Investigations into Relationship Between Significant Wave Height and Mean Wave Period in Nearshore Straits Under Typhoon Conditions

Nearshore waterways, such as straits, channels, and estuaries, accommodate various types of coastal transportation infrastructure [1]. Large waves generated by typhoons in the Pacific Ocean or hurricanes in the Atlantic Ocean [2] cause excessive hydrodynamic loads that threaten structural safety and significantly shorten the lifetime of such structures [3]; for example, severe damage to coastal bridges was reported during Hurricane Ivan in 2004 [4], Hurricane Katrina in 2005 [5], and Hurricane Ike in 2008 [6]. Therefore, a reliable assessment of wave conditions in nearshore waterways prone to tropical cyclones is highly important for designing typhoon-resistant coastal and sea-crossing infrastructure and mitigating the impacts of potential disasters.

In the current design practice, site-specific wave conditions corresponding to a specific range of mean return periods (MRPs), such as 1, 5, 50, and 100 years, are used to characterize storm-induced wave loads for the design and performance assessment of coastal and offshore structures [7–10]. To determine the correlation among different wave parameters, it is necessary to perform a rigorous statistical analysis of geographically referenced long-term measured or simulated wave data. Therefore, dependence analysis and database information are two fundamental factors in estimating typhoon-induced design wave conditions.

The data from in situ buoys are typically geographically sparse due to a lack of instrumentation and are limited in temporal coverage because of the relatively infrequent occurrence of typhoons. The data collected during measurement campaigns are normally contaminated with missing values and outliers due to a low signal-to-noise ratio, instrumental failures, or transmission errors during extreme events [11,12]. Therefore, data obtained via numerical hindcasting have become a common supplemental source for performing dependence analysis and long-term extrapolation [13].

Early definitions of the dependence between the wave height and period were characterized by either an empirical scatter table or a joint probability distribution of the relevant parameters [13–15]. The former definition provides the cumulative occurrences of the wave parameters over different intervals, which is fundamental for joint probability formulation. However, detailed values for specific sea states are not always available if only an associated scatter table is provided without a conversion method [16]. The latter definition is based on both mathematical and probabilistic techniques, which are used to estimate the correlations; thus, this approach is infeasible for the early phases of design at a site with insufficient data. Consequently, this definition is generally used as a rule of thumb in real-time practical circumstances [17]. To avoid certain difficulties, the current design standards and recommended practices for coastal and offshore structures [7–10] stipulate regression-based internal and external parametrizations of governing variables. The internal scheme relates the bulk measures (extreme/significant/mean) within each defining characteristic (height or period), e.g., $T¯∝Tp$ or $Hmax∝Hs$⁠. The external approach expresses the relationship between the bulk measures (extreme/significant/mean) of height and period, e.g., $T¯∝Hscoeff$⁠. This provides a convenient basis for the gross estimation of the expected or variation range of design characteristics in early design stages in areas with insufficient data. Following this approach, scholars have established internal [18–20] and external [21–23] empirical relationships for different locations worldwide. A comparison of the recommended and proposed relationships reveals high sensitivity to geographic and metocean factors. Such variability has been shown to have a significant effect on the performance of marine structures that are dynamically sensitive, i.e., structures with natural frequencies very close to the frequencies of waves [24]. This highlights the importance of defining site-specific internal and external relationships for coastal waterways prone to typhoon metocean conditions, where dynamically sensitive coastal infrastructure, such as large and long-span sea-crossing bridges [3], are planned or currently in operation.

The objective of this study is to identify nearshore straits susceptible to typhoons by developing and validating the internal and external relationships between wave height and period. The remainder of the article is organized as follows. Section 2 introduces detailed information for two operational sites. Section 3 provides the numerical background, configuration for the studied areas, setup, and validation of the coupled SWAN (Simulating WAves Nearshore) + ADCIRC (ADvanced CIRCulation) model as well as the measured data and their postprocessing procedure. Section 4 presents the methods used to determine the internal and external relationships between bulk parameters and comparisons with the relations developed in the literature and recommended by design codes. Additionally, the validation of the developed relationships is performed, and their application in the design process is explored. The summary and conclusions are given in Sec. 5.

