CN112765802B - Method for evolving water wave waveform based on high-order water wave model - Google Patents
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Abstract
A method for evolving a water wave waveform based on a high-order water wave model comprises the steps of constructing the water wave model, carrying out logarithmic transformation, constructing a test function f, evolving the water wave model and determining parameters. The invention considers a complex actual water wave model, combines the waveform characteristics of periodic waves and solitary waves, evolves a high-order high-dimensional water wave model, obtains corresponding parameters and a corresponding symbolic representation form, and shows the physical characteristics of a new waveform corresponding to the model in the motion process and the propagation process. Through simulation experiments, the invention realizes the combination of theoretical analysis and practical problems, enriches the display of the interaction of long waves corresponding to the models, can be used for analyzing the physical characteristics of long-wave models in practical problems and expands the application range of the long waves.
Description
Technical collar city
The invention belongs to the technical field of waveform processing, and particularly relates to determination of an evolution water wave waveform of a high-order water wave model.
Background
In real life, research on a nonlinear high-dimensional high-order water wave model relates to a plurality of natural disciplines such as biology, hydraulics, physics, mechanics, computer disciplines and the like. In the research process, students find that a high-dimensional water wave model can be simulated according to the natural state of water waves, parameters of a high-order high-dimensional water wave model are determined through the evolution of a known wave pattern, a mathematical model of the water waves can be obtained according to parameter analysis, and actual problems are abstracted into the mathematical model which is favorable for research and analysis. The energy change and the propagation state of high-dimensional high-order water waves can be better observed through the research on the parameters of the mathematical model. Therefore, the research of high-dimensional high-order water wave models is gradually valued by scientists. Previous studies on this type of model were limited to periodic or soliton wave patterns alone. The periodic wave graph can well show the propagation process and physical properties of the periodic wave, but only can show the properties of the periodic wave, and the analysis of multiple angles cannot be achieved. Similarly, the graph of the solitary wavelets can only show the wave change and the propagation process when the solitary wavelets collide, has certain limitation, cannot analyze more complex waveforms, and cannot adapt to more complex natural conditions. For the wave pattern of the low-dimensional water wave model, the process of wave propagation is difficult to show due to low dimension, which is not beneficial to the research and analysis of waves.
Disclosure of Invention
The technical problem to be solved by the invention is to overcome the defects of the prior art and provide a method for evolving a water wave waveform based on a high-order water wave model, which is simple, high in operation speed and high in accuracy.
The technical scheme adopted for solving the technical problems comprises the following steps:
(1) Construction of water wave model
A2 + 1-dimensional 4-order nonlinear water wave model is constructed according to formula (1):
αu yt +βu xxxy +γu x u xy +δu y u xx ＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and a space variable x and a space variable y x Is the first order partial derivative of u with respect to x, u y Is the first order partial derivative of u with respect to y, u t Is the first-order partial derivative of u with respect to t, u xx Is the second order partial derivative of u with respect to x, u yy Is the second order partial derivative of u with respect to y, u tt Is the second order partial derivative of u with respect to t, u xxxx Is the fourth order partial derivative of u with respect to x, u yyyy Is the fourth order partial derivative, u, of y tttt Is the fourth order partial derivative of u with respect to t, α, β, γ, ε are rational numbers, the ratio of α to β, γ, ε is 1:1 to 5: -5 to-1: -5 to-1.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf) x (2)
f＝f(x,y,t)
f is a differentiable function with respect to the variables x, y, t, and N is an even number.
(3) Construction of a checking function f
The test function f is constructed as equation (3):
ξ i ＝k i x+w i y+c i t，
wherein a is i Is amplitude, k i Is the wave velocity in the x direction, w i Wave velocity in the y direction, c i As the frequency of the wave, i e [1, 2, 3 ]]，a i ,k i ,w i ,c i The value range of (A): a is i ∈[-10,10],k i 、w i 、c i ∈[-10,0)∪(0,10]。
(4) Evolution of water wave models
Obtaining an arbitrary function u with respect to the parameter a according to equation (2) i ,c i ,w i ,k i The symbol expression of (a):
the formula (4) is to simulate a water wave model by the determined interaction form of the periodic wave and the solitary wave, and find out when a non-blasting waveform exists by combining with the specific waveform display of the actual water wave generation and transmission.
