Wave making method for Peregrine respiratory wavelets
Through special wave-making methods, wave parameters and wave-making signals are determined, which successfully solved the problem that the prior art is difficult to simulate Peregrine breathing sub-waves, and achieved high-precision wave-making effect.
Patent Information
- Application Number
- CN202510163902.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-14
- Publication Date
- 2025-04-25
AI Technical Summary
The prior art is difficult to effectively simulate and generate Peregrine respiratory sub-waves, and traditional wave-making methods are not suitable for such nonlinear extreme waves.
A wavemaking method specifically used for Peregrine breathing sub-wave is proposed. By determining wave parameters, inputting wavemaking signals, measuring and recording wave surface time calendar, adjusting focus position and wave height, achieving high-precision wavemaking.
The efficient and accurate simulation of the Peregrine respiratory sub-wave is achieved, and the focus position and wave height reach the target value, and the wavemaking results are consistent with the analytical values.
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Figure CN119880342A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of marine environment simulation, and in particular relates to a wave-making method for Peregrine breathing sub-waves. Background Art
[0002] Freak waves, sometimes also called vicious waves, are nonlinear extreme waves that have characteristics not possessed by regular waves, such as extremely large wave crests and wave heights, concentrated energy, and accidental occurrences. They are highly destructive and often lead to catastrophic consequences. Usually, a freak wave is defined as a wave with a maximum wave height greater than twice the significant wave height.
[0003] The freak wave model based on the modulation instability principle is currently the most widely used type of nonlinear freak wave model. The modulation instability of Stokes waves can be explained by the nonlinear Schrödinger equation. Equation, NLS equation), Davey-Stewartson system, Korteweg-de Vries (KdV) equation, Kadomtsev-Petviashvili equation, etc. Among them, in the field of waves, the NLS equation is the most widely used. There are many breather solutions of the NLS equation, including the Peregrine breather solution, the Kuznetsov-Ma breather solution, the Akhmediev breather solution, etc. The Akhmediev breather solution shows obvious spatial periodicity, the Kuznetsov-Ma breather solution shows obvious temporal periodicity, and the Peregrine breather solution has no temporal or spatial periodicity. Due to the non-periodic characteristics of spatiotemporal distribution, the Peregrine breather solution is most suitable for simulating extreme waves, and is used as a theoretical prototype for simulating freak waves in many studies. Freak waves (Peregrine breather waves) based on the Peregrine breather solution are a typical type of nonlinear extreme waves with obvious characteristics such as large wave crests, large wave heights, and concentrated energy.
[0004] Based on Ma's research, Peregrine derived the Peregrine breather solution of the NLS equation for the first time, which is an analytical solution. Dysthe and Trulsen's analysis showed that the Peregrine breather solution is an effective analytical model for simulating steep wave motion. Chabchoub et al. observed the wave surface time history of the Peregrine-type breather solution in a laboratory physical water tank, which was in good agreement with the analytical solution, proving the existence of the Peregrine breather solution in the field of surface waves. Chabchoub et al., Shemer and Alperovich, Onorato et al. observed the Peregrine breather wave through experiments and studied its basic characteristics. Hu Zhe, Klein and other scholars have simulated the dynamic behavior and basic characteristics of Peregrine breathing waves through numerical methods, which are very similar to the experimental results. Chabchoub tracked the evolution of Peregrine breathing waves in the JONSWAP (The JointNorth Sea Wave Project) wave field through experimental methods. The experiment shows that breathing waves can exist and continue to develop in random irregular wave fields.
[0005] Existing research shows that the Peregrine breather solution is an effective model for simulating freak waves and studying the interaction between waves and structures, and can be effectively simulated and generated through experimental and numerical methods.
[0006] So far, although there are relatively many studies on Peregrine breathing wavelets, few of them focus on the wave-making methods of Peregrine breathing wavelets. Traditional methods for irregular wave generation, such as the optimization approach of wave sequences and the phase-amplitude iteration scheme, are mostly applicable to irregular wave sequences generated by linear superposition, but are not suitable for the simulation generation of such Peregrine breathing wavelets.
[0007] Due to the nonlinear effects in the wave propagation process, the Peregrine breathing wavelets generated by experimental wave generation often have certain differences from the analytical values in terms of actual focusing position, maximum peak height, etc.