Two operational nearshore straits along the east coast of China were selected as the study areas, as shown in Fig. 1. The Pingtan Strait, where a 16.32-km-long sea-crossing cable-stayed bridge connects Fuzhou city to Pingtan Island, was selected as the “reference” site to determine the relationship between the wave height and period. The Xihoumen Strait, which contains a newly constructed 2.664-km sea-crossing cable-stayed suspension bridge, was selected as the “validation” site to examine and validate the relationships developed at the reference site. The seabed at the selected straits is rugged and complex, with slopes and valleys. Both sites are surrounded by many islands and reefs, which implies a limited fetch in most directions. Figure 2 shows that the dominant fetch window at the validation site is oriented between west‒northwest and north‒northeast, whereas it is oriented between east and north‒northeast at the reference site. This makes the validation site more sheltered against remotely propagated destructive typhoon-generated waves from the western Pacific Ocean; hence, a less severe wave environment is present at the validation site than at the reference site under certain conditions (see Sec. 4.3). Overall, wave generation and propagation processes and, as a result, wave characteristics can be largely affected by the complex seabed and geographical terrain differences in the study areas.

In this section, two sets of short-term measured and long-term simulated wave datasets are presented. The objective is to use these data to develop and validate internal and external relationships between wave height and period so that one can define typhoon-generated design mean wave conditions in nearshore straits with limited fetch and shallow water conditions on the east coast of China.

The first set of data includes short-term in situ measurements taken at both sites. These data are used to establish internal relationships between height- and period-defining parameters. On the basis of the historical typhoon information provided by the Fujian Marine Forecasting Observatory² and China National Meteorological Center,³ more than 100 typhoons have influenced the studied areas [13,25,26], 5 of which significantly affected the selected sites during the measurement campaigns. These events were Typhoon Soudelor (No. 201513) and Dujuan (No. 201522) at the Pingtan Strait (solid track lines in Fig. 3) and Typhoon In-Fa (No. 202106), Typhoon Chanthu (No. 202114), and Typhoon Muifa (No. 202212) at the Xihoumen Strait (dashed track lines in Fig. 3). The time histories of the water surface elevation were obtained from two field wave measurement campaigns that were conducted in 2015 at the Pingtan Strait [11] and between 2021 and 2022 at the Xihoumen Strait. These data were recorded hourly for 17 min with a sampling interval of 0.5 s by SBY2-1 ultrasonic wave gauges, which have spatial and temporal measurement accuracies of ±0.2 m and ±0.25 s, respectively. The time histories were preprocessed through an iterative preprocessing framework developed in a previous study [11], which included error data cleaning, baseline trend removal, and missing data reconstruction. The wave data were subsequently obtained through spectral and zero-crossing analyses. The pre- and postprocessing procedure is briefly summarized in Fig. 4.

The second set of data is wave data from historical typhoons simulated via a coupled wave and circulation model, SWAN + ADCIRC [27]. The simulated data encompass a total of 58 typhoons [13] that affected the Pingtan Strait from 1990 to 2018 (Fig. 5(a)) and 49 typhoons [25,26] that affected the Xihoumen Strait from 1987 to 2018 (Fig. 5(b)); notably, these events either crossed or passed through these straits. These data are used to describe the external relationship between the mean wave period and the significant wave height. Next, the theoretical background and details of the numerical simulations are presented, with a focus on the SWAN and ADCIRC modeling schemes, the coupled simulation procedure, and the model configuration.

The left-hand side includes the kinematic terms that simulate the spatial propagation of wave action density in the speed of wave group velocity, $(cg,λ,cg,φ)$⁠, during which frequency $(cg,σ)$ and direction $(cg,θ)$ dispersion occur. The right-hand side contains the source/sink terms $S(σ,θ;λ,φ,t)$⁠, which simulate the total wave energy density generated by wind; dissipated due to whitecapping, bottom friction, and depth-induced breaking; and redistributed owing to nonlinear quadruplet and triad interactions between waves. This conserved form allows the integration of currents and water levels into the numerical framework, thus accounting for their effects on waves. More detailed specifications regarding the theoretical and numerical backgrounds can be found in the SWAN scientific and technical documentation [29] and the relevant literature therein.

