Method and system for calculating the schroedinger wave dispersion relation of a saturated porous medium metasurface

Abstract

The present application relates to the technical field of physical acoustics, in particular to a saturated porous medium metasurface Scholte wave dispersion relation calculation method and system, the method comprises the following steps: obtaining the expression of solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of Scholte wave; obtaining the expression of fluid displacement component and fluid normal stress; obtaining the expression about the relative motion of resonator; according to all expressions, stress and displacement continuity boundary conditions are applied at the interface between saturated porous medium and fluid layer to obtain the characteristic equation of dispersion characteristics, and the characteristic equation is solved to obtain the dispersion curve. The present application provides the explicit dispersion relation of saturated porous medium metasurface Scholte wave, improves the calculation efficiency and prediction accuracy of complex interface wave propagation, can guide the optimization design of metasurface resonator array, and provides key theoretical tools and technical support for the design of high-performance acoustic functional devices and structures.

Description

Technical Field

[0001] This invention relates to the field of physical acoustics, and in particular to a method and system for calculating the dispersion relation of Schulte waves on a metasurface of saturated porous media. Background Technology

[0002] Schulte-Stoneley waves are a general term for a class of elastic guided waves that exist at the interface between two closely contacting media. Among them, those propagating at fluid-solid interfaces (such as the seabed) are called Schulte waves, a special case of Stoneley waves in the fluid-solid case. These 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 fields of marine engineering, geological exploration, and underwater acoustics, accurate prediction and control of interface wave propagation are crucial for improving the accuracy of acoustic detection equipment and optimizing the vibration reduction and noise reduction performance of underwater structures. However, existing technologies for characterizing the interaction between Schulte waves and elastic half-space surface resonant arrays still have significant shortcomings. Specifically, due to the lack of efficient and accurate analytical models, it is difficult to accurately calculate and predict the complex coupling behavior between waves and structures. This leads to a general reliance on numerical trial and error or empirical estimation in the design of real soil environments such as saturated porous media and nonlinear metasurfaces, resulting in technical bottlenecks such as low computational efficiency, poor prediction accuracy, and long design cycles. Therefore, it is necessary to explore an efficient and accurate method for calculating the dispersion relationship of Schulte waves on nonlinear metasurfaces in saturated porous media. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides a method and system for calculating the Schulte wave dispersion relationship of saturated porous media metasurfaces.

[0005] To achieve the above objectives, in a first aspect, the present invention provides a method for calculating the dispersion relation of Schulte waves on a saturated porous medium metasurface. The method includes the following steps: obtaining expressions for the 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; for non-viscous acoustic fluids, introducing a scalar potential function to describe fluid motion according to the fundamental governing equations of fluid dynamics, thereby obtaining expressions for the fluid displacement components and fluid normal stress; based on Newton's second law, considering the nonlinear stiffness of the spring, obtaining an expression for the relative motion of a discrete-mass nonlinear spring resonator; applying stress and displacement continuity boundary conditions at the interface between the saturated porous medium and the fluid layer according to all the expressions, thereby obtaining the characteristic equation of the dispersion characteristics of the coupled system; and solving the wave number and frequency in the characteristic equation using numerical methods to finally obtain the dispersion curve of the coupled system. This invention improves the computational efficiency and prediction accuracy of complex interface wave propagation by providing explicit dispersion relations for saturated porous metasurface Schulte waves. It can guide the optimized design of metasurface resonator arrays and provide key theoretical tools and technical support for the design of high-performance acoustic functional devices and structures.

[0006] Optionally, the process of obtaining the expressions for the 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:

[0007] 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;

[0008] 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.

[0009] 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.

[0010] 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.

[0011] Optionally, the governing equations for the solid skeleton displacement field and the fluid relative displacement field are respectively:

[0012]

[0013]

[0014] 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.

[0015] Optionally, 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:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] 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].

[0022] Optionally, the expressions for the fluid displacement component and the fluid normal stress are respectively:

[0023]

[0024]

[0025] 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.

[0026] Optionally, the step of 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:

[0027] 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.

[0028] 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.

[0029] Optionally, the expression for the relative motion of the discrete-mass nonlinear spring resonator is:

[0030]

[0031] 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.

[0032] Optionally, when the fluid layer has a finite thickness, the characteristic equation is:

[0033]

[0034] 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 wavenumber. 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.

[0035] Optionally, when the fluid layer is a semi-wireless space, the characteristic equation is:

[0036]

[0037] 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 wavenumber. 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.

[0038] Secondly, the present invention provides a system for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface. The system 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 provided by the present invention.

[0039] The present invention has at least the following beneficial effects:

[0040] 1. This method establishes an explicit dispersion relation model for Schulte waves on nonlinear metasurfaces in saturated porous media, which improves the computational efficiency and prediction accuracy for wave propagation at complex interfaces.

