1 Introduction

The wave period and wavelength fluctuate based on the prevailing local and remote weather conditions. The wave period affects vessel stability and the safety of maritime operations. Longer-period waves tend to have smoother crests and troughs, making navigation and ship operations more manageable. Short-period waves, on the other hand, can be choppier and more challenging to navigate, especially for smaller vessels. Hence, knowing the wave period helps ship captains, maritime authorities, and port operators make informed decisions regarding navigation, port operations, and the safety of vessels. Waves with longer lengths tend to have a deeper influence on nearshore processes, such as sediment transport and beach erosion, and coastal morphology. Since the wavelength is directly related to the energy carried by waves, understanding the wavelength also helps in predicting the impact of waves on coastal areas and designing appropriate coastal protection measures. Waves can have different combinations of wavelength and amplitude, leading to variations in energy.

The interaction between waves and structures depends on the relative size of the structure compared to the wavelength. Hence, in coastal engineering projects, such as the design of breakwaters, seawalls, and harbour structures, knowledge of wavelength is crucial. Depending on the application, the dominant wave period for a given sea state is defined by several approaches. The different wave periods estimated from zero crossing analysis and spectral analysis are used in various studies. Knowledge of the relationship between various wave periods helps coastal managers and researchers estimate the required wave period from the known values of other periods. Joint probability distribution of wave heights and wave periods is used for assessing the period for a given wave height (Longuet-Higgins 1983). The bivariate lognormal distribution proposed by Ochi (1978) is widely used to describe the joint characters of wave heights and periods (Zhao et al. 2022). Based on the long-term statistics of the waves in the China Sea and the Northwest Pacific Ocean, Fang and Hogben (1982) proposed a modified joint distribution model. Stansell et al. (2004) proposed a joint distribution function for waves in extreme storms and for moderately broad-banded spectra.

Higher wave steepness induces white capping and wave breaking in deep seas and is the primary sink of energy in the ocean (Perlin et al. 2013). Wave steepness increases in coastal seas since the wave height increases and wavelength decreases. When wave steepness enters the Stokes limit of 0.1428 tanh (kd) (Stokes 1880), the wave becomes unstable and breaks at water depth (d). According to Mendes and Oliveira (2021), the steepness values of the most severe storms in the northeast Atlantic are higher than the yearly average. Steep waves contain more energy and are stronger, they can destroy the coastline faster and more severely than less steep waves (Kraus and Larson 1988). Hence, the steepness of waves can have a major impact on coastal erosion (Chen et al. 2023).

Bay of Bengal (BoB), a semi-closed tropical basin, is the largest bay in the world. The northwestern BoB, located in the northeastern part of the Indian Ocean, is influenced by the seasonal monsoon winds, with the southwest (SW) monsoon occurring from June to September and the northeast (NE) monsoon prevailing from October to December. These monsoons generate different wind patterns, which influence the waves. During the SW monsoon, strong winds from the southwest produce longer period waves, while the NE monsoon generates shorter period waves due to winds from the northeast. During the transition periods (May and October) between the southwest and northeast monsoons, the wind patterns are more variable. As a result, the wave characteristics during these months vary as well. Wave periods may extend slightly, and longer-period swell waves generated by distant storms or swells may propagate into the region, resulting in larger and more consistent waves. The coastal areas of BoB are prone to tropical cyclones, which are intense low-pressure systems characterized by strong winds and high waves. These cyclones cause waves with significant wave heights (Hs) up to 9 m in the BoB (Amrutha et al. 2014) and cause storm surges and coastal flooding. The occurrence of tropical cyclones in BoB has an annual bimodal structure, with high occurrences in May and October–November (Balaguru et al. 2014). In the northwestern BoB, particularly during the winter, swell waves originating from the Indian Ocean or the southern BoB propagate into the region. Shorter period waves typically result from local wind activity, while longer period waves are generated by distant storms.