The simulations were performed on the basis of an integrated SWAN and ADCIRC scheme [27]. Unstructured finite element mesh grids were used in both models so that information can be exchanged between these two models without interpolation, enhancing both accuracy and efficiency (see Fig. 6). The coupling in each computational interval is carried out through a three-step sequential dynamic procedure. First, the ADCIRC model is initiated and run with meteorological drivers, astronomical boundary conditions and geographical data (coastline, topography, bathymetry, etc.). Second, the water levels and currents computed with ADCIRC along with the initial inputs are passed to the SWAN model to update all the relevant physical processes in the action balance equation and obtain the wave spectra. Third, the wave radiation stress gradients calculated on the basis of the wave energy density spectra from SWAN are transferred back to ADCIRC to update and drive circulation calculations for the next computational interval.

The drivers of the coupled model are geographical data (coastline, bathymetry, and topography), unstructured mesh grids, oceanographic boundary conditions, spatially and temporally free surface winds, and atmospheric pressures. The modeled domain for the reference site spans from 18.6°N to 30.9°N latitude and from 114.2°E to 127.5°E longitude, encompassing the whole Taiwan Strait and extending outward to the western North Pacific. The modeled domain extends from 25°N to 41°N latitude and from 115°E to 127°E longitude for the validation site, encompassing the Yellow Sea, the Bohai Sea, and the East China Sea, which opens up to the western North Pacific. The simulated domains were selected to be sufficiently large to account for the appropriate formation of remotely propagated swell waves from typhoons inside the computational domain but far from the studied straits.

The bathymetric and topographic models were configured with the NOAA ETOPO1 (1 arc-minute global relief model), which was enhanced by nearshore bathymetry data from the China Maritime Safety Administration [13] and considering a minimum water depth of 4 m [30] to improve the quality in the nearshore zone and provide a correction for tides. Flexible unstructured triangular mesh grids were employed. The resolutions of the meshes were approximately 50 m and 150 m in the shallow water coastal regions and gradually increased to approximately 32 km and 60 km toward the open ocean at the reference and validation sites, respectively. Thus, fine-resolution meshes were established in the studied straits and near the surrounding islands, accounting for the complex geometry throughout both regions that can influence wave generation, dissipation, and propagation processes in SWAN, as well as the accurate exchange of wave radiation stress gradients in ADCIRC. This approach also provides a well-balanced tradeoff between computational accuracy and convergence efficiency. The mesh grid of the modeled domain for the reference site included 110,788 elements and 56,928 nodes, and that for the reference site contained 70,144 elements and 37,064 nodes. Figure 7 shows the mesh grids for the respective simulated domains.

The oceanographic boundary conditions were set on the basis of the most significant astronomical tide constituents (K1, K2, M2, N2, O1, P1, Q1, and S2) from the TPXO9_atlas model [31]. Meteorological driving forces were established via empirical wind and pressure fields. The empirical pressure field was set according to the Holland model [32]. The empirical wind field was defined on the basis of a combination of parametric and background wind fields to improve the parametric wind field far from the typhoon center. This hybrid wind field reflects both the central and peripheral wind field characteristics of typhoons and was obtained through a distance-based superposition method [33]. The parametric wind field was assumed to be composed of translational and gradient components. The translational component was calculated on the basis of the Miyazaki Masae model of geostrophic wind velocity [34]. The gradient component was estimated according to the modified dynamic Holland model [35] incorporating the empirical formula proposed in a previous study [36,37] for the maximum wind speed radius and the Holland model [32] to calculate the B parameter. The surface gradient wind at 10 m above sea level was converted to the wind at the top of the atmospheric boundary layer, with a boundary layer adjustment factor of 0.9 [13,25]. The key inputs for the parametric wind field were set on the basis of adjusted historical typhoon data retrieved from the China Meteorological Administration [13], and the background wind field used to enhance the quality and precision of the wind field was set on the basis of the cross-calibrated multiplatform reanalysis product.

