Fitting method of wave spectrum mathematical expression of constrained secondary gravity wave generated by short wave group

Abstract

The invention provides a fitting method of a wave spectrum mathematical expression of a constrained secondary gravity wave generated by a short wave group, and relates to the field of ocean wave hydrodynamic force and ocean engineering, and the method comprises the steps: improving an existing constrained secondary gravity wave spectrum model, namely increasing a surge wave height item and a spectrum peak rising item, and obtaining a mathematical expression of the wave spectrum of the constrained secondary gravity wave; and fitting the spectrum of the constrained secondary gravity wave by using an improved spectrum model of the constrained secondary gravity wave to obtain a mathematical expression of the spectrum of the constrained secondary gravity wave, thereby solving the problem that the spectrum mathematical expression fitted by the existing spectrum model of the constrained secondary gravity wave cannot accurately describe the spectrum of the constrained secondary gravity wave, and improving the accuracy of the spectrum of the constrained secondary gravity wave. The method is suitable for spectrum fitting of constrained secondary gravity waves generated by a short wave group.

Description

Technical Field

[0001] The present invention relates to the fields of ocean wave hydrodynamics and ocean engineering, and particularly to a fitting method for the spectral mathematical expression of the trapped sub-gravity waves generated by short-wave groups. Background Art

[0002] Typical ocean waves have a frequency distribution between 0.04 Hz and 1 Hz, and these waves are called gravity waves. However, waves with a frequency range distribution between 0.004 Hz and 0.04 Hz also account for a considerable part of the energy in the ocean. Their frequencies are much lower than those of common gravity waves, and they are called sub-gravity waves.

[0003] Sub-gravity waves are a main form of waves in the nearshore sea area, and are closely related to phenomena such as wave run-up at the shoreline, nearshore sediment movement, the formation of special-shaped coasts, and harbor oscillations. Therefore, it is particularly important to study the wave characteristics of sub-gravity waves.

[0004] According to the differences in generation mechanisms and propagation characteristics, sub-gravity waves are further divided into two categories: trapped sub-gravity waves and free sub-gravity waves. Among them, trapped sub-gravity waves are locked-phase long waves generated by the nonlinear interaction of short-wave groups and propagating at the same phase velocity as the short-wave groups, and can be calculated through the characteristic parameters of the short-wave groups.

[0005] As the waves propagate towards the shore, the energy of the sub-gravity waves increases under the action of wave nonlinearity, resulting in changes in the wave spectrum morphology. The existing wave spectrum model of trapped sub-gravity waves is: , where, , represents the existing wave spectrum of trapped sub-gravity waves, represents the acceleration due to gravity, represents the wave number, represents the frequency. When using the existing wave spectrum model of trapped sub-gravity waves to fit the wave spectrum of trapped sub-gravity waves, the wave spectrum of trapped sub-gravity waves cannot be accurately described at positions such as the spectral peak and the high-frequency side of the wave spectrum. Summary of the Invention

[0006] The technical problem to be solved by the present invention: The present invention provides a fitting method for the spectral mathematical expression of the trapped sub-gravity waves generated by short-wave groups, which solves the problem that the spectral mathematical expression fitted by the existing wave spectrum model of trapped sub-gravity waves cannot accurately describe the wave spectrum of trapped sub-gravity waves.

[0007] The technical solution adopted by the present invention to solve the above technical problem: A fitting method for the spectral mathematical expression of the trapped sub-gravity waves generated by short-wave groups, comprising the following steps: S1. Obtain the short-wave group wave time series of the sea area to be analyzed; S2. Calculate the wave surface of the constrained sub - gravity waves generated by the short - wave group based on the short - wave group wave time series using the second - order nonlinear theory; S3. Conduct a frequency - domain analysis of the wave surface of the constrained sub - gravity waves to obtain the wave spectrum of the constrained sub - gravity waves; S4. Fit the wave spectrum of the constrained sub - gravity waves using the improved wave spectrum model of the constrained sub - gravity waves to obtain the mathematical expression of the wave spectrum of the constrained sub - gravity waves. The improved wave spectrum model of the constrained sub - gravity waves is: , where, represents the wave spectrum of the constrained sub - gravity waves, represents the first fitting coefficient, represents the swell wave height, represents the second fitting coefficient, represents pi, represents the gravitational acceleration, represents the wave number, represents the frequency, represents the constrained sub - gravity wave spectrum peak elevation term, , represents the constrained sub - gravity wave spectrum peak elevation factor, represents the period corresponding to the lower limit of the constrained sub - gravity wave frequency, represents the spectral peak width factor.

[0008] Furthermore, in S2, the calculation formula for calculating the wave surface of the constrained sub - gravity waves generated by the short - wave group using the second - order nonlinear theory is: , where, represents the wave surface of the constrained sub - gravity waves, represents the frequency of the th component generating the constrained sub - gravity waves, represents the spectral peak frequency of the short - wave group, represents the frequency of the th component generating the constrained sub - gravity waves, represents the combined wave surface of the th component and the th component generating the constrained sub - gravity waves, , represents the forced amplitude of the th component, represents the forced amplitude of the th component, represents the phase of the th component, represents the phase of the th component, represents the interaction coefficient between the th component and the , represents the acceleration due to gravity, represents the wavenumber of the th component, wavenumber of the th component, original frequency of the th component, original frequency of the th component, wavenumber of the difference frequency term between the th component and the , represents a coefficient, , represents the hyperbolic cosine function, represents the water depth.