[0008] At the same time, there are relatively few experimental measured data on the existing Peregrine breathing wavelet, and there is a lack of relevant experimental simulation data. Summary of the invention
[0009] Aiming at the problem that the traditional wave-making method for irregular waves is not suitable for the simulation generation of such Peregrine breathing wavelets, a wave-making method for Peregrine breathing wavelets is proposed.
[0010] The technical solution of the present invention is:
[0011] A wave-making method for Peregrine breathing wavelets comprises the following steps:
[0012] Step 1: Determine the wave parameters: At the beginning, determine the relevant wave parameters of the target Peregrine breathing sub-wave, including the carrier amplitude A 0 , carrier wave number k, water depth h, focusing position x 0 , focus time t 0, we can get the wavefront history η(x 0 , t), and then obtain the wave height resolution value H at the focus position A =η max (x 0 ,t)-η min (x 0 ,t); Set the target focus position to a constant X 0 , let X 0 =x 0 ; Set the initial value of the wave height adjustment parameter Factor to 1.0; According to the Peregrine breathing sub-solution, the wave surface time history of this wave can be uniquely determined;
[0013] Step 2: Input the wave signal: Based on the determined wave parameters, the wave height time history η(0,t) at the inlet wave boundary can be obtained by the Peregrine breathing sub-solution, where t is time. After multiplying it by the parameter Factor, the wave height form of the wave signal η inlet (t) inputting into a wave maker to generate waves;
[0014] Step 3: Measure and record the target focus position X 0 and the two places before and after it (X 0 –△x,X 0 +△x) is used to analyze and determine the nonlinear evolution of the Peregrine breathing wavelet;
[0015] Step 4, focus position judgment: judge the actual focus position of the generated breathing wavelet according to the axisymmetry of the measured wavefront time history, and the relative front-back relationship with the target focus position; according to the evolution characteristics of the Peregrine breathing wavelet, if the target focus position X 0 The wavefront time history curve at is approximately axisymmetric about a certain moment, which means that the actual focusing position is the target focusing position X 0 If the axial symmetry of the wavefront time history curve before the target focusing position is more obvious, it means that the actual focusing position is before the target focusing position, and x needs to be appropriately increased. 0 If the axial symmetry of the wavefront time history curve after the target focusing position is more obvious, it means that the actual focusing position is after the target focusing position, and x needs to be appropriately reduced. 0 value; adjust x 0 After the value is large or small, return to step 2 to regenerate the wave signal;
[0016] Step 5: Target wave height determination: Determine the target focus position X 0 The wave height H measured at E Is it close to the analytical value H? A ; If the target focus position X0 The relative value of the difference between the wave height at and the analytical value Error (Error = |H E -H A | / H A ) is less than the actual set value, the judgment is "yes", then the height of the generated deformed wave reaches the target height; if the difference between the two is large, it is necessary to adjust the parameter Factor value appropriately and return to step 2 to regenerate the wave signal;
[0017] Step 6. Successfully generate the target wave: If the steps "focus position judgment" and "target wave height judgment" are both "yes", the Peregrine breathing sub-wave is successfully generated, and its focus position and wave height both reach the target values.
[0018] Furthermore, in step 1, the parameter carrier wave number k can also be replaced by carrier frequency ω and carrier wavelength L.
[0019] Furthermore, in step 1, the target focus position X is determined 0 It is a constant, non-variable, and once determined, it will not change during the entire wave-making process.
[0020] Furthermore, in step 1, the first-order solution of the wave surface height of the Peregrine breathing sub-wave is:
[0021]
[0022] Where i represents a complex unit, x represents a one-dimensional spatial coordinate along the wave propagation direction, t represents time, and α and β are parameters. The carrier envelope A(x, t) is calculated by the Peregrine breathing wavelet, ψ = kx–ωt, k and ω satisfy the dispersion relation ω 2 =gk tanh(kh), g is the acceleration due to gravity.
[0023] Furthermore, in step 2, the inlet wave-generating boundary is at x=0, where x is a one-dimensional spatial coordinate along the wave propagation direction.
[0024] Furthermore, in step 3, the value of △x is recommended to be in the range of 5% to 16% of the carrier wavelength.