The coupled model was executed in nonstationary mode with spherical coordinates. In SWAN, the wave action densities were computed in 36 equally spaced propagation directions, 34 logarithmically spaced frequency bins between 0.04 Hz and 1 Hz, and with a time step of 10 min. The governing equation was integrated in time, spatial space, and spectral space via an advantageous implicit upwind finite difference scheme known as the backward space, backward time scheme, which is simple, robust, and economical for modeling waves in coastal regions. The solution was considered to have converged if 90% of the vertices in the modeled domain converged after a maximum of 10 iterations. The convergence criteria for terminating the iterative procedure were set on the basis of the default absolute and relative changes in the significant wave height as well as its normalized curvature of the iteration curve. Additionally, the default directional refraction limiter was used to avoid potential local instabilities from spreading throughout the simulated domain [38]. The parameterization of the wind input and whitecapping dissipation terms in the SWAN model was performed on the basis of Janssen's (WAM-Cycle4) formulation with default values for linear and exponential growth terms [39–42], as this approach has been shown to minimize the combined error in the prediction of the significant wave height and zero-upcrossing wave period under cyclonic metocean conditions [43–45]. The nonlinear quadruplet and triad wave‒wave interactions were computed via a fully explicit per-sweep computational scheme, the discrete interaction approximation method [46] and the lumped triad approximation method [47], respectively, with relevant default tuning values. Depth-induced wave breaking was specified via the BJ78 model with a breaker index of 0.73 and a dissipation rate proportionality coefficient of 1 [48]. The bottom friction was also included in the computations via the JONSWAP formulation [49], with a bottom friction coefficient of $0.067m2/s3$⁠, which is typically recommended for waves in storms [50,51]. The threshold depth was set equal to 0.1 m and any positive depth smaller than this value was discarded and replaced with 0.1 m. All other physical and numerical settings were set to the default values. Further details are available in the SWAN scientific and technical documentation [29]. The time step for integration was set to 10 min. In ADCIRC, the lateral eddy viscosity coefficient was set to a constant of $5m/s2$⁠. The hybrid bottom friction relationship was used to allow the bottom friction to change with respect to water depth. The relationship was formulated on the basis of the minimum bottom friction coefficient, breaking depth, a parameter that reflects the tendency of the hybrid bottom friction relationship to approach the upper or lower limit when the water depth is greater than or less than the breaking height, and a parameter that reflects the rate at which the friction factor increases with decreasing water depth. These parameters were set to 0.0022, 1 m, 10, and 1/3, respectively, in this study. Barotropic dynamics were included, but baroclinic effects have been shown to be relatively small during typhoon events [52]. Finite amplitude and advective effects were also accounted for in the simulations. The computational time step for the ADCIRC model was also set to 2 s and 1.5 s for the reference and validation sites, respectively.

The internal relationships between height- and period-defining parameters were determined through linear regression analyses of measured data. Tables 1 and 2 present the ensembled average ratios of the wave height- and period-related characteristic measures and their corresponding linear fitting coefficients and prediction confidence bounds. Table 1 indicates a strong correlation between the height parameters and the mean wave height, with the significant wave height displaying the highest correlation, as it is directly obtained from the mean wave height. A comparison of the results with the current design practice (see Table 3) reveals that wave height-related measures provide a very close estimate of actual values, with less than 2% underestimation, compared with those based on the Chinese Code of the Hydrology for Harbor and Waterway [10]. Similar slight underestimations of wave characteristic heights have also been reported in offshore and coastal areas of China, e.g., the South China Sea [61,62], Zhoushan Island [63], Hangzhou Bay [64], and Sanmen Bay [65]. In addition, a comparison with the literature in different sea areas indicated close agreement with the empirical estimates derived on the basis of the analysis of field wave data. In this case, the ratio $Hs/H¯$⁠, with an overall mean and standard deviation of 1.59 and 0.06, respectively, ranges from 1.40–1.75, and the ratio $He/Hs$⁠, with an overall mean and standard deviation of 1.65 and 0.26, respectively, lies mainly within the range of 1.1–2.4 [18]. Similar observations from field wave data at different locations of the Pacific and Atlantic coasts have been reported by other scholars [66–69]. Compared with the theoretical estimates based on the assumption of the Rayleigh distribution of wave heights, a very close prediction of significant wave height, $Hs=1.6H¯$⁠, yet fairly comparable estimates of extreme wave height, $He=(1.9∼3.9)H¯$ and $He=(1.2∼2.5)Hs$⁠, were observed [20]. This discrepancy might be due to the deviation of the wave height distribution from the Rayleigh distribution in shallow waters [70,71]. On the basis of the relationships between representative wave periods, estimations are relatively higher than the prescribed values in the design practice. This may be partly due to the strong influence of swell waves on the overall wave statistics at the reference site [11,72]. Similarly, a positive and high bias for wave period characteristic measures has also been observed in other coastal areas of China, e.g., Zhoushan Island [63] and Sanmen Bay [65], whereas observations in the southern offshore areas of China [61,62] have indicated negative yet similar highly biased estimations. In addition, a comparison with field measurements at several locations with less limited fetch and water depths deeper than that at the reference site [18,22,23,73] indicated a lower range of variations or expected values for the extreme (maximum) wave period, $Te=(0.54∼1.82)T¯$⁠, and significant wave period, $Ts=(0.9∼1.4)T¯$⁠, whereas a close average value for $Ts/Tp≅0.8$ was reported by Ref. [23]. Interestingly, the maximum wave period at the reference site can be considered approximately equal to the significant wave period, i.e., $Te/Ts≅1.1$⁠, which is somewhat consistent with the assumption of Ref. [19] that characteristic periods do not vary much for large waves. Contrary to what was observed for the wave characteristic heights, the discrepancies in the characteristic wave periods from the recommended estimations indicate relatively high sensitivity of the wave period to the complex physical processes in nearshore straits, such as land shedding, shallow water effects, wave diversion, fetch restrictions, and other factors.

