Fitting method for mathematical expression of spectrum of constrained subgravity waves generated by short-wave group

Abstract

The present invention provides a method for fitting a mathematical expression of a spectrum of constrained sub-gravity waves generated by a short-wave group, which relates to the fields of ocean wave hydrodynamics and marine engineering. The method improves an existing constrained sub-gravity wave spectrum model, namely, adds a surge wave height term and a spectrum peak rise term. The spectrum of the constrained sub-gravity wave is fitted using the improved constrained sub-gravity wave spectrum model to obtain a mathematical expression of the spectrum of the constrained sub-gravity wave. This solves the problem that the mathematical expression of the spectrum fitted by the existing constrained sub-gravity wave spectrum model cannot accurately describe the spectrum of the constrained sub-gravity wave. The present invention is suitable for fitting the spectrum of constrained sub-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 in particular to a method for fitting a mathematical expression of a spectrum of constrained sub-gravity waves generated by a short-wave group. Background Art

[0002] Typical ocean wave frequencies range from 0.04 Hz to 1 Hz, and these waves are called gravity waves. However, waves with frequencies between 0.004 Hz and 0.04 Hz also account for a significant portion of the ocean's energy. Their frequencies are much lower than those of common gravity waves, and they are called sub-gravity waves.

[0003] Infragravity waves are a major wave form in nearshore waters, closely linked to wave runup at the shoreline, nearshore sediment movement, the formation of unique coastline morphologies, and harbor oscillations. Therefore, studying the characteristics of infragravity waves is particularly important.

[0004] Based on the differences in generation mechanism and propagation characteristics, subgravity waves are divided into two categories: constrained subgravity waves and free subgravity waves. Constrained subgravity waves are phase-locked long waves generated by the nonlinear interaction of short-wave groups and propagated at the same phase velocity as the short-wave group. They can be calculated through the characteristic parameters of the short-wave group.

[0005] As the waves propagate towards the shore, the nonlinear effect of the waves increases the energy of the sub-gravity waves, causing the wave spectrum to change. The existing constrained sub-gravity wave spectrum model is: ,in, , represents the existing constrained subgravity wave spectrum, represents the acceleration due to gravity, represents the wave number, When fitting the constrained subgravity wave spectrum using the existing constrained subgravity wave spectrum model, the constrained subgravity wave spectrum cannot be accurately described at the spectral peak and high-frequency side. Summary of the Invention

[0006] Technical problem solved by the present invention: The present invention provides a method for fitting the mathematical expression of the spectrum of constrained subgravity waves generated by a short-wave group, which solves the problem that the mathematical expression of the spectrum fitted by the existing constrained subgravity wave spectrum model cannot accurately describe the constrained subgravity wave spectrum.

[0007] The present invention solves the above technical problems by adopting a technical solution: a method for fitting the mathematical expression of the spectrum of constrained subgravity waves generated by a short-wave group, comprising the following steps:

[0008] S1. Obtain the short wave group wave time series of the sea area to be analyzed;

[0009] S2. Based on the short wave group wave time series, the second-order nonlinear theory is used to calculate the wavefront of the constrained sub-gravity wave generated by the short wave group;

[0010] S3. performing frequency domain analysis on the wavefront of the constrained subgravity wave to obtain the spectrum of the constrained subgravity wave;

[0011] S4. Fitting the spectrum of the constrained subgravity wave using an improved spectral model of constrained subgravity waves to obtain a mathematical expression of the spectrum of the constrained subgravity wave, wherein the improved spectral model of constrained subgravity waves is: ,in, represents the spectrum of constrained infragravity waves, represents the first fitting coefficient, Indicates the swell wave height, represents the second fitting coefficient, represents pi, represents the acceleration due to gravity, represents the wave number, Indicates frequency, represents the constrained subgravity wave spectrum peak rise term, , represents the constrained subgravity wave spectrum peak raising factor, represents the period corresponding to the lower limit of the subgravity wave frequency, Indicates the peak width factor.

[0012] Furthermore, in S2, the second-order nonlinear theory is used to calculate the wavefront of the constrained sub-gravity waves generated by the short-wave group. The calculation formula used is: ,in, represents the constrained subgravity wavefront, Indicates the generation of the constrained subgravity wave The frequency of the components, Indicates the spectrum peak frequency of the shortwave group, Indicates the generation of the constrained subgravity wave The frequency of the components, Indicates the generation of the constrained subgravity wave components and The combined wavefront of the components, , Indicates the The forced amplitude of each component, Indicates the The forced amplitude of each component, express The phase of the components, Indicates the The component phases, Indicates the components and The interaction coefficients between the components, , represents the acceleration due to gravity, Indicates the The wave number of the component, Indicates the The wave number of the component, Indicates the The original frequency of the component, Indicates the The original frequency of the component, Indicates the The first component and the The wave number of the difference frequency term of the components, , represents the coefficient, , represents the hyperbolic cosine function, Indicates water depth.