[0009] Furthermore, in S3, the frequency domain analysis of the constrained sub - gravity wave surface is processed by Fourier transform. The formula for Fourier transform processing is: , where represents the wave surface of the constrained sub - gravity wave, represents the wave surface of the constrained sub - gravity wave within time t, represents half of the total sampling duration, represents the total number of samples within time t, represents the number of sampling times, , , , represents the sampling time interval, represents the th sampling - corresponding time.

[0010] Advantages of the present invention: The present invention provides a fitting method for the mathematical expression of the wave spectrum of constrained sub - gravity waves generated by short - wave groups. By improving the existing wave spectrum model of constrained sub - gravity waves, that is, adding the swell wave height term and the spectral peak elevation term, and using the improved wave spectrum model of constrained sub - gravity waves to fit the wave spectrum of the constrained sub - gravity waves, a mathematical expression of the wave spectrum of the constrained sub - gravity waves is obtained. Thus, the problem that the mathematical expression of the wave spectrum fitted by the existing wave spectrum model of constrained sub - gravity waves cannot accurately describe the wave spectrum of constrained sub - gravity waves is solved. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 is a schematic flow chart of a fitting method for the mathematical expression of the wave spectrum of constrained sub - gravity waves generated by short - wave groups provided by the present invention; Figure 2 is a schematic diagram of the wave surface of the constrained sub - gravity wave in an embodiment of the present invention; Figure 3is the wave spectrum of the constrained sub-gravity waves in the embodiments of the present invention; Figure 4 is the comparison chart of the original fitting spectrum and the improved fitting spectrum in the embodiments of the present invention; Figure 5 is the comparison chart of the fitting wave height and the measured wave height in the embodiments of the present invention. Specific embodiments

[0012] In view of the problem that the wave spectrum mathematical expression fitted by the existing constrained sub-gravity wave spectrum model cannot accurately describe the constrained sub-gravity wave spectrum, the present invention proposes a fitting method for the wave spectrum mathematical expression of the constrained sub-gravity waves generated by short wave groups, as Figure 1 shown, including the following steps: S1. Obtain the short wave group wave time series of the sea area to be analyzed.

[0013] S2. Based on the short wave group wave time series, use the second-order nonlinear theory to calculate the wave surface of the constrained sub-gravity waves generated by the short wave group.

[0014] Specifically, the calculation formula for using the second-order nonlinear theory to calculate the wave surface of the constrained sub-gravity waves generated by the short wave group is: , where represents the wave surface of the constrained sub-gravity waves, represents the frequency of the th component that generates the constrained sub-gravity waves, represents the spectral peak frequency of the short wave group, represents the frequency of the th component that generates the constrained sub-gravity waves, represents the combined wave surface of the th component and the th component that generates the constrained sub-gravity waves, , represents the forced amplitude of the th component, represents the forced amplitude of the th component, represents the phase of the th component, represents the phase of the th component, represents the interaction coefficient between the th component and the , represents the gravitational acceleration, represents the wave number of the th component, represents the wave number of the th component, represents the The original frequency of the component, denotes the original frequency of the component, and the wavenumber of the difference frequency term between the component and the , denotes the coefficient, , denotes the hyperbolic cosine function, and the water depth.

[0015] S3. Perform a frequency domain analysis on the constrained sub-gravity wave surface to obtain the wave spectrum of the constrained sub-gravity wave.

[0016] Specifically, the frequency domain analysis of the constrained sub-gravity wave surface is processed using Fourier transform, and the formula for Fourier transform processing is: , where denotes the wave surface of the constrained sub-gravity wave, denotes the wave surface of the constrained sub-gravity wave within time t, denotes half of the total sampling duration, denotes the total number of samples within time t, denotes the number of samplings, , , , denotes the sampling time interval, denotes the th sampling corresponding time.

[0017] S4. Fit the wave spectrum of the constrained sub-gravity wave using the improved wave spectrum model of the constrained sub-gravity wave to obtain the mathematical expression of the wave spectrum of the constrained sub-gravity wave.

[0018] Specifically, the improved wave spectrum model of the constrained sub-gravity wave is: , where denotes the wave spectrum of the constrained sub-gravity wave, denotes the first fitting coefficient, denotes the swell wave height, denotes the second fitting coefficient, denotes pi, denotes the gravitational acceleration, denotes the wavenumber, denotes the frequency, denotes the constrained sub-gravity wave spectrum peak elevation term, , denotes the constrained sub-gravity wave spectrum peak elevation factor, denotes the period corresponding to the lower limit of the constrained sub-gravity wave frequency, denotes the spectrum peak width factor.

[0019] Example: Taking the wave data obtained from real observations in a certain sea area as an example, the sampling frequency is 4 Hz, the sampling duration is 4096 s, and the total amount of data is 1355. Thus, a short-wave group wave time series of the sea area to be analyzed is obtained.