The comparisons show that the use of the prescribed internal relationships in current practice and the literature provides conservative estimates of wave conditions, i.e., taller heights and shorter periods that could increase the wave loadings on intensity-dependent and dynamically sensitive marine structures. Figures 9(a)–9(c) depict a nearly linear dependence among the wave height-related parameters, and a trend that becomes more scattered for moderate to high sea states. A relatively scattered trend over the entire range of waves is also observed among the wave period-related characteristic measures in Figs. 9(d)–9(f), except for the significant wave period, which is directly estimated from the mean wave period. Moreover, the standard prescribed relationships vary at the lower edge of the data, providing a lower bound for the dependence on the characteristic periods, which again highlights the conservatism of these recommendations compared with measurements. A comparison with the lower 99% confidence bounds of the fitted curves further confirms this finding. These relationships provide an average basis for estimating variables based on the values of other variables and could be essential if one has access to only certain design-defining variables, e.g., significant wave height or peak wave period are available, while an equivalent form, like extreme (maximum) wave height or extreme (maximum) wave period, is needed for the desired analysis form or design situation. This application is illustrated in Sec. 4.4. It should be noted that the duration of the measurements, the accompanying uncertainties, and the nonintercept linear fitting process chosen for simplicity may impact the degree of accuracy of the presented results.

Therefore, estimating the $μln(T¯)$ on the basis of the wave height is critically important for providing a reliable wave period range for extreme conditions, i.e., large MRPs for which the stability and integrity of a structure are at great risk. In the third step, a physically interpretive dependence function is fitted to $E(T¯|E[Hs])$⁠. Several functions, including a two- or three-parameter power function, $fμln(T¯)(x)=c1+c2xc3$⁠, an exponential function, $fμln(T¯)(x)=c1+c2ec3x$⁠, or an arctangent function, $fμln(T¯)(x)=c1×arctan[c2(x+c3)]$⁠, have been used to model the conditionality [9]. Although the latter expressions may provide reliable estimates for extreme wave conditions [76], the power function seems to be the most physically consistent with actual conditions [77], expressing a generic yet physically interpretive form of wave steepness, s, if the shape parameter is fixed and set as $T¯=2π/gsHs$⁠, where $[Hs]=m$ and $[g]=m/s−2$ result in $[T¯]=s$⁠. On the basis of such a physically consistent dependence function, several empirical functions of the relationship between the significant wave height and corresponding mean wave period under extreme wave conditions have been developed in the literature [21–23], whereas others are recommended in current design standards, such as the DNV [8] and the Chinese Code of the Hydrology for Harbor and Waterway [10]. The relationships considered in this study are presented in Table 4. The selected cases mainly represent wave environments in regional seas the northeast Asia that are close to those in this study. The function for establishing these relations is reported mainly in the form of $Ts=aHsb$⁠, where a and b are the scale and shape parameters, respectively [10,21–23]. This necessitates the internal conversion between significant and mean wave periods to define the physically interpretive conditionality in the third step and estimate the expected mean wave period given a significant wave height via Eq. (18). This conversion was performed based on the internal relationships that were presented in the relevant studies, as summarized in Table 4.

