A transfer matrix-based method for solving the reflection coefficient of a vertical elastic plate

Abstract

The application provides a vertical elastic plate reflection coefficient solving method based on a transfer matrix, comprising the following steps: establishing a water-plate-water three-layer model; calculating a normal wave number and a normal characteristic impedance; constructing a state vector and a transfer matrix; applying an interface continuity condition; solving a complex reflection coefficient and a complex transmission coefficient; and being used for parameter scanning analysis again; and calculating a result of the reflection coefficient changing with upper and lower boundary plate thicknesses; and being used for quickly solving local reflection and transmission responses corresponding to a fixed tangential wave number component under the condition that upper and lower ends of the vertical elastic plate are respectively in contact with upper and lower boundary plates.

Description

Technical Field

[0001] This invention relates to the fields of underwater acoustics engineering, waveguide acoustics, structural acoustics, and multilayer media acoustic scattering calculation, specifically to a method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix. Background Technology

[0002] In underwater acoustic waveguides, ship compartments, confined water tank experimental models, and other fluid channels with limited height, vertical elastic plates, partitions, reinforcing plates, or functional plates often form plate-fluid-plate coupled structures together with upper and lower boundary plates. When sound waves propagate in such structures and encounter vertical elastic plates, phenomena such as reflection, transmission, propagation within the plate layers, and local phase delay occur. The reflection coefficient of the vertical elastic plate is related not only to the plate parameters and thickness but also to the incident angle, frequency, tangential wavenumber, and upper and lower boundary conditions.

[0003] Traditional reflection calculation methods typically use the plane wave incident angle as the independent variable for analysis. This method is suitable for free-field or simple interface problems, but in underwater acoustic waveguide structures, the acoustic field often needs to be uniformly described using tangential wavenumbers. For example, an obliquely incident plane wave can correspond to a single tangential wavenumber, the near field of a point source can be expanded into a continuous tangential wavenumber spectrum, and waveguide modes can correspond to a discrete set of tangential wavenumbers. Therefore, within a unified framework, the fundamental module that truly needs to be repeatedly called is the single-spectrum reflection coefficient solver under given frequency and tangential wavenumber conditions.

[0004] While existing full-field finite element methods can handle complex structures, they are computationally intensive and difficult to rapidly scan within parameters such as frequency, incident angle, plate thickness, and boundary thickness. Although partially analytical or semi-analytical models can provide reflection results, they may not be clearly organized into standard modules that can be repeatedly used for continuous spectrum integration and discrete mode superposition. Therefore, it is necessary to propose a standardized calculation method that uses a fixed tangential wavenumber as input, a transfer matrix as the core, and complex reflection and transmission coefficients as outputs.

[0005] This invention provides a specific solution to the local water-plate-water three-layer scattering problem within a unified tangential wavenumber framework. Compared to general unified modeling methods, this invention focuses on solving the local transfer matrix under a fixed tangential wavenumber condition. Compared to point source near-field methods and waveguide mode methods, this invention does not directly handle continuous spectrum integrals or discrete mode superpositions, but instead provides the single-spectrum reflection response required by both. Summary of the Invention

[0006] The purpose of this invention is to solve the above-mentioned problems and provide a method for solving the reflection coefficient of a vertical elastic plate based on the transfer matrix. This method is used to quickly solve the local reflection and transmission responses corresponding to a fixed tangential wavenumber component under the condition that the upper and lower ends of the vertical elastic plate are in contact with the upper and lower boundary plates, respectively.

[0007] To achieve the above objectives, the present invention adopts the following technical solution.

[0008] A method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix includes the following steps: establishing a three-layer model of water-plate-water; calculating the normal wave number and normal characteristic impedance; constructing the state vector and transfer matrix; applying the interface continuity condition; solving for the complex reflection coefficient and complex transmission coefficient; and then using it for parameter scanning analysis.

[0009] Furthermore, the model includes a left-side fluid medium, a vertical elastic plate layer, and a right-side fluid medium, which together constitute a water-plate-water three-layer scattering unit.

[0010] Furthermore, given the angular frequency and tangential wavenumber, the total wavenumber is first determined based on the sound velocity of each medium layer, and then calculated based on the relationship between the normal wavenumbers.

