CN102880768A - Method for calculating acoustic scattering coefficient of periodic structure plate - Google Patents

Abstract

The present invention proposes a method for calculating the acoustic scattering coefficient of a periodic structural plate. First, a system equation suitable for calculating the sound pressure of a periodic structural plate node is derived; on the other hand, the periodic structural plate and a reference plate of the same size are modeled with nodes , and then use the moving least squares method to construct the shape function. After combining the two to obtain the sound pressure difference at the node, the sound pressure at the receiving point can be obtained, and then the directional scattering coefficient and the average scattering coefficient can be obtained. The invention introduces the gridless method into the numerical calculation of the acoustic scattering coefficient of the periodic structure, avoiding a series of problems caused by the existence of the grid in the traditional numerical method. The invention has good adaptability, and can locally increase the node density when the upper limit of calculation frequency needs to be increased without re-dividing the model. Compared with the measurement experiment, it is verified that the present invention has higher precision, and thus has broad application prospects in the numerical calculation of the scattering coefficient of the periodic structure.

Description

A Method for Calculating Acoustic Scattering Coefficient of Periodic Structural Plates

technical field

The invention relates to the technical field of acoustic scattering, in particular to a method for calculating the acoustic scattering coefficient of a periodic structural plate, which is applied to calculating the acoustic scattering coefficient of a periodic structural plate of any shape.

Background technique

The periodic structure is a special interface sound scattering structure, which refers to the structure in which the geometric shape of the material is arranged according to the periodic rules in the macroscopic space. The periodic structure has already existed in form and is widely used because of its good sound diffusion performance. It is mostly used in concert halls, recording studios, theaters and other places that require high sound field diffusion. The key to the application of periodic structures is the acquisition of their scattering coefficients. There are usually two ways to obtain the acoustic scattering coefficient, one is the measurement experiment method, and the other is the numerical calculation method. For measurement experiments, it was not until 2004 that an international standard was established, that is, the reverberation chamber turntable method. This method still has many aspects to be improved, and in the face of a wide variety of periodic structures, experimental measurement requires a lot of manpower and material resources. In addition, it is impossible to measure the periodic structure in the design stage, so it is very necessary to study the numerical calculation method of the scattering coefficient. However, due to theoretical limitations, it is difficult to calculate the acoustic energy reflected by the specular surface, so the development of numerical calculation has been relatively slow. In 2000, E. Mommertz proposed that a flat structure with the same size as the periodic structure can be used to replace the periodic structure when calculating the specular reflected sound energy, thus developing a numerical algorithm for the scattering coefficient. Since then, the numerical calculation of the scattering coefficient has gained rapid development and more and more attention.

At present, the numerical calculation of the scattering coefficient mainly adopts grid-based methods such as the boundary element method. Due to the existence of the grid, these methods have some inherent defects: firstly, the pre-processing is difficult, and the grid model must be generated before the calculation, so that the data preparation The workload is large, especially for more complex structural models, which are prone to distorted meshes; second, this type of method generally uses low-order polynomials as the interpolation function of the sound pressure function, and it is impossible to give a good solution to the problem of higher-frequency sound wave propagation. The approximation of ; Third, the calculation result is not smooth and continuous, and smoothing post-processing is required.

Contents of the invention

technical problem to be solved

In order to solve the problems in the prior art, the present invention proposes a method for calculating the acoustic scattering coefficient of a periodic structure plate.

Meshless method is a new type of numerical calculation method developed in the field of mechanics in recent years. It uses a group of independent nodes to discretely solve the area, and directly constructs shape functions with the help of discrete points, so that it can completely or partially Eliminate the grid. Compared with the finite element method and boundary element method, the meshless method has advantages in the following aspects: first, the pre-processing is simple, because only the nodes need to be used to describe the model, and there is no need to consider the topological relationship between nodes; The gridless method can obtain a banded sparse matrix, which is suitable for solving large-scale scientific and engineering problems; third, it has good adaptability, and can flexibly increase the number of nodes or increase the order of the interpolation function in high error areas; fourth, it can provide continuous , A flexible shape function, so that the calculation results are smooth and continuous, without smoothing post-processing.

This method is based on the meshless method, combined with the idea of boundary element calculation of scattering coefficients, first deduces the system equations suitable for the calculation of sound pressure at the joints of periodic structural plates; Modeling is carried out, and then the shape function is constructed by using the moving least square method. After combining the two to obtain the sound pressure difference at the node, the sound pressure at the receiving point can be obtained, and then the directional scattering coefficient and the average scattering coefficient can be obtained.

