near-bed rubble ∗ mounds (Van Gent and Wallast, 2001; Tørum et al., 2010; Wallast and van Gent, 2002), and the protection of rock slopes and gravel beaches (van der Meer and Pilarczyk, 1987). However, studies on the stability of isolated rocks on comparatively ﬂat bottoms are lacking. Present common design approaches based on stability criteria, such as the stability number, Shields number, and mobility number, cannot be applied to such conditions because they assume stone layers with a particular weight, shape, and density and do not explicitly evaluate the eﬀect of frictional resistance on stability. In the current design criteria for artiﬁcial reefs in Japan, the critical stable mass of rocks in surf zones is given by the following formula from Akeda et al. (1992). Mcr = CUm6 (1) where Mcr is the critical stable mass (in kg) of rocks, C is an experimental coeﬃcient depending on the density of the rocks deployed, and Um is the wave-induced maximum peak velocity (in m/s). This formula assumes that the hydraulic load and friction with respect to the rocks Corresponding author. E-mail address: [email protected]ﬀrc.go.jp (S. Kawamata). https://doi.org/10.1016/j.coastaleng.2018.08.005 Received 10 December 2017; Received in revised form 12 July 2018; Accepted 2 August 2018 Available online 04 August 2018 0378-3839/ © 2018 Elsevier B.V. All rights reserved. Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Considering the probabilistic nature of rock stability, the damage ratio rd was deﬁned as the relative number of quarry rocks ‘damaged’ under given physical conditions. The mobility of stones on a relatively ﬂat bed can be expressed simply by the ratio of the maximum drag force (proportional to ρUm2Dn502) to the frictional force (μ50(ρs − ρ)gDn503), i.e., are proportional to the square of the velocity and the submerged weight, respectively, similar to the equation proposed by Isbash (1936) for steady ﬂows. Akeda et al. (1992) found the C value to be 25 for solitary rocks by performing a scale wave ﬂume experiment with rounded pebbles on a smooth, ﬂat cement mortar bed. They determined the C value considering the performance in terms of the cost of constructing an artiﬁcial rock reef by deﬁning the motion threshold as the maximum velocity at which 10% of the test stones begin to move (Akeda et al., 1992). Equation (1) is convenient for practical design but is too simpliﬁed to evaluate the eﬀects of factors other than Um, such as the mass density, friction, and velocity asymmetry. A critical problem is that the calculated stable mass for isolated rocks is often considerable. For example, for Um = 3 m/s, Eq. (1) with C = 25 gives Mcr ≈ 18 Mg (or t). It is unclear whether such a large stable mass is reasonable in practice. The objective of this study was to develop an empirical formula for predicting the probability of the stability of solitary quarry rocks on relatively ﬂat beds under the eﬀects of waves in shallow waters, including the eﬀects of the asymmetry of the wave oscillatory velocity, the coexisting ﬂow, the friction coeﬃcient between the rocks and the seabed, and the mass density of the rocks. To do so, three diﬀerent laboratory experiments were conducted. The ﬁrst was a common scale experiment using a wave ﬂume, conducted to analyze the stability of the solitary crushed stones placed on diﬀerent roughness beds under non-breaking and breaking wave conditions. The second was similar to the ﬁrst but was conducted with lightweight replicas of the crushed stones. Finally, a circulating water channel (CWC) experiment was conducted to examine the stability of the same crushed stones as in the ﬁrst case, under sinusoidal oscillatory ﬂows with or without currents. The periods of the oscillatory ﬂow were the same as those of sea waves, thus corresponding to conditions under which stones smaller than the actual quarry rocks are placed under the eﬀects of full-scale waves or with Keulegan–Carpenter numbers (KC) higher than that in the ﬁeld. The laboratory experiments were conducted under the assumption that rocks are placed on ﬂat hard substrates, such as closely packed cobble and boulder beds or ﬂat bedrock, without overlying sand. This is the least stable condition of the rocks because the presence of a sand layer increases the stability of the rocks; however, waves frequently wash away the overlying thin layer of sand. In addition, an empirical method for predicting the friction coeﬃcient was also developed to enable practical use of the proposed formula. Finally, a ﬁeld stability test was performed on quarry rocks at shallow coastal sites to demonstrate whether the developed method provides a reasonable prediction of the minimum stable mass, compared to the previous formula based solely on Um. ψ1 = Um2 μ50 ΔgDn50 (2) where Dn50 = (M50/ρs) with ρ and ρs being the mass densities of water and stones, respectively, and where M50 is the median mass of the stones; μ50 is the median friction coeﬃcient between the stones and the bed; Δ = (ρs/ρ − 1), deﬁning the submerged speciﬁc density of the stones; and g is the acceleration due to gravity. Note that if the friction coeﬃcient in Eq. (2) is omitted under an implicit assumption of the constant friction coeﬃcient, then the ψ1 is equivalent to the well-known “mobility number” for sediment particles (Brebner, 1980; Nielsen, 1992) or the frequently used “mobility parameter” for rubble-mound materials (Van Gent and Wallast, 2001; Tørum et al., 2010; Wallast and van Gent, 2002). However, the hydrodynamic force is due to not only the drag, which is proportional to the square of the velocity u, but also the inertia force, which is proportional to the acceleration of the ﬂuid a. Therefore, the hydrodynamic force is at its maximum before the velocity reaches its maximum. Thus, there might be a better alternative to ψ1. In the study, the maximum semi-velocity amplitude Ua and the maximum acceleration amax (Fig. 1) were compared to Um in terms of its explanatory power in determining the damage ratio of stones on a given substratum. Two additional mobility indices can be deﬁned using Ua and amax as the ratios of ρUa2Dn502 and ρamaxDn503, respectively, to μ50(ρs − ρ)gDn503: 1/3 ψ2 = Ua2 μ50 ΔgDn50 (3) ψ3 = amax μ50 Δg (4) The successive trough and crest velocities in the wave cycle with maximum peak-to-peak velocity amplitude are denoted as umin and umax, respectively (thus Ua = (umax − umin)/2). In case of a sinusoidal velocity variation, amax = 2πumax/T, and therefore, if Um = umax (as in most test cases), the ratio of ψ1 to ψ3 is proportional to KC, which is deﬁned as umaxT/Dn50, where T is the period of an individual oscillation cycle. However, when the velocity asymmetry increases, the above equation with respect to amax may be invalid. Instead, umax/Tzp can be used as a better index for amax, where Tzp is the zero-to-peak period. Accordingly, the ratio of ψ1 to ψ3 can be assessed as follows. 