Formulating Wave Overtopping at Vertical and Sloping Structures with Shallow Foreshores Using Deep-Water Wave Characteristics

Introduction

Methods

Description of Data Sets

A comprehensive collection of physical and numerical model data sets, with different foreshore slopes, structure slopes, offshore wave conditions, and relative water depths, were considered in this study. Data sets with measurements both offshore and at the structure toe were used to assess the influence of the foreshore on incident wave conditions, while data sets with overtopping measurements for normally incident waves over smooth sloping or vertical structures that met the criteria for very shallow conditions (h_toe/H_m0,deep ≤ 1) were used in the development of a new overtopping formulas.

The reference, identifier (ID), number of tests, and range of parameters for each data set are provided in Tables 1–3. Each of the selected data sets followed Fig. 1, in terms of layout, with uniformly sloping foreshore slopes under irregular, long-crested, and shore-normal wave attack. Rather than providing a lengthy and exhaustive description here, we focus on two data sets: (1) the Goda et al. (1975) data set (DS-802)—which makes up the bulk of the available vertical wall data; and (2) the Lashley et al. (2020a) data set (XB)—which, unlike the others, is comprised of purely numerical model results. For detailed descriptions of the other data sets, the reader is pointed to the references listed in Tables 1–3 for the details related to each individual test setup. The data sets with an ID prefix DS refer to tests obtained from the CLASH project (Crest Level Assessment of Coastal structures by full-scale monitoring, neural network prediction, and hazard analysis on permissible wave overtopping) database, see also EurOtop (2018). Note that some of the tests within the CLASH database were listed as confidential, therefore no references for these tests were available. Additionally, it should be noted that 1:1,000 foreshore slope in the CLASH database refers to a completely flat foreshore in reality.

Reference	ID	No. of tests	sm−1,0	htoe/Hm0,deep	cot (m)
—	Sal	13	0.018–0.039	1.53–5.19	30
Smith et al. (2003)	Smith	114	0.017–0.052	0.36–5.96	30; 100
Lashley et al. (2020a)	XB	384	0.006–0.041	0.05–5	10; 25; 50; 100; 250; 500; 1,000
Coates et al. (1997)	HR	161	0.008–0.058	0.46–4.63	10; 20; 30; 50
—	DS-005	15	0.016–0.028	1.4–1.67	100
—	DS-111	81	0.002–0.036	1.27–5.08	50
Van der Meer and de Waal (1993)	DS-221	148	0.011–0.059	0.97–6.87	100; flat
—	DS-307	18	0.042–0.043	0.98–2.28	250
—	DS-330	12	0.046–0.062	1.87–3.38	100
Pullen and Allsop (2004)	DS-509	18	0.048–0.09	1.08–1.60	50
—	DS-916	47	0.015–0.048	1.34–2.65	30

Reference	ID	No. of tests	som−1,0	htoe/Hm0,deep	$\cot (m)$	$\cot (\alpha )$	Rc/Hm0,deep
Altomare et al. (2016)	13–116	90 (60)	0.015–0.063	−0.06–0.22	35	3	0.40–0.82
	00–025	21 (17)	0.039–0.041	−0.14–0.25	35	2	0.24–0.67
	00–142	17	0.013–0.036	0.25–0.5	35	3	0.22–0.52
	13–168	42 (15)	0.007–0.018	0–0.86	50	2	0.26–2.55
Goda (2009) and Tamada et al. (2002)	Tam	198 (37)	0.019–0.049	0–0.71	10; 30	3; 5; 7	0.31–1.50
Van Gent (1999)	DS-226	97	0.018–0.053	0.32–2.56	100; 250	2.5; 4	1.1–2.9

Reference	ID	No. of tests	som−1,0	htoe/Hm0,deep	$\cot (m)$	Rc/Hm0,deep
Goda et al. (1975)	DS-802	127 (44)	0.005–0.044	−0.75–1	10; 30	0.40–1.92
Herbert (1993)	DS-028	24	0.018–0.049	0.49–0.97	10; 100	0.48–2.07
Bruce et al. (2002)	VOWS	10	0.027–0.072	0.81–0.99	10; 50	0.76–1.65

It is important to note that by using data sets collected from different facilities around the world, additional differences may arise due to model effects and different measurement techniques. Therefore, a certain degree of spread in the compiled data set is to be expected and accepted as inherent in the approach taken.

