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Empirical relationship between significant wave period and wave energy period in the coastal waters of China
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Yuhuan Xue1, Chuanjiang Huang1, 2, 3, *, Gang Wang1, 2, 3, Dejun Dai1, 2, 3, Fangli Qiao1, 2, 3
Acta Oceanologica Sinica | 2025, 44(1) : 50 - 58
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Acta Oceanologica Sinica | 2025, 44(1): 50-58
• Physical Oceanography, Marine Meteorology and Marine Physics •
Empirical relationship between significant wave period and wave energy period in the coastal waters of China
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Yuhuan Xue1, Chuanjiang Huang1, 2, 3, *, Gang Wang1, 2, 3, Dejun Dai1, 2, 3, Fangli Qiao1, 2, 3
Affiliations
  • 1 First Institute of Oceanography, Ministry of Natural Resources, Qingdao 266061, China
  • 2 Laboratory for Regional Oceanography and Numerical Modeling, Qingdao Marine Science and Technology Center, Qingdao 266237, China
  • 3 Key Laboratory of Marine Science and Numerical Modeling, Ministry of Natural Resources, Qingdao 266061, China
Published: 2025-01-25 doi: 10.1007/s13131-024-0002-3
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Significant wave period is an important parameter in coastal and offshore engineering design. Traditional spectral wave models do not directly calculate this parameter, which means that it needs to be estimated from the spectral periods using empirical formulas. The wave energy period is one of the wave periods directly output by many wave models and is often used in studies of wave energy. This study investigated the relationship between significant wave period and wave energy period using wave data measured at three stations in the coastal waters of China. The observations recorded at these stations in the South China Sea, the East China Sea, and the Bohai Sea covered a wide range of surface wave conditions. Analysis indicated that the ratio of significant wave period to wave energy period is closely related to the Goda peakedness parameter of the wave spectra. Therefore, we proposed an empirical formula in which significant wave period is a function of wave energy period and the Goda peakedness parameter. Evaluation results showed that the performance of this formula is substantially better than that of fitting formulas that use constant coefficients.

surface waves  /  significant wave period  /  wave energy period  /  wave spectral parameter  /  wave spectra
Yuhuan Xue, Chuanjiang Huang, Gang Wang, Dejun Dai, Fangli Qiao. Empirical relationship between significant wave period and wave energy period in the coastal waters of China[J]. Acta Oceanologica Sinica, 2025 , 44 (1) : 50 -58 . DOI: 10.1007/s13131-024-0002-3
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1 Introduction
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Surface waves have notable impact on coastal and offshore structures (Goda, 2010; Xu et al., 2022; Sun et al., 2023) and on processes at the air-sea interface (Drennan et al., 2003; Huang and Qiao, 2021). Due to the limitations of observation conditions and engineering costs, numerical wave models are often used to estimate wave parameters in ocean engineering design. Although wave models have made considerable progress in the past few decades (Cavaleri et al., 2007; Roland and Ardhuin, 2014), traditional wave models have been developed using energy balance analysis (Yuan and Huang, 2012). Consequently, such models simulate the spectral distribution of wave energy rather than the sea surface height, from which almost all wave parameters are then derived by the spectral analysis.
Significant wave period (T s), defined as the mean period of one-third of the highest waves, is an important parameter used to characterize waves. However, because this parameter is obtained from the zero-crossing analysis of sea surface heights, it is not predicted by traditional spectral wave models (Chun and Suh, 2018). Therefore, it often needs to be estimated from the spectral periods using empirical formulas.
In order to describe different statistical and physical characteristics of waves, there are various commonly used spectral period parameters, including the peak period T p, zero-crossing period T m02 $ (=\sqrt{{m}_{0}/{m}_{2}}) $, mean period T m01 $ (={m}_{0}/{m}_{1}) $, and wave energy period T m–1,0 $ (={m}_{-1}/{m}_{0}) $, where $ {m}_{n}={\displaystyle\int }_{0}^{\infty }{f}^{n}S\left(f\right){\mathrm{d}}f $ is the n-th moment of the wave power spectral density function S(f), and f is the frequency (Cuadra et al. 2016; ECMWF, 2021). Many studies have investigated the relationship between different wave periods, most of which were based on simple linear analysis (Holthuijsen, 2007; Li, 2007; Suh et al., 2010; Huang et al., 2016; Ahn, 2021; Kumar and Mandal, 2022), but there were also some studies suggesting that their relationship is related to the spectral characteristics of waves (Goda, 2010; Chun and Suh, 2018). Overall, there are significant differences in the relationship between different wave periods in these studies (Li, 2007; Goda, 2010; Huang et al., 2016), and then there is still much room for studies of wave periods.