Technical solutions

Technical scheme of the present invention is:

The method for calculating the acoustic scattering coefficient of a periodic structural plate is characterized in that it comprises the following steps:

Step 1: Divide the periodic structural plate to be calculated into a periodic structural plate node model containing n uniformly distributed nodes; establish a flat reference plate, which has the same shape as the planar projection of the periodic structural plate, and place the reference plate It is also divided into a reference plate node model containing n uniformly distributed nodes;

Step 2: Evenly set no less than 100 sound source points and receiving points on a hemispherical surface with the geometric center of the periodic structural plate node model as the center of the sphere. Twice the maximum distance from the edge of the structural plate; also set the same position and number of sound source points and receiving points on a hemispherical surface with the geometric center of the reference plate node model as the center of the sphere;

Step 3: Using the moving least squares method, establish the shape functions of the periodic structural plate joint model and the reference plate joint model;

Step 4: Calculate the system equations corresponding to the periodic structural plate node model and the reference plate node model respectively, and obtain the sound pressure difference p _d at the nodes on the periodic structural plate node model and the reference plate node model;

The system equation is obtained by discrete Helmholtz differential equation, and the form of the system equation is:

(C+D)·p _d ＝F

In the formula, C and D are coefficient matrices of order n×n, p _d is a vector of order n×1, which refers to the sound pressure difference of all nodes on the periodic structural plate node model or reference plate node model, and F is a vector of order n×1 The loading matrix of ; where

c(Pi)＝α/4π，i＝1,...,n

c(Pi)=α/4π, i=1,...,n

α is the solid angle of the surface at node _Pi ;

${\mathrm{D.}}_{\mathrm{im}}={\∫}_{\mathrm{\Γ}}{N}_{P_{i}Q}(\frac{\∂G(P_i,Q)}{{\∂\mathrm{no}}_Q}+\mathrm{\β}\frac{\∂G\left(P_{i}Q\right)}{\∂{\mathrm{no}}_{P_i}{\∂\mathrm{no}}_Q})\mathrm{d\Γ},$ m=1,...,n

$f_i={\mathrm{j\ρ\ωq}}_{\mathrm{\ω}}\left(r_0\right)[G(P_i,r_0)+\frac{j}{k}\frac{\∂G(P_i,r_0)}{{\∂\mathrm{no}}_{P_i}}],$ i=1,...,n

表示P_i与声场中任意一点Q之间的距离，且Q点与声源点和接受点不重合；

为周期结构板节点模型或参考板节点模型的形函数中对应P_i及Q的元素，n_Q表示周期结构板节点模型或参考板节点模型上经过点Q的法向向量，

表示周期结构板节点模型或参考板节点模型上经过点P_i的法向向量；Γ表示周期结构板节点模型或参考板节点模型的表面边界；ρ为空气密度，ω为圆频率，k为波数，j为虚数单位；r₀表示声源点，q_ω(r₀)表示声源点r₀的声源强度；Both G(P _i ,Q) and G(P _i ,r ₀ ) are Green's functions, specifically expressed as

Indicates the distance between P _i and any point Q in the sound field, and the Q point does not coincide with the sound source point and the receiving point;

is the element corresponding to P _i and Q in the shape function of the periodic structural slab joint model or the reference slab joint model, n _Q represents the normal vector passing through point Q on the periodic structural slab joint model or the reference slab joint model,

Indicates the normal vector passing through the point P _i on the periodic structural plate joint model or the reference plate joint model; Γ indicates the surface boundary of the periodic structural plate joint model or the reference plate joint model; ρ is the air density, ω is the circular frequency, and k is the wave number , j is the imaginary unit; r ₀ represents the sound source point, q _ω (r ₀ ) represents the sound source intensity of the sound source point r ₀ ;

Step 5: Using the following formula and the sound pressure difference p _d at the upper node of the periodic structural plate node model and the reference plate node model obtained in step 4, respectively calculate the sound pressure at the receiving point corresponding to the periodic structural plate node model and the reference plate node model :

p _ω (R)＝-B·p _d

p_ω(R)表示接受点R处的声压，N_RQ为周期结构板节点模型或参考板节点模型的形函数中对应R及Q的元素，G(R,Q)为格林函数；in

p _ω (R) represents the sound pressure at the receiving point R, N _RQ is the element corresponding to R and Q in the shape function of the periodic structural plate node model or the reference plate node model, and G(R,Q) is the Green’s function;

Step 6: Calculate the directional scattering coefficient of each sound source point according to the following formula:

分别代表声源点相对于周期结构板节点模型几何中心的俯仰角与方位角；s为周期结构板节点模型中接收点的数量；θ_l和

分别代表第l个接收点的俯仰角与方位角；

是周期结构板节点模型所对应的第l个接收点的声压；

是参考板节点模型所对应的第l个接收点的声压；*代表复共轭；Where: θ and

represent the pitch angle and azimuth angle of the sound source point relative to the geometric center of the periodic structural plate node model; s is the number of receiving points in the periodic structural plate node model; θ _l and

represent the elevation angle and azimuth angle of the lth receiving point respectively;

is the sound pressure of the lth receiving point corresponding to the periodic structural plate node model;

is the sound pressure of the lth receiving point corresponding to the node model of the reference plate; * stands for complex conjugate;

Step 7: After obtaining the directional scattering coefficients of all sound source points, calculate the average scattering coefficient δ _r according to the following formula:

Beneficial effect

The invention proposes to introduce the gridless method into the numerical calculation of the acoustic scattering coefficient of the periodic structure, effectively avoiding a series of problems caused by the existence of the grid in the traditional numerical method. By dividing the model into nodes, periodic structures of arbitrary shapes can be calculated. The invention has good adaptability, and can locally increase the node density when the upper limit of calculation frequency needs to be increased without re-dividing the model. Compared with the measurement experiment, it is verified that the present invention has higher precision, and thus has broad application prospects in the numerical calculation of the scattering coefficient of the periodic structure.