Goda et al. (1975) Data Set

Goda et al. (1975) carried out a series of experiments in 1973 and 1974 with irregular waves (Bretschneider–Mitsuyasu type spectrum), with H_1/3 = 0.15 m and T_1/3 = 1.7, 2.3, and 2.8 s, overtopping a vertical structure fronted by 1:10 and 1:30 sloping foreshores. Incident wave heights were estimated with a technique to resolve incident and reflected waves using two simultaneous wave records. The overtopping rate was obtained as the average of three measurements, each for 200 waves.

The original data set comprised consisted of 205 tests with conditions at the toe varying from shallow to emergent, where the toe of the structure was initially dry (h_toe ≤ 0). Here, we analyze a subset of these tests with h_toe/H_m0,deep ≤ 1 (DS-802, Table 3).

Goda et al. (1975) noted that during the physical tests, the amplitude of the IG waves [estimated at the time using Eq. (1)] and the level of wave-induced setup were higher than those typically observed at a real coast. This was attributed to the re-reflection of waves between the vertical wall and the wave generation paddle, the magnitude of which varied depending on tan (m) and h_toe/H_m0,deep. This modification was based on engineering judgment, not measurements.

To compensate for this, Goda et al. (1975) increased the measured h_toe—and by relation reduced R_c—by the values shown in Table 4 based on engineering judgment, not measurements. These corrected values were then used to derive the design diagrams presented in Goda (2000) which have since been adopted in the technical standard for the design of port and harbor facilities in Japan. To be consistent with this analysis, we have applied the same correction to the original data here.

htoe (cm)	tan (m) = 1:30			tan (m) = 1:10
	T1/3 = 1.7 s	T1/3 = 2.3 s	T1/3 = 2.8 s	T1/3 = 1.7 s	T1/3 = 2.3 s	T1/3 = 2.8 s
22.5	1.5	1.5	1.5	1.5	1.5	1.5
15.0	1.5	1.5	1.5	1.5	1.5	1.5
10	1.5	1.5	1.5	1.5	1.5	1.5
5	1.5	1.5	1.5	1.5	2	2
0	1.5	1.5	1.5	2	2.5	2.5
−5	1.5	1.5	1.5	2	2.5	2.5
−10	—	—	—	2	2.5	2.5

Lashley et al. (2020a) Data Set

After verifying the numerical model's ability to accurately simulate wave transformation under very shallow conditions, Lashley et al. (2020a) carried out 672 simulations in prototype scale—using the XBeach nonhydrostatic numerical model (Kingsday release)—of irregular waves [Joint North Sea Wave Project (JONSWAP) type with a peak enhancement factor of 3.3] propagating over an idealized structure-foreshore profile (Fig. 1). In these simulations, H_m0,deep, T_p, wave directional spreading, h_toe, m, bottom friction, width of vegetated cover, and structure slope (α) were systematically varied, following a one-[factor]-at-a-time (OAT) approach. In the present study, we analyze a part of this data set, without the influence of vegetation, directional spreading, bottom friction, or wave reflection (i.e., with the structure removed). In each simulation, H_m0,toe, T_m−1,0,toe, wave-induced setup at the structure toe $(\overline{\eta })$, and the relative magnitude of the IG waves, defined as the ratio of IG to SS waves at the structure toe (H_m0,IG,toe/H_m0,SS,toe), were assessed.

In each simulation, the offshore water depth was set to four times H_m0,deep, in line with the definition of deep water proposed by Hofland et al. (2017). The numerical model was configured such that the numerical parameter governing the maximum breaking wave steepness was set to 0.5; a smooth flume bottom was represented using a Manning roughness coefficient of 0.012 s/m^1/3; and a cross-shore grid spacing varying from a maximum of 100-grid cells per deep-water wavelength (offshore) to 1 m at the toe.

Results and Discussion

Foreshore Effect on Nearshore Conditions

In order to establish accurate wave overtopping formulas based on deep-water wave characteristics, the effects of shallow foreshores on wave conditions at the toe of the structure need to be accurately parameterized. In this section, we demonstrate that the foreshore effects on nearshore conditions can be accurately modeled as functions of h_toe/H_m0,deep, tan (m) and s_om−1,0. The nearshore processes considered are, namely: (1) the change in significant wave height due to shoaling and breaking; and (2) the increase in static and dynamic wave setup (i.e., the magnitude of IG waves).