(5) Determining parameters
Step (4), when the long-wave waveform appears in the non-blasting model, finding out the relation existing among partial parameters, assigning the parameters without the relation in a determined range, and obtaining a when the parameters are determined to be consistent with the actual long-wave waveform i ,k i ,w i ,c i Specific values of the parameters and the final form of equation (4).
In formula (1) of step (1) of constructing the water wave model of the present invention, the optimum ratio of α to β, γ, and ∈ is 1:1: -3: -3.
In the formula (2) of the logarithmic conversion step (2), N is an even number of-10 to 14.
In the formula (2) of the logarithmic transformation step (2) of the present invention, the value of N is preferably-2.
In formula (3) of step (3) of constructing the test function of the present invention, c i Value of 1,a 1 The value of 0,a 2 Value of 1,a 3 The value is-6.25, k 1 The value is 1,k 2 The value of k is 0.5 3 The value is 0.1,w 1 Value of 1,w 2 The value is 0.2,w 3 The value was 0.04.
The invention combines the period method and the soliton method to generate a new waveform, can better adapt to variable and complex practical application conditions, and can well research the high-dimensional long-wave model by a wide-field method and a direct planning method.
The invention provides a simulation method based on a high-dimensional water wave model, namely a field method and a direct fitting method, which are simpler and can be completed with the assistance of special simulation software aiming at analyzing various existing waveforms of the existing ships, oceans, climates and the like. Through the value taking of the discrete parameters, the physical characteristics of the new waveform corresponding to the model in the motion process and the propagation process are shown, and the combination of theoretical analysis and practical problems is realized. The invention enriches the display of the interaction of the long waves corresponding to the model and expands the application range of the long waves.
Drawings
FIG. 1 is a flowchart of example 1 of the present invention.
FIG. 2 is a three-dimensional view of a 2+1 dimensional 4-order water wave model modeled with a first set of parameters when t is-4.
FIG. 3 is a three-dimensional view of a 2+1 dimensional 4-order water wave model modeled with a first set of parameters when t is 0.
FIG. 4 is a three-dimensional view of a 2+1 dimensional 4-order water wave model modeled with a first set of parameters at t of 4.
Detailed Description
The present invention will be described in further detail with reference to the following drawings and examples, but the present invention is not limited to these examples.
Example 1
Taking the known BLMP long-wave model as an example, the method for evolving a water wave waveform based on the high-order water wave model of the present embodiment comprises the following steps (see fig. 1):
(1) Constructing a water wave model
A2 + 1-dimensional 4-order nonlinear water wave model is constructed according to formula (1):
αu yt +βu xxxy +γu x u xy +δu y u xx ＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and space variables x and y x Is the first order partial derivative of u with respect to x, u y Is the first order partial derivative of u with respect to y, u t Is the first order partial derivative of u with respect to t, u xx Is the second order partial derivative of u with respect to x, u yy Is the second order partial derivative of u with respect to y, u tt Is the second order partial derivative of u with respect to t, u xxxx Is the fourth order partial derivative of u with respect to x, u xyyy Is the fourth order partial derivative, u, of y tttt Is the fourth order partial derivative of u with respect to t, alpha, beta, gamma, epsilon are rational numbers, the ratio of alpha to beta, gamma, epsilon is 1:1 to 5: -5 to-1: -5 to-1, the ratio of α to β, γ, ε being 1:1: -3: -3.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf) x (2)
f＝f(x,y,t)
wherein f is a differentiable function related to variables x, y, t, N is an even number, N takes on an even number of-10 to 14, and N takes on a value of-2 in this embodiment.
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξ i ＝k i x+w i y+c i t，
wherein a is i Is amplitude, k i Is the wave velocity in the x direction, w i Wave speed in the y direction, c i As the frequency of the wave, i e [1, 2, 3 ]]，a i ∈[-10,10],k i 、w i 、c i ∈[-10,0)∪(0,10]The values of this embodiment are: c. C i Take 1,a 1 Take 0,a 2 Take 1,a 3 Take-6.25,k 1 Take 1,k 2 Take 0.5,k 3 Take 0.1,w 1 Take 1,w 2 Take 0.2,w 3 0.04 is taken.