Kumar et al. (2014) reported that the seasonal average value of Hs varied from 0.8 m in the NE monsoon season to 1.3 m in the SW monsoon along the northwestern BoB. Based on the data from 8 to 13 October 2013 during the TC Phailin, Amrutha et al. (2014) reported that at 50 m water depth in the northwestern BoB, a maximum wave height (Hmax) of 13.5 m is measured. Patra et al. (2016), based on data for the years 1994, 2008–2009, and 2013–2014, observed that the waves of Gopalpur are predominantly from the south during the SW monsoon and south-southeast for the rest of the year. Patra and Bhaskaran (2017) analysed the wave climate in the head BoB using a combination of satellite altimeter data and numerical models and observed significant seasonal and interannual variations in the wave climate of the head BoB. Based on the wave measurements carried out from February 2013 to December 2015 and the ERA-Interim reanalysis wave data from 1979 to 2015, Anjali Nair et al. (2018) described the spectral wave characteristics off Gopalpur in northwestern BoB. Anjali Nair et al. (2018) observed two well-differentiated wave systems, one young sea from the east-southeast and another more developed swell from the south, leading to bimodality in the wave spectrum in the northwestern BoB. Along the Indian shelf seas, the annual mean wave power is minimal at sheltered regions of the east coast with an annual mean wave power of 2.0 kW/m and a maximum of the northern west coast (13.7 kW/m) (Amrutha and Kumar 2022).

Freak waves are waves that are higher than that are predicted for a particular sea state (Bitner-Gregersen et al. 1998, Gramstad et al. 2018). The presence of freak waves can be a major cause of ship accidents and hence the data of freak waves has a prior importance. Many oceanographers have examined the probability and mechanism of freak wave occurrences, as well as highlighted significant environmental conditions that lead to freak wave creation, to limit the possibility of marine mishaps. Several theories are explained about the causes and development of freak waves in the open ocean. Glejin et al. (2014), analysed the existence of freak waves in the coastal region of Ratnagiri in the eastern Arabian Sea. A freak wave is identified when the maximum wave height (Hmax)-to-significant wave height (Hs) ratio exceeds 2–2.2 (Muller et al. 2005). According to various researchers, the mechanism of trapping a wave by an opposing current generates a freak wave (Lavrenov 1998; Shrira and Slunyaev 2014). Haver (2001) reported that the ratio of maximum wave height (Hmax) to significant wave height (Hs) that is most likely to be exceeded in 1 out of 100 situations is roughly 2.0. Amrutha and Kumar (2017) noted that the Hmax/Hs ratio ranges from 1.50–1.62 for several areas in the nearshore waters of the eastern Arabian Sea during the SW monsoon.

The main goal of this article is to analyse the various wave periods, their associated wavelengths, and steepness during monsoon months in the northeastern Bay of Bengal and to analyse the characteristics of freak waves that occurred. The influence of tides and currents on waves is not considered in the present study. This paper is structured as follows: Sect. 2 describes about data and methods used in the study. Section 3 provides detailed analyses of wave period, wavelength, wave steepness, and freak waves. Section 4 discusses the results and Sect. 5 summarizes the findings.

2 Data and methods

The study is based on measured wave data collected with a 0.9 m diameter Datawell Directional Waverider Mk III buoy (Datawell 2009) at Gopalpur along the northeastern BoB from 1 January to 31 December 2015, as well as 1 June to 31 August 2013, 2017, and 2018. The wave data are measured by the authors under the project funded by the Earth System Science Organization, Ministry of Earth Sciences, Government of India through the Indian National Centre for Ocean Information Services, Hyderabad. Figure 1 provides the geographical location (latitude 19.2657° N, longitude 84.94834° E) of the buoy. At the location of the buoy, the water depth is 15 m. The buoy is fixed to the seafloor by a 15-m-long 35-mm-diameter rubber cord and a 25-m-long 16-mm-diameter polypropylene rope. The buoy data are sampled at a frequency of 3.84 Hz and logged at 1.28 Hz and for the period selected, there are no data gaps. According to Datawell (2009), heave is measured with a 3% accuracy and a 1 cm resolution. The zero up-crossing approach is used to calculate the wave height (H) and period (T) (Tucker 1991). In a 30-min record, the average periods of 10% and 33%, i.e., T1/10 and T1/3, are calculated and employed in the analysis. The maximum wave height (Hmax) and the corresponding period (THmax) are analysed in terms of a 30-min period. By dividing the duration of the record by the number of crests in the record, the crest period (Tc) is estimated. Tavg is the proportion of a record's length to its total number of zero-up crossings.

