Saturated porous medium metasurface Portrait wave frequency dispersion relation calculation method and system

Abstract

The invention relates to the technical field of physical acoustics, in particular to a saturated porous medium metasurface Portrait wave frequency dispersion relation calculation method and system, and the method comprises the following steps: obtaining expressions of solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of Portrait waves; obtaining an expression of a fluid displacement component and a fluid normal stress; obtaining an expression about the relative motion of the resonator; according to all the expressions, stress and displacement continuity boundary conditions are applied to the interface of the saturated porous medium and the fluid layer, a characteristic equation of frequency dispersion characteristics is obtained and solved, and a frequency dispersion curve is obtained. According to the method, the explicit frequency dispersion relation of the saturated porous medium metasurface portrait waves is provided, the calculation efficiency and prediction precision of complex interface wave propagation are improved, the optimal design of the metasurface resonator array can be guided, and a key theoretical tool and technical support are provided for design of high-performance acoustic functional devices and structures.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of physical acoustics, and particularly to a method and system for calculating the dispersion relation of saturated porous medium metasurface Scholte waves. BACKGROUND

[0002] Scholte-Stoneley waves are a general term for a type of elastic guided wave that exists at the interface between two closely contacting media. Among them, the wave propagating at the fluid-solid interface (such as the seabed) is called Scholte wave, which is a special case of Stoneley wave in the fluid-solid case. Such waves propagate along the interface, with energy concentrated near the interface, and their amplitude decays exponentially in the direction perpendicular to the interface.

[0003] In the technical fields of ocean engineering, geological exploration, and underwater acoustics, accurate prediction and control of the propagation of interface waves are crucial for improving the accuracy of acoustic detection equipment and optimizing the vibration and noise reduction performance of underwater structures. However, the existing technology still has obvious deficiencies in characterizing the interaction between Scholte waves and elastic half-space surface resonant arrays. Specifically, due to the lack of efficient and accurate analysis models, it is difficult to accurately calculate and predict the complex coupling behavior of waves and structures, leading to widespread reliance on numerical trial and error or empirical estimates in the design involving real soil environments such as saturated porous media and nonlinear metasurfaces. This results in low computational efficiency, poor prediction accuracy, and long design cycles, among other technical bottlenecks. Therefore, it is necessary to explore a method for efficiently and accurately calculating the dispersion relation of nonlinear metasurface Scholte waves on saturated porous media. SUMMARY

[0004] In view of the defects in the prior art, the present application provides a method and system for calculating the dispersion relation of saturated porous medium metasurface Scholte waves.

[0005] In order to achieve the above object, the first aspect of the present application provides a saturated porous medium metasurface sholter wave dispersion relation calculation method, the method comprising the following steps: obtaining expressions of solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of sholter wave based on Biot theory and Helmholtz vector decomposition theorem; for incompressible acoustic fluid, introducing a scalar potential function to describe fluid motion according to the basic control equation of fluid dynamics, and then obtaining expressions of fluid displacement component and fluid normal stress; based on Newton's second law, considering the nonlinear stiffness of the spring, obtaining the expression about the relative motion of the discrete mass nonlinear spring resonator; according to all the expressions, the stress and displacement continuity boundary conditions are applied at the interface between the saturated porous medium and the fluid layer, and then the characteristic equation of the dispersion characteristics of the coupled system is obtained; the wave number and frequency in the characteristic equation are solved by using a numerical method, and finally the dispersion curve of the coupled system is obtained. The present application provides an explicit dispersion relation of saturated porous medium metasurface sholter wave, improves the calculation efficiency and prediction accuracy of complex interface wave propagation, can guide the optimization design of metasurface resonator array, and provides a key theoretical tool and technical support for the design of high-performance acoustic functional devices and structures.