2. Deﬁnition of damage and its governing variables K C = 4u max Tzp/ Dn50 It is diﬃcult to accurately deﬁne the motion threshold of stones resting on bed roughness elements under the eﬀects of waves with or without currents. In this study, considering the process of movement and the design and construction practices of artiﬁcial reefs, ‘signiﬁcant movement’ or ‘damage’ was deﬁned as the shifting of a stone entirely out of its initially occupied area after placing it on the bed at a random position, but with the largest ﬂat surface oriented downward to make the stone more stable. Such an orientation can readily occur via toppling due to ﬂows, even if the stones are initially placed in unstable postures. As the oscillatory velocity increases, stones placed on the irregular surfaces of the bed shake at ﬁrst, occasionally with a slight slide, and then distinctly move in a sudden manner by either sliding, rolling, or both. Most of the initial movements cease at once, indicating that they are primarily attributed to the initially unstable positions after placement. Additionally, natural ﬂat beds have irregular surfaces. Therefore, an unexpectedly low ﬂow velocity may lead to a small shift even in large rocks. Thus, the above deﬁnition of the movement relative to the size of the stones is preferred to an absolute deﬁnition. The mobility index among ψ1, ψ2, and ψ3 that is most closely related to the damage ratio will be used to predict the stone stability. However, if there are considerable systematic deviations in the relationship between the best explanatory index and the damage ratio, the residual components may be a function of KC. An attempt will be then made to establish a better predictor to determine the damage ratio by multiplying the candidate by possible functions of KC. (5) 3. Laboratory model experiments 3.1. Experimental setups and procedures Stone stability was examined under various physical conditions (Table 1) using a wave ﬂume and a CWC at the National Research Institute of Fisheries Engineering. First, scale model experiments were conducted to determine the stability of the stones under the eﬀects of waves in a 70 m long, 0.7 m wide, and 2.2 m deep wave ﬂume with a smooth sloping bottom made of cement mortar (Fig. 2). A 210 cm long 258 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 1. Schematic example of velocity and acceleration time-series showing the deﬁnition of the characteristic velocity quantities. Although in this example the wave cycle with Um diﬀers from that with Ua, both waves were the same in most test cases (thus Um = umax). stones. They were tested by placing them apart from each other on the bed in slightly diﬀerent positions but with the largest surface area oriented downward and the largest projected side area oriented normal to the wave direction. Subsequently, 7–13 waves were generated using a piston-type wavemaker with given periods and amplitudes, and ‘signiﬁcantly moved’ stones were counted (Fig. 4). The orientation of the stones helped in testing the lowest potential stability in the given direction, considering the uncontrolled horizontal direction of the rocks relative to the variable wave directions. For wave periods (T) of 2.0, 2.5, 3.0, and 3.5 s, the wave height was increased by 1 or 2 cm from values at which no stones were ‘damaged’, up to the wave maker capacity. Although numerous studies have examined rock stability under irregular waves, a model for predicting stability was developed in the current work without irregular wave experiments by using a diﬀerent approach. The previous studies assumed that the probability of instability due to random waves could be represented with central location statistics, such as signiﬁcant wave height and average wave period, so that neither the highest wave height nor the detailed waveform of velocity vs. time was taken into consideration. In the current research, it is assumed that whether waves are random or not, the damage ratio due to a wave train can be determined solely by a single wave exerting a maximum hydrodynamic force on the rocks. Thus, only the maximum and 70 cm wide test bed was set on top of a gentle slope (1:50) area (i.e., cement mortar bed, referred to as bed CM) and then on a layer of rounded coarse gravel (the median surface grain size B50 = 34.2 mm) or medium gravel (B50 = 8.5 mm) was closely packed and glued using a silicon sealant onto 5-mm thick polyvinyl chloride boards (referred to as beds CG1 and MG1, respectively). The test beds helped to simulate diﬀerent surface roughnesses of the natural seabed. The still water depth at the center of the test bed was 56 cm. The model scale was set to 1:15, providing the largest scale model that could generate waves large enough to damage the scale model with a 1 t rock even on coarse gravel beds with the highest friction. One hundred crushed granite stones (Fig. 3) were used as scale models of irregularly shaped quarry rock weighing approximately 1 t to empirically determine the damage ratio under the given physical conditions. The main characteristics of the stones include masses ranging from 232 to 358 g with M50 = 293 g, ρs = 2.62 g/cm3, and Dn50 = 4.82 cm. In addition, a set of lightweight duplicates (M50 = 161.5 g and ρs = 1.37 g/cm3) of the 100 crushed stones was used, prepared by modeling each stone using gypsum and then casting a mixture of ceramic mortar (Eagle 8, Eaglevision Co., Tokyo) and lightweight foam aggregates (TS sand, Sekisui Plastics Co. Ltd., Osaka, Japan) into the gypsum mold. The 100 test stones were divided into 10 sets, each comprising 10 259 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Table 1 Experimental conditions. Case name T (s) Number of