Fig. 1
figure 1

Map showing the study area. Data used in the study are from the buoy located in the northeastern Bay of Bengal. The colour shade is the depth in metres

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Spectral analysis helps in characterizing wave conditions, identifying wave components, and understanding the complex nature of wave systems. The fast Fourier transform calculates the wave spectrum of the surface elevation from the measured heave data every 30 min. The obtained wave spectrum has a resolution of 0.005 Hz between 0.025 Hz and 0.1 Hz, and then it is 0.01 Hz till 0.58 Hz (Datawell 2009). The wave parameters used in the study and derived from the wave spectra are significant wave height (Hs), energy wave period (Te), wave period corresponding to the mean frequency of the spectrum (Tm01), integral wave period (Ti), mean zero-crossing period (Tm02), Tdw2, Tdw1, and average wave period (Tm–1,1). The spectral peak period (Tp) is the wave period corresponding to the maximum spectral energy density.

By following the wave dispersion relation, the wavelength (λ) corresponding to different wave periods is determined from the wave period and the sea depth. The occurrence percentage of wave period is calculated with 0.5 s bin (0–0.5, 0.5–1,…) and 5 m bin (0–5, 5–10,…) for its corresponding wavelength. The 99 percentiles for June–August are taken as an extreme value index and are computed for all the wave periods. The interannual variations of wave period and wavelength are estimated based on the 99, 90 percentiles and mean values from the available data at Gopalpur. The 99 and 90 percentile wave period and its corresponding wavelength are calculated from the available half-hourly time series data. Wave steepness is estimated as the ratio of the wave height to the corresponding wavelength.

The joint probability distribution of wave heights and wave periods is examined with the bivariate lognormal distribution proposed by Ochi (1978) and the Fang and Hogben model (1982). The criteria used to estimate the freak wave events are (i) ratio of Hmax/H1/3 >  = 2.1 and (ii) ratio of Hc /H1/3 >  = 1.25, where Hmax, H1/3, and Hc are defined as maximum wave height, and significant wave height and crest height of waves. Wave power (P) per unit circular area of the sea surface is calculated from spectral data using the below expression:

$$ P = \rho g\mathop \smallint \limits_{0}^{2\pi } \mathop \smallint \limits_{0}^{\infty } C_{{\text{g}}} \left( {f,d} \right) S\left( {f,\theta } \right){\text{d}}f {\text{d}}\theta , $$
(1)

where ρ is seawater density (kg/m3), d is the water depth (m), g is the gravitational acceleration (m/s2), S(f,θ) is the directional wave spectral energy density (m2/Hz/deg), θ is the wave direction (deg), and Cg(f,d) is the wave group velocity (m/s) and is estimated using linear wave theory as given below (Coastal Engineering Manual 2008).

3 Results and discussion

3.1 Distribution of wave period

The probability of exceedance of the half-hourly average of various wave periods over different years from June to August is presented in Fig. 2. Tp provides information on the dominance of swells or wind-seas at a particular location. From hours to decades, waves exhibit fluctuations, and as a result, Tp also fluctuates. The annual average Tp for all the years in the study is up to 11.2 s. In most BoB locations, the average Tp is about 10–11 s during the peak monsoon months (June–August), when winds are significantly stronger (> 12 m/s) than in other months (George and Kumar 2021). The short-period waves (T < 6 s) are caused by wind-seas, long-period waves (T > 8 s) are dominated by swell and intermediate-period waves (6 s ≤ T ≤ 8 s) are caused by the swell (Bromirski et al. 2005). ~ 1% of Tp are less than 6 s and 86% are with values greater than 8 s, which indicates a swell dominance at Gopalpur from June to August. Swell dominant regions have a larger value for Tp and relatively low wind speed indicating an inverse relationship between those parameters.

Fig. 2
figure 2

Distribution of various wave periods at Gopalpur for 4 years during the monsoon months June to August

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In the coastal waters of Gopalpur, the 30-min mean values of the mean wave period range between 4 and 10.3 s (Fig. 2b), while 42% of Tm02 values are shorter than 6 s. Statistical parameters of wave periods are listed in Table 1. The integral period (Ti) goes up to a maximum value of 15.9 s (Fig. 2c) with an annual average value of 9.5 s. Half of the records showed that the crest period (Tc) is shorter than 4 s in the northeastern BoB. A maximum value for the parameters Tp, and THmax occurs in 2015 and 2018, and the other parameters reach a maximum value in the year 2013. Tm01 is the parameter used to obtain the magnitude of Stokes drift (Stokes 1880) transport in the deep waters and the average value of Tm01 is 7.1 s in the present study (Table 1). The maximum value of Tm02 during non-monsoon months is up to 12.7 s, while during monsoon months, the value is about 8.5 s, showing a decrease in the values of Tm02 during monsoon months. The 99-percentile values of Tdw2 and Tdw1 are 11.3 s and 16.5 s. Standard deviation values of THmax, T1/10, and T1/3 are 2.1, 1.4, and 1.2 s (Table 1). When the spectral form is unknown, the energy wave period (Te) can be calculated from other variables, such as the mean or peak period (Contestabile et al. 2015), since the values of Te play a significant role in offshore power calculations. The mean values of Tavg and Te are 6.6 and 8.8 s. The 90-percentile values of THmax, Tavg, T1/3, and Te are 12.1, 7.7, 10.1, and 10.3 s (Table 1).