(4) Evolution of water wave models
Obtaining a long wave u with respect to a parameter a according to equation (2) i ,c i ,w i ,k i The symbol expression of (c):
the formula (4) is to simulate a water wave model by the determined interaction form of the periodic wave and the solitary wave, and find out when a non-blasting waveform exists by combining with the specific waveform display of the actual water wave generation and transmission.
(5) Determining parameters
Finding out partial parameters when the non-blasting model has long-wave waveform in step (4)Existing relation, assigning the parameters without relation in the determined range, and obtaining a when determining that the wave form is consistent with the actual long wave form i ,k i ,w i ,c i The specific values of the parameters and the final form of equation (4) (see fig. 2, 3, 4).
Example 2
Taking a known long-wave model as an example, the method for evolving a water wave waveform based on a high-order water wave model of the embodiment comprises the following steps:
(1) Constructing a water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to formula (1):
αu yt +βu xxxy +γu x u xy +δu y u xx ＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and a space variable x and a space variable y x Is the first order partial derivative of u with respect to x, u y Is the first order partial derivative of u with respect to y, u t Is the first order partial derivative of u with respect to t, u xx Is the second order partial derivative of u with respect to x, u yy Is the second order partial derivative of u with respect to y, u tt Is the second-order partial derivative of u with respect to t, u xxxx Is the fourth order partial derivative of u with respect to x, u yyyy Is the fourth order partial derivative, u, of y tttt Is the fourth order partial derivative of u with respect to t, alpha, beta, gamma, epsilon are rational numbers, the ratio of alpha to beta, gamma, epsilon is 1:1 to 5: -5 to-1: -5 to-1, the ratio of α to β, γ, ε being 1:1: -5: -5.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf) x (2)
f＝f(x,y,t)
wherein f is a differentiable function related to variables x, y, t, N is an even number, N takes on an even number of-10 to 14, and N takes on a value of-10 in this embodiment.
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξ i ＝k i x+w i y+c i t，
wherein a is i Is amplitude, k i Is the wave velocity in the x direction, w i Wave speed in the y direction, c i As the frequency of the wave, i e [1, 2, 3 ]]，a i ,k i ,w i ,c i The value range of (A): a is a i ∈[-10,10],k i 、w i 、c i ∈[-10,0)∪(0,10]A of the present embodiment i The value is 0.1,k i 、w i 、c i Is 10.
The other steps were the same as in example 1. To obtain a i ,k i ,w i ,c i Specific values of the parameters and the final form of equation (4).
Example 3
Taking a known long-wave model as an example, the method for evolving a water wave waveform based on a high-order water wave model of the embodiment comprises the following steps:
(1) Construction of water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to formula (1):
αu yt +βu xxxy +γu x u xy +δu y u xx ＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and space variables x and y x Is the first order partial derivative of u with respect to x, u y Is the first order partial derivative of u with respect to y, u t Is the first order partial derivative of u with respect to t, u xx Is the second order partial derivative of u with respect to x, u yy Is the second order partial derivative of u with respect to y, u tt Is the second order partial derivative of u with respect to t, u xxxx Is the fourth order partial derivative of u with respect to x, u yyyy Is the fourth order partial derivative, u, of y tttt Is the fourth order partial derivative of u with respect to t, alpha, beta, gamma, epsilon are rational numbers, the ratio of alpha to beta, gamma, epsilon is 1:1 to 5: -5 to-1: -5 to-1, the ratio of α to β, γ, ε being 1:5: -1: -1.
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf) x (2)
f＝f(x,y,t)
where f is a differentiable function for variables x, y, and t, N is an even number between-10 and 14, and N in this embodiment is 14.