sub-cases Wave ﬂume experiment W1 2.0 8 2.5 7 3.0 7 3.5 8 W2 2.0 5 2.5 5 3.0 7 3.5 6 W3 2.0 7 2.5 6 3.0 7 3.5 7 W4 2.0 11 2.5 10 3.0 10 3.5 8 Circulating water channel experiment C1 10.0 19 C2 10.0 19 C3 8.0 13 C4 12.0 13 Hmax a (cm) 27.7–39.6 25.1–39.5 30.5–42.6 25.9–41.7 29.6–42.5 31.6–45.8 26.6–50.8 26.7–50.8 23.4–41.5 27.0–44.8 26.5–50.1 21.8–49.8 11.6–28.8 10.5–24.1 11.6–27.6 10.9–19.5 Um (cm/s) umax umax − umin KC Re = umaxDn50/ν (104) Test bedb Test stonesc μ50d 44.4–57.9 51.1–59.2 57.4–63.6 45.7–63.4 43.2–53.4 52.3–59.3 47.5–66.6 52.9–75.8 34.2–51.7 45.1–56.9 45.2–62.3 41.5–70.6 17.0–37.0 23.0–48.9 22.2–45.8 20.4–37.1 0.49–0.57 0.53–0.56 0.62–0.68 0.58–0.60 0.49–0.55 0.51–0.56 0.43–0.62 0.54–0.58 0.46–0.57 0.48–0.60 0.50–0.62 0.52–0.57 0.43–0.55 0.54–0.58 0.54–0.68 0.52–0.58 16.6–17.3 19.8–23.1 20.2–28.5 18.6–20.8 13.5–18.8 14.9–24.9 12.7–20.7 21.7–25.3 12.5–18.4 11.9–22.0 17.2–21.6 20.1–26.8 6.0–15.1 10.8–19.7 12.6–20.3 11.8–18.5 2.03–2.57 2.48–2.86 2.79–3.17 2.17–2.95 1.87–2.45 2.43–2.81 2.16–2.95 2.45–3.51 1.66–2.58 2.19–2.77 2.07–2.90 2.09–3.36 0.77–1.81 1.13–2.40 1.09–2.25 1.03–1.78 CG1 CGS 0.86 MG1 CGS 0.70 CM CGS 0.62 MG1 LWD 0.74 40.8–131.4 38.3–138.4 62.2–118.4 76.2–149.5 0.48–0.52 0.66–0.69 0.48–0.67 0.55–0.73 78.6–285.6 78.7–281.8 84.8–191.8 215.1–364.5 1.97–6.34 1.85–7.21 3.00–5.71 3.67–7.21 MG2 MG2 CG2 CG2 CGS CGS CGS CGS 0.72 0.81 a Hmax: Average of the highest wave heights measured within the test section. CG1: large rounded coarse gravel bed (B50 = 34.2 mm), MG1: rounded medium gravel bed (B50 = 8.5 mm), CM: smooth cement mortar bed, MG2: subangular medium gravel bed (B50 = 8.9 mm), CG2: small rounded coarse gravel bed (B50 = 26.5 mm). c CGS: 100 crushed granite stones (M50 = 293.0 g and ρs = 2.62 g/cm3), LWD: 100 lightweight duplicates (M50 = 161.5 g and ρs = 1.37 g/cm3). d μ50: median of friction coeﬃcients shown in Fig. 8. b characteristic velocities of wave trains were used for analysis. Note that the waves produced in this study were variable rather than regular (as described below), so the maximum characteristic velocities of interest diﬀered from the mean wave velocities. The near-bed velocity was recorded using three-dimensional Nortek Vectrino acoustic Doppler velocimeters (ADVs). The sensing volumes of the ADVs were set to be 7 cm above the bed surface to allow for stable velocity measurement. In the preliminary ADV measurements, the vertical velocity proﬁle between 12.5 and 80 mm above the bottom was nearly constant, even for the roughest bed (i.e., bed CG1), but with more frequent contamination by spike noise closer to the bottom. The velocity records were obtained at 50 Hz in three locations: in the midcross-section of the test-bed section, and 75 cm oﬀshore and inshore from there. Each velocity time series was low-pass ﬁltered at a cutoﬀ frequency of 2 Hz to determine Um as the maximum velocity in the wave train and three characteristic quantities of the oscillatory velocity, being Ua, Tzp, and amax, for the individual wave (deﬁned by the zero down-crossing method) with the maximum peak-to-peak velocity amplitude. Despite a regular piston motion, the resultant wave trains were not completely regular and, in many cases, showed signiﬁcant Fig. 3. The set of 100 crushed granite stones used for laboratory experiments. Fig. 2. Wave ﬂume dimensions (in meters) and location of measurement instruments in the test section on the slope. WHM: resistance type wave gauge. 260 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 4. Examples of top views of 10 crushed stones deployed over a test bed before and after wave action in the wave ﬂume experiment. The crushed stones indicated by circles are judged to be ‘damaged’. patterns of the oscillatory ﬂows: symmetric (umax ≈ –umin) and asymmetric (umax ≈ –2umin) oscillatory ﬂows with a constant oscillation period of 10 s on bed MG2; and asymmetric oscillatory ﬂows with constant oscillation periods of 8 s and 12 s, and a stepwise increase in only umax but nearly constant umin on bed CG2. The asymmetric oscillatory ﬂows had a net transport, which helped in simulating the combined eﬀect of the waves and currents. The freestream velocity was measured 20 cm above the bed at the center of the test section using an ADV at 10 Hz. ﬂuctuations due to long-period oscillations (Fig. 5). Thus, the wave with Um did not always coincide with that with Ua or amax. The values of Um, Ua, amax, and Tzp obtained using the three ADVs were averaged for analysis. The stone stability in an oscillatory ﬂow with or without current was examined using the same set of crushed stones as in the CWC (Fig. 6). The CWC is equipped with a computer-controlled, reversible impeller, using which an arbitrarily time-varying velocity was produced in a fully enclosed 3 m (length) × 0.4 m (width) × 0.4 m (height) test section. However, unlike the wave ﬂume, near-sinusoidal oscillatory ﬂows with stable velocity amplitudes of > 1 m/s can be produced for longer wave periods (≥8 s) but not for similarly shorter periods because of the cavitation at the impeller. The crushed stones in such periods of oscillatory ﬂows relative to the orbital amplitude of the ﬂuid are smaller than the actual quarry rocks, thus obtaining a KC value higher than expected without increasing the relative eﬀect of viscosity on the hydrodynamic forces. Therefore, if an empirical formula can describe our experimental data overall, it can be expected to be applicable over a wide range of ﬁeld conditions through interpolation. The inner bottom surface of the test section was covered using subangular medium gravel (MG2) or rounded coarse gravel (CG2) bed. The critical velocities at which crushed stones were initially ‘damaged’ were examined for all the crushed stones (selected in sets of six or fewer) as follows: (1) The stones were arranged in the same manner as in the wave ﬂume experiment in a test-bed area located 50 cm from both ends and 10 cm from the side walls of the test section; (2) Oscillatory ﬂows were generated with predetermined time variations and a constant period but with an increase in the velocity amplitude and mean ﬂow velocity (Fig. 7); and (3) The initial displacement time of each stone was recorded to later determine the characteristic ﬂow quantities (Um, Ua, amax, and Tzp) of the oscillation cycle at the time