Table 1 Statistical parameters of different wave periods and corresponding wavelengths for the months June, July and August for different years (2013, 2015, 2017, and 2018)
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Figure 3a, b presents the percentage of occurrence of various wave periods during monsoon and non-monsoon months. In 2015, during the monsoon months, the percentage of Tp < 6 s was about 1.6%, while in non-monsoon months, the value is 9.3% and Tp > 8 s is around 81.3% in monsoon months and 88% in non-monsoon months. The percentage occurrence of Tc is 70.8% in between 3 and 4 s during monsoon (Fig. 3a), while 44% occurrence is seen during non-monsoon months (Fig. 3b). An occurrence of 40% is seen for T1/10 and Te, in between 8 and 9 s. During non-monsoon, the occurrence of Te > 12 s is 6%, while there is no occurrence of Te > 12 s during monsoon months. The significant wave period (T1/3) plays a significant role in the design of coastal structures (Goda 2000) and the standard deviation of T1/3 is 1.2 s.

Fig. 3
figure 3

Percentage occurrence of different wave period (a, b) and its corresponding wavelength (c, d) for monsoon months (June–August) and non-monsoon months for 2015

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3.2 Relationship between different wave periods

Scatter plots of different parameters with Te are presented in Fig. 4a–h and the results indicate that only Tm–1,1, T1/3 and Ti are almost close to the values of Te. The scatter plot of Te with Tp (Fig. 4e), Tdw1 (Fig. 4f), and THmax shows a wide scatter (Fig. 4g). T1/3 and Te are similar parameters (Goda 2000), and in the absence of Te, Tp is used to compute the wave power (Contestabile et al. 2015). Since Tp values are significantly higher than Te in the study area, computing wave power using Tp will result in a large overestimation of wave power. According to Hofland et al. (2017), several wave periods, including Tp and the primary zero-crossing period (T1/3), are associated with a range of coastal activities for typical spectral shapes and deep-sea conditions.

Fig. 4
figure 4

Inter comparison of Te with various wave periods for the 1 year wave data in 2015; a Tm–1,1, b Tm02, c Tm01, d T1/3, e Tp, f Tdw1, g THmax, h Ti,. Scatter plots of i T1/3 with THmax, j Tm02 with T1/3, k Hs with Hmax, and l Hs with H1/3

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The maximum wave height (Hmax) in the coastal waters of Gopalpur is 5.5 m from June to August which is much less than the values observed during the tropical cyclone. The northern BoB at 21.00° N, 89.00° E experienced a greater Hmax of ~ 17.9 m during the passage of tropical storm Aila (Kumar et al. 2020). The variation of Hs with Hmax is presented in Fig. 4k and it shows Hmax is 1.65 times Hs. The monthly average Hs varies from 1.37 to 1.73 m in different years. The scatter plot between THmax and T1/3 shows that THmax has a wide range compared to T1/3 (Fig. 4i), while T1/3 is equal to 1.36 times Tm02 (Fig. 4j). Te estimates from Tp are lower (Te = 0.786 Tp) due to the presence of numerous wave systems, on comparison with the Te estimates from the JONSWAP spectrum (Te = 0.9Tp) and the Pierson–Moskowitz spectrum (Te = 0.86Tp). The deep-water ratio of Te/Tp ~ 0.9 for a single-peaked spectrum is frequently utilized in engineering to know the wave period near a structure's toe from a known offshore wave period Tp (Hofland et al. 2017). The slope value of Te with Ti is close to 1, showing a linear relationship between both variables. Figure 4d indicates the relationship between Te and T1/3, Te is about 1.03 times Tp, and both play a significant role in the design of coastal structures. During the cyclone Phailin (8–13 October 2013), waves off Gopalpur showed a mean wave period ranging from 6.6 to 8.6 s, and Tp ranging between 8.4 and 18.4 s (Amrutha et al. 2014). According to Chun and Ahn's (2017) model, with a maximum value of Hs around 5.35 m, the predicted and measured values for T1/3 at Hamada during a storm period are 9.79 and 9.58 s. George and Kumar (2021) reported that the maximum monthly average Tp value over the BoB is 15 s and throughout the year in southeastern BoB, the occurrence of Tp > 12 s is larger. Despite the BoB being open to waves from the south IO and SO, significant longitudinal variation in Tp is seen. In the BoB, the values of Tp vary from a minimum of 2.7 s to a maximum of 21 s with an average value of 9.9 s (George and Kumar 2021).