On this basis, a comparison with the simulated data, as shown in Fig. 10, was first performed to examine the extent to which practical relationships are applicable in the case of typhoon-affected nearshore straits with similar geographical and metocean conditions in the East China Sea. Figure 10 shows that all of the proposed formulas tend to coincide with low $(Hs<0.6m)$ and medium $(0.6m<Hs<1.5m)$ wave height trends up to approximately $Hs=2.6m$ and deviate from the data as the wave height increases, except for the one recommended by DNV-RP-H103 [8]. This is because these formulas were developed mostly for extreme wave conditions along coastal regions in the western Pacific Ocean, where moderately limited fetch incorporates long-period swells into locally generated wind waves. Such a combined wave system was also reported in the study area in previous works [11,72], within which high waves were associated with a narrow range of periods, whereas low $(Hs<0.6m)$ and medium $(0.6m<Hs<1.5m)$ waves were associated with a relatively wide range of periods.

Figure 10 also shows that the lower bound of high waves tends to be linear, which indicates a strong correlation among the variables owing to the breaking of waves beyond a certain steepness. However, none of the recommended relationships effectively model the lower bound of the data, meaning that there might be sea states in nearshore strait areas with the same wave height but a lower wave period that can be discarded. According to Ref. [78], the same wave height with a lower wave period can induce a larger wave load on an offshore jacket-type structure. Moreover, the fundamental natural period of coastal infrastructure in nearshore straits usually ranges from 2 to 5 s, which makes the assessments of structural dynamics sensitive to the selection of wave periods. This was demonstrated in an earlier study [24], in which the use of a generic dependence relation led to the exclusion of critical sea states within the lower bound of the period and resulted in a less conservative design that was not consistent with the language used on specifications or industrial practice. Therefore, the abovementioned points highlight the importance of characterizing the upper and lower bounds of the data. Additionally, they emphasize the potential uncertainties that could lead to damage or uncertainty in the ultimate design of infrastructure in nearshore straits along East China Sea coastlines if practical and literature-based external relationships are used without adequate justifications.

After the efficiency of the standard-recommended and literature-proposed relationships was evaluated, new scale and shape parameters for the external relationship were derived on the basis of the described probabilistic approach. To this end, the logarithmic conversions of the mean wave periods were sampled within a bin size of a one-meter significant wave height (first step). The binned data were then used for probability model fitting, assuming a log-normal distribution (second step). The mean values and standard deviations within each group were calculated and fitted by a power function through the least squares method (third step). Two scenarios were considered to establish a conditional power function relationship. In the first one, the shape parameter was set to 0.5, so that the form of the function was consistent with the theory and specifications, $T¯∝Hs$⁠, whereas in the second one, it was set as an unknown value to be estimated. This modeling approach supports further comparisons on the basis of the suitable form of the dependence function. Figure 11 shows that both scenarios provide a close estimate of the lower bound (LB), whereas using a predefined shape parameter (Fig. 11(a)) results in an upper bound (UB) that is underestimated for low $(Hs<0.6m)$ and medium $(0.6m<Hs<1.5m)$ sea states up to approximately $Hs=2.6m$ and relatively poor approximations for extreme waves. Therefore, for the lower bound that is of particular concern in nearshore strait areas, one can use either of the following formulas: $T¯Fitted,1LB=1.95Hs$ or $T¯Fitted,2LB=1.74Hs0.6$⁠. For the upper bound, however, it is recommended that the result of the second scenario shown in (Fig. 11(b)), i.e., $T¯Fitted,2UB=5.96Hs0.09$⁠, be considered because it displays high agreement with the measurements within all ranges of the data. From the first scenario (Fig. 11(a)), it can also be concluded that the wave steepness $2πHs/(gT¯2)$ is approximately limited to $steepness≤1/23$⁠, whereas on average, $steepness≈1:11.63$⁠, which is very close to the value recommended by DNV-RP-C205 [9].