[0025] Furthermore, in step 4 and step 5, the Error can be set to 2.0% according to the wave-making accuracy requirement.
[0026] Furthermore, the carrier envelope A(x,t) is calculated by the Peregrine breathing sub-solution:
[0027]
[0028] Among them, A is related to time t and space x, A = A(x, t), which represents the carrier envelope in the distorted wave model; A 0 is a constant, indicating the carrier amplitude; ω is the carrier frequency; α and β are parameters.
[0029] Furthermore, the parameters α and β are expressed as:
[0030]
[0031]
[0032] Where h is the water depth.
[0033] Furthermore, under the deep water condition kh>3, equations (7) and (8) can be simplified as follows:
[0034]
[0035] .
[0037] The beneficial effects of the present invention are:
[0038] The present invention proposes a wave-making method dedicated to Peregrine breathing wavelets, which improves the wave-making efficiency while ensuring the high accuracy of the wave-making results. Based on this method, a series of Peregrine breathing wavelets with different wave heights and periods are generated in a deep water experimental pool, and the wave-making results are in good agreement with the analytical values. This method realizes the accurate and efficient simulation of Peregrine breathing wavelets in numerical pools and experimental pools.
[0039] Based on the unique nonlinear evolution characteristics of Peregrine breathing wavelets, this paper proposes an accurate and efficient wave-making method dedicated to Peregrine breathing wavelets, which is suitable for numerical and experimental wave-making of this type of waves. Compared with general wave-making methods, this wave-making method can focus the Peregrine breathing wavelets at a specific position. BRIEF DESCRIPTION OF THE DRAWINGS
[0040] Figure 1 The figure is a wave-making flow chart of the Peregrine breathing wavelet of the present invention;
[0041] Figure 2 Diagram for the Peregrine breather. DETAILED DESCRIPTION
[0042] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.
[0043] The present invention proposes an accurate and efficient wave-making method dedicated to Peregrine breathing sub-waves, which is suitable for numerical and experimental wave-making of this type of waves. The specific wave-making process is as follows: Figure 1 shown.
[0044] In the wave making process, the specific steps are described in detail as follows.
[0045] (1) Determine the wave parameters: At the beginning, determine the relevant wave parameters of the target Peregrine breathing sub-wave (carrier amplitude A 0 , carrier wave number k (or carrier frequency ω, carrier wavelength L), water depth h, focusing position x 0 , focus time t 0 ), from which the wavefront history η(x 0 , t), and then obtain the wave height resolution value H at the focus position A =η max (x 0 ,t)-η min (x 0 ,t); Set the target focus position to a constant X 0 , let X 0 =x 0 ; Set the initial value of the wave height adjustment parameter Factor to 1.0 (the parameter Factor is used to adjust the wave height later). According to the Peregrine breathing sub-solution, the wave surface history of this wave can be uniquely determined. Note that the target focus position X determined in this step 0 It is a constant (non-variable) and once determined, it will not change during the entire wave-making process.
[0046] (2) Inputting wave-making signal wave-making: Based on the determined wave parameters, the wave surface height time history η(0, t) (t is time) at the inlet wave-making boundary (i.e., x = 0, x is the one-dimensional spatial coordinate along the wave propagation direction) can be obtained through the Peregrine breathing sub-solution. After multiplying it by the parameter Factor, the wave-making signal η in the form of wave surface height is obtained. inlet (t) Input the wave maker to generate waves.
[0047] (3) Measure and record the target focus position X 0 and the two places before and after it (X 0 –△x,X 0+△x) is used to analyze and determine the nonlinear evolution of the Peregrine breathing wavelet. The value of △x is recommended to be in the range of 5% to 16% of the carrier wavelength.
[0048] (4) Focus position judgment: The actual focus position of the generated breathing wavelet is judged based on the axial symmetry of the measured wavefront time history, as well as the relative front-back relationship with the target focus position. According to the evolution characteristics of the Peregrine breathing wavelet, if the target focus position X 0 If the wavefront time history curve at the position is approximately axisymmetric about a certain moment (this item is judged as "yes"), it means that the actual focus position is the target focus position X 0 If the axial symmetry of the wavefront time history curve before the target focusing position is more obvious, it means that the actual focusing position is before the target focusing position, and x needs to be appropriately increased. 0 If the axial symmetry of the wavefront time history curve after the target focusing position is more obvious, it means that the actual focusing position is after the target focusing position, and x needs to be appropriately reduced. 0 Value. Adjust x 0 After the value is obtained, return to step (2) to regenerate the wave signal.