[0006] Optionally, the obtaining expressions of solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of sholter wave based on Biot theory and Helmholtz vector decomposition theorem comprises the following steps: determining the basic expressions of the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress and the pore pressure under the action of sholter wave; establishing control equations of the solid skeleton displacement field and the fluid relative displacement field in the saturated porous medium under the plane strain condition according to Biot theory; using Helmholtz vector decomposition theorem to decompose the solid skeleton displacement field and the fluid relative displacement field into a combined form of scalar potential and vector potential, and combining the control equations to obtain wave equations of P wave and S wave decoupling; letting the solution of the wave equation be a simple harmonic wave form, and then combining the basic expressions to obtain the expressions of the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress and the pore pressure.

[0007] Optionally, the control equations of the solid skeleton displacement field and the fluid relative displacement field are respectively: wherein, and is the Lame constant of the solid skeleton, E is the compressive modulus of the saturated porous medium, M is the inverse of the coupling modulus of the saturated porous medium, u is the solid skeleton displacement field, φ is the fluid relative displacement field, ρ is the total density of the saturated porous medium, t is time, ρf is the fluid density, m is a parameter relating fluid density to pore geometry, b is a viscous coupling parameter between solid and fluid.

[0008] Optionally, the expressions of the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress and the pore pressure are in turn: wherein, uis the solid skeleton displacement in the X-axis direction, x and z are horizontal and vertical coordinates respectively, i is the imaginary unit, k is the wave number, , and are the amplitudes of the potential functions, , and are the wave numbers of the potential functions, ω is the angular frequency, t is time, uis the solid skeleton displacement in the Z-axis direction, , and are the ratios of the potential function amplitudes, φxis the fluid relative displacement in the X-axis direction, φzis the fluid relative displacement in the Z-axis direction, σ is the total stress, and are reference symbols, and are the Lame constants of the solid skeleton, E is the compressive modulus of the saturated porous medium, and are the wave numbers of the P1 and P2 waves in the saturated porous medium, M is the inverse of the coupling modulus of the saturated porous medium, τ is the solid shear stress, kS is the wave number of the S wave, p is the pore pressure.

[0009] Optionally, the expressions of the fluid displacement component and the fluid normal stress are respectively: wherein, is the fluid displacement component in the X-axis direction, i is the imaginary unit, k is the wave number, and is the amplitude, x and z are the horizontal and vertical coordinates respectively, is a parameter related to the wave number and the fluid pressure wave velocity, is the fluid displacement component in the Z-axis direction, is the Lame constant of the solid skeleton, is the fluid normal stress, is the bulk modulus of the incompressible fluid, is the fluid density, is the fluid pressure wave velocity, is the angular frequency, and t is the time.

[0010] Optionally, the expression about the relative motion of the discrete mass nonlinear spring resonator is obtained based on Newton's second law, considering the nonlinear stiffness of the spring, comprising the following steps: For the discrete mass nonlinear spring resonator arranged at the interface between the saturated porous medium and the fluid layer, the dynamic balance equation thereof is established according to Newton's second law, considering the nonlinear stiffness of the spring; The nonlinear term in the dynamic balance equation is processed by using the first harmonic balance method, to obtain the expression about the relative motion of the discrete mass nonlinear spring resonator.

[0011] Optionally, the expression about the relative motion of the discrete mass nonlinear spring resonator is: wherein, is the mass of the discrete mass nonlinear spring resonator, and is a reference symbol, is the angular frequency, and Y is the harmonic amplitude, , and is the amplitude corresponding to each potential function, i is the imaginary unit, and k is the wave number, and is the wave number corresponding to the potential function, is the natural frequency of the discrete mass nonlinear spring resonator, is the nonlinear coefficient.

[0012] Optionally, when the fluid layer is of finite thickness, the characteristic equation is: wherein, are reference symbols, , , , , , , , , , , , , and are reference symbols, M is the inverse of the coupling modulus of the saturated porous medium, is the fluid layer thickness, is the compression modulus of the saturated porous medium, , and are the ratio of potential function amplitudes, is a parameter related to the wave number and the fluid pressure wave velocity, is the angular frequency, is the porosity, and are the wave numbers of two different types of P-waves, is the fluid pressure wave velocity, is the mass of the discrete mass nonlinear spring resonator, A is the unit effective area of the discrete mass nonlinear spring resonator, is the fluid density, k is the wave number, is the wave number of S-wave, is the Lame constant of the solid skeleton, and are the wave numbers corresponding to each potential function, i is the imaginary unit, and are the amplitudes, Y is the harmonic amplitude.