of 'damage'. Lowpass ﬁltering at a cutoﬀ frequency of 0.5 Hz was applied to the velocity data prior to the determination of the quantities. Unlike the wave ﬂume experiment, once the stones began to displace, they travelled much further compared to their sizes. Consequently, some stones came into contact or collided with other displaced stones before moving with the ﬂow. In such cases, the stones were re-examined. The stability of the crushed stones was examined in four cases with diﬀerent time-variation 3.2. Determination of friction coeﬃcients The friction coeﬃcients for the test stones on each test bed were determined from the friction angles θ as tan θ. The friction angles were measured with a portion of the test bed ﬁxed on a tilting frame. The 100 test stones were each individually placed on the sample bed at random positions but with the same orientation as in the above experiments. Subsequently, the frame was tilted very slowly until the stones moved more than one stone diameter. The maximum tilting angle before movement was measured to the nearest 0.1° using a digital inclinometer, which was attached to the tilting frame. The measurement was repeated three times for each stone by changing its position. Fig. 8 shows the cumulative frequencies of the friction coeﬃcients for the test stones resting on the test beds for all cases. The median and standard deviation of the coeﬃcient of friction on the even surface of bed CM in W3 were the lowest, as expected. However, for a small percentage of stones (e.g., 10%), the coeﬃcients of friction on beds MG1 and MG2 in W2, W4, R1, and R2, which had higher surface roughnesses, were lower than that measured on bed CM. 3.3. Analysis of explanatory variables for damage ratio The relationships described by (1) Um vs. rd, (2) Ua vs. rd, and (3) amax vs. rd were compared to determine the best explanatory variable for the damage ratio. It should be noted that the damage ratio calculated from the 100 samples varies to some extent because of the relatively small number of samples. For example, when 10 stones are ‘damaged’ among 100 samples, rd = 10%; however, assuming the same 261 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 5. Examples of the velocity time series observed in the wave ﬂume experiment, where x is the inshore distance from the center of the test-bed section. 4. Development of prediction model probability of damage among the individual stones, a binomial distribution suggests that the 95% conﬁdence interval of the true rd ranges from 4.9% to 17.6% (Zar, 1999). Therefore, the 95% binomial conﬁdence intervals were calculated to check the variability of the damage ratio. Furthermore, for all the cases, the three dimensionless mobility indices ψ1, ψ2, and ψ3 were compared. If even the best-correlated mobility index failed to adequately account for the damage ratio, a new dimensionless quantity was established by modifying the mobility index using an empirically determined function of KC. 4.1. Model for stability of rocks Fig. 9 shows the damage ratio with respect to the characteristic ﬂow quantities Um, Ua, and amax for each case. The solid line represents a ﬂexible and robust Bayesian spline regression (Ishiguro and Arahata, 1982) to show a trend expected to monotonically increase. Although any characteristic ﬂow quantities did not have a regression line that 262 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 6. Schematic of the circulating water channel. Fig. 8. Cumulative frequency of the friction coeﬃcient μ for each combination of test stones and test beds. The abbreviations in the legend are listed in Table 1. passed through all the 95% conﬁdence intervals of the data points, the damage ratio, overall, was closely related to Um and Ua, and less so to amax. On closer inspection, however, it was apparent that in case W1 with highest velocity skew umax/(umax − umin) in the wave ﬂume experiment (Table 1), the damage ratio remained closely related to Ua but not to Um. Fig. 10 shows the variation in the damage ratio using the mobility indices ψ1, ψ2, and ψ3 for all cases. There were large diﬀerences between cases, particularly between the wave ﬂume and CWC data for ψ3. In contrast, the diﬀerences were minimal for ψ2. Furthermore, the slopes of the imaginary trend lines in the individual cases for ψ2 are similar to those of a sigmoid curve, whereas the slopes of the curves in the individual cases for ψ1 and ψ3 decrease with the increase in the magnitude Fig. 7. Time variations in the velocity generated in the four cases of the CWC experiment. Graphs on the right show enlarged views of the rectangular portions indicted in the graphs on the left. 263 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. factor for ψ2 that ﬁt the data, using the Marquardt nonlinear leastsquares algorithm. Consequently, it was found that the damage ratio could be even better represented by the following sigmoid function of the dimensionless predictor ϕ, which is a function of KC and ψ2: of the indices. In terms of a predictor for the damage ratio, however, the remaining deviations in ψ2 are still considerable, especially between the wave ﬂume and CWC data. The deviations in the correlation between ψ2 and rd might be attributed to the incomplete description of the maximum hydrodynamic force with ρUa2Dn502. A correction factor was therefore introduced to establish a better predictor than ψ2. From previous studies (Chakrabarti and Armbrust, 1987; Grace and Zee, 1981; Keulegan and Carpenter, 1958), the ratio of the maximum wave force to ρUa2Dn502 can be expected to be a decreasing function of KC. This is consistent with an apparent tendency of the damage threshold for ψ2 in each case to be larger with larger values of KC (see Table 1 and Fig. 10). Therefore, several functions of KC were tested to determine a correction rd = exp { −exp[−b (ϕ − c )]} (6a) with ϕ = (α − ln K C ) ψ2 (6b) where α = 8.66, b = 0.608, and c = 4.17 are experimental constants determined using the nonlinear least squares regression (R2 = 0.849, 1 standard error (SE) = 9.5%, and n = 183). Fig. 11 shows the scatter Fig. 9. Variation in the damage ratio rd with respect to Umax, Ua, and amax for each case. Error bars indicate the 95% conﬁdence intervals, and solid lines indicate spline regression lines. 