Significant Wave Height

Under very shallow conditions (h_toe/H_m0,deep ≤ 1), two trends in Fig. 2 become evident: (1) H_m0,toe/H_m0,deep decreases linearly as h_toe/H_m0,deep decreases; and (2) H_m0,toe/H_m0,deep increases as cot (m) becomes steeper (visible by the marker shading in Fig. 2). The area where 1 < h_toe/H_m0,deep ≤ 1.5 appears to be a transition region where the foreshore—represented by h_toe/H_m0,deep and tan (m)—shows a minor influence on H_m0,toe/H_m0,deep.

In addition, Fig. 3 indicates that the influence of the deep-water wave steepness (s_om−1,0) on H_m0,toe/H_m0,deep decreases as h_toe/H_m0,deep decreases, made evident by the reduced scatter at lower h_toe/H_m0,deep values. Note that s_om−1,0 is used here in place of s_op in accordance with the current standard, where T_m−1,0 is used in place of T_p or T_1/3 (EurOtop 2018); for conversion, we take T_m−1,0 = T_p/1.1.

For conditions where h_toe/H_m0,deep ≤ 1, the following expression holds with R² = 0.84 and SCI = 0.18 (Fig. 4):

$${\displaystyle \frac{H_{m0,\mathrm{toe}}}{H_{m0,\mathrm{deep}}}=M\cdot {\displaystyle \frac{h_{\mathrm{toe}}}{H_{m0,\mathrm{deep}}}+C}}$$

(16)

where

$$M=0.35\cdot {\displaystyle \frac{\tan {(m)}^{0.10}}{s_{om-1,0}^{0.20}}}$$

(17)

and,

$$C=0.95\cdot \tan (m{)}^{0.15}-0.30$$

(18)

Eqs. (16)–(18) were derived based on the observed linear relationship between H_m0,toe/H_m0,deep and h_toe/H_m0,deep where the slope (M) and intercept (C) of the relationship are dependent on tan (m) and s_om−1,0. The exponents of each term were then obtained using a trial-and-error approach to minimize scatter in the data (Table 1). It should be highlighted that H_m0,toe/H_m0,deep ≠ 0 when h_toe/H_m0,deep = 0 due to the influence of static and dynamic setup (IG waves), discussed in the following section and also highlighted by Goda (2000). However, experience suggests that care should be taken for cases with h_toe/H_m0,deep = 0, as the bed may intermittently become dry.

As noted in the “Description of Data Sets” section of the “Methods”, some scatter in Fig. 4 is to be expected due to model effects and differences in measurement techniques between data sets. In particular, the low-frequency cut-off used to calculate H_m0 could significantly influence results under very shallow conditions where IG waves dominate.

Through Eq. (16), the wave height at the structure may be estimated using parameters that are usually either known or estimated without difficulty: the offshore wave height, offshore steepness, foreshore slope, and relative water depth at the structure toe. This is also directly in line with the work of Hofland et al. (2017) who showed that the relative wave period at the toe (T_m−1,0,toe/T_m−1,0,deep) can be empirically modeled, with reasonable accuracy, as a function of h_toe/H_m0,deep and tan (m). The main disadvantage of such an empirical approach is the assumption of a straight (uniform) foreshore slope. However, this disadvantage is seen as minor compared with the use of numerical models that do not include IG waves in very shallow water, e.g., SWAN.

Given the differences in the approach of Goda (2000)—namely the use of H_1/3 and not H_m0, and the treatment of the IG waves as an increase in mean water level versus directly including them in the wave height estimate, as done here—our comparison of Eqs. (3) and (16) is purely qualitative. Both equations capture the linear relationship between h_toe/H_m0,deep and H_m0,toe/H_m0,deep and the increase in H_m0,toe/H_m0,deep with steeper slopes. The main difference between the two approaches is the treatment of the deep-water wave steepness (s_om−1,0). Eq. (3) shows parallel lines for different values of s_om−1,0, while Eq. (16) converges as h_toe/H_m0,deep decreases (Fig. 3). This convergence was observed in the data (Fig. 3) and is due to the depth-limited nature of shallow water waves. That is, as the water depth becomes shallower, the influence of s_om−1,0 decreases and the magnitude of H_m0,toe is now governed by h_toe.