[0013] Optionally, when the fluid layer is semi-infinite space, the characteristic equation is: wherein, and are reference symbols, , , , , , , , , , , , and are reference symbols, M is the inverse of the coupling modulus of the saturated porous medium, is the compression modulus of the saturated porous medium, , and are the ratio of potential function amplitudes, is a parameter related to the wave number and the fluid pressure wave speed, is the angular frequency, is the porosity, and are the wave numbers of P1 and P2 waves in the saturated porous medium, is the fluid pressure wave speed, is the mass of the discrete mass nonlinear spring resonator, A is the unit effective area of the discrete mass nonlinear spring resonator, is the fluid density, k is the wave number, is the wave number of S wave, is the Lame constant of the solid skeleton, and are the wave numbers corresponding to each potential function, i is the imaginary unit, is the amplitude coefficient, Y is the harmonic amplitude.

[0014] In a second aspect, the present application provides a saturated porous medium metasurface Scholte wave dispersion relation calculation system, which comprises a data acquisition device, a data output device, a processor and a storage, the storage comprises a computer readable storage medium, the computer readable storage medium stores a computer program, the computer program comprises program instructions, the program instructions enable the processor to realize the saturated porous medium metasurface Scholte wave dispersion relation calculation method provided by the present application when executed by the processor.

[0015] The present application has at least the following beneficial effects: 1. The method establishes an explicit dispersion relation model of nonlinear metasurface Scholte wave on a saturated porous medium, improves the calculation efficiency and prediction accuracy of complex interface wave propagation.

[0016] 2. The analysis framework established by the method can efficiently and accurately represent the hybrid behavior of the fundamental mode near the collective resonance frequency, and reveal key phenomena such as surface wave leakage to the fluid domain.

[0017] 3. The method can guide the optimization design of metasurface resonator array, and provide key theoretical tools and technical support for the design of high-performance acoustic functional devices and structures.

[0018] 4. The system is provided to adapt to the method, thereby improving the practicability of the method and facilitating the popularization of the method. BRIEF DESCRIPTION OF DRAWINGS

[0019] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings needed in the embodiments will be briefly introduced as follows. It should be understood that the following drawings only show some embodiments of the present application, and therefore should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can also be obtained without creative labor on the basis of these drawings.

[0020] Figure 1 The flowchart of the saturated porous medium metasurface Scholte wave dispersion relation calculation method of the embodiments of the present application; Figure 2 The framework diagram of the saturated porous medium metasurface Scholte wave dispersion relation calculation system of the embodiments of the present application. DETAILED DESCRIPTION

[0021] The specific embodiments of the present application will be described in detail below. It should be noted that the embodiments described herein are only used for illustration and do not limit the present application. In the following description, a large number of specific details are set forth in order to provide a thorough understanding of the present application. However, it is obvious to those skilled in the art that the present application does not have to be implemented with these specific details. In other examples, in order to avoid obscuring the present application, well-known circuits, software or methods are not specifically described.

[0022] Throughout the specification, the reference to "one embodiment", "an embodiment", "one example" or "an example" means that the specific features, structures or characteristics described in connection with the embodiment or example are included in at least one embodiment of the present application. Therefore, the phrases "in one embodiment", "in an embodiment", "one example" or "an example" appearing throughout the specification do not necessarily refer to the same embodiment or example. In addition, specific features, structures or characteristics can be combined in one or more embodiments or examples in any appropriate combination and / or subcombination. In addition, those skilled in the art should understand that the drawings provided herein are for illustrative purposes and the drawings are not necessarily drawn to scale.