264 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 9. (continued) plots of rd vs. ϕ along with the regression Equation (6). The KC value ranges from 6.0 to 364.5 so that the KC correction factor, 8.66 – ln KC, varies from 6.87 to 2.76. The correction factor decreases in an unbounded manner for higher KC (smaller stones or longer period waves); however, it should be noted that the KC value at which the factor reaches 0 is extremely high (approximately 5770). Overall, the regression results show a reasonable agreement with the experimental results, with reduced discrepancy between the wave ﬂume and CWC data. The physical model assumes that the predominant forces determining rock stability are the gravity and the inertial force, which is proportional to the velocity squared. In small physical models, however, the viscous force proportional to the velocity can be large relative to the inertial force, leading to a scale eﬀect. The scale eﬀect may be evaluated by a Reynolds number, which represents, in physical terms, the ratio of the inertial force to the vicious force. The Reynolds number Re is deﬁned as UmDn50/ν, where ν is the kinematic viscosity of the ﬂuid, ranging from 8 × 103 to 7.2 × 104 (Table 1). As discussed later, these Reynolds numbers are high enough to expect correct scaling of the forces. If the critical damage ratio rd, cr is considered as the maximum damage ratio for practical design, the corresponding critical value ϕcr can be obtained as follows. ϕcr = c − ln (−ln rd, cr )/ b If rd, 265 cr = 0.1, (7) which has typically been applied in design in Japan, Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 10. Relationships between the three mobility indices and the damage ratio. Fig. 12. Relationship between the ratio of the stone diameter to the median surface bed diameter and the median friction angle. D = Dn50 in the present study. The horizontal dashed line represents the value for the smooth bed CM, which is assumed to be the asymptote of the regression model. Fig. 11. Relationship between ϕ and damage ratio rd. The curved line indicates the results obtained using the nonlinear least squares regression Equation (6a). The red circle with an error bar indicates the result of the ﬁeld test oﬀ the Shiwagi coast, where the error bar indicates the 95% conﬁdence interval. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the Web version of this article.) 4.2. Model for friction coeﬃcients The friction coeﬃcient needs to be estimated in order to apply the above model to determine the stability of rocks. As it is diﬃcult to measure the friction coeﬃcient of quarry rocks on the seabed in situ, it was estimated instead from the ratio of the quarry rock size to the bed grain size as follows. Kirchner et al. (1990) showed that the median friction angle of a single stone on a rough bed could be described using the following empirical relationship. Eq. (7) gives ϕcr = 2.80. The minimum stable rock mass corresponding to the critical damage ratio can be obtained from the following equation by substituting Eqs. (3) and (5) into Eq. (6b). 2 ⎛α − ln u max Tzp ⎞ Ua = ϕcr D μ ΔgD cr cr ⎝ ⎠ 50 ⎜ ⎟ (8) θ50 = 55. 2(D / B50)−0.307 (10) and Mcr = ρs Dcr3 where θ50 is the median friction angle in degrees, D is the diameter of the stone, and B50 is the median surface grain size of the bed. They measured the intermediate axis of randomly selected grains using a caliper to determine B50. However, such a method cannot be applied in a ﬁeld environment. Alternatively, B50 was determined from the cumulative area-weighted frequency of the surface grains with short axis diameters, which can be easily determined by processing photographic (9) where Dcr and Mcr are the nominal diameter and mass of the rocks at the critical damage ratio, respectively. The value of Dcr can be obtained from Eq. (8) using an iterative calculation procedure, such as the Newton's method. 266 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. tide station, Awayuki (33°46′N, 134°36′E), was 1.84 m. images of the seabed. For D = Dn50, a considerable diﬀerence was found between the experimental data and Eq. (10) (Fig. 12). The cause of this diﬀerence is unknown, though there was a slight diﬀerence in the way the stones were placed; Kirchner et al. (1990) placed individual test stones on a horizontal bed by dropping them from above to randomize not only the location but also the orientation of the individual stones, whereas in the current work, individual stones were placed on a horizontal bed at random positions but with a speciﬁed orientation. In addition, they pointed out that Eq. (10) could underestimate the friction angles for a smooth, ﬂat bed (thus ultimately resulting in a high value of Dn50/B50), because θ50 should approach some non-zero value close to that of bed CM. Therefore, a modiﬁed empirical model was introduced to predict θ50, with an asymptotic value equal to the median friction angle of bed CM (θ50 = 32.5°). θ50 = 32.5 + β exp(−γ Dn50 / B50) 5.2. Observation procedure and results The ﬁeld observations began on July 29, 2015, aiming to determine the critical velocity at which the quarry rocks began to move due to typhoon waves. An MSR Electronics data logger (MSR145) was attached to each quarry rock to sense its three-axis accelerations at 5 Hz such that it would record accelerations that were 0.5 G (where 1 G = 9.80665 m/s2) higher or lower than the initial value, along with the time. At the same time, a wave meter (WH-501, IO Technic Co. Ltd., Machida, Tokyo) with a pressure gauge and a two-dimensional electromagnetic velocimeter (EMV) was placed on the seabed near the quarry rocks to measure the pressure and velocities at 2 Hz for 20 min every 1 h. The pressure and velocity sensors were approximately 57 and 66 cm above the seabed, respectively. The height at which the velocity measurement was taken relative to the water depth, 0.66/9.6 ≈ 0.07, was smaller than that in the wave ﬂume experiment, 7/56 = 0.125, so that the velocity measurement could be regarded as the near-bed velocity. However, prior to the observation, 2 of the 10 quarry rocks had already fallen on their sides (thus they were moved outside their initially occupied areas and ‘damaged’; Table 2) because of severe waves (hereafter referred to as Storm Waves I) associated with Typhoon 1511, which passed