3.3 Joint distribution of wave height and wave period

Once the significant wave height is determined using long-term distribution, there arises the problem of associating it with a mean wave period or peak period. The scatter plots of Tp with Hs indicate that waves with Hs more than 2.5 m are associated with Tp between 10 and 12 s (Fig. 5a). The waves with Hs more than 2 m have Tm02 between 6 and 8 s (Fig. 5b). Similarly, waves with height more than 4 m are with period between 8 and 12 s (Fig. 5c). Joint distribution of Hs and Tm02 shows that distribution of wave periods is broader than that of the wave heights, since the waves measured are mainly swells (Fig. 5d). Goda (2000) observed that distribution of wave periods is narrower than that of wave heights in wind seas. Both the models of the joint distribution of Hs and Tm02 considered in the study indicate that the models are in good agreement with the overall trend of the measured data (Fig. 5e, f). Liu et al. (2020) also observed that bivariate lognormal distribution and the Fang and Hogben model are in good agreement with the measured wave data in the coastal waters of the South China Sea.

Fig. 5
figure 5

Left panel shows the scatter plots of wave height and period. a Peak wave period and significant wave height, b mean wave period and significant wave height, c period and height of individual waves. Right panel is the contour plots showing the joint distribution of significant wave height and mean wave period during the year 2015 d measured data, e Fang and Hogben model, f Bivariate lognormal distribution

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3.4 Interannual variations in the wave period

The distribution of different wave periods of individual waves from June to August for different years is analysed. The analysis shows that higher wave periods (Tp > 8 s) are greater for the year 2013 and lower period waves (Tp < 6 s) are more in 2015. The monthly average wave spectrum indicates that the wave spectra show two peaks (Fig. 6). Most of the spectral energy density of the sea surface height variance during June to August including both the spectral peaks are in the low-frequency bands corresponding to swells with Tp ranging from 8.33 to 14.28 s.

Fig. 6
figure 6

Monthly average wave spectrum in different years. a June, b July and c August

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The 99-percentile values of Tm02 are up to 15% (Fig. 7a). Mean values of Tm02 for all the years of study are approximately the same (~ 6 s), while the 99-percentile values are high for the year 2013. In 2018, the mean value of Tc was 3.8 s, with the 99 and 90 percentile values reaching a maximum of 5.3 s and 4.6 s. The 99-percentile values are lower for the year 2015 except for Tc, THmax, and Tp. A maximum seasonal mean value of 9.5, 9.3, and 8.9 s for THmax, T1/10, Te is observed in the year 2017. The 90-percentile values of Tm01 and T1/3 are 8.5 and 10.6 s for 2017. Interannual variations in the mean values of Tavg (Fig. 7c) are less (< 5%).

Fig. 7
figure 7

Interannual variations in 99-percentile, 90-percentile and mean values of different wave periods (a–c) and corresponding variations in wavelength (d–f) in June to August 2013, 2015, 2017 and 2018

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3.5 Distribution of wavelength

For various years, the probability of exceedance of wavelength corresponding to different wave periods is estimated and presented in Fig. 8 at Gopalpur and the statistical parameters are presented in Table 1. Mean values of λTp and λTm02 are 123.7 and 56.7 m with a minimum-to-maximum value of 29.4 to 263.9 m for λTp (Fig. 8a) and 25.4 to 112.6 m for λTm02 (Fig. 8b). The wavelength based on the period of the highest wave ranged from 23.7 to 231.4 m (Fig. 8 h). The percentage occurrence plots of wavelength based on wave period during monsoon and non-monsoon months are presented in Fig. 3c, d. The standard deviation values of λTi and λTm01 are 17.1 and 14.3 m, respectively, and when compared with other parameters, λTp exhibits a large standard deviation. For waves with Tp > 5 s, the wavelength based on peak wave period (λTp) varies from 42.2 to 263.9 m. Wavelength based on crest period (λTc) ranges between 9.4 and 53 m with a mode value of 19 m (Fig. 8e). The values of the wavelength corresponding to λTdw2 and λTdw1 go up to a maximum of 155.6 m (Fig. 8f) and 277.6 m (Fig. 8g) and a minimum value of 39 m and 30.5 m, while the standard deviation of λTdw1, λTHmax is ~ 29 m. A maximum occurrence of 18.91% (Fig. 3c) is observed in between 85 and 90 m during monsoon months and about 1.67% (Fig. 3d) is observed for λTp for the same interval during non-monsoon months. During non-monsoon months, for higher wavelength (150–155 m), the occurrence of λTp is higher (15.90%) compared to that during the monsoon (5%). The values for λTc vary up to 53 m and show a peak in occurrence percentage for values between 10 and 20 m (Fig. 3c, d). The wavelength based on the average wave period (λTavg) and significant wave period (λT1/3) shows a mean value of 61 and 90.4 m for the study period. Interannual variation in the 90-percentile of the average wavelength is up to 9.84%. Wavelength based on significant wave period (T1/3) is less than 90 m during 51% of the time. The occurrence of λT1/10 is maximum in between 80 and 85 m with a percentage of 16.3 during monsoon and 4.14% during non-monsoon months. Around 17.2% is seen for λTe in between 95 and 100 m during monsoon months. Variations are seen in the occurrence percentage of wavelength associated with different wave periods during monsoon and non-monsoon months.

Fig. 8
figure 8

Distribution of various wavelength corresponding to different wave periods for 4 years during the monsoon months June–August

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3.6 Interannual variations in the wavelength corresponding to various wave periods

The interannual variations of wavelength based on different wave periods for four different years are shown in Fig. 7d–f. Figure 7d shows a decrease in 99-percentile values for the year 2015 except for λTHmax, λTp, and λTc. Most of the time, higher values for 90-percentiles are observed during 2017 for various parameters. The wavelength based on average wave period (λTavg) is mostly concentrated in a range between 28.8 and 135.3 m and the 90-percentile and mean values show an interannual variation of up to 10% and 7% (Fig. 7e, f). The 99-percentile values of λTavg are around 104.8 m for the year 2013. Interannual variations in the 90-percentile value of λTp are up to 9% (Fig. 7e).

The 90-percentile and the mean value of λTm02 are 76 and 59.4 m, and the 90-percentile and mean values are higher for the years 2013 and 2018. For all the years, the values of λTc are concentrated in the range of 9.4 to 53 m. A maximum 99-percentile value of 226, 172.4, 157.2, and 165.1 m is observed for λTdw1, λT1/10, λT1/3, and λTi. The mean values of λTm01 are 71.5 m and higher in 2018 and λTHmax are 101.7 m with higher values in 2017. The values of λTe vary between 46.9 and 174.6 m with a maximum mean value of 94.5 m in 2017 and the 90-percentile value (106.7 m) is lower for 2015. Interannual variations in mean values of λTe are only up to 7% (Fig. 7f).

3.7 Monthly variations in wave period and its corresponding wavelength during 2015

Monthly variations of wave period and its corresponding wavelength are presented in Fig. 9 for the year 2015. Earlier in this study, interannual variation and the distribution of all parameters during the monsoon months (June, July, and August) for different wave periods and the corresponding wavelength are analysed. In this section, the analysis of average, 99-percentile, and 90-percentile of different parameters throughout a year is studied. Figure 9 shows a rise in various wave periods and the corresponding wavelength except for the parameters Tp and λTp during October, November, and December, which might be caused by the influence of northeast monsoons or swells, or distant storms.

Fig. 9
figure 9

Monthly variation of various wave period and its corresponding wavelength during 2015

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Strong winds from the southwest provide longer period waves during the southwest monsoon, whereas winds from the northeast cause shorter period waves during the northeast monsoon. The analysis has shown that the values of Tp > 8 s during the northeast monsoon season are 91%. During the northeast monsoon period, even though short-period waves are present, the results show the presence of more long-period waves which might be caused due to the presence of swells and distant storms during this period. About 51% of values of wavelength corresponding to Tp are more than 100 m during the monsoon months and 87% of the values are higher than 100 m during non-monsoon months. The 99-percentile values of Tc and λTc range from 4 to 5.4 s and 39 to 44 m for various months. The average values for 2015 for the parameters Tm02, Tm01, Ti, and THmax are 6, 6.9, 9.2, and 8.7 s. The 90-percentile values of λTp, λTdw1, λTavg, and λTe are 194.8, 106, 85, and 122.4 m.