[0013] Furthermore, in S3, the frequency domain analysis of the constrained subgravity wave surface is performed using Fourier transform processing. The formula for Fourier transform processing is: ,in, represents the wavefront that constrains the subgravity wave, represents the wavefront of the constrained subgravity wave within time t, Represents half of the total sampling time, represents the total number of samples within time t, Indicates the number of sampling times, , , , represents the sampling time interval, Indicates the The time corresponding to the subsampling.

[0014] Beneficial effects of the present invention: The present invention provides a method for fitting a mathematical expression of a spectrum of constrained subgravity waves generated by a short-wave group. By improving an existing constrained subgravity wave spectrum model, that is, adding a surge wave height term and a spectrum peak rise term, the spectrum of the constrained subgravity wave is fitted using the improved constrained subgravity wave spectrum model to obtain a mathematical expression of the spectrum of the constrained subgravity wave. In this way, the problem that the mathematical expression of the spectrum fitted by the existing constrained subgravity wave spectrum model cannot accurately describe the constrained subgravity wave spectrum is solved. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 This is a flow chart of a method for fitting a mathematical expression of a spectrum of constrained subgravity waves generated by a short-wave group provided by the present invention;

[0016] Figure 2 Schematic diagram of the wavefront of a constrained subgravity wave in an embodiment of the present invention;

[0017] Figure 3 is the spectrum of the constrained subgravity wave in the embodiment of the present invention;

[0018] Figure 4 is a comparison diagram of the original fitting spectrum and the improved fitting spectrum in an embodiment of the present invention;

[0019] Figure 5 3 is a comparison diagram of the fitted wave height and the measured wave height in an embodiment of the present invention. DETAILED DESCRIPTION

[0020] Aiming at the problem that the spectral mathematical expression fitted by the existing constrained subgravity wave spectrum model cannot accurately describe the constrained subgravity wave spectrum, the present invention proposes a fitting method for the spectral mathematical expression of the constrained subgravity wave generated by the short wave group, such as Figure 1 As shown, the following steps are included:

[0021] S1. Obtain the shortwave group wave time series of the sea area to be analyzed.

[0022] S2. Based on the short wave group wave time series, the second-order nonlinear theory is used to calculate the wavefront of the constrained sub-gravity wave generated by the short wave group.

[0023] Specifically, the calculation formula used to calculate the wavefront of constrained sub-gravity waves generated by short-wave groups using the second-order nonlinear theory is: ,in, represents the constrained subgravity wavefront, Indicates the generation of the constrained subgravity wave The frequency of the components, Indicates the spectrum peak frequency of the shortwave group, Indicates the generation of the constrained subgravity wave The frequency of the components, Indicates the generation of the constrained subgravity wave components and The combined wavefront of the components, , Indicates the The forced amplitude of each component, Indicates the The forced amplitude of each component, express The phase of the components, Indicates the The component phases, Indicates the components and The interaction coefficients between the components, , represents the acceleration due to gravity, Indicates the The wave number of the component, Indicates the The wave number of the component, Indicates the The original frequency of the component, Indicates the The original frequency of the component, Indicates the The first component and the The wave number of the difference frequency term of the components, , represents the coefficient, , represents the hyperbolic cosine function, Indicates water depth.

[0024] S3. Perform frequency domain analysis on the constrained subgravity wave wavefront to obtain the constrained subgravity wave spectrum.

[0025] Specifically, the frequency domain analysis of the constrained subgravity wave front is performed using Fourier transform processing, and the Fourier transform processing formula is: ,in, represents the wavefront that constrains the subgravity wave, represents the wavefront of the constrained subgravity wave within time t, Represents half of the total sampling time, represents the total number of samples within time t, Indicates the number of sampling times, , , , represents the sampling time interval, Indicates the The time corresponding to the subsampling.

[0026] S4. Fitting the spectrum of the constrained subgravity waves using the improved spectrum model of the constrained subgravity waves to obtain a mathematical expression of the spectrum of the constrained subgravity waves.

[0027] Specifically, the improved spectral model for constraining subgravity waves is: ,in, represents the spectrum of constrained infragravity waves, represents the first fitting coefficient, Indicates the swell wave height, represents the second fitting coefficient, represents pi, represents the acceleration due to gravity, represents the wave number, Indicates frequency, represents the constrained subgravity wave spectrum peak rise term, , represents the constrained subgravity wave spectrum peak raising factor, represents the period corresponding to the lower limit of the subgravity wave frequency, Indicates the peak width factor.

[0028] Example:

[0029] Taking the wave data obtained from actual observations in a certain sea area as an example, the sampling frequency is 4 Hz, the sampling time is 4096 s, and the total amount of data is 1355. In this way, the short-wave group wave time series of the sea area to be analyzed is obtained.