On the basis of in situ observation data, laboratory data, and simulation data, Li (2007) argued that the relationship between T s and the spectral periods calculated from the negative moment of the spectrum was more stable compared to other spectral periods. Using wave data from the Sea of Japan, Chun and Suh (2018) found that the formula using T m–1,0 produced the best fitting result for T s. The T m–1,0, also known as the wave energy period (T e), is one of the mean wave periods directly output by many wave models (WW3DG, 2019; ECMWF, 2021) and is often used in studies on wave energy (Bouferrouk et al., 2016; Cuadra et al., 2016). It is more stable than the peak period, especially for sea states where wind waves and swells coexist, and it is less sensitive than other commonly used wave periods (such as T m02 and T m01) to the high-frequency cutoff of the spectrum (Jiang et al., 2022; Anju et al., 2023; Muraleedharan et al., 2023).
In this study, we examined the relationship between T s and T e using wave data measured at three stations in the coastal waters of China, and we proposed an empirical formula for calculating T s using T e. The remainder of the paper is structured as follows. The data and methods used in the study are described in Section 2. The derived empirical formula and the evaluation results are presented in Section 3. Finally, our conclusions are provided in Section 4.
2 Data and methods
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2.1 Wave data
The observations were conducted in the South China Sea (SCS), East China Sea (ECS), and Bohai Sea (BS) (Fig. 1), where the surface waves were measured through acoustic surface tracking using acoustic Doppler current profilers (ADCPs) fixed to bottom-mounted frames. The observation station in the SCS is at 21.44°N, 111.39°E (Table 1). The location is approximately 6.5 km from the nearest shore, with a mean water depth of 16 m. The observational period extended from February 23, 2017, to August 1, 2017, during which a 1 000 kHz ADCP (Nortek Signature 1000) was deployed to measure surface waves. Its sampling time was set to 1024 s per hour, and the output wave spectrum was in the range of 0.02–0.99 Hz with a resolution of 0.01 Hz.
The observation station in the ECS is at 27.68°N, 121.35°E. The location is approximately 25 km from the coast, with a mean water depth of 28 m. The observational period extended from June 4, 2017, to September 13, 2017. The observation station in the BS is at 38.31°N, 118.91°E. The location is approximately 19 km from the coast, with a mean water depth of 19 m. The observational period extended from December 19, 2022, to March 31, 2023. At the ECS and BS stations, 1 000 kHz ADCPs (Nortek AWAC) were used to measure surface waves. The sampling time was also set to 1 024 s per hour, but the wave spectral range was 0.02–1.99 Hz with a resolution of 0.01 Hz.
2.2 Evaluation method
Following previous studies (Jiang et al., 2022; Bujak et al., 2023), the metrics of bias, root mean square error (RMSE), and correlation coefficient (COR) were used to evaluate the performance of the empirical formulas for the relationship between different wave periods. The formulas for calculation of these metrics can be expressed as follows:
$ \mathrm{Bias}=\frac{1}{N}\sum _{i=1}^{N}({P}_{i}-{O}_{i}), $
$ \mathrm{RMSE}=\sqrt{\frac{1}{N}\sum _{i=1}^{N}{({P}_{i}-{O}_{i})}^{2}}, $
$ \mathrm{COR}=\frac{\displaystyle\sum _{i=1}^{N}{(P}_{i}-{\overline {P})(O}_{i}-\overline{O})}{\left[\sqrt{\displaystyle\sum _{i=1}^{N}{({P}_{i}-\overline{P})}^{2}}\right]\left[\sqrt{\displaystyle\sum _{i=1}^{N}{({O}_{i}-\overline{O})}^{2}}\right]}, $
where O represents the observed wave period, P represents the wave period modelled using empirical formulas, and N represents the number of available observation data points.