[0011] Furthermore, the first medium region contains incident waves and reflected waves, the vertical elastic plate layer contains forward waves and backward waves, and the second medium region contains transmitted waves.

[0012] Furthermore, sound pressure and normal velocity are selected as state variables to construct a state vector. Wave propagation within the vertical elastic plate is represented by the plate transfer matrix, that is, it is mapped from the state at the left end to the state at the right end. When the vertical elastic plate is approximated by an equivalent fluid plate, the transfer matrix is ​​determined by the plate normal wave number, the plate normal characteristic impedance, and the plate thickness. When a higher-order elastic plate model is used, the dimension of the state vector is expanded.

[0013] Furthermore, pressure continuity and normal velocity continuity conditions are applied at different interfaces. Since each layer has the same tangential phase factor, a set of linear equations with respect to the normal amplitude is obtained. By combining the incident-reflection field on the left, the plate layer transfer matrix relationship, and the transmission field on the right, the complex reflection coefficient and complex transmission coefficient can be obtained. The results are functions of frequency, tangential wavenumber, plate thickness, and material parameters.

[0014] Furthermore, when the tangential wavenumber is less than the total wavenumber of the medium, the normal wavenumber is a real number; when the tangential wavenumber is greater than the total wavenumber of the medium, the normal wavenumber is an imaginary or complex number.

[0015] Furthermore, when the incident angle or frequency changes, the tangential wavenumber, normal wavenumber, normal impedance, and plate phase delay are updated to obtain the result of the change in reflection coefficient.

[0016] The positive effects of the present invention's method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix are: (1) The problem of local reflection of vertical elastic plate is organized into a standardized water-plate-water transfer matrix model, which is convenient for program implementation and repeated use.

[0017] (2) It can process the propagation spectrum and evanescent spectrum components at the same time, improving its applicability in near-field angular spectrum and waveguide mode scenarios.

[0018] (3) It can quickly scan parameters such as incident angle, frequency, vertical plate thickness and upper and lower boundary plate thickness.

[0019] (4) It can be used in conjunction with the unified framework of tangential wavenumber, the near-field angle spectrum method of point source, the waveguide mode expansion method and the system equivalent reflection evaluation method. Attached Figure Description

[0020] Figure 1 This is a schematic diagram of the local water-plate-water three-layer scattering model of the present invention; Figure 2 A schematic diagram of the wave field composition of incident wave, reflected wave, traveling wave and transmitted wave within the plate with a fixed tangential wave number component; Figure 3 A schematic diagram illustrating the relationship between the normal wavenumber and normal characteristic impedance of each layer; Figure 4 The curve shows the calculated reflection coefficient of a vertical 304 steel plate as a function of the incident angle. Figure 5 The curve shows the calculated reflection coefficient of a vertical 304 steel plate as a function of frequency. Figure 6 The curves show the calculated reflection coefficient of a vertical 304 steel plate as a function of the thickness of the upper and lower boundary plates. Figure 7 This is a flowchart of the transfer matrix solution for this invention. Detailed Implementation

[0021] The following describes a specific implementation of the present invention's method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix, with reference to the accompanying drawings. However, it should be noted that the specific implementation is not intended to limit the specific implementation of the present invention. Any similar structures or variations employing the present invention should be included within the scope of protection of the present invention. The following description of the embodiments refers to the accompanying drawings, which are used to illustrate specific embodiments in which the present invention can be implemented. Directional terms mentioned in the embodiments, such as "up," "down," "front," "back," "left," "right," "top," and "bottom," are only for reference to the accompanying drawings. Therefore, the directional terms used are for illustrating and understanding the present invention, and not for limiting the present invention.

[0022] See Figures 1 to 7First, a local water-plate-water three-layer scattering unit is established. This scattering unit includes a left-side fluid medium, a vertical elastic plate layer, and a right-side fluid medium. The left-side fluid medium is denoted as medium 1, the vertical elastic plate layer as medium 2, and the right-side fluid medium as medium 3. The thickness of the vertical elastic plate layer is d, with its upper end in contact with the upper boundary plate and its lower end in contact with the lower boundary plate; the direction perpendicular to the surface of the vertical elastic plate is defined as the x-direction, and the direction extending along the plate surface is defined as the z-direction.