near Shiwagi on July 16, 2015. Severe breaking waves (hereafter referred to as Storm Waves II) occurred again during August 22–23, 2015, because of two strong typhoons (Typhoons 1515 and 1516) 1100–1700 km away from the study site in the Paciﬁc Ocean. The signiﬁcant wave height reached 4.5 m, with a spectral peak period of 17.4 s. The measurement of the distances between the quarry rocks and the benchmarks set on 10 t armor blocks (2.6 m × 2.6 m × 1.5 m) placed approximately 2–4 m away from the rocks, and the top-view images of the quarry rocks (Fig. 14) show no shift or only a slight shift in the rocks after Storm Waves II, with the maximum displacement of the rocks being 8 cm (Table 2). Although the quarry rocks should have become more stable because of the sideways toppling of the two rocks during Storm Waves I, the rocks were displaced a little further because of Storm Waves II. It can be inferred from the above facts that if the quarry rocks had ﬁrst encountered Storm Waves II, the damage to the rocks would have been similar to that inﬂicted by Storm Waves I, or in other words, the damage ratio would have been 0.2. As wave oscillatory velocities during Storm Waves II exceeded the maximum measurable velocity of the EMV (3 m/s), the velocity time series was estimated from the pressure data using linear theory spectral transformation (Guza and Thornton, 1980). Fig. 15 shows the time variation in the wave oscillatory velocity estimates when maximum peak-to-peak velocity amplitude occurred during Storm Waves II. The estimated wave oscillatory velocity largely agrees with the measured velocity component in the principle wave direction, which is deﬁned as the direction in which the variance in the velocity component is maximum. In addition, the EMV may output a lower value because of biofouling on the probe, whereas the pressure measurement is hardly aﬀected by biofouling. Thus, the wave oscillatory velocity time series during Storm Waves II can be expected to be reasonably estimated from the pressure. Fig. 15 shows that umax = 3.53 m/s, umin = −1.47 m/s, and Tzp = 2.0 s, therefore, Ua = 2.50 m/s. (11) where β and γ are experimental constants. To determine B50, the topview images of the test beds except for bed CM were considered. Subsequently, the horizontal short-axis diameters (assumed to be identical to the intermediate-axis because sediment grains are normally settled such that the shortest-axis is directed vertically) and surface areas of 100–160 randomly sampled grains were analyzed for each bed using the AreaQ image processing software (Kawamata, 2011). The nonlinear least squares regression of Eq. (11) performed on the current data gave β = 10.8 and γ = 0.230 with R2 = 0.916, SE = 0.809, and n = 5. The regression, shown in Fig. 12, is statistically signiﬁcant (F1,3 = 33.2, P = 0.010). 5. Comparison with a ﬁeld experiment 5.1. Test rocks and site A ﬁeld experiment was conducted to determine the rock stability of 10 samples of granite quarry rocks at an exposed site oﬀ the Paciﬁc coast of Shiwagi, Minami, Tokushima Prefecture, Japan (33°46.91′ N, 134°36.18′ E). The rocks were visually classiﬁed as “2–3 t class” rubbles, which have been generally used to create seaweed beds in Tokushima Prefecture. Their masses, measured using a 40-t load cell, ranged from 1.16 to 3.01 t with M50 = 2.04 t (Table 2). As in the laboratory experiments, each quarry rock was placed on the ground with its maximum surface area oriented downward. An inverted U-shaped reinforcing steel bar with a diameter of 19 mm was embedded and ﬁxed using chemical anchors onto the top surface of the rock. The rocks were then placed by hanging then from the inverted U-shaped bar at intervals of approximately 2.5 m on the seabed oﬀ the Shiwagi coast on December 25, 2014. The deployment site was approximately 9.6 m below mean sea level and was a relatively ﬂat rocky bottom covered with a thin layer of sand (Fig. 13). The mean spring tidal range at the nearest Table 2 Observations of the quarry rocks in the ﬁeld test. Rock No. Mass (t) Beginning situationa Displacementb (cm) 1 2 3 4 5 6 7 8 9 10 3.01 2.64 2.61 1.93 1.31 1.72 1.16 2.14 1.47 2.84 no change fell on its side no change fell on its side no change no change no change inclined no change no change 2 1 2 1 8 2 2 1 1 0 5.3. Comparison between measurement and prediction The mass density of the quarry rocks was determined to be 2592 ± 4.8 kg/m3 (mean ± 1 SE, n = 4) using a water immersion method with the help of debris samples (dry mass in the range of 1.513–2.836 kg) collected from the same stockpile as the rocks. Hence, Dn50 = (M50/ρs)1/3 = 0.923 m. The short surface grain diameters of the seabed in the experimental site oﬀ the Shiwagi coast was measured by processing the photographic images of the seabed. The results show that B50 = 52 cm (n = 89). The substitution of these values into Eq. a Posture at the onset of the observation period (July 29, 2015 after Storm Waves I). b Change in the measurements from benchmarks on September 4, 2015 (after Storm Waves II). 267 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 13. Underwater images taken one month after deploying the 10 quarry rocks on the test site oﬀ the Shiwagi coast. The term “Nos.” indicates the identiﬁcation numbers of the rocks. Fig. 14. Top-view images of the 10 quarry rocks taken before (July 29, 2015; upper image) and after (September 4, 2015; lower image) Storm Waves II. The two arrows in the pair of images indicate the same direction and position of the rocks. (11) gives θ50 = 39.2°, and thus μ50 = tanθ50 = 0.81. Assuming ρ = 1025 kg/m3, the observed maximum velocity oscillation with umax = 3.53 m/s, Tzp = 2.0 s, and Ua = 2.5 m/s leads to KC = 4umaxTzp/ Dn50 = 30.6 and ψ2 = Ua2/(μ50ΔgDn50) = 0.559, and thus ϕ = 2.93. This ϕ-value was plotted against the previously mentioned rd-value (0.2) in Fig. 11 as a red circle. The small sample size (n = 10) led to a large 95% conﬁdence interval for the true rd (indicated by an error bar), which was calculated based on the binomial distribution. Equation (6) and the above ϕ-value predicts an rd = 0.12, which is close to the observed value. The critical stable masses for rd,cr = 0.1 (0.2) can be predicted under the same external force conditions as above. Equation (8), when solved using the Newton's