[0030] Based on the short wave group wave time series, the second-order nonlinear theory is used to calculate the wavefront of the constrained sub-gravity wave generated by the short wave group. The obtained constrained sub-gravity wavefront is as follows Figure 2 As shown, the horizontal axis represents time and the vertical axis represents the wave surface.

[0031] The frequency domain analysis of the constrained subgravity wave surface is performed using Fourier transform processing to obtain the constrained subgravity wave spectrum, such as Figure 3 As shown in the figure, 0.004 Hz represents the starting frequency of the constrained subgravity wave, and 0.04 Hz represents the cutoff frequency of the constrained subgravity wave. Figure 3 It can be seen that the spectrum of the constrained subgravity waves corresponding to 0.004 Hz to 0.04 Hz has a relatively uniform expression, so it can be described using the mathematical expression of the spectrum of the constrained subgravity waves.

[0032] The existing constrained subgravity wave spectrum model is used to fit the constrained subgravity wave spectrum to obtain the original fitting spectrum. The improved constrained subgravity wave spectrum model is used to fit the constrained subgravity wave spectrum to obtain the improved fitting spectrum. The original fitting spectrum and the improved fitting spectrum are compared, as shown in FIG. Figure 4 As shown in the figure, it can be found that the improved constrained subgravity wave spectrum model is superior to the existing constrained subgravity wave spectrum model in terms of spectral peak position and high-frequency side of the wave spectrum, and can better describe the energy distribution of constrained subgravity waves in the frequency domain.

[0033] Further verification was performed by fitting area and fitting wave height. , calculate the fitted area, where represents the fitting area, The mathematical expression of the constrained subgravity wave spectrum fitted by the improved constrained subgravity wave spectrum model is represented. Indicates the maximum frequency. The calculation formula for wave height is: , Indicates the fitted wave height, the comparison between the fitted wave height and the measured wave height is as follows Figure 5 As shown in the figure, it can be found that the fitted wave height is in good agreement with the measured wave height.

[0034] In summary, the improved constrained subgravity wave spectrum model is used to fit the constrained subgravity wave spectrum, which can accurately describe the constrained subgravity wave spectrum.

Claims

1. A method for fitting the mathematical expression of the spectrum of constrained subgravity waves generated by a short-wave group, characterized in that: The following steps are involved: 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, the second-order nonlinear theory is used to calculate the wavefront of the constrained sub-gravity wave generated by the short wave group; S3. performing frequency domain analysis on the wavefront of the constrained subgravity wave to obtain the spectrum of the constrained subgravity wave; S4. Fitting the spectrum of the constrained subgravity wave using an improved spectral model of constrained subgravity waves to obtain a mathematical expression of the spectrum of the constrained subgravity wave, wherein the improved spectral model of constrained subgravity waves is: ,in, represents the spectrum of constrained infragravity waves, represents the first fitting coefficient, Indicates the swell wave height, represents the second fitting coefficient, represents pi, represents the acceleration due to gravity, represents the wave number, Indicates frequency, represents the constrained subgravity wave spectrum peak rise term, , represents the constrained subgravity wave spectrum peak raising factor, represents the period corresponding to the lower limit of the subgravity wave frequency, Indicates the peak width factor.

2. The method for fitting the mathematical expression of the spectrum of constrained subgravity waves generated by the short-wave group according to claim 1, characterized in that: In S2, the second-order nonlinear theory is used to calculate the wavefront of the constrained sub-gravity waves generated by the short-wave group. The calculation formula is: ,in, represents the constrained subgravity wavefront, Indicates the generation of the constrained subgravity wave The frequency of the components, Indicates the spectrum peak frequency of the shortwave group, Indicates the generation of the constrained subgravity wave The frequency of the components, Indicates the generation of the constrained subgravity wave components and The combined wavefront of the components, , Indicates the The forced amplitude of each component, Indicates the The forced amplitude of each component, express The phase of the components, Indicates the The component phases, Indicates the components and The interaction coefficients between the components, , represents the acceleration due to gravity, Indicates the The wave number of the component, Indicates the The wave number of the component, Indicates the The original frequency of the component, Indicates the The original frequency of the component, Indicates the The first component and the The wave number of the difference frequency term of the components, , represents the coefficient, , represents the hyperbolic cosine function, Indicates water depth.

3. The method for fitting the mathematical expression of the spectrum of constrained subgravity waves generated by the short-wave group according to claim 1 is characterized in that: In S3, Fourier transform is used to perform frequency domain analysis on the constrained subgravity wave surface. The formula for Fourier transform is: ,in, represents the wavefront that constrains the subgravity wave, represents the wavefront of the constrained subgravity wave within time t, Represents half of the total sampling time, represents the total number of samples within time t, Indicates the number of sampling times, , , , represents the sampling time interval, Indicates the The time corresponding to the subsampling.

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