[0023] It should be noted in advance that in an alternative embodiment, in addition to making independent descriptions, the same symbols or letters appearing in all formulas have the same meaning and value.

[0024] In an alternative embodiment, please refer to Figure 1 The present application provides a saturated porous medium metasurface Scholte wave dispersion relation calculation method, which comprises the following steps: S1, obtaining expressions of solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of Scholte wave based on Biot theory and Helmholtz vector decomposition theorem.

[0025] The step S1 specifically comprises the following steps. S11, determining basic expressions of the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress and the pore pressure under the action of Scholte wave.

[0026] Specifically, in the embodiment, the Scholte wave can excite horizontal (X-axis direction) and vertical (Z-axis direction) displacements of the saturated porous medium, and the soil is taken as an example, wherein the horizontal and vertical displacements of the solid skeleton and the fluid relative to the solid in the horizontal and vertical directions can be expressed as: wherein, is the solid skeleton displacement in the X-axis direction, and are scalar potential and vector potential of the solid skeleton respectively, and x and z are horizontal and vertical coordinates respectively, is the solid skeleton displacement in the Z-axis direction, is the fluid relative displacement in the X-axis direction, is the fluid relative displacement in the Z-axis direction, and are scalar potential and vector potential of the fluid respectively.

[0027] Further, the total stress, the solid shear stress and the pore pressure are expressed by displacement, and are in turn: wherein, is the total stress, and are Lame constants of the solid skeleton, , E is Young's modulus, is Poisson's ratio, is the compression modulus of the saturated porous medium, , , is the soil skeleton bulk modulus, is the soil particle bulk modulus, is the solid shear stress, and M is the inverse of the coupling modulus of the saturated porous medium, , porosity, fluid bulk modulus, pore pressure.

[0028] The above seven relationships are the basic expressions of the solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of the Scholte wave in the embodiment.

[0029] S12, according to the Biot theory, the control equation of the solid skeleton displacement field and the fluid relative displacement field in the saturated porous medium under the plane strain condition is established.

[0030] Specifically, in the embodiment, since there is pore water in the saturated porous medium, it is assumed that the pore size is much smaller than the Scholte wave length. At the same time, in order to ensure the effectiveness of the periodic structure, the Scholte wave length is also much larger than the lattice constant of the discrete mass nonlinear spring resonator array. On the basis of the above conditions, the control equation of the solid skeleton displacement field and the fluid relative displacement field in the saturated porous medium under the plane strain condition is established according to the Biot theory, and the control equation is specifically: wherein, the gradient operator, , the solid skeleton displacement field, the fluid relative displacement field, the total density of the saturated porous medium, t is time, the fluid density, the soil particle density, m is the relationship parameter between the fluid density and the pore geometry, , the tortuosity, b is the viscous coupling parameter between the solid and the fluid, , the fluid viscosity, the permeability coefficient.

[0031] S13, the solid skeleton displacement field and the fluid relative displacement field are decomposed into a combination form of scalar potential and vector potential using the Helmholtz vector decomposition theorem, and the wave equation of P wave and S wave decoupling is obtained combined with the control equation.

[0032] Specifically, in the embodiment, the solid skeleton displacement field and the fluid relative displacement field are decomposed into a combination form of scalar potential and vector potential using the Helmholtz vector decomposition theorem, and the solid skeleton displacement field and the fluid relative displacement field are decomposed into a combination form of scalar potential and vector potential, which is specifically: Substitute the two expressions into the control equation of step S12, and the wave control equation of P wave decoupling and the wave control equation of S wave decoupling are obtained, which are as follows: S14, let the solution of the wave equation be a simple harmonic wave form, and then obtain the expression of the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress and the pore pressure by combining the basic expression.

[0033] Specifically, in the embodiment, it is assumed that the solution of the wave equation obtained in step S13 is a simple harmonic wave form, which satisfies: wherein, , , and are the amplitudes of , , and respectively, and are the wave vectors of P wave and S wave respectively, r is the position vector, i is the imaginary unit, is the angular frequency.