method, leads to Dcr = 0.974 (0.769) m and thus Mcr = 2.40 (1.18) t. The predictions agree well with a wide range of test rock masses, compared to the minimum stable mass of the isolated rocks determined using the conventional method, i.e., Eq. (1) with C = 25, at 46.0 t. 6. Discussion Most of the previous methods for predicting the minimum stable stone mass are based on the assumption that the maximum hydrodynamic force acting on the stones is proportional to the square of the 268 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. Fig. 15. Time variation in the pressure-estimated orbital velocity (solid line) and measured velocity component in the principal wave direction (circles) for a maximum velocity at the test site oﬀ the Shiwagi coast. The time series data were obtained from 01:00 am, August 22, 2015. “regular” waves or oscillatory ﬂows with or without currents, wherein the damage (if any) to individual stones is determined using only the largest peak-to-peak velocity amplitude in the experienced cycles of oscillatory ﬂow. The wave trains generated in wave ﬂumes sometimes included far larger waves, and Ua sometimes occurred in a wave different from that with the maximum velocity Um (thus umax < Um). Nevertheless, the damage ratio was closely related to Ua or half the maximum peak-to-peak velocity amplitude, supporting the above assumption. The experiments in the present study employed the largest stones possible to minimize the scale eﬀect and therefore did not use stones of diﬀerent size groups. The experiments were conducted instead with diﬀerent larger scales of oscillatory ﬂows. In general, the maximum force coeﬃcient for an object in an oscillatory ﬂow decreases with the increase in KC (Chakrabarti and Armbrust, 1987; Grace and Zee, 1981; Keulegan and Carpenter, 1958). This was consistent with the result indicating that the predictor ϕ for the damage ratio of stones was a decreasing function of KC. Additionally, KC can be deﬁned, from a physical perspective, as the ratio of the moving distance of a water particle in a half cycle of the oscillatory ﬂow to the object diameter. The reasonable goodness-of-ﬁt of the resulting model, i.e., Eq. (6), may support the assertion that the model is applicable over a wide range of stone sizes. With respect to the scale eﬀect on the rock stability, no apparent dependence of the damage ratio on the Reynolds number was found in the laboratory tests. The scale eﬀect due to the viscous force may become negligible at a high Reynolds number. Although no studies regarding the critical Reynolds number for isolated rocks have been conducted, the Reynolds number in this study (8 × 103–7 × 104) was higher than the lower limit of the critical values for the armor layer of rubble-mound structures, which range from 6 × 103–4 × 105 (Hughes, 1993). The stability of the armor layer depends on the scale eﬀect due to the viscous force of ﬂow through the porous structure (Hughes, 1993). This leads to the expectation that no scale eﬀects would occur in experimental tests on isolated rocks. The Reynolds number, deﬁned as umaxDn50/ν, was 3.3 × 106 for the maximum wave-induced velocity estimate from the ﬁeld test. This value was higher than the Reynolds number determined in the laboratory test by approximately two orders of magnitude. However, a reasonable agreement was found between the predicted stability and the observed stability in the ﬁeld. Therefore, crest velocity, but independent of the preceding trough velocity (Van Gent and Wallast, 2001; Tørum et al., 2010; Wallast and van Gent, 2002; Lorang, 2000; Goto et al., 2009). However, with the increase in velocity asymmetry, the semi-velocity amplitude Ua exhibited a more explanatory power for the stability of the stones than the maximum crest velocity Um, suggesting that the hydrodynamic forces increase not only with the crest velocity but also with the preceding trough velocity. This is consistent with the hydrodynamic force measurements conducted on marine pipelines, wherein the maximum force in a given velocity half cycle has been found to strongly depend on the magnitude of the velocity in the preceding half cycle because of the wake eﬀect (Lambrakos et al., 1987). The force coeﬃcients normalized with the square of the crest velocity substantially increase with the increase in the absolute ratio of the preceding trough velocity to the crest velocity. The wake eﬀect has been modeled to more accurately predict forces due to waves, even with superimposed steady current, acting on marine pipelines (Lambrakos et al., 1987). However, it is diﬃcult to develop a “wake force” model of irregularly shaped stones because it requires introducing numerous additional hydrodynamic coeﬃcients and determining the coeﬃcients from direct measurements of the forces acting on an object. To avoid such diﬃculty in modeling the hydrodynamic forces, the developed model given by Eq. (6) simply takes Ua as a representative velocity instead of Um, leading to a substantial improvement over the predictions obtained using the conventional method. The above ﬁnding is of great importance in the establishment of a reasonable formula for rock stability under asymmetric velocity ﬂow due to shallow waves because the minimum stable mass is fundamentally proportional to the sixth power of the representative velocity. For example, in the ﬁeld test, taking Ua = 2.5 m/s as the representative velocity instead of the maximum velocity (= 3.53 m/s) leads to the reduction of the minimum stable mass by a factor of (2.5/ 3.53)6 = 0.13. However, this ﬁnding is contrary to the well-known Morison equation, which expresses a time-varying wave force as the sum of the inertia force proportional to a and the drag force proportional to u2. Thus, for example, if the velocity skew umax/(umax − umin) increases with an increase in umax and an even more substantial decrease in |umin| (thus amax increases but Ua decreases), then the maximum force prediction from the Morison equation should increase while that predicted using Ua will decrease. The proposed model is based on the stability test conducted under 269 Coastal Engineering 140 (2018) 257–271 S. Kawamata et al. obtained using the conventional method, Eq. (1), was considerably overestimated. For the proper application of Eq. (6), it is necessary