3 Empirical formulas and evaluation results
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3.1 Surface wave conditions
China’s coastal waters are affected by monsoons, cold fronts, typhoons, and other factors that create complex and variable surface wave conditions. The SCS is controlled by the Southeast Asian monsoon system. Our observations in this region, conducted in February–August, were affected first by the winter monsoon and then by the summer monsoon. Overall, the observed surface waves were dominated by swells, with significant wave heights in the range of 0.3–1.5 m and T s in the range of 3–7 s. However, during the observational period, the passage of two severe tropical storm processes resulted in a long T s of 8.9 s on June 12 and a large significant wave height of 2.4 m on July 16 (Figs 2a and b).
Our observations in the ECS were obtained in summer, when the sea area is susceptible to the influence of typhoons. During our observation period, owing to the influence of Typhoon Nesat, the maximum significant wave height exceeded 5 m, and the maximum T s exceeded 12 s (Figs 3a and b).
The BS is a shallow, semi-enclosed marginal sea where the surface waves are dominated by wind waves. During our observational period, approximately 81% of the valid data segments had a significant wave height of less than 1 m and T s of less than 5 s. However, owing to the influence of winter cold fronts, there were also some observational segments with significant wave heights of larger than 3 m and T s of larger than 7 s (Figs 4a and b).
Previous studies typically used linear fitting with constant coefficients to examine the relationship between different wave periods (Li, 2007; Huang et al., 2016; Ahn, 2021). Figure 5 shows scatter plots of T s and T e measured at the three stations, together with their linear fitting formulas in the form of T s = αT e, where α is a constant. It is evident that the coefficient of the linear fitting of these two parameters is not the same at each of the three stations. In the SCS, dominated by swells, the value of coefficient α is 0.95, whereas it is 0.99 in the BS. This finding is similar to that of previous studies on fitting wave periods, where different fitting coefficients are usually obtained when using different observational datasets (Li, 2007; Ahn, 2021). It means that the constant coefficient fitting method has a certain dependence on local wave characteristics, which to some extent limits the applicability of this fitting method.
3.2 Empirical formula derived from wave data in the SCS
Although T s compares well with T e for each of the three observation datasets (Figs 2b, 3b, and 4b), the ratio of the two parameters (i.e., T s/T e) varies greatly, both spatially and temporally, in the range of 0.40–1.07 (Figs 2c, 3c, and 4c). This variation might be related to the wave spectral characteristics at the three stations. The Goda peakedness parameter (see ECMWF, 2021), which is an important parameter often used to represent wave spectral characteristics (Fairley et al., 2020; Le Merle et al., 2021), can be calculated directly using the wave spectrum as follows:
$ {Q}_{{\mathrm{p}}}=\frac{2}{{m}_{0}^{2}}{\int }_{0}^{\infty }f{S}^{2}\left(f\right){\mathrm{d}}f. $
This parameter is a measure of the sharpness of the wave spectrum, where larger Q p usually correspond to narrower spectra (or more sharply peaked spectra), and vice versa. Compared to other commonly used wave bandwidth parameters, it is more reliable because of the stable quantities such as m 0 and S(f) used in its calculation, and it is independent of the high-frequency cutoff choice of the spectrum (Prasada Rao, 1988). It has been used to fit the relationship between T s and the peak period by Chun and Suh (2018) and to predict wave runup on beaches by Bujak et al. (2023).
Figure 6a shows the relationship between the ratio of T s to T e and Q p derived from the wave data obtained in the SCS. Overall, this ratio varies approximately exponentially with Q p. It is slightly larger than 1 when Q p is greater than 2 and decreases rapidly with Q p when the latter is less than 2. Therefore, on the basis of the least squares method, we proposed the following empirical regression model:
$ {T}_{{\mathrm{s}}}=\left[1.02-2.8\;\mathrm{exp}(-2.49{Q}_{{\mathrm{p}}})\right]{T}_{{\mathrm{e}}}. $
The performance of this formula and that of the constant coefficient linear fitting formula (see Fig. 5a) are respectively shown in Figs 7a and d. For wave data in the SCS, the bias, RMSE, and COR between the T s modelled using the linear fitting formula and the observations are 0.052 s, 0.360 s, and 0.930, respectively (Table 2). However, when using Eq. (5), the bias and RMSE decreased to 0.010 s and 0.230 s, respectively, while the COR increased to 0.970, i.e., all markedly improved relative to the linear fitting results.