[0023] Given an angular frequency ω and a tangential wavenumber kz, the total wavenumbers of each medium layer are k1, k2, and k3, respectively. The normal wavenumber in the i-th medium layer is determined by the normal wavenumber relationship: , where k xi Let k be the normal wavenumber in the i-th layer of the medium. This relationship shows that for a fixed tangential wavenumber k... z The original two-dimensional local scattering problem can be transformed into a one-dimensional propagation problem along the x-direction. If k z If it is less than the total wavenumber of the corresponding medium, then k xi Let k be a real number, corresponding to the propagation spectral component; if k z If it is greater than the total wavenumber of the corresponding medium, then k xi It is an imaginary or complex number, corresponding to the evanescent spectral component.

[0024] For a fluid layer or equivalent fluid layer, its normal characteristic impedance can be expressed using the normal characteristic impedance expression: , where ρ i Let be the density of the i-th layer of medium. For propagation wave conditions, the normal characteristic impedance can also be written in a form related to the equivalent propagation angle: , where c i Let θi be the sound velocity of the i-th layer of medium, and θi be the equivalent propagation angle corresponding to the angle between the i-th layer and the normal.

[0025] In the fluid region on the left, the sound pressure field consists of a unit incident wave and a reflected wave, and can be expressed as: Where R is the complex reflection coefficient to be determined. In the middle vertical elastic plate layer, the sound pressure field consists of forward and backward waves, which can be expressed as: Where A and B are the complex amplitudes of the forward and backward waves within the vertical elastic plate, respectively. In the fluid region on the right, the sound pressure field can be represented as a transmitted wave: Where T is the complex transmission coefficient to be determined. Since each layer has the same tangential phase factor... Therefore, when matching the interface, the common tangential phase factor can be canceled, and the problem is further reduced to a one-dimensional linear problem with respect to the normal magnitude and normal velocity.

[0026] To construct the transfer matrix, we choose the sound pressure p and the normal velocity v. n As a fundamental state variable, a state vector is defined: The state vector of the vertical elastic plate layer from x = 0 to x = d satisfies the transfer matrix relationship: Where M2 is the transfer matrix of the vertical elastic plate layer. If the vertical elastic plate layer is approximated by an equivalent fluid plate, then M2 can be written as: .

[0027] Where Z2 is the characteristic impedance of the plate layer in the normal direction. If an elastic plate model, a multilayer composite plate model, or a state-space model is used, the state vector can also be extended to a higher-dimensional form, but the basic idea of ​​connecting the left and right ends of the plate layer through the transfer matrix remains unchanged.

[0028] At the two interfaces x = 0 and x = d, pressure continuity and normal velocity continuity conditions are applied respectively: ; By combining the above interface continuity condition with the plate-layer transfer matrix relationship, the complex reflection coefficient R and complex transmission coefficient T can be obtained. The results can be expressed as: .

[0029] Therefore, the output of this invention is a single-spectrum reflection response under conditions of fixed tangential wavenumber and fixed frequency. This response can be directly used for oblique incidence plane wave reflection analysis, and can also serve as the basic input for continuous spectrum integration of point sound sources, discrete mode superposition of waveguides, and system reflection evaluation.

[0030] Step 1: Establish a local three-layer scattering model. For example... Figure 1 As shown, a local region near the vertical elastic plate inside the underwater acoustic waveguide is selected and represented as a three-layer scattering unit consisting of a fluid medium on the left, a vertical elastic plate layer, and a fluid medium on the right. The thickness of the vertical elastic plate layer is d, the left interface position is x = 0, and the right interface position is x = d. The upper end of the vertical elastic plate is in contact with the upper boundary plate, and the lower end is in contact with the lower boundary plate. Since this method focuses on the local scattering calculation under a fixed tangential wavenumber, the local model focuses on describing the normal propagation, reflection, and transmission relationships along the x-direction.

[0031] Step 2: Given the tangential wavenumber, calculate the normal wavenumber. For example... Figure 3 As shown, given the angular frequency ω and the tangential wave number k z Then, the total wavenumber k is determined based on the sound velocity of each medium layer. z Then calculate k based on the normal wavenumber relationship. xiThis step transforms the propagation problem in each layer into a normal propagation problem along the x-direction. If the normal wavenumber is real, the spectral component is the propagation spectral component; if the normal wavenumber is imaginary or complex, the spectral component is the evanescent spectral component. The use of the same transfer matrix process to solve for both the propagation and evanescent spectra is a key reason why this invention can serve near-field angular spectra and waveguide mode scenarios.