[0020] Based on the short-wave group wave time series, the second-order nonlinear theory is used to calculate the wave surface of the constrained sub-gravity waves generated by the short-wave group. The obtained wave surface of the constrained sub-gravity waves is as Figure 2 shown. The horizontal axis represents time, and the vertical axis represents the wave surface.

[0021] The wave spectrum of the constrained sub-gravity waves is obtained by performing frequency-domain analysis on the wave surface of the constrained sub-gravity waves using Fourier transform processing, as Figure 3 shown. Among them, 0.004 Hz represents the starting frequency of the constrained sub-gravity waves, and 0.04 Hz represents the cut-off frequency of the constrained sub-gravity waves. It can be seen from Figure 3 that the wave spectrum of the constrained sub-gravity waves corresponding to 0.004 Hz to 0.04 Hz has a relatively unified expression form. Therefore, the wave spectrum mathematical expression of the constrained sub-gravity waves can be used for description.

[0022] The wave spectrum of the constrained sub-gravity waves is fitted using the existing constrained sub-gravity wave spectrum model to obtain the original fitting spectrum. The wave spectrum of the constrained sub-gravity waves is fitted using the improved constrained sub-gravity wave spectrum model to obtain the improved fitting spectrum. By comparing the original fitting spectrum and the improved fitting spectrum, as Figure 4 shown, it can be found that the improved constrained sub-gravity wave spectrum model is superior to the existing constrained sub-gravity wave spectrum model both in the spectral peak position and the high-frequency side of the wave spectrum, and can better describe the energy distribution of the constrained sub-gravity waves in the frequency domain.

[0023] Further verification is carried out through the fitting area and the fitting wave height. Using , the fitting area is calculated, where represents the fitting area, represents the mathematical expression of the wave spectrum of the constrained sub-gravity waves fitted by the improved constrained sub-gravity wave spectrum model, represents the maximum frequency. The calculation formula for the wave height is: , represents the fitting wave height. The comparison between the fitting wave height and the measured wave height is as Figure 5 shown, and it can be found that the fitting wave height is in good agreement with the measured wave height.

[0024] In summary, using the improved constrained sub-gravity wave spectrum model to fit the wave spectrum of the constrained sub-gravity waves can accurately describe the wave spectrum of the constrained sub-gravity waves.

Claims

1. A fitting method for the spectral mathematical expression of sub-gravity waves constrained by short-wave groups, characterized in that, Including the following steps: S1. Obtain the short-wave group wave time series of the sea area to be analyzed; S2. Based on the short-wave group wave time series, use the second-order nonlinear theory to calculate the wave surface of the constrained sub-gravity waves generated by the short-wave group; S3. Conduct frequency domain analysis on the wave surface of the constrained sub-gravity waves to obtain the wave spectrum of the constrained sub-gravity waves; S4. Fit the spectrum of the constrained sub-gravity wave using an improved wave spectrum model of the constrained sub-gravity wave to obtain a mathematical expression for the spectrum of the constrained sub-gravity wave. The improved wave spectrum model of the constrained sub-gravity wave is as follows: , where represents the spectrum of the constrained sub-gravity wave, represents the first fitting coefficient, represents the swell wave height, represents the second fitting coefficient, represents pi, represents the acceleration due to gravity, represents the wave number, represents the frequency, represents the constrained sub-gravity wave spectrum peak elevation term, , represents the constrained sub-gravity wave spectrum peak elevation factor, represents the period corresponding to the lower limit of the constrained sub-gravity wave frequency, represents the spectral peak width factor.

2. The fitting method for the spectral mathematical expression of the constrained sub-gravity waves generated by short-wave groups according to claim 1, characterized in that, In S2, the calculation formula for the wave surface of the short-wave group generating the bound sub-gravity wave using the second-order nonlinear theory is as follows: , where represents the wave surface of the bound sub-gravity wave, represents the frequency of the th component generating the bound sub-gravity wave, represents the spectral peak frequency of the short-wave group, represents the frequency of the th component generating the bound sub-gravity wave, represents the combined wave surface of the th component and the th component generating the bound sub-gravity wave, , represents the forced amplitude of the th component, represents the forced amplitude of the th component, represents the phase of the component, represents the phase of the th component, represents the th component and the th component, the interaction coefficient between the components, , represents the gravitational acceleration, represents the th component wave number, represents the th component wave number, represents the th component original frequency, represents the th component original frequency, represents the th component and the th component, the wave number of the difference frequency term, , represents the coefficient, , represents the hyperbolic cosine function, represents the water depth.

3. The fitting method of the spectral mathematical expression of the constrained sub-gravity waves generated by short-wave groups according to claim 1, characterized in that, In S3, the frequency-domain analysis of the constrained sub-gravity wave surface is processed by Fourier transform, and the formula for Fourier transform processing is: , where represents the wave surface of the constrained sub-gravity wave, represents the wave surface of the constrained sub-gravity wave within time t, represents half of the total sampling duration, represents the total number of samples within time t, represents the number of sampling times, , , , represents the sampling time interval, represents the time corresponding to the

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