to determine three characteristic quantities of the wave-induced near-bed velocity, Ua, umax, and Tzp, as well as μ50. The three velocity quantities could be predicted using scale model experiments or numerical models that can simulate irregular wave trains, and μ50 could be predicted using photogrammetry of the surface geometry of the seabed using Eq. (11). The reasonable prediction of rock stability using Eq. (6), however, would still require a novel, reliable method to predict Ua and its associated parameters umax and Tzp under the action of random waves. it might be concluded that the developed equations are valid for predicting the stability of isolated quarry rocks in the ﬁeld. The equations for rock stability, Eqs. (6) and (11), assume rocks on relatively ﬂat beds whose roughness height is apparently lower than the rocks. If the bed roughness increases relative to the rocks size, the rocks might be more deeply embedded and thus more stabilized. In such a case, stabilization might also be enhanced by the accompanying decrease in rock protrusion (thus a decrease in drag). The empirical Eq. (6) also assumes a single isolated rock or an array of rocks spaced sparsely enough to eliminate ﬂow interference. By analogy with the maximum wave force on a slender pile group (Bonakdar et al., 2015), the wave force on a rock within a more closely spaced rock array can change from that on a single isolated rock, depending on the spacing relative to the rock diameter, the Keulegan–Carpenter number, rock arrangement, etc. In the ﬁeld test, quarry rocks were arrayed in a 5 × 2 arrangement and directed obliquely oﬀshore. The relative spacing was approximately 1.7 times the rock size, which was considerably closer than in the laboratory tests. Nevertheless, the interference eﬀect in the ﬁeld test was likely small not only because the change in the maximum force due to the side-by-side or oblique (referred to as “staggered” in reference (Bonakdar et al., 2015)) arrangement of slender piles is typically less than 10% for a relative spacing greater than 1.5, but also because the low height of the rocks may enable over-the-top ﬂow, resulting in less of an eﬀect on the surrounding side ﬂows than long vertical cylinders like piles that force all ﬂow around the sides. The developed method is more complex than the conventional method, requiring three characteristic quantities of near-bed oscillatory velocity (Ua, umax, and Tzp) as well as the median friction coeﬃcient between the quarry rock and the seabed. With respect to the friction coeﬃcient, an empirical equation for the friction angle, Eq. (11), was proposed in the present study. As the bed surface grain size can be easily obtained from photogrammetry of the surface geometry of the seabed, the friction angle and therefore the friction coeﬃcient can be estimated. Previously, ﬂow quantities have been studied as characteristic properties, describing the asymmetry of near-bed orbital velocities under shoaling and breaking waves (Dibajnia et al., 2001; Elfrink et al., 2006; Tajima and Madsen, 2002). However, no empirical formulae have been validated in terms of predicting the waveform from the maximum peak-to-peak velocity amplitude (thus Ua) under irregular wave trains. Reliable prediction of Ua is particularly important for reasonable prediction of rock stability using Eq. (6) because the minimum stable mass is proportional to Ua6 and because Ua might not be as large as umax due to the eﬀect of the sea bottom on the waveform in shallow waters. Further study is required to address this issue. At the moment, the only available techniques for waveform prediction may be scale model experiments and numerical models, such as the extended Boussinesq-type model BOUSS-2D (Nwogu and Demirbilek, 2001) and the non-hydrostatic model SWASH (Zijlema et al., 2011), which can eﬃciently compute the propagation and transformation of waves over large-scale coastal areas from oﬀshore to shore, and can accurately reproduce the variations in waves and wave-driven currents. Acknowledgments This study was performed within the framework of the research project “The sophistication of the methods for designing facilities of ﬁshing ports and grounds,” funded by the Fisheries Infrastructure Development Project, Japan Fisheries Agency. We thank Akihiko Moriguchi for assistance with laboratory work. We also thank Editage (www.editage.jp) for English language editing. We are grateful to two anonymous reviewers for their useful comments that helped improve the manuscript. List of symbols acceleration of ﬂuid maximum acceleration of ﬂuid in the individual oscillation cycle with Ua b experimental constant in Eq. (6) B50 median bed surface grain size c experimental constant in Eq. (6) C coeﬃcient of critical stable mass in Eq. (1) D size of stone on the bed Dcr critical value of Dn50 Dn50=(M50/ρs)1/3 median nominal diameter of stone g acceleration due to gravity KC Keulegan–Carpenter number, deﬁned by Eq. (5) M50 median stone mass Mcr critical stable mass of stone rd damage ratio rd,cr critical damage ratio R2 coeﬃcient of determination Re=umaxDn50/v Reynolds number u time varying velocity umax maximum velocity at the crest in the individual oscillation cycle with Ua umin minimum velocity at the trough in the individual oscillation cycle with Ua Ua maximum semi-velocity amplitude Um maximum peak velocity T period of individual oscillation cycle Tzp zero-to-peak period in the individual oscillation cycle with Ua α experimental constant in Eq. (6b) β experimental constant in Eq. (11) γ experimental constant in Eq. (11) Δ=ρs/ρ − 1 submerged speciﬁc density of the stone θ friction angle θ50 median friction angle μ=tanθ friction coeﬃcient μ50 median friction coeﬃcient ν kinematic viscosity of the ﬂuid ρ mass density of the ﬂuid ρs mass density of the stone ψ1 mobility index deﬁned by Eq. (2) ψ2 mobility index deﬁned by Eq. (3) ψ3 mobility index deﬁned by Eq. (4) a amax 7. Conclusions The results of the wave ﬂume and CWC experiments show that the damage ratio of isolated rocks was most closely related to ψ2 based on Ua, and not to ψ1 based on Um. The ﬁnding is particularly important for the accurate prediction of stable rock mass under the eﬀects of asymmetric waves in shallow waters. 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