3.3 Validation of the formula using wave data from the ECS and the BS
The sea conditions in the ECS and BS are different to those in the SCS (Figs 2–4), but the relationship of the ratio of T s to T e with Q p in the ECS and BS is fairly consistent with that in the SCS (Figs 6b and c). In the ECS, the bias, RMSE, and COR between the T s modelled using the linear fitting formula (Fig. 5b) and the observed data are 0.136 s, 0.54 s, and 0.88, respectively (Fig. 7b). However, when using Eq. (5) including Q p, the bias and RMSE between them decreased to 0.003 s and 0.29 s, respectively, and the COR increased to 0.97 (Fig. 7e). In the BS, the bias, RMSE, and COR between the T s modelled using the linear fitting formula (Fig. 5c) and the observed data are 0.093 s, 0.29 s, and 0.96, respectively (Fig. 7c). However, when using Eq. (5), the bias and RMSE between them decreased to −0.004 s and 0.18 s, respectively, and the COR increased to 0.99 (Fig. 7f). These improvements demonstrate the applicability of our derived formula [i.e., Eq. (5)] to the coastal waters of China.
3.4 Comparison with previous fitting formulas
Using in situ observations from the BS, laboratory data, and simulation data, Li (2007) derived a linear empirical formula for fitting T s using T e, which can be expressed as follows:
$ {T}_{{\mathrm{s}}}=1.035{T}_{{\mathrm{e}}}. $
For our three sets of wave observations, the ratio of T s to T e is in the range 0.40–1.07. Therefore, although the fitting coefficient of Li (Li, 2007) is greater than ours (Fig. 5), it is still within the range of our observed ratios.
Using wave data from the Sea of Japan, Chun and Suh (2018) estimated T s as follows:
$ {T}_{{\mathrm{s}}}=0.76{T}_{{\mathrm{e}}}^{1.11}. $
Figures 7g–l show scatter plot comparisons between the T s modelled by the above two formulas and our measured T s in the SCS, ECS, and BS. The results show that the performance of our derived formula [i.e., Eq. (5)] is substantially better than that of the other two formulas in relation to our wave data obtained in the coastal waters of China (Table 2).
4 Conclusions
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The wave period, together with the wave height, is an important parameter in ocean engineering and physical oceanography. Compared to other wave spectral periods, the wave energy period is relatively stable and less sensitive to high-frequency cutoff of the spectrum, and it is also one of the wave periods directly output by many wave modes. In this study, we investigated the relationship between the significant wave period and wave energy period using wave data measured at three stations in the SCS, ECS, and BS.
In our observational data, although the two wave periods might appear similar, the ratio between them varies greatly. Therefore, traditional linear fitting using constant coefficients does not provide robust and universally applicable results. We found that this ratio is closely related to the Goda peakedness parameter of wave spectra. Therefore, we proposed an empirical formula for their relationship. Evaluation results showed that the performance of this proposed formula is notably better than that of both traditional linear fitting using constant coefficients and several empirical formulas presented in previous studies.
In addition to the wave energy period, other spectral periods are commonly used in wave studies (Cuadra et al., 2016). Although their performance is not as robust as that of the wave energy period, they do represent the spectral characteristics of waves from different aspects. In future studies, we will examine the relationship between significant wave period and these other spectral periods.
Funding
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  • The National Natural Science Foundation of China under contract(41821004)
  • Basic Scientific Fund for National Public Research Institutes of China under contract(2020Q08)
  • Laoshan Laboratory under contract(LSKJ202201600)
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Year 2025 volume 44 Issue 1
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Article Info
doi: 10.1007/s13131-024-0002-3
  • Receive Date：2024-07-03
  • Online Date：2025-10-27
  • Published：2025-01-25
Article Data
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History
  • Received：2024-07-03
  • Accepted：2024-11-30
Funding
The National Natural Science Foundation of China under contract(41821004)
Basic Scientific Fund for National Public Research Institutes of China under contract(2020Q08)
Laoshan Laboratory under contract(LSKJ202201600)
Affiliations
    1 First Institute of Oceanography, Ministry of Natural Resources, Qingdao 266061, China
    2 Laboratory for Regional Oceanography and Numerical Modeling, Qingdao Marine Science and Technology Center, Qingdao 266237, China
    3 Key Laboratory of Marine Science and Numerical Modeling, Ministry of Natural Resources, Qingdao 266061, China

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* Huang Chuanjiang, E-mail:
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