[0034] Substitute formula 1 into the wave control equation of P wave decoupling, and the control equation of P wave is obtained as follows: wherein, is a symbol, , is the wave number of P wave in the saturated porous medium.

[0035] The condition for the control equation of P wave to have two non-zero solutions is that the determinant of its coefficients is zero, and the two non-zero solutions are as follows: wherein, and are the velocities of P1 wave and P2 wave in the saturated porous medium respectively, and are the wave numbers of P1 wave and P2 wave in the saturated porous medium respectively, and are coefficients, , .

[0036] Similarly, substitute formula 2 into the wave control equation of S wave decoupling, and the control equation of S wave is obtained as follows: The condition for the governing equation of S-wave to have a non-zero solution is that the determinant of its coefficient matrix is zero, and the non-zero solution is where, is the S-wave velocity in saturated porous media, is the wave number of S-wave.

[0037] For the harmonic surface wave propagating along the X-axis direction, if the exponential decay characteristic in the Z-axis direction is considered, then , , and The four potential functions can be written in the form of complex functions as follows: where, , and are the amplitudes of the potential functions, , and are the wave numbers of the potential functions, , and are the ratios of the amplitudes of the potential functions, , , , , , , k is the wave number.

[0038] Substituting the complex function form of the four potential functions into the basic expression given in step S11, the expressions of the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress and the pore pressure can be obtained, which are in turn: where, and are the reference symbols set for the convenience of writing.

[0039] S2. For non-viscous acoustic fluids, based on the fundamental governing equations of fluid dynamics, a scalar potential function is introduced to describe the fluid motion, thereby obtaining expressions for the fluid displacement components and the fluid normal stress.

[0040] Specifically, in this embodiment, for non-viscous acoustic fluids, the governing equations satisfy: in, For inviscid fluids, the bulk modulus is... , For fluid pressure wave velocity, For fluid displacement vector, , This represents the fluid displacement component along the X-axis. This represents the fluid displacement component along the Z-axis.

[0041] By introducing scalar potential , The governing equations for nonviscous acoustic fluids can be written as: Under harmonic propagation, assume the scalar potential in the fluid layer is: in, and For amplitude, These are parameters related to wave number and fluid pressure wave velocity. Substituting this formula into From this, the expression for the fluid displacement component can be obtained as follows: Given the fluid-saturated porous medium coupling, the expression for the fluid normal stress is derived as follows: in, For fluid normal stress.

[0042] S3. Based on Newton's second law, considering the nonlinear stiffness of the spring, obtain an expression for the relative motion of the discrete mass nonlinear spring resonator.

[0043] Specifically, step S3 includes the following steps: S31. For the discrete mass nonlinear spring resonator arranged at the interface between the saturated porous medium and the fluid layer, considering the nonlinear stiffness of the spring, its dynamic equilibrium equation is established according to Newton's second law.

[0044] Specifically, in this embodiment, the dynamic balance equation considering the discrete-mass nonlinear spring resonator coupled to the interface between the saturated porous medium and the flow layer is: in, The mass of a discrete-mass nonlinear spring resonator. Let be the linear stiffness of the spring. For the nonlinear stiffness of the spring, Let y represent the second derivative of y, where y represents the mass displacement of the discrete-mass nonlinear spring resonator. Vertical displacement of the soil free surface The relative motion between them .

[0045] Vertical displacement of soil free surface The following relationship must be satisfied: in, The referential symbols were set up for ease of writing. .

[0046] Assuming relative motion of the resonators The harmonic solution is expressed in the form of: Where Y is the harmonic amplitude.

[0047] The dynamic equilibrium equations can then be written in the following form: S32. The nonlinear terms in the dynamic balance equation are processed using the first harmonic balance method to obtain an expression for the relative motion of the discrete mass nonlinear spring resonator.