3.8 Distributions of wave steepness and wavelength

The distribution of wave steepness and wavelength at Gopalpur for different years during the monsoon months of June, July, and August is presented in Fig. 10. The wave steepness at the study region varied from 0.0002 to a maximum value of 0.2170 and for minimum steepness values, the wavelength ranges from 79 to 178 m. An average steepness value of ~ 0.018 is noticed during the entire year. A few waves with a period less than 1.7 s, height less than 1 m, and wavelength ranging between 4 and 4.5 m, exceed the wave-breaking limit. The most common values of wave period (T) range from 4 to 6.5 s. The shift in monsoon intensity causes significant variations in the temporal distribution of wave steepness over the eastern Arabian Sea, and the coastal waters along the central west coast of India are similar in terms of changes in wave steepness over a distance of around 325 km, but the southeastern Arabian Sea exhibits significant differences (Anju and Kumar 2023).

Fig. 10
figure 10

Distribution of a wave steepness and b wavelength for different years during monsoon months June, July, and August

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At Gopalpur, the values of the wavelength range from 4 to 283 m with the highest value of wavelength in 2017. For waves with wavelength < 5 m, the steepness values exceed the wave-breaking limit (0.1428 tanh kd). Higher values of wavelength ranging between 272 and 283 m are observed and its corresponding wave steepness varies from 0.002 to 0.006. Most of the time waves with higher steepness values have wavelengths ranging between 4 and 7 m. The maximum wavelength varies between 94 and 99 m for waves with a height of more than 5 m and their wave period is ~ 9 s for different years.

3.9 Variations in wave power

In 2015, Hs varied from 0.3 to 3.3 m with higher waves during the SW monsoon (June–August) and during the cyclone (Fig. 11a). The peak wave period ranged from 2.9 to 22 s with an annual mean value of 12.2 s (Fig. 11b). 50% of the time, Tp is between 10 and 14.3 s. The mean wave direction of 50% of the data is from 152 to 163° (Fig. 11b). For the study location, large fetch (> 1600 km) is available in the directional sector from 150° to 190° and hence, long period waves reach the measurement location from this directional sector. Since the nearshore bathymetry and the coastline have an inclination of ~ 48°to the east, the waves after refraction will approach the coast close to 138°. At the measurement location, the wave power ranges up to 45 kW/m in the monsoon period, while the annual mean wave power is 5 kW/m (Fig. 11d). 50% of the time, the wave power is between 2 and 7 kW/m, and 25% of the time, the wave power is more than 7 kW/m.

Fig. 11
figure 11

Time series plots of a significant wave height, b peak wave period and mean wave direction, c energy period and d wave power for the year 2015

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3.9.1 Analysis of selected freak wave events

This section analyses the freak wave events in Gopalpur in 2015. Freak waves with H1/3 greater than 1.5 m are only considered since freak waves with low wave heights do not pose problems to marine operations. Out of 17,519 data, it is observed that as per Hmax/Hs criteria around 37 freak wave events occurred and 11 events occurred as per the second criteria. Among this, by applying all the conditions, there are 5 freak wave events, and the characteristics of waves during these freak wave events are listed in Table 2. For the freak wave events, the values H1/3 vary from 1.5 to 2.1 m, steepness varies between 0.035 and 0.052 and the wavelength varies between 76.1 and 98.2 m. Among the freak waves that occurred, the highest H1/3 (2.1 m) and Hmax (4.5 m) are in June.

Table 2 Characteristics of selected freak waves occurred during 2015
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According to Shrira and Slunyaev (2014), opposing currents trap surface gravity waves and the nonlinear dynamics associated with these waves significantly increase the probability of generating freak waves. The crest to significant wave height ratio is more for freak wave event 3. The freak wave event numbered 4 (Table 2) has a high abnormality index of about 2.22 and the steepness is higher for the freak wave event 3 (Table 2). There are three events of freak waves with values of Tp greater than 8 s with the Hs values being 1.7, 2.1, and 1.5 m, which indicates that the major amount of energy is contributed by long-period swells and the other two events have Tp between 6 and 8 s, and the energy is contributed by intermediate period waves. On analysing freak wave events, the values of Tp vary between 7.6 and 9.2 s so the presence of intermediate and long-period waves is not significant.