[0049] (5) Target wave height judgment: Determine the target focus position X 0 The wave height H measured at E Is it close to the analytical value H? A Here, it is considered that if the target focus position X 0 The relative value of the difference between the wave height at and the analytical value Error (Error = |H E -H A | / H A ) is less than 2.0% (the error criterion can be adjusted according to specific needs), and the judgment is "yes", then the height of the generated deformed wave reaches the target height; if the difference between the two is large, it is necessary to adjust the parameter Factor value appropriately and return to step (2) to regenerate the wave signal.
[0050] (6) Successfully generating the target wave: If the answers to both the step “focus position determination” and the step “target wave height determination” are “yes”, the Peregrine breathing sub-wave is successfully generated, and its focus position and wave height both reach the target values.
[0051] Example verification
[0052] Nine Peregrine breathing wavelet conditions are selected for experimental wave generation, as shown in Table 1. Among them, conditions H1~H5 are breathing wavelets with different carrier amplitudes and the same carrier wavelength, conditions L1~L5 are breathing wavelets with the same carrier amplitude and different carrier wavelengths, and conditions H4 and L2 are the same conditions. All breathing wavelets are generated in a still water environment and are non-breaking waves.
[0053] Table 1 Experimental wave-making conditions of Peregrine breathing wavelet
[0054]
[0055] In all working conditions, the maximum peak height, trough depth, wave height of the experimentally generated Peregrine breathing sub-wavelet and its comparison with the analytical solution are shown in Table 2. It can be found that compared with the analytical solution, the wave height error of all working conditions is less than 2.0%, indicating that the wave making method of the present invention can achieve the expected low error and high precision wave making.
[0056] Table 2 Peregrine breathing wavelet experimental data and its comparison with analytical values
[0057]
[0058] The following is the theoretical model of the Peregrine respiratory wavelet which is the theoretical basis of the present invention:
[0059] 1.1 Modulation instability and nonlinearity of Stokes waves equation
[0060] When a Stokes wave train with amplitude a propagates in a deep water environment with fundamental frequency ω, under the condition of small low-frequency sideband disturbance, the carrier frequency ω and the sideband wave frequency ω(1±δ) (where δ satisfies k is the carrier wave number) is very close, the carrier energy will be transferred to the sideband wave, and under the nonlinear effect, the amplitude of the sideband wave will increase exponentially. This phenomenon is called modulational instability (also known as Benjamin-Feir instability) of the wave.
[0061] In the wave domain, this modulation instability can be described by the nonlinear Schrödinger equation (NLS equation), which is expressed as
[0062]
[0063] This equation is about the carrier envelope A(x,t); where i represents a complex unit, x represents a one-dimensional spatial coordinate along the wave propagation direction, t represents time, α and β are parameters (specific expressions are given later), and c g is the wave group speed.
[0064] Equation (1) is sometimes also written as equation (2) about the carrier envelope A(ξ,t), where ξ and t are independent variables, and the conversion relationship between the coordinates ξ and x is ξ = xc g ·t.
[0065]
[0066] Equations (1) and (2) can be obtained by performing coordinate transformation on the dimensionless NLS equation (3).
[0067] iq T +q XX +2q 2 q=0(3)
[0068] Similarly, taking equation (1) as an example, coordinate transformation T = -αt, X = ξ = xc g t, We can get equation (3): where k is the carrier wave number.
[0069] The following describes the derivation process of transforming the coordinates of equation (3) to obtain equation (1).
[0070] Coordinate transformation: t = -1 / α·T, x = X + c g t,
[0071] In formula (3),
[0072]
[0073]
[0074]
[0075]
[0076] Substituting the above results into formula (3) and sorting them out, we can get formula (1).
[0077] Similarly, coordinate transformation is performed on equation (3): t = -1 / α·T, ξ = X, We can get formula (2).
[0078] 1.2 Peregrine breather solution of the NLS equation
[0079] Nonlinear The equation (NLS equation) has multiple solutions, among which the Peregrine breather solution is most suitable for simulating extreme waves.