The observed convergence is also supported by the linear wave theory, which states that waves become less (frequency) dispersive as the water depth becomes shallower—that is, the influence of the wave period (and by extension s_om−1,0) on nearshore wave conditions decreases as h_toe/H_m0,deep decreases. This is also made evident in Fig. 5 by the decrease in scatter with shallower water depths. Of particular note is the correspondence between h_toe/H_m0,deep ≤ 1 and the definition of shallow water according to linear wave theory, where the ratio of the local water depth to wavelength (h_toe/L_toe) < 1/20 (Fig. 5), and L_toe is obtained by solving the well-known dispersion relationship.

Static and Dynamic Wave Setup

In addition to the relative wave height and period at the toe, h_toe/H_m0,deep and tan (m) serve as descriptors for both the magnitude of relative (static) wave setup [$\overline{\eta }/H_{m0,\mathrm{deep}}$, Fig. 6(a)] and the dynamic wave setup—represented by the relative magnitude of the IG waves at the toe [H_m0,IG,toe/H_m0,SS,toe, Fig. 6(b)]—generated due to SS waves shoaling and breaking over the foreshore. Furthermore, both quantities appear to reach their maximum because h_toe/H_m0,deep approaches zero (Fig. 6). The trend in $\overline{\eta }/H_{m0,\mathrm{deep}}$ is also supported by the work of Goda (2000) which was obtained by digitizing and interpolating between the curves of Fig. 3.25 of the same reference, although XBeach predictions are consistently lower than that of Goda (2000). These differences are likely due to the definition of $\overline{\eta }$: Goda (2000) refers to $\overline{\eta }$ at the shoreline, i.e., the mean water level in the swash zone (where $\overline{\eta }$ reaches its maximum), while XBeach estimates of $\overline{\eta }$ were taken in the surf zone (Lashley et al. 2020a).

As Eq. (1) does not consider the influence of the foreshore slope, it is more valuable to assess the best-fit trend of H_m0,IG,toe—predicted using Eqs. (1) and (2)—normalized by H_m0,SS,toe (predicted by XBeach). Remarkably, the best-fit trend of Eqs. (1) and (2) agrees well with the XBeach model results [Fig. 6(b)].

One important takeaway is that the influence of the foreshore only becomes significant once h_toe/H_m0,deep ≤ 1; that is, all of the nearshore processes considered here: change in wave height (Figs. 2 and 3), change in wave period (Hofland et al. 2017), wave setup [Fig. 6(a)] and shift in energy to low frequencies [Fig. 6(b)] show high correlations with h_toe/H_m0,deep and tan (m) when h_toe/H_m0,deep ≤ 1. So, in short, if h_toe/H_m0,deep > 1, then the foreshore may be neglected in the analysis and the EurOtop (2018) approach is practical. However, if h_toe/H_m0,deep ≤ 1, wave shoaling and breaking become significant and a more accurate approach would be that of Goda et al. (1975).

Development of Overtopping Formulas

In the “Foreshore Effect on Nearshore Conditions” section of the “Results and Discussion,” we demonstrated and discussed how the effects of the foreshore on nearshore conditions can be accurately represented as functions of h_toe/H_m0,deep and tan (m). This suggests that wave overtopping, which is typically estimated as a function of nearshore parameters (at the toe of the structure), may actually be represented as a function of deep-water parameters when h_toe/H_m0,deep ≤ 1. In line with the approach of Goda et al. (1975), also published in Goda (2000), we propose the following for vertical structures:

$${\displaystyle \frac{q}{\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}}=f\left({\displaystyle \frac{h_{\mathrm{toe}}}{H_{m0,\mathrm{deep}}},{\displaystyle \frac{R_c}{H_{m0,\mathrm{deep}}},s_{om-1,0},\tan (m)}}\right)}$$

(19)

Likewise, for sloping structures:

$$\begin{array}{r}{\displaystyle \frac{q}{\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}}=f\left({\displaystyle \frac{h_{\mathrm{toe}}}{H_{m0,\mathrm{deep}}},{\displaystyle \frac{R_c}{H_{m0,\mathrm{deep}}},s_{om-1,0},\tan (m),\tan (\alpha )}}\right)}\end{array}$$

(20)