[0041] 2. 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.

[0042] 3. 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.

[0043] 4. A system adapted to the method is provided, which can improve the practicality of the method and facilitate its promotion. Attached Figure Description

[0044] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of this application and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0045] Figure 1 This is a flowchart illustrating the method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface according to an embodiment of the present invention.

[0046] Figure 2 This is a schematic diagram of the framework of the Schulte wave dispersion relation calculation system for saturated porous medium metasurfaces according to an embodiment of the present invention. Detailed Implementation

[0047] Specific embodiments of the present invention will now be described in detail. It should be noted that the embodiments described herein are for illustrative purposes only and are not intended to limit the invention. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the invention. However, it will be apparent to those skilled in the art that these specific details are not necessary to practice the invention. In other instances, well-known circuits, software, or methods have not been specifically described to avoid obscuring the invention.

[0048] Throughout this specification, references to "an embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in connection with that embodiment or example is included in at least one embodiment of the invention. Therefore, the phrases "in an embodiment," "in an embodiment," "an example," or "an example" appearing in various places throughout the specification do not necessarily refer to the same embodiment or example. Furthermore, specific features, structures, or characteristics can be combined in one or more embodiments or examples in any suitable combination and / or sub-combination. Moreover, those skilled in the art will understand that the illustrations provided herein are for illustrative purposes and are not necessarily drawn to scale.

[0049] It should be noted in advance that, in one alternative embodiment, except for independent descriptions, the same symbols or letters appearing in all formulas have the same meaning and value.

[0050] In one optional embodiment, please refer to Figure 1 This invention provides a method for calculating the Schulte wave dispersion relation of a saturated porous medium metasurface, the method comprising the following steps:

[0051] S1. Based on Biot theory and Helmholtz vector decomposition theorem, obtain the expressions for the solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of Schulte waves.

[0052] Specifically, step S1 includes the following steps:

[0053] S11. Determine the basic 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.

[0054] Specifically, in this embodiment, the Schulte wave can excite horizontal (X-axis direction) and vertical (Z-axis direction) displacements in a saturated porous medium. Taking soil as an example, the displacements of the solid skeleton in the horizontal and vertical directions, as well as the fluid-to-solid fluid displacements in the horizontal and vertical directions, can be expressed as follows:

[0055]

[0056]

[0057] in, This represents the displacement of the solid skeleton along the X-axis. and Let x and z be the scalar and vector potentials of the solid framework, respectively, and x and z be the horizontal and vertical coordinates, respectively. This represents the displacement of the solid skeleton along the Z-axis. Let X be the relative displacement of the fluid in the X-axis direction. Let Z represent the relative displacement of the fluid along the Z-axis. and These are the scalar potential and vector potential of the fluid, respectively.

[0058] Furthermore, the total stress, solid shear stress, and pore pressure are expressed in terms of displacement, as follows:

[0059]

[0060]

[0061]

[0062] in, For the total stress, and Let be the Lamé constant of the solid framework. , E is Young's modulus. Poisson's ratio, The compressive modulus of a saturated porous medium. , , The bulk modulus of the soil skeleton. The bulk modulus of soil particles. Let be the shear stress in the solid, and M be the reciprocal of the coupling modulus of the saturated porous medium. , Porosity For fluid bulk modulus, This refers to pore pressure.

[0063] The above seven relationships are the basic expressions for the solid skeleton displacement, fluid relative displacement, total stress, solid shear stress and pore pressure under the action of Schulte waves in this embodiment.

[0064] S12. Based on Biot theory, establish the governing equations for the displacement field of the solid skeleton and the relative displacement field of the fluid in a saturated porous medium under plane strain conditions.

[0065] Specifically, in this embodiment, since pore water exists in the saturated porous medium, it is assumed that the pore size is much smaller than the Schulte wave wavelength. Simultaneously, to ensure the effectiveness of the periodic structure, the Schulte wave wavelength must also be much larger than the lattice constant of the discrete mass nonlinear spring resonator alignment. Based on the above conditions, the governing equations for the solid skeleton displacement field and the fluid relative displacement field in the saturated porous medium under plane strain conditions are established according to Biot theory. The governing equations are specifically as follows:

[0066]

[0067]

[0068] in, For gradient operators, , For the displacement field of the solid skeleton, This represents the relative displacement field of the fluid. The total density of the saturated porous medium. t is time. For fluid density, Let be the soil particle density, and m be the parameter relating fluid density and pore geometry. , denoted as tortuosity, and b as the viscous coupling parameter between the solid and the fluid. , For fluid viscosity, is the permeability coefficient.

[0069] S13. Using the Helmholtz vector decomposition theorem, the displacement field of the solid skeleton and the relative displacement field of the fluid are decomposed 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.