[0080] Peregrine first derived the Peregrine breathersolution of the NLS equation. The expression of the Peregrine breathersolution of the NLS equation (3) is
[0081]
[0082] By scaling X→aX, T→a 2 T, q p →q p / a, formula (4) can be transformed into
[0083]
[0084] Pick At the same time, substitute the coordinate transformation T = -αt, X = ξ = xc g t, Note that p With -q p The same solution to equation (3) gives
[0085]
[0086] Among them, A is related to time t and space x, A = A(x, t), which represents the carrier envelope in the distorted wave model; A 0 is a constant, indicating the carrier amplitude; ω is the carrier frequency; parameters α and β are expressed as
[0087]
[0088]
[0089] Where h is the water depth. Under deep water conditions (kh>3), equations (7) and (8) can be simplified to
[0090]
[0091]
[0092] 1.3 Peregrine breathing wavelet
[0093] Next, we introduce the Peregrine breathing wavelet based on the Peregrine breathing solution (6). By performing a multi-scale expansion of the Euler equation, we can obtain the first-order solution of the velocity potential and wave height.
[0094]
[0095]
[0096] The carrier envelope A(x,t) is calculated by equation (6): represents the average flow velocity potential, z represents the vertical coordinate, ψ=kx–ωt, k and ω satisfy the dispersion relation ω 2 =gk tanh(kh), g is the acceleration due to gravity (the value is 9.81m / s 2).
[0097] From this, we can get the horizontal velocity component u, vertical velocity component v and total flow field pressure value p of the distorted wave (Peregrine breathing wave) based on the Peregrine breathing sub-solution respectively:
[0098]
[0099]
[0100]
[0101] Where ρ is the liquid density, which is 1000 kg / m in the laboratory water pool. 3 , seawater is taken as 1025kg / m 3 .
[0102] Based on equations (12) to (15), the numerical and experimental wave generation of the Peregrine breathing wavelet can be easily performed. If the focus position of the Peregrine breathing wavelet (i.e., the position where the maximum peak height appears) is set to x 0 , the focusing moment (i.e. the moment when the maximum peak height appears) is t 0 , then the variables x and t in the above formula should be replaced by (xx 0 ) and (tt 0 ).
[0103] Based on equations (6), (9), (10) and (12), it is found that the Peregrine breathing wavelet has the following basic characteristics:
[0104] (1) The maximum wave height only appears once at time t = 0 and space x = 0 (one-breath characteristic);
[0105] (2) When x = 0, t = 0, A = 3A 0 , that is, the maximum wave height is always three times the carrier amplitude;
[0106] (3) After determining the wave parameters (carrier amplitude A 0 , carrier frequency ω or carrier wave number k, focusing position x 0 and focusing time t 0 ), the analytical wave surface of the Peregrine breathing wavelet in deep water is uniquely determined. This is extremely convenient and highly repeatable for wave simulation and generation, and is an important advantage of the Peregrine breathing wavelet model.
[0107] The wave steepness of the Peregrine breathing wavelet is defined as ε 0 = kA 0 , where k is the carrier wave number, A0 is the carrier amplitude. Experimental and numerical simulation results show that the critical breaking wave steepness of Peregrine breathing sub-wave is ε b is 0.12, that is, ε 0 >ε b When the wave breaks, it appears.
[0108] The above-mentioned embodiment only expresses one implementation mode of the present invention, and its description is relatively specific and detailed, but it cannot be understood as limiting the scope of the invention patent. It should be pointed out that for ordinary technicians in this field, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be based on the attached claims.