Vertical Structures

The data sets listed in Table 3 show a negative exponential relationship between relative discharge and relative freeboard (Fig. 7). This typical relationship is evident by the linear increase in $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$ (logarithmic y-axis) with decreasing freeboard (see light to dark gradations in shading). On the contrary, $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$ shows a positive and much more dynamic relationship with h_toe/H_m0,deep. Visual inspection of the data and original Goda et al. (1975) design diagrams revealed three distinct regimes: (1) a very shallow regime (0.5 ≤ h_toe/H_m0,deep ≤ 1), where a steep linear relationship exists between $\log \left(q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}\right)$ and h_toe/H_m0,deep; (2) an extremely shallow or emergent regime (h_toe/H_m0,deep ≤ 0.1), where the relationship is much gentler—suggesting a reduced dependence; and (3) a transition region between the two regimes (0.1 < h_toe/H_m0,deep < 0.5), where the rate of reduction $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$ with change in h_toe/H_m0,deep is maximum. Notably, this transition region is centered on the threshold between very and extremely shallow water, h_toe/H_m0,deep = 0.3 (Hofland et al. 2017).

As noted in the “Foreshore Effect on Nearshore Conditions” section and Fig. 6, both static wave setup and dynamic setup—that is, the slow periodic variations in mean water level due to IG waves—increase considerably as h_toe/H_m0,deep decreases. These static and periodic changes in mean water level equate to a reduction in R_c/H_m0,deep which results in higher-than-expected overtopping for cases with h_toe/H_m0,deep ≤ 0.1, as also observed by Goda et al. (1975).

Based on the aforementioned trends, the proposed formulas for vertical structures each have the following basic form:

$${\displaystyle \frac{q}{\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}}=a\cdot \exp \left(-b\cdot {\displaystyle \frac{R_c}{H_{m0,\mathrm{deep}}}+c\cdot {\displaystyle \frac{h_{\mathrm{toe}}}{H_{m0,\mathrm{deep}}}}}\right)}$$

(21)

where the coefficients a, b, and c are each a function of s_om−1,0 and m. For very shallow cases (Regime 1), with 0.5 ≤ h_toe/H_m0,deep ≤ 1:

$$a_1=0.90\cdot {\displaystyle \frac{\tan {(m)}^{2.05}}{s_{om-1,0}^{0.20}}}$$

(22)

$$b_1=5.10\cdot {\displaystyle \frac{s_{om-1,0}^{0.25}}{\tan {(m)}^{0.15}}}$$

(23)

and

$$c_1=0.70\cdot {\displaystyle \frac{s_{om-1,0}^{0.10}}{\tan {(m)}^{0.55}}}$$

(24)

For extremely shallow or emergent cases (Regime 2), with h_toe/H_m0,deep ≤ 0.1:

$$a_2=0.09\cdot {\displaystyle \frac{\tan {(m)}^{2.35}}{s_{om-1,0}^{1.25}}}$$

(25)

$$b_2=5.40\cdot {\displaystyle \frac{s_{om-1,0}^{0.30}}{\tan {(m)}^{0.45}}}$$

(26)

and

$$c_2=0.75\cdot {\displaystyle \frac{s_{om-1,0}^{0.50}}{\tan {(m)}^{0.60}}}$$

(27)

For the transition between the two regimes (0.1 < h_toe/H_m0,deep < 0.5), the user should interpolate exponentially:

$$y_3=y_2\cdot {\left({\displaystyle \frac{y_1}{y_2}}\right)}^{(x_3-x_2)/(x_1-x_2)}$$

(28)

where x and y are stand-ins for h_toe/H_m0,deep and $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$, respectively; and the subscripts 1, 2, and 3 refer to the first, second, and transition (third) regimes, respectively.

Qualitatively speaking, Eqs. (21)–(28) match both the data and the original Goda et al. (1975) design curves—which were obtained by digitizing the diagrams as presented in Goda (2000) (Figs. 7 and 8). Additionally, although Eqs. (21)–(28) were developed for conditions where h_toe/H_m0,deep ≤ 1, the overtopping discharge for cases with 1 < h_toe/H_m0,deep ≤ 1.5 may also be well-estimated by taking the predicted maximum of Eqs. (21)–(28) (i.e., for h_toe/H_m0,deep = 1) (Figs. 7 and 8). It should also be noted that most of the data used to derive the EurOtop (2018) formulas for vertical structures start in this shallow range and extend to deeper conditions (h_toe/H_m0,deep > 1). This further highlights the significance of the Goda et al. (1975) data set and the need for formulas—such as Eqs. (21)–(28)—for shallower conditions.