[0032] Step 3: Wavefield representation for each region. For example... Figure 2 As shown, the fluid region on the left contains incident and reflected waves, the vertical elastic plate layer in the middle contains forward and backward waves, and the fluid region on the right contains transmitted waves. The sound field expressions in all three regions have the same tangential phase factor. The difference lies in the normal wave number, impedance, and propagation direction. Using the above wavefield expression, the unknowns can be unified as the reflection coefficient R, transmission coefficient T, and wave amplitudes A and B within the plate layer. Since the tangential phase factor remains consistent at the interface, the interface continuity condition can ultimately be written as a system of linear equations concerning the normal amplitude.

[0033] Step 4: Construction of State Vector and Transfer Matrix. Sound pressure and normal velocity are selected as state variables to construct the state vector Q(x). Wave propagation within the vertical elastic plate can be represented by the plate transfer matrix M2, which maps from the left-hand state Q(0) to the right-hand state Q(d). When the plate is approximated using an equivalent fluid plate, the transfer matrix is ​​determined by the plate normal wave number, the plate normal characteristic impedance, and the plate thickness d. When using a higher-order elastic plate model, the dimension of the state vector can be expanded, but the solution path of "state vector—transfer matrix—interface continuity condition" is still maintained.

[0034] Step 5: Solving for interface continuity conditions and reflection coefficients. Apply pressure continuity and normal velocity continuity conditions at the two interfaces, x = 0 and x = d, respectively. Since all layers have the same tangential phase factor, the reflection coefficient can be canceled during interface matching. This yields a system of linear equations relating only to the normal amplitude. By simultaneously solving the incident-reflected field on the left, the transfer matrix relationship between the plates, and the transmitted field on the right, the complex reflection coefficient R and the complex transmission coefficient T can be obtained. Both R and T are functions of frequency, tangential wavenumber, plate thickness, and material parameters.

[0035] Step Six: Unified Processing of Propagation and Evanescent Spectra. This invention is applicable to both propagation and evanescent spectra. When |k z When |k| is less than the total wavenumber of the medium, the normal wavenumber is a real number; when |k| is less than the total wavenumber of the medium, the normal wavenumber is a real number; z When the normal wavenumber is greater than the total wavenumber of the medium, it is an imaginary or complex number. In near-field point source problems, the evanescent spectrum may have a significant impact on local reflection results. Therefore, this method does not exclude it, but instead incorporates it into the solution within the framework of complex normal wavenumber and complex impedance.

[0036] Step 7: Calculation results of the reflection coefficient as a function of the incident angle. For example... Figure 4 As shown, under the condition of a vertical 304 steel plate thickness of 10mm and a given frequency, the water-plate-water three-layer transfer matrix model established by this invention can be used to obtain the calculation results of the reflection coefficient as a function of the incident angle. Figure 4 It can be seen that as the incident angle gradually changes from a small angle to a larger oblique incident angle, the reflection coefficient exhibits a law of change with the angle. This change essentially originates from the tangential wavenumber k. z The change; when k z When changing, the normal wavenumber k of each layer xi Normal impedance Z i Furthermore, the phase delay within the plate layers changes, leading to a change in the reflection coefficient. This result demonstrates that the present invention can be used to analyze the variation of the reflection performance of a vertical elastic plate with the incident angle under oblique incidence conditions.

[0037] Step 8: Calculation results of reflection coefficient as a function of frequency. (e.g.) Figure 5 As shown, under the conditions of a fixed vertical 304 steel plate thickness and several typical incident angles, the method of this invention can be used to obtain the calculation results of the reflection coefficient as a function of frequency. Figure 5 It is evident that the reflection coefficient exhibits a significant frequency dependence, with different curves corresponding to different incident angles. As the frequency changes, the total wavenumber, normal wavenumber, and phase delay within each layer also alter, resulting in fluctuations, peaks, or troughs in the reflection coefficient curve. This result demonstrates that the present invention is applicable not only to single-frequency calculations but also to wideband reflection characteristic scanning.