Claims
1. A wave-making method for Peregrine breathing wavelet, characterized in that: The steps include: Step 1, determine the wave parameters: At the beginning, determine the relevant wave parameters of the target Peregrine breathing wavelet, including carrier amplitude A0, carrier wave number k, water depth h, focus position x0, and focus time t0. From this, the wavefront time history η(x0,t) of the analytical solution of the Peregrine breathing wavelet at the focus position can be calculated, and then the wave height analytical value H at the focus position can be obtained. A =η max (x0,t)-η min (x0, t); Set the target focus position to a constant X0, and let X0 = x0; Set the initial value of the wave height adjustment parameter Factor to 1.0; According to the Peregrine breathing sub-solution, the wave surface time history of the wave can be uniquely determined; Step 2: Input the wave signal: Based on the determined wave parameters, the wave height time history η(0,t) at the inlet wave boundary can be obtained by the Peregrine breathing sub-solution, where t is time. After multiplying it by the parameter Factor, the wave height form of the wave signal η inlet (t) inputting into a wave maker to generate waves; Step 3, measure and record the wavefront time history at the target focus position X0 and the two positions before and after it (X0–△x, X0+△x) to analyze and determine the nonlinear evolution of the Peregrine breathing wavelet; Step 4, focus position judgment: judge the actual focus position of the generated breathing wavelet according to the axial symmetry of the measured and recorded wavefront history, and the relative front-back relationship with the target focus position; according to the evolution characteristics of the Peregrine breathing wavelet, if the wavefront history curve at the target focus position X0 is approximately axially symmetric about a certain moment, it means that the actual focus position is the target focus position X0; if the axial symmetry of the wavefront history curve before the target focus position is more obvious, it means that the actual focus position is before the target focus position, and the x0 value needs to be appropriately increased; if the axial symmetry of the wavefront history curve after the target focus position is more obvious, it means that the actual focus position is after the target focus position, and the x0 value needs to be appropriately reduced; after adjusting the x0 value, return to step 2 to regenerate the wave signal; Step 5: Target wave height determination: Determine the wave height H measured at the target focus position X0 E Is it close to the analytical value H? A ; If the relative value of the difference between the wave height at the target focus position X0 and the analytical value Error = |H E -H A | / H A If the value is less than the actual set value, the judgment is "yes", and the height of the generated deformed wave reaches the target height; if the difference between the two is large, it is necessary to adjust the parameter Factor value appropriately and return to step 2 to regenerate the wave signal; Step 6. Successfully generate the target wave: If the answer to both the step "focus position judgment" and the step "target wave height judgment" are "yes", the Peregrine breathing sub-wave is successfully generated, and its focus position and wave height both reach the target values.
2. The wave-making method for Peregrine breathing wavelet according to claim 1, characterized in that: In step 1, the parameter carrier wave number k can also be replaced by carrier frequency ω and carrier wavelength L.
3. The wave-making method for Peregrine breathing wavelet according to claim 1, characterized in that: In step 1, the determined target focus position X0 is a constant, a non-variable, and once determined, it does not change during the entire wave-making process.
4. The wave-making method for Peregrine breathing wavelet according to claim 1, characterized in that: In step 1, the first-order solution of the wave surface height of the Peregrine breathing sub-wave is: Where i represents a complex unit, x represents a one-dimensional spatial coordinate along the wave propagation direction, t represents time, and α and β are parameters. The carrier envelope A(x, t) is calculated by the Peregrine breathing wavelet, ψ = kx–ωt, k and ω satisfy the dispersion relation ω 2 =gk tanh(kh), g is the acceleration due to gravity.
5. The wave-making method for Peregrine breathing wavelet according to claim 1, characterized in that: In step 2, the inlet wave-generating boundary is at x=0, where x is the one-dimensional spatial coordinate along the wave propagation direction.
6. The wave-making method for Peregrine breathing wavelet according to claim 1, characterized in that: In step 3, the value of △x is recommended to be in the range of 5% to 16% of the carrier wavelength.
7. The wave-making method for Peregrine breathing wavelet according to claim 1, characterized in that: In step 4 and step 5, according to the wave-making accuracy requirement, the relative value Error of the difference between the wave height at the target focus position X0 and the analytical value can be set to 2.0%.
8. The wave-making method for Peregrine breathing wavelet according to claim 4, characterized in that: The carrier envelope A(x,t) is calculated by the Peregrine breath sub-solution: Among them, A is related to time t and space x, A = A (x, t), which represents the carrier envelope in the distorted wave model; A0 is a constant, which represents the carrier amplitude; ω is the carrier frequency; α and β are parameters.
9. The wave-making method for Peregrine breathing wavelet according to claim 8, characterized in that: The parameters α and β are expressed as: Where h is the water depth.
10. The wave-making method for Peregrine breathing wavelet according to claim 9, characterized in that: Under deep water conditions kh>3, equations (7) and (8) can be simplified as: