Modeling of the group structure of waves in the problems of the impact of waves on a ship

. The variability of the group structure sea surface waves is analyzed by numerical simulation. The analysis is carried out on the basis of an analytical model that describes the group structure of nonlinear waves. The characteristics of the wave group are controlled by two parameters, the group height factor (GFH) and the group length factor (GLF). With a fixed wave energy, the maximum wave height is determined only by GFH and does not depend on the GLF. It is shown that if GFH varies from 0.1 to 0.9, then at a fixed wave energy, the maximum wave height in the group increases by one and a half times. As the GLF increases, the number of high-height waves successively affecting the ship increases.


Introduction
The parametric roll is the result of the wave sequence effect.It is usually investigated by numerical simulation [1].The result depends on how well the wave is set.The wave is defined as regular (harmonic), irregular (superposition of harmonic waves with a random phase), and as a wave with a group structure.The last representation is the closest to a real sea wave.Irregular waves correspond to a linear process, while sea waves are weakly nonlinear [2,3].
The group structure of sea surface waves is observed at all stages of their development [4].It manifests itself in the form of alternating sequences of waves of high and low altitude.The impact of grouped waves on ships and coastal structures is greater than the impact of regular waves [5][6][7].Also, the group structure is associated with the generation of infrasound by the sea surface [8].
The main parameters characterizing the group structure of sea waves in the analysis of their impact on the ship are the groupiness factor of heights (GFH) and the group length factors (GLF) [5,9].GFH characterizes changes in the relative height of waves within a group.GLF defines the length of the group measured in the periods of the dominant wave (waves with a frequency corresponding to the peak of the spectrum).
In this paper, the analysis is carried out on the basis of the analytical model representing the group structure [10].The model describes a nonlinear wave process [11].

Roll motion
The roll motion of ship in the first approximation can be described by a linear differential equation of the second order where  is the roll angle, t is time,  is a coefficient that determines the attenuation of the roll motion,  is a frequency of free roll oscillations, ) (t F is a term describing the effect of waves on the ship. If the ship is on calm water (there are no waves), then   , equation (1) becomes homogeneous.If the ship is affected by waves of length  , then [12]     where x ~is the reduction coefficient, the value of which is determined by the ratio of the size of the ship and the wavelength, is a slope of the surface created by the wave.
In higher-order approximations, the equation describing the side pitching becomes nonlinear [5,7].For a narrow-band wave spectrum, it is shown that the random wave excitation moment is proportional to . Thus, in order to correctly describe the roll motion of ship, it is necessary to correctly set the wave profile.

Group structure of waves
The group structure is inherent in all types of sea surface waves and manifests itself in the form of alternating waves of high and low heights.The group structure of waves is usually represented as where a is amplitude multiplier;   and the carrier wave is represented as a harmonic where g is the gravitational acceleration.
In the case of irregular waves, the wave profile can be defined as a superposition of N harmonic waves [7]     where . It is assumed that the wave components do not interact with each other and satisfy the dispersion relation (5).
For a large N, according to the central limit theorem, equation ( 6) describes the Gaussian process [13].For him, skewness 3  and excess kurtosis 4  are zero.This statement does not take into account that sea waves are a weakly nonlinear process, the wave profile is asymmetric, respectively, in the general case 6) describes an irregular waves, this waves does not have a group structure.It should be noted that the group structure cannot arise in a linear wave field described by equation ( 6) and has a strong dispersion [14].The group structure can arise only due to the nonlinearity of wave processes [15,16], including due to the Benjamin-Fair instability [17,18].

Group structure simulation
Usually, two independent parameters are used to describe the magnitude of the wave grouping.These parameters are defined as where A  and A are the standard deviation and average of the wave envelope, respectively, where p f and pA f are the frequency of the peak of the wave spectrum and the spectrum of the wave envelope, respectively.For further analysis, we will use the analytical model of the wave profile proposed in [10], in which the wave envelope is given as where parameters 1  and 2  determine the shape of the envelope and its length.The carrier wave is given as  In the construction of Fig. 1, a dimensionless time  is introduced, measured in the periods of the dominant wave.Also, a wave profile is constructed in dimensionless units.Here and further, normalization is used for the wave profile, according to which the variance     is equal to one.

 
) cos( , ) To correctly compare different types of waves, it is necessary that they have the same energy.This is provided by normalization, which leads the variance of the wave to one.
Sea waves are not a Gaussian process.According to the data of wave measurements carried out in different regions of the World Ocean, the values of the skewness of surface  In the case of a well-defined group structure, the wave energy is concentrated in the waves located in the middle of the group, while their amplitude significantly exceeds the level corresponding to the energy of regular waves.Fig. 3 shows the changes in the wave profile for two values of GFH.Calculations were carried out at the length of the group 7 .7 GLF  .tends to 2.83 Although the GLF does not affect the maximum wave height in the group, in [7] concluded that changes in the GLF have more obvious effects on the parametric roll.The explanation is given in Fig. 5.For small values of the GLF, one wave of high height affects the ship, for large values of the GLF, several waves of high height consistently affect the ship.

Conclusion
Grouped surface waves have a more intense effect on the vessel than regular or irregular ones.The intensity of the impact can be parameterized by two characteristics: the height group factor (GFH) and the length group factor (GLF).If there is a change in GFH from 0.1 to 0.9, then at a fixed energy, the maximum wave height in the group increases by one and a half times.If the GLF increases, then the number of high-height waves consistently affecting the ship increases.


are the wave number and frequency of the carrier wave.The wave number and frequency are related to each other by a dispersion relation, which for gravitational waves in deep water has the form

E3S
Web of Conferences 431, 05007 (2023) ITSE-2023 https://doi.org/10.1051/e3sconf/202343105007whereparameter 0  determines the asymmetry of the carrier wave.This model allows us to obtain a wave profile with a given skewness of the statistical distribution of surface elevations.The model also takes into account the difference between the phase ph C fact that, as shown in Fig.1, the wave profiles in neighboring groups differ.

Fig. 1 .
Fig. 1.Wave profile     . In the presence of abnormal waves (killer waves) in the coastal zone, values were recorded 3 The effect of the asymmetry of the carrier wave is illustrated in Fig.2.In an asymmetric wave asim height is 15 % higher than the crest height reg  .