[0038] Step Nine: Calculation results of the reflection coefficient as a function of the thickness of the upper and lower boundary plates. (For example...) Figure 6 As shown, under fixed vertical plate thickness, frequency, and incident conditions, by using the thicknesses of the upper and lower boundary plates as parameters, the calculation results of the reflection coefficient of the vertical 304 steel plate as a function of the thicknesses of the upper and lower boundary plates can be obtained. Figure 6 It is evident that the thickness of the upper and lower boundary plates affects the equivalent boundary load or equivalent impedance condition in the underwater acoustic waveguide, thereby further influencing the reflection coefficient of the local vertical elastic plate. This result demonstrates that the present invention can be used to analyze the impact of boundary plate thickness variations on local reflection characteristics and provide a calculation basis for boundary parameter selection and structural design.

[0039] The process of solving the transfer matrix. (e.g.) Figure 7 As shown, the solution process of this invention includes: establishing a local three-layer scattering model; inputting the frequency and tangential wavenumber; calculating the normal wavenumber and normal impedance of each layer; establishing the plate state vector and transfer matrix; applying the interface continuity condition; obtaining the complex reflection coefficient and complex transmission coefficient; and outputting the single-spectrum reflection results for subsequent continuous spectrum integration, discrete mode superposition, or system evaluation.

[0040] In the above process, plate thickness, frequency, tangential wavenumber, and material parameters, as input parameters, collectively affect the complex reflection coefficient. Specifically, plate thickness primarily affects the phase delay within the plate layers, frequency affects the total wavenumber and impedance of each layer, tangential wavenumber affects the normal wavenumber component, and material parameters affect the wave velocity and impedance of the plate layers. This invention allows for scanning of these parameters to obtain R(k) z The variation law of R(k), ω). z ω) can be further used as input for the continuous angular spectrum integral of a point source, as input for waveguide discrete mode reflection calculation, and as the basis for evaluating the system's equivalent reflection coefficient and power reflection ratio.

Claims

1. A method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix, characterized in that, Includes the following steps: Establish a three-layer model of water-plate-water; Calculate the normal wave number and normal characteristic impedance; Construct the state vector and transfer matrix; Apply interface continuity conditions; Solve for the complex reflection coefficient and complex transmission coefficient; It is then used for parameter scanning analysis.

2. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, The model includes a fluid medium on the left, a vertical elastic plate layer, and a fluid medium on the right, which together constitute a water-plate-water three-layer scattering unit.

3. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, Given the angular frequency and tangential wavenumber, the total wavenumber is first determined based on the sound velocity of each medium layer, and then calculated based on the relationship between the normal wavenumbers.

4. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, The fluid region on the left contains incident waves and reflected waves, the vertical elastic plate layer contains forward waves and backward waves, and the fluid region on the right contains transmitted waves.

5. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, Sound pressure and normal velocity are selected as state variables to construct a state vector. Wave propagation within the vertical elastic plate is represented by the plate transfer matrix, which maps from the state at the left end to the state at the right end. When the vertical elastic plate is approximated by an equivalent fluid plate, the transfer matrix is ​​determined by the plate normal wave number, the plate normal characteristic impedance, and the plate thickness. When a higher-order elastic plate model is used, the dimension of the state vector is expanded.

6. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, At different interfaces, pressure continuity and normal velocity continuity conditions are applied respectively. Since each layer has the same tangential phase factor, a system of linear equations with respect to normal amplitude is obtained. By combining the incident-reflection field on the left, the plate layer transfer matrix relationship, and the transmission field on the right, the complex reflection coefficient and complex transmission coefficient can be obtained. The results are functions of frequency, tangential wavenumber, plate thickness, and material parameters.

7. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, When the tangential wavenumber is less than the total wavenumber of the medium, the normal wavenumber is a real number; when the tangential wavenumber is greater than the total wavenumber of the medium, the normal wavenumber is an imaginary or complex number.

8. The method for solving the reflection coefficient of a vertical elastic plate based on a transfer matrix according to claim 1, characterized in that, When the incident angle or frequency changes, the tangential wavenumber, normal wavenumber, normal impedance, and plate phase delay are updated to obtain the result of the change in reflection coefficient.

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