The results of the analysis of the simulated data were also compared with data obtained on the basis of the specifications and previous studies, as shown in Fig. 12. Compared with the first scenario-driven relationships, the relation in the Chinese Code of the Hydrology for Harbor and Waterway (JTS) [10] yields results that closely follow the upper bound curve, with slight underestimations, whereas the DNV-RP-H103 relation [8] slightly overestimates the expected value of this scenario and hence may be applicable for determining mean wave conditions in nearshore areas prone to typhoon metocean conditions in the East China Sea. Among the proposed formulas in the literature, the lower and upper bounds of the relationships that were developed by Ref. [22] yield close yet slightly high estimations of the expected and upper bounds in the first scenario-driven relationships. A comparison with the second scenario-driven relationships shows that the DNV-RP-H103 [8] model and lower bound of Ref. [22] model seem to yield slightly lower estimates than the expected values in the second scenario for low $(Hs<0.6m)$ and medium $(0.6m<Hs<1.5m)$ sea states, whereas they yield higher estimates than the expected values in the second scenario in cases with extreme sea states.

The close agreement but slight over- or underestimation of values with the standard-recommended and literature-proposed relationships is due mainly to potential differences in depth and site-specific geography or metocean conditions. For example, the formulas presented by Ref. [23] were derived on the basis of hindcasting data along the east coast of the Korean Peninsula and the west coast of Japan, areas with greater depths and more uniform and less limited fetch conditions than those at the reference site. Similarly, the formulas of Ref. [22] were developed on the basis of hindcasting data from the Korean Peninsula area for $Hs>3m$⁠, which explains the similarity between the estimations of this model and those obtained based on the first scenario-driven relationships in cases with $Hs=3m$⁠. Such differences also exist in Refs. [8,10], where an averaged formula in Ref. [10] and greater depths and more uniform and less limited fetch in Ref. [8] could explain the slight under- and overestimations, respectively.

Since the relationships presented in Secs. 4.1 and 4.2 were developed on the basis of data from the Pingtan Strait, to assess the efficiency of these relationships in similar nearshore straits, the developed internal and external relation functions were compared to measured and simulated datasets from the Xihoumen Strait. The comparisons of internal relationships are presented in Table 5, wherein, in most cases, the relationships yield positively biased yet very close estimates. The bias errors for the wave height-related parameters are generally less than ∼3%, whereas those for the wave period-related variables are less than ∼14%. This positive bias is not unexpected, as the Xihoumen Strait is much more limited in terms of fetch, as shown in Fig. 2; as a result, less severe waves than those in the Pingtan Strait are encountered in the Xihoumen Strait. A data-over-threshold analysis of the data collected at the Xihoumen Strait revealed that the change in biases for wave height-related measures is insignificant, up to 1.2%, whereas the discrepancies between wave period-related measures of observed data and developed relationships can be reduced by ∼2%–5%. This confirms the dominance of less severe waves in the Xihoumen Strait, with only up to an approximately 10% discrepancy between the actual data and the data obtained from wave period-related relationships. It should be noted that the data above various thresholds were analyzed because typhoon and nontyphoon data were unavailable at the validation site. The external relationships are compared in Fig. 13. Reasonable agreement between the simulated data from the Xihoumen Strait and the data-based external relationships was observed. Like those based on internal relationships, the results obtained on the basis of external relationships are positive yet slightly biased in terms of the lower and upper bounds, although the bias of the upper bound seems to increase in extreme sea states. These comparisons increase the confidence that the proposed relationships in Secs. 4.1 and 4.2 can reasonably reproduce typhoon-generated mean wave conditions in nearshore straits. However, one should consider the potential uncertainties that may influence the calculations, especially under extreme wave conditions, when used in the design of infrastructure in nearshore straits in the East China Sea.

After defining the relationship between the mean wave period and the significant wave height and establishing the relationships between wave height- and period-related statistical characteristic measures, a logical question is, can the newly developed formulas be applied in the design and performance assessment of infrastructure located in nearshore strait areas? In practice, different combinations of height and period definitions, depending on the regular/irregular linear/nonlinear wave theory, are used to simulate the wave field. These combinations are not all available, meaning that one can usually only access certain combinations of design-defining variables, e.g., significant wave height or peak wave period, whereas an equivalent form of this combination, e.g., extreme (maximum) wave height or extreme (maximum) wave period, might be needed for a certain analysis or design situation.

