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

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 htoe/Hm0,deep, tan (m) and som−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 HeightUnder very shallow conditions (htoe/Hm0,deep ≤ 1), two trends in Fig. 2 become evident: (1) Hm0,toe/Hm0,deep decreases linearly as htoe/Hm0,deep decreases; and (2) Hm0,toe/Hm0,deep increases as cot (m) becomes steeper (visible by the marker shading in Fig. 2). The area where 1 < htoe/Hm0,deep ≤ 1.5 appears to be a transition region where the foreshore—represented by htoe/Hm0,deep and tan (m)—shows a minor influence on Hm0,toe/Hm0,deep.In addition, Fig. 3 indicates that the influence of the deep-water wave steepness (som−1,0) on Hm0,toe/Hm0,deep decreases as htoe/Hm0,deep decreases, made evident by the reduced scatter at lower htoe/Hm0,deep values. Note that som−1,0 is used here in place of sop in accordance with the current standard, where Tm−1,0 is used in place of Tp or T1/3 (EurOtop 2018); for conversion, we take Tm−1,0 = Tp/1.1.For conditions where htoe/Hm0,deep ≤ 1, the following expression holds with R2 = 0.84 and SCI = 0.18 (Fig. 4): (16) Hm0,toeHm0,deep=M⋅htoeHm0,deep+C where (17) M=0.35⋅tan(m)0.10som−1,00.20 and, (18) C=0.95⋅tan(m)0.15−0.30Eqs. (16)–(18) were derived based on the observed linear relationship between Hm0,toe/Hm0,deep and htoe/Hm0,deep where the slope (M) and intercept (C) of the relationship are dependent on tan (m) and som−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 Hm0,toe/Hm0,deep ≠ 0 when htoe/Hm0,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 htoe/Hm0,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 Hm0 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 (Tm−1,0,toe/Tm−1,0,deep) can be empirically modeled, with reasonable accuracy, as a function of htoe/Hm0,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 H1/3 and not Hm0, 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 htoe/Hm0,deep and Hm0,toe/Hm0,deep and the increase in Hm0,toe/Hm0,deep with steeper slopes. The main difference between the two approaches is the treatment of the deep-water wave steepness (som−1,0). Eq. (3) shows parallel lines for different values of som−1,0, while Eq. (16) converges as htoe/Hm0,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 som−1,0 decreases and the magnitude of Hm0,toe is now governed by htoe.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 som−1,0) on nearshore wave conditions decreases as htoe/Hm0,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 htoe/Hm0,deep ≤ 1 and the definition of shallow water according to linear wave theory, where the ratio of the local water depth to wavelength (htoe/Ltoe) < 1/20 (Fig. 5), and Ltoe is obtained by solving the well-known dispersion relationship.Static and Dynamic Wave SetupIn addition to the relative wave height and period at the toe, htoe/Hm0,deep and tan (m) serve as descriptors for both the magnitude of relative (static) wave setup [η¯/Hm0,deep, Fig. 6(a)] and the dynamic wave setup—represented by the relative magnitude of the IG waves at the toe [Hm0,IG,toe/Hm0,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 htoe/Hm0,deep approaches zero (Fig. 6). The trend in η¯/Hm0,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 η¯: Goda (2000) refers to η¯ at the shoreline, i.e., the mean water level in the swash zone (where η¯ reaches its maximum), while XBeach estimates of η¯ 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 Hm0,IG,toe—predicted using Eqs. (1) and (2)—normalized by Hm0,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 htoe/Hm0,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 htoe/Hm0,deep and tan (m) when htoe/Hm0,deep ≤ 1. So, in short, if htoe/Hm0,deep > 1, then the foreshore may be neglected in the analysis and the EurOtop (2018) approach is practical. However, if htoe/Hm0,deep ≤ 1, wave shoaling and breaking become significant and a more accurate approach would be that of Goda et al. (1975).



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