Quantitatively, Eqs. (21)–(28) are unbiased with minor scatter, yielding an ${\overline{x}}_G$ value of 1.02 and $\sigma ({\overline{x}}_G)$ of 1.68. If we consider wave overtopping to be normally distributed, then 90% of the predicted overtopping discharge would be located within a range of values between 0.43 and 2.40 times the measured overtopping discharge. Furthermore, Fig. 9 shows that the ratio of the predicted [using Eqs. (21)–(28)] to measured overtopping discharge (q_pred/q_meas) is relatively uniform for varying values of R_c/H_m0,deep, h_toe/H_m0,deep, tan (m) and s_om−1,0. This condition, referred to as homoscedasticity, suggests that the accuracy of Eqs. (21)–(28) is not dependent on any single parameter. Fig. 9 also shows the parameter ranges used in the derivation of Eqs. (21)–(28), and hence, the ranges within which they can be reliably applied.

Sloping Structures

Using the same approach as outlined in the previous section “Vertical Structures” and the data sets listed in Table 2, the formulas derived for sloping structures with shallow foreshores follow:

$${\displaystyle \frac{q}{\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}}=d\cdot \exp \left(-e\cdot {\displaystyle \frac{R_c}{H_{m0,\mathrm{deep}}}+f\cdot {\displaystyle \frac{h_{\mathrm{toe}}}{H_{m0,\mathrm{deep}}}}}\right)}$$

(29)

For very shallow cases (Regime 1), with 0.5 ≤ h_toe/H_m0,deep ≤ 1:

$$d_1=1.90\cdot s_{om-1,0}^{1.15}$$

(30)

$$e_1=7.40\cdot {\displaystyle \frac{s_{om-1,0}^{0.60}}{\tan {(m)}^{0.25}\cdot \tan {(\alpha )}^{0.60}}}$$

(31)

and,

$$f_1=0.70\cdot {\displaystyle \frac{\tan {(m)}^{0.80}}{s_{om-1,0}^{0.80}}}$$

(32)

For extremely shallow or emergent cases (Regime 2), with h_toe/H_m0,deep ≤ 0.1:

$$d_2=1.35\cdot \tan (m{)}^{0.35}\cdot s_{om-1,0}^{0.85}$$

(33)

$$e_2=3.75\cdot {\displaystyle \frac{s_{om-1,0}^{0.70}}{\tan {(m)}^{0.70}\cdot \tan {(\alpha )}^{0.60}}}$$

(34)

and,

$$f_2=0.20\cdot {\displaystyle \frac{s_{om-1,0}^{0.35}}{\tan {(m)}^{1.30}}}$$

(35)

with exponential interpolation between the two regimes (0.1 < h_toe/H_m0,deep < 0.5).

Eqs. (29)–(35) agree reasonably well with the data for sloping structures (Fig. 10) which show similar trends to those observed for vertical structures (Fig. 8). Figs. 11 and 12 also show no trend of increasing (or decreasing) scatter with changes in R_c/H_m0,deep, h_toe/H_m0,deep, tan (m), s_om−1,0, or tan (α), suggesting that homoscedasticity is maintained. It should be noted that due to the very small scale of the Tamada et al. (2002) data—with 0.027 m ≤ H_m0,deep ≤ 0.068 m—there is larger inherent scatter in the observed overtopping discharge, compared with the other data sets (Figs. 11 and 12). Nevertheless, the Tamada et al. (2002) data set has been considered reliable because it was previously used to derive overtopping formulas (Goda 2009; Mase et al. 2013; Yuhi et al. 2020) and draw design diagrams [Figs. 5.8–5.10 in Goda (2010)].

Compared with the existing EurOtop (2018) approach [Eqs. (7)–(11)], Eqs. (29)–(35) show higher accuracy and a wider range of applicability. With ${\overline{x}}_G=1.01$ and $\sigma ({\overline{x}}_G)=1.90$, Eqs. (29)–(35) may be considered both accurate and unbiased, and if we consider wave overtopping to be normally distributed, then 90% of the predicted overtopping discharge would be located within a range of values between 0.35 and 2.92 times the measured overtopping discharge. On the contrary, the EurOtop (2018) formulas show a negative bias $({\overline{x}}_G=0.68)$ and much larger scatter $[\sigma ({\overline{x}}_G)=3.84]$—scatter which increases as the foreshore slope becomes steeper (Fig. 13). This is to be expected since the formulas were derived mainly for relatively mild foreshore slopes (Altomare et al. 2016). If we only consider the cases with $\cot (m)>35$, the accuracy of Eqs. (7)–(11) increases significantly (${\overline{x}}_G=1.11$ and $\sigma ({\overline{x}}_G)=2.11$), thereby confirming their inapplicability to steeper foreshore slopes.