To illustrate the engineering application of the developed relationships for estimating typhoon-generated mean wave conditions in nearshore straits, an integrated framework of internal and external relationships was established, as summarized in Fig. 14 and as follows:

• Given any wave height-related characteristic variable, such as the extreme (maximum) wave height, $He$⁠, or average wave height, $H¯$⁠, the internal relationships in Table 1 provide a basis for estimating the significant wave height, $Hs$⁠.

• The external relationships presented in Sec. 4.2 provide a predictive model to determine the average/mean wave period, $T¯$⁠, with the upper and lower boundaries corresponding to the estimated significant wave height, $Hs$⁠.

• Given the mean wave period, other wave period-related characteristic variables, such as the extreme (maximum) wave period, $Te$⁠; significant wave period, $Ts$⁠; and peak wave period, $Tp$⁠, can be estimated via the internal relationships in Table 2.

• This process can be used inversely if one desires wave height-related characteristics for a specific wave period-related range.

A relationship representing the external dependence between the mean wave period and significant wave height was established to provide the basis for estimating the mean wave conditions in nearshore straits affected by typhoons. This relationship was developed via long-term numerically simulated typhoon-generated waves in the relatively shallow waters of the Pingtan Strait on the east coast of China. The simulations included 58 typhoons that hit this site between 1990 and 2018; these events were validated in comparison to wave data from a measurement campaign that was conducted in 2015 in the Pingtan Strait. The developed relationships, including the lower and upper bounds and mean curves, were established through a probabilistic method using a physically interpretive function that represents a generic form of wave steepness: $T¯=aHsb$⁠. The internal relationships between the representative wave height and wave period variables were also determined by performing a regression analysis of the wave data obtained from the aforementioned measurements. This approach provides a statistical basis within which one can use external relationships to obtain basic characteristic measures of mean wave conditions, namely, the mean wave period or significant wave height, and then exchange them with desired relevant representative variables for any design and analysis purpose via internal relationships. The developed relationships were then tested and compared to practice-recommended and literature-proposed external relationships. Moreover, the results were compared to field wave data from typhoon-affected straits and numerically simulated typhoon-generated wave data from the Xihoumen Strait. The major findings of the study are as follows:

1. Linear regression analyses of wave height- and period-related characteristic measurements revealed that the prescribed internal relationships in current practice and in the literature provide conservative estimates of typhoon-generated mean wave conditions in the Pingtan Strait, indicating that taller heights and shorter periods could be used in design calculations if the recommended internal relationships were used.

2. The preceding analysis of the simulated data revealed that among the empirical relationships given in specifications and the literature, the combination of the external relationships suggested by the Chinese Code JTS-145 [10] and the formula recommended by the DNV Code H103 [8] provides a close estimate of the typhoon-generated mean wave conditions in the Pingtan Strait. Similarly, the lower bound of the formulas in Ref. [22] provides a reasonable estimate of the expected extreme waves at the reference site.

3. None of the external relationships recommended on the basis of design standards and the literature satisfactorily predict either the lower bound, $T¯=1.95Hs$⁠, or the upper bound, $T¯=5.96Hs0.09$⁠, of typhoon-generated wave conditions in the Pingtan Strait.

Notably, even though the newly derived relationships from the reference site agree well with the measured and simulated data at the validation site, the generality of the developed internal and external relationships is limited to the East China Sea. Therefore, the use of derived equations beyond the described environmental conditions and water depths in this study must be validated. Moreover, the presented results might be affected by statistical, modeling, and climatic uncertainties. The assumptions, limitations, and methodologies that were used in the derivation process should therefore be considered when the established relationships are used. In future studies, the applicability of these relationships to other coastal areas of China could be explored through studies of infrastructure in nearshore straits prone to tropical storms and the assimilation of numerical models.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52222804 and 52221002).

There is no conflict of interest

Some or all data and codes that support the findings of this study are available from the corresponding author upon reasonable request.