4 Discussion

In practice, when the wave parameters are obtained from a numerical model, T1/3 and THmax are not available in the absence of time series data. Hence, it is necessary to know the relationship of THmax and T1/3 with Tm02. Coastal Engineering Manual (2008) and Goda (2000) relate THmax is equivalent to T1/3 and is equal to 1.1 to 1.3 times Tm02. For the study area, the best fit slope is 1.36 which is slightly outside the range of 1.1–1.3. James and Panchang (2022) reported that for waves off the UK coast, T1/3 is 1.43 times Tm02. For the study area, THmax is not equivalent to T1/3 and there is a large scatter for THmax compared to values of T1/3. At Gopalpur, the annual average Tmo2 for all the years is up to 6.2 s and the annual average Tp is about 11.2 s. According to Mendes and Oliveira (2021), the average Tm02 values at Faro, Nazare, and Leixoes fall within the range of 6–7 s, varying from a minimum of 3 s to a maximum of 15 s. Remya et al. (2020) reported a general decrease in mean wave period across most of the Indian Ocean suggesting that, while swells dominate the Indian Ocean, wind seas are becoming more important during June–August if a positive IOD, La Nina, and a positive phase of SAM co-occur or develop independently. Zhou et al. (2020) observed a weak correlation (0.47) between Tp and Tmean, and the Tp-to-Tmean ratio was determined to be 2.04 and T1/3 and Tm02 displayed a significant linear correlation with Tmean, while Tmax and Tp exhibited a less pronounced linear correlation with Tmean.

The average wave steepness value for Gopalpur in the present study is about 0.018. Glejin et al. (2013) observed that the steepness is greater than 0.25 for an ocean dominated by wind sea. In the Indian Ocean, the steepness ranges between 0.01 and 0.25 when the sea is dominated by young swell (Majumder et al. 2022). In the deep waters of the Norwegian continental shelf, the average wave steepness for wind-sea falls within the 0.031–0.046 range, while for combined swell and wind-sea, the values span a broader range of 0.0054–0.125 (Myrhaug 2018). Mendes and Oliveira (2021) reported an approximate average value of steepness as 0.031 in the deep waters off the coast of Portugal. The steepness of the highest waves varied from 0.048 to 0.082, with the highest value observed at the northern location and the lowest at the southern location in the Arabian Sea (Anju and Kumar 2023).

A total of five freak wave events out of 37 events were analysed in the present study in the year 2015. In the South China Sea, between May and August, the probability of occurrence of observed freak waves falls within the range of 0.03–0.53% (Ji et al. 2022). Over a 6-year period, wave measurements taken at 11 locations in the southern North Sea reveal that the frequencies of freak waves remain consistently uniform across seasons and show no correlation between stations (Teutsch et al. 2020).

5 Conclusion

The variations in the wave steepness, wave period, and wavelength of ocean surface waves in the coastal waters, which are crucial for assessing hazards, ensuring safe navigation, designing coastal infrastructure, and studying beach dynamics are examined. The various wave periods measured at 15 m water depth are analysed and their corresponding wavelength is also presented. The values of the peak wave period (Tp) indicate that most of the time a swell dominance is seen at the coastal waters of Gopalpur. The percentage of Tp > 8 s is about 86%, whereas only 1% of waves are with periods less than 6 s in the monsoon. The energy wave period (Te) plays an important role in describing wave energy and Te > 12 s is more in 2018. Half of the record shows that waves with wavelengths based on crest period (λTc) are less than 21 m. The analysis of the relationship of Te with various wave periods shows that Te is in good relation with Tm–1,1, T1/3, and Ti. Wavelength based on significant wave period (λT1/3) is less than 90 m during 51% of the time at Gopalpur. The values of λTp exhibit a large standard deviation. The interannual variation of mean values of λTavg is about 7%. The variations in the mean values of Tavg are up to 5%. During 2015, due to the presence of swells and distant storms large presence of long-period waves are observed. Analysis of the monthly variations for 2015, shows an increase in different parameters during the post-monsoon period. Among the five freak wave events, freak waves with higher maximum wave height (Hmax) and significant wave height (H1/3) are observed in June. In three freak wave events, the values of Tp are higher than 8 s, and the values for five events vary between 7.6 and 9.2 s. A higher abnormality index is noted for freak wave event 4, with a value of 2.22. The results have practical implications, aiding in the selection of a suitable wave energy converter for the study location and the analysis of freak waves provides information regarding abnormal wave characteristics. Comprehending the variations in wave characteristics on monthly, annual, and interannual scales is vital for a range of applications, including coastal engineering and navigation.