The new formulas proposed here [Eqs. (29)–(35)] may therefore be seen as an attractive alternative to Eqs. (7)–(11), when obtaining nearshore parameters is either impractical or would otherwise result in unwanted uncertainty. Furthermore, as the existing EurOtop (2018) formulas were mainly developed for relatively mild slopes $(\cot (m)\ge 35)$; Eqs. (29)–(35)—which consider foreshore slopes as steep as 1:10—may prove further advantageous.

Physical Explanation of the Proposed Empirical Coefficients

The coefficients of Eqs. (21)–(35) were established based on the observed influence of s_om−1,0, tan (α) and tan (m) on the mean overtopping discharge, while the values of their exponents were obtained using a trial-and-error approach to minimize scatter. Although the coefficients were the result of empirical fitting, their functional forms are in line with existing approaches which make use of a breaker parameter—which combines s_om−1,0, with either tan (α) (Van Gent 1999) or tan (m) (Bruce et al. 2004). The main difference here is the direct inclusion of both tan (m) and tan (α) compared with existing approaches that use a single imaginary or equivalent slope (Altomare et al. 2016; Mase et al. 2013).

Eqs. (21)–(35) show that $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$ increases as s_om−1,0 decreases (Fig. 14), consistent with the notion that longer waves yield larger overtopping volumes (EurOtop 2018). Likewise, the formulas also show that $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$ increases as the foreshore and structure slopes become steeper (Fig. 15), consistent with the early findings of Goda et al. (1975) and Owen (1980), respectively.

Considering the full range of structure slopes considered (2 ≤ cot (α) ≤ 7, Table 2), Figs. 14 and 15 show that $q/\sqrt{g\cdot H_{m0,\mathrm{deep}}^3}$ is maximum at cot (α) = 2, but then reduces significantly for vertical structures (cot (α) = 0). Based on the work of Victor and Troch (2012) and Section 5.3.3 of EurOtop (2018) for less shallow water—who found that q indeed decreased from cot (α) = 2 to cot (α) = 0—interpolation between the formulas for vertical structures [Eqs. (21)–(28)] and those for sloping structures [Eqs. (29)–(35)] is recommended for very steep sloping structures (0 < cot (α) < 2). Curves for cot (α) = 1, obtained using this approach, are also shown in Figs. 14 and 15.

Conclusions

Data Availability Statement

Acknowledgments

Notation

C_ηη

wave-variance density (m²/Hz);

g

gravitational constant of acceleration (m²/s);

H_1/3

significant wave height based on zero-crossing analysis (m);

H_m0

significant wave height, based on spectral moments = $4\sqrt{{\int }_0^{\mathrm{\infty }}{C}_{\eta \eta }df}$ (m);

H_m0,deep

significant wave height offshore in deep water (m);

H_m0,toe

significant wave height at the structure toe (m);

h_toe

initial water depth at the structure toe (m);

L_m−1,0

wave length in deep water based on the spectral wave period (m);

m

foreshore slope angle (°);

q

mean wave overtopping discharge (m³/s/m);

R²

coefficient of determination (—);

R_c

crest freeboard (m);

R_u2%

2% exceedance wave run-up (m) with respect to the number of incident waves;

s_om−1,0

deep-water wave steepness based on the spectral wave period (—);

s_op

deep-water wave steepness based on the peak wave period (—);

T_1/3

significant wave period (s) ≈ T_p/1.04;

T_m−1,0,deep

spectral wave period in deep-water (s) ≈ T_p/1.1;

T_m−1,0,toe

spectral wave period at the structure toe (s);

T_p

peak wave period in deep water (s);

${\overline{x}}_G$

geometric mean (—);

α

structure slope angle (°);

δ

equivalent slope, following Altomare et al. (2016) (—);

$\overline{\eta }$

wave-induced setup (m);

ξ_m−1,0

breaker index (Iribarren number) based on the spectral wave period and significant wave height at the structure toe (—); and

$\sigma ({\overline{x}}_G)$

geometric standard deviation (—).

References