[0048] Specifically, in this embodiment, by using Euler's formula and the first harmonic balance method to approximately ignore the low-frequency harmonics generated by nonlinearity, the dynamic balance equation can be transformed into the following expression for the relative motion of the discrete mass nonlinear spring resonator: in, Y is a referential symbol. The natural frequency of the discrete-mass nonlinear spring resonator. , These are nonlinear coefficients. It can be observed that when At this time, the discrete mass nonlinear spring resonator degenerates into a linear one.

[0049] S4. Based on all the above expressions, stress and displacement continuity boundary conditions are applied at the interface between the saturated porous medium and the fluid layer to obtain the characteristic equation of the dispersion characteristics of the coupled system.

[0050] Specifically, in this embodiment, based on the analysis of steps S1 to S3, when the fluid layer has a finite thickness, the Schulte wave dispersion relation of the nonlinear metasurface on the saturated porous medium can be obtained by executing the first set of boundary conditions. The first set of boundary conditions is as follows: in, For fluid layer thickness, For fluid stress, , This represents the resonator stress.

[0051] Since the fluid does not support shear stress, therefore on its free surface Only the first boundary condition in the first set of boundary conditions is satisfied at that location, i.e. At the saturated porous media-fluid interface At this point, the continuity of normal stress and vertical displacement is applied. The first set of boundary conditions neglects any direct kinematic coupling between the resonator and the fluid. This simplification is reasonable when the size of the discrete-mass nonlinear spring resonator is significantly smaller than the Schulte wave wavelength.

[0052] By , , , , , Substituting the expression for the relative motion of the discrete-mass nonlinear spring resonator into the first set of boundary conditions, we can obtain the characteristic equation for the dispersion characteristics of the coupled system composed of a solid, a fluid, and a discrete-mass nonlinear spring resonator when the fluid layer has a finite thickness. Its matrix form is as follows: in, All are referential symbols. , , , , , , , , , A is the effective area per unit of the discrete-mass nonlinear spring resonator. The fluid density is given.

[0053] When the fluid layer is a semi-wireless space, i.e. In this case, the volume can be modeled as a half-space, and this can be achieved by assuming... To rewrite the scalar potential in the fluid layer, thereby avoiding Unbounded solution: Therefore, the relevant dispersion relation can be obtained by simplifying the first set of boundary conditions. The simplified boundary conditions are as follows: By , , , , , Substituting the expression for the relative motion of the discrete-mass nonlinear spring resonator into the simplified boundary conditions, we can obtain the characteristic equation for the dispersion characteristics of the coupled system composed of a solid, a fluid, and a discrete-mass nonlinear spring resonator when the fluid layer is a semi-wireless space. Its matrix form is as follows: S5. The wave number and frequency in the characteristic equation are solved using numerical methods to obtain the dispersion curve of the coupled system.

[0054] Specifically, in this embodiment, for the characteristic equation when the fluid layer has a finite thickness, its non-zero solution is found by applying a determinant of a 7×7 matrix equal to zero. The dispersion curve of the interaction between the Schulte wave and the nonlinear metasurface located at the interface between the fluid layer and the saturated porous medium is derived, i.e., k and The relationship curve is shown. Since analytical solutions for coupled systems are difficult to obtain, Newton's iteration method is used to calculate the roots. Similarly, for the characteristic equation when the fluid layer is a semi-infinite space, its non-zero solutions can be found by applying a 6×6 matrix with a zero determinant. The corresponding dispersion curve is then obtained. The parameter settings for the calculation are shown in Table 1.

[0055] Table 1 Parameter Settings It should be noted that in some cases, the actions described in the specification can be performed in different orders and still achieve the desired results. In this embodiment, the order of steps is given only to make the embodiment clearer and easier to explain, and not to limit it.

[0056] In one optional embodiment, please refer to Figure 2To improve the practicality of this method and facilitate its promotion, this invention also provides a system for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface. The system includes: a data acquisition device 1, a data output device 2, a processor 3, and a storage device 4. The storage device 4 includes a computer-readable storage medium storing a computer program. The computer program includes program instructions, which, when executed by the processor 3, cause the processor 3 to perform the contents described in steps S1 to S5.