(3) Construction of a checking function f
The test function f is constructed according to equation (3):
ξ i ＝k i x+w i y+c i t, wherein a i Is amplitude, k i Is the wave velocity in the x direction, w i Wave velocity in the y direction, c i As the frequency of the wave, i e [1, 2, 3 ]]，a i ,k i ,w i ,c i The value range of (A): a is i ∈[-10,10],k i 、w i 、c i ∈[-10,0)∪(0,10]A of the present embodiment i Has a value of 10,k i 、w i 、c i Is 0.1.
The other steps were the same as in example 1. To obtain a i ,k i ,w i ,c i Specific values of the parameters and the final form of equation (4).
In order to verify the beneficial effects of the invention, the inventor adopts the method for determining the optimal evolution waveform of the high-order water wave model in the embodiment 1 of the invention to carry out simulation experiments, and the experimental conditions are as follows:
the results of simulation experiments in example 1 are shown in fig. 2, 3, and 4, where fig. 2 is a three-dimensional view of u when the spatial variable t is-4, fig. 3 is a three-dimensional view of u when the spatial variable t is 0, and fig. 4 is a three-dimensional view of u when the spatial variable t is 4. As can be seen from fig. 2, when the space variable t takes-4, the two waves collide, the amplitude decreases, and the energy of the waves decreases. As can be seen from fig. 3, when the spatial variable t takes 0, the two waves still collide, and the propagation direction does not change. As can be seen from fig. 4, when the spatial variable t takes 4, the wave continues to propagate from the original position along the direction, and the energy does not change. Test results show that the long wave propagation process can be completely shown in the embodiment 1 of the invention, and the method in the embodiment 1 is proved to be good in simulation of the model.
Claims (4)
1. A method for evolving a water wave waveform based on a high-order water wave model is characterized by comprising the following steps:
(1) Constructing a water wave model
Constructing a 2+ 1-dimensional 4-order nonlinear water wave model according to formula (1):
αu yt +βu xxxy +γu x u xy +δu y u xx ＝0 (1)
u＝u(x，y，t)
where u is a long wave formed by a time variable t and two space variables x and y x Is the first order partial derivative of u with respect to x, u y Is the first order partial derivative of u with respect to y, u t Is the first order partial derivative of u with respect to t, u xx Is the second-order partial derivative of u with respect to x, alpha, beta, gamma and delta are rational numbers, and the ratio of alpha to beta, gamma and delta is 1: 1-5: 5-1;
(2) Logarithmic transformation
The following logarithmic transformation is performed according to equation (2):
u＝N(lnf) x (2)
f＝f(x，y，t)
where f is a differentiable function with respect to the variables x, y, t, and N is an even number;
(3) Construction of a test function f
The test function f is constructed according to equation (3):
ξ i ＝k i x+w i y+c i t，
wherein a is i Is amplitude, k i Is the wave velocity in the x direction, w i Wave velocity in the y direction, c i The frequency of the wave, i is 1, 2, 3, a i ，k i ，w i ，c i The value range of (A): a is a i ∈[0，10]，k i 、w i 、c i ∈(0，10]；
(4) Evolution of water wave models
Obtaining an arbitrary function u with respect to the parameter a according to equation (2) i ，c i ，w i ，k i The symbol expression of (a):
the formula (4) is that a water wave model is simulated by the determined interaction form of periodic waves and solitary waves, and when non-blasting waveforms exist is found out by combining with the specific waveform display of actual water wave generation and transmission;
(5) Determining parameters
Finding out the relation among partial parameters when the non-blasting model has long wave form, assigning the parameters without relation in the determined range, and obtaining a when the parameters are consistent with the actual long wave form i ，k i ，w i ，c i Specific values of the parameters and the final form of equation (4).
2. The method for evolving a water wave waveform based on a higher-order water wave model as claimed in claim 1, wherein: in the formula (1) of the step (1) of constructing the water wave model, the ratio of alpha to beta, gamma and delta is 1: 3.
3. The method for evolving water wave waveforms based on higher-order water wave models of claim 1, wherein: in the formula (2) in the logarithmic transformation step (2), N takes an even number of-10 to 14.
4. The method for evolving water wave waveform based on higher-order water wave model according to claim 1 or 3, wherein: in the formula (2) of the logarithmic transformation step (2), N takes the value of-2.
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