[0070] Specifically, in this embodiment, 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, as follows:

[0071]

[0072]

[0073] Substituting these two expressions into the control equations of step S12, we obtain the wave control equations for P-wave decoupling and S-wave decoupling, as follows:

[0074]

[0075]

[0076] S14. 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.

[0077] Specifically, in this embodiment, it is assumed that the solution to the wave equation obtained in step S13 is in the form of a simple harmonic wave, satisfying:

[0078]

[0079]

[0080] in, , , and They are respectively , , and amplitude, and Here, r represents the wave vectors of the P-wave and S-wave, respectively; r is the position vector; and i is the imaginary unit. ω is the angular frequency.

[0081] Substituting Equation 1 into the wave control equation for P-wave decoupling, we obtain the control equation for the P-wave as follows:

[0082]

[0083] in, It is a symbolic representation. , denoted as the wavenumber of the P-wave in a saturated porous medium.

[0084] The governing equations of the P-wave have two non-zero solutions if their coefficient determinant is zero. These two non-zero solutions are:

[0085]

[0086] in, and These are the velocities of wave P1 and wave P2 in a saturated porous medium, respectively. and Let P1 and P2 be the wave numbers in a saturated porous medium. and For coefficients, , .

[0087] Similarly, substituting Equation 2 into the wave control equation for S-wave decoupling, we obtain the control equation for the S-wave as follows:

[0088]

[0089] The governing equations of the S-wave have a non-zero solution if and only if the determinant of its coefficient matrix is ​​zero. This non-zero solution is:

[0090]

[0091] in, The wave velocity of S-waves in saturated porous media. Let be the wave number of the S-wave.

[0092] For a harmonic surface wave propagating along the X-axis, if we consider its exponential decay characteristic in the Z-axis direction, then , , and These four potential functions can be written in the following complex function form:

[0093]

[0094]

[0095]

[0096]

[0097] in, , and Let be the amplitude corresponding to each potential function. , and Let be the wave number corresponding to each potential function. , and The ratio of the amplitudes of the potential function. , , , , , , where k is the wave number.

[0098] Substituting the complex function forms of the four potential functions into the basic expressions given in step S11, we can obtain the expressions for the solid skeleton displacement, fluid relative displacement, total stress, solid shear stress, and pore pressure, which are as follows:

[0099]

[0100]

[0101]

[0102]

[0103]

[0104] in, and These are reference symbols designed to facilitate writing.

[0105] 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.

[0106] Specifically, in this embodiment, for non-viscous acoustic fluids, the governing equations satisfy:

[0107]

[0108] 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.

[0109] By introducing scalar potential , The governing equations for nonviscous acoustic fluids can be written as:

[0110]

[0111] Under harmonic propagation, assume the scalar potential in the fluid layer is:

[0112]

[0113] 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:

[0114]

[0115] Given the fluid-saturated porous medium coupling, the expression for the fluid normal stress is derived as follows:

[0116]

[0117] in, For fluid normal stress.

[0118] 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.

[0119] Specifically, step S3 includes the following steps:

[0120] 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.

[0121] 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:

[0122]

[0123] 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 .

[0124] Vertical displacement of soil free surface The following relationship must be satisfied:

[0125]

[0126] in, The referential symbols were set up for ease of writing. .

[0127] Assuming relative motion of the resonators The harmonic solution is expressed in the form of:

[0128]

[0129] Where Y is the harmonic amplitude.

[0130] The dynamic equilibrium equations can then be written in the following form:

[0131]

[0132] 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.

[0133] 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:

[0134]

[0135] 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.

[0136] 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.

[0137] 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:

[0138]

[0139] in, For fluid layer thickness, For fluid stress, , This represents the resonator stress.

[0140] 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.

[0141] 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:

[0142]

[0143] in, All are referential symbols. , , , , , , , , , A is the effective area per unit of the discrete-mass nonlinear spring resonator. The fluid density is given.

[0144] 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:

[0145]

[0146] Therefore, the relevant dispersion relation can be obtained by simplifying the first set of boundary conditions. The simplified boundary conditions are as follows:

[0147]

[0148] 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:

[0149]

[0150] S5. The wave number and frequency in the characteristic equation are solved using numerical methods to obtain the dispersion curve of the coupled system.

[0151] 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.

[0152] Table 1 Parameter Settings

[0153]

[0154] 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.

[0155] In one optional embodiment, please refer to Figure 2 To 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.

[0156] 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.

[0157] 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 SPOD relation of a saturated porous medium metasurface, characterized in that, Includes the following steps: Determine the fundamental expressions for the displacement of the solid skeleton, the relative displacement of the fluid, 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 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. 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; 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 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].

3. 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.

4. 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.

5. 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.

6. 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 wavenumber. 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.

7. 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 wavenumber. 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.

8. 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 1-7.

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