[0057] In summary, this invention has at least the following beneficial effects: The method establishes an explicit dispersion relation for nonlinear metasurface Schulte waves on saturated porous media, improving the computational efficiency and prediction accuracy for complex interface wave propagation, and is particularly suitable for real soil environments composed of saturated porous media and fluid layers; the analytical framework established by this method can efficiently and accurately characterize the hybridization behavior of the fundamental mode near the collective resonant frequency and reveal key phenomena such as surface wave leakage into the fluid domain; this method can guide the optimized design of metasurface resonator arrays, providing key theoretical tools and technical support for the design of high-performance acoustic functional devices and structures; and it provides a system compatible with the method, thereby improving its practicality and facilitating its promotion.

[0058] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A method for calculating the Schulte wave dispersion relation of a metasurface in saturated porous media, characterized in that, Includes the following steps: Based on Biot theory and Helmholtz vector decomposition theorem, expressions for solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of Schulte waves are obtained; For nonviscous acoustic fluids, based on the fundamental governing equations of fluid dynamics, a scalar potential function is introduced to describe the fluid motion, thereby obtaining expressions for the fluid displacement components and the fluid normal stress. Based on Newton's second law, considering the nonlinear stiffness of the spring, we obtain an expression for the relative motion of a discrete mass nonlinear spring resonator. Based on all the aforementioned expressions, stress and displacement continuity boundary conditions are applied at the interface between the saturated porous medium and the fluid layer, thereby obtaining the characteristic equation of the dispersion characteristics of the coupled system. The wave number and frequency in the characteristic equation are solved using numerical methods, and the dispersion curve of the coupled system is finally obtained.

2. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, The method for obtaining expressions for solid skeleton displacement, fluid relative displacement, total stress, solid shear stress, and pore pressure under the action of Schulte waves based on Biot theory and Helmholtz vector decomposition theorem includes the following steps: Determine the fundamental expressions for the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress, and the pore pressure under the action of Schulte waves; Based on Biot theory, the governing equations for the displacement field of the solid skeleton and the relative displacement field of the fluid in a saturated porous medium are established under plane strain conditions. The Helmholtz vector decomposition theorem is used to decompose the displacement field of the solid skeleton and the relative displacement field of the fluid into a combination of scalar potential and vector potential. Combined with the control equation, the wave equation for decoupling P-wave and S-wave is obtained. Let the solution of the wave equation be in the form of a simple harmonic wave, and then combine it with the basic expression to obtain the expressions for the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress, and the pore pressure.

3. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 2, characterized in that, The governing equations for the solid skeleton displacement field and the fluid relative displacement field are as follows: in, and Let be the Lamé constant of the solid framework. The compressive modulus of the saturated porous medium. Here, M is the gradient operator, and M is the reciprocal of the coupling modulus of the saturated porous medium. Let be the displacement field of the solid skeleton. Let be the relative displacement field of the fluid. The total density of the saturated porous medium is given by t, where t is time. Let be the fluid density, m be the parameter relating the fluid density to the pore geometry, and b be the viscous coupling parameter between the solid and the fluid.

4. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, The expressions for the solid skeleton displacement, the fluid relative displacement, the total stress, the solid shear stress, and the pore pressure are as follows: in, Let x represent the displacement of the solid skeleton along the X-axis, where x and z are the horizontal and vertical coordinates, respectively, i is the imaginary unit, and k is the wave number. , and Let be the amplitude corresponding to each potential function. , and Let be the wave number corresponding to each potential function. Let t be the angular frequency and t be the time. The displacement of the solid skeleton in the Z-axis direction. , and The ratio of the amplitudes of the potential function. The relative displacement of the fluid in the X-axis direction. The fluid relative displacement in the Z-axis direction. The total stress is... and As a referential symbol, and Let be the Lamé constant of the solid framework. The compressive modulus of the saturated porous medium. and Let P1 and P2 be the wave numbers in the saturated porous medium, and M be the reciprocal of the coupling modulus of the saturated porous medium. The shear stress of the solid is... The wave number of the S-wave. The pore pressure is [value missing].

5. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, The expressions for the fluid displacement component and the fluid normal stress are as follows: in, Let i be the fluid displacement component in the X-axis direction, i be the imaginary unit, and k be the wave number. and Let x be the amplitude, and z be the horizontal and vertical coordinates, respectively. These are parameters related to wave number and fluid pressure wave velocity. The fluid displacement component in the Z-axis direction. Let be the Lamé constant of the solid framework. The fluid normal stress, For inviscid fluids, the bulk modulus is... For fluid density, For fluid pressure wave velocity, Let t be the angular frequency and t be the time.

6. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, The method for obtaining an expression for the relative motion of a discrete-mass nonlinear spring resonator based on Newton's second law and considering the nonlinear stiffness of the spring includes the following steps: For the discrete mass nonlinear spring resonator arranged at the interface between the saturated porous medium and the fluid layer, considering the nonlinear stiffness of the spring, its dynamic equilibrium equation is established according to Newton's second law. The nonlinear terms in the dynamic balance equation are processed using the first harmonic balance method to obtain an expression for the relative motion of the discrete mass nonlinear spring resonator.

7. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, The expression for the relative motion of the discrete mass nonlinear spring resonator is: in, The mass of the discrete-mass nonlinear spring resonator is... and As a referential symbol, Y is the angular frequency, and Y is the harmonic amplitude. , and Let i be the amplitude corresponding to each potential function, i be the imaginary unit, and k be the wave number. and Let be the wave number corresponding to the potential function. The natural frequency of the discrete-mass nonlinear spring resonator. These are nonlinear coefficients.

8. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, When the fluid layer has a finite thickness, the characteristic equation is: in, All are referential symbols. , , , , , , , , , , , , and All of these are descriptive symbols, where M is the reciprocal of the coupling modulus of the saturated porous medium. For fluid layer thickness, The compressive modulus of the saturated porous medium. , and The ratio of the amplitudes of the potential function. These are parameters related to wave number and fluid pressure wave velocity. Angular frequency, Porosity and The wave numbers of two different types of P waves, For fluid pressure wave velocity, Let A be the mass of the discrete-mass nonlinear spring resonator, and let A be the effective area per unit of the discrete-mass nonlinear spring resonator. Where is the fluid density and k is the wave number. The wave number of the S-wave. Let be the Lamé constant of the solid framework. and Let be the wave number corresponding to each potential function, and i be the imaginary unit. and Let Y be the amplitude, and Y be the harmonic amplitude.

9. The method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to claim 1, characterized in that, When the fluid layer is a semi-wireless space, the characteristic equation is: in, and All are referential symbols. , , , , , , , , , , , and All of these are descriptive symbols, where M is the reciprocal of the coupling modulus of the saturated porous medium. The compressive modulus of the saturated porous medium. , and The ratio of the amplitudes of the potential function. These are parameters related to wave number and fluid pressure wave velocity. Angular frequency, Porosity and Let P1 and P2 be the wave numbers in a saturated porous medium. For fluid pressure wave velocity, Let A be the mass of the discrete-mass nonlinear spring resonator, and let A be the effective area per unit of the discrete-mass nonlinear spring resonator. Where is the fluid density and k is the wave number. The wave number of the S-wave. Let be the Lamé constant of the solid framework. and Let be the wave number corresponding to each potential function, and i be the imaginary unit. is the amplitude coefficient, and Y is the harmonic amplitude.

10. A system for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface, characterized in that, The aforementioned system for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface includes: a data acquisition device, a data output device, a processor, and a storage device. The storage device includes a computer-readable storage medium storing a computer program. The computer program includes program instructions, which, when executed by the processor, cause the processor to implement the method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface as described in any one of claims 1-9.

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