Description of drawings

Figure 1: Sinusoidal periodic structure;

Figure 2: Schematic diagram of sound source and receiver settings;

Figure 3: Comparison error between scattering coefficient and measurement experiment;

Figure 4: Flow chart of the method of the present invention.

Detailed ways

Now in conjunction with example, accompanying drawing, the present invention will be further described:

The model of this embodiment is a sinusoidal periodic structure, as shown in Figure 1. Each sine cycle has a length of 0.177m and a height of 0.051m. The periodic structural plate is square with a side length of 3m.

The steps of this embodiment include the following steps:

Step 1: Divide the periodic structural plate to be calculated into a periodic structural plate node model containing n uniformly distributed nodes; establish a flat reference plate, which has the same shape as the planar projection of the periodic structural plate, and place the reference plate It is also divided into a reference plate node model containing n uniformly distributed nodes;

Step 2: Evenly set 100 sound source points and receiving points on a hemispherical surface with the geometric center of the periodic structural plate node model as the center of the sphere. Can cover the hemisphere more comprehensively. The diameter reduction of the hemisphere is not less than twice the maximum distance from the geometric center of the periodic structural plate to the edge of the periodic structural plate to meet the far-field condition; the same position is also set on a hemispherical surface with the geometric center of the reference plate node model as the center of the sphere and the number of sound source points and sink points. The pitch angle and horizontal angle of each sound source point relative to the node model represent the incident direction, and the pitch angle and horizontal angle of each receiving point relative to the node model represent the receiving direction. The schematic diagram is shown in Figure 2.

Step 3: Using the moving least squares method, respectively establish the shape functions of the periodic structural plate joint model and the reference plate joint model. The following describes the process of constructing the shape function by the moving least squares method according to the public literature:

The moving least squares method can apply lower-order basis functions to obtain shape functions with higher continuity and consistency. In this approach, an approximation of a field function u(x) at a point can be expressed as:

${\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>$

是计算点x的邻域范围内各节点的坐标，为基函数向量，m为基函数的个数，a(x)＝[a₁(x),a₂(x),...a_m(x)]为待定系数向量。通常可以使用单项式基函数做运算，在三维空间中常用的线性及二次单项式基函数分别为：in

is the coordinates of each node within the neighborhood of the calculation point x, is the basis function vector, m is the number of basis functions, a(x)=[a ₁ (x), a ₂ (x),...a _m (x)] is the undetermined coefficient vector. Usually, monomial basis functions can be used for calculations. The commonly used linear and quadratic monomial basis functions in three-dimensional space are:

p(x)=[1,x,y,z] ^T ,m=4

p(x)=[1,x,y,z,x ² ,xy,y ² ,yz,z ² ,xz] ^T ,m=10

该函数只在一个有限区域（支撑域）内不为零，在区域之外为零，在三维情况下，权函数的支撑域通常为球形。常用的权函数有高斯函数、样条函数等。选定权函数后，就可以求得近似函数在节点处的误差加权平方和：After discretizing the solution domain with nodes, define a weight function at each node

This function is non-zero only in a limited area (support domain), and is zero outside the area. In the three-dimensional case, the support domain of the weight function is usually spherical. The commonly used weight functions are Gaussian function, spline function and so on. After the weight function is selected, the error weighted sum of squares of the approximate function at the node can be obtained:

$\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; [</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

Let J take the minimum value, that is

$\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; J</span>}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}\underset{\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}<span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}}<span\; class="notranslate">\; )</span>$

$=0$ j=1,2,...,m

After sorting, the following formula can be obtained:

A(x)a(x)=B(x)u

The meanings of A(x) and B(x) in the formula are:

$\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; I</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; w</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$

A(x) can be obtained from formula (41), and it can be substituted into formula (37) to get:

${\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; A</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; u</span>}$

Finally, the shape function N can be obtained.

Step 4: Calculate the system equations corresponding to the periodic structural plate node model and the reference plate node model respectively, and obtain the sound pressure difference p _d at the nodes on the periodic structural plate node model and the reference plate node model;

The system equation is obtained by discrete Helmholtz differential equation, and the form of the system equation is:

(C+D) p _d = F (1)

In the formula, C and D are coefficient matrices of order n×n, p _d is a vector of order n×1, which refers to the sound pressure difference of all nodes on the periodic structural plate node model or reference plate node model, and F is a vector of order n×1 The loading matrix of ; where

c(P_i)＝α/4π，i＝1,...,n

c(P _i )=α/4π, i=1,...,n

α is the solid angle of the surface at node _Pi ;

${\mathrm{D.}}_{\mathrm{im}}={\&Integral;}_{\mathrm{\&Gamma;}}{N}_{P_{i}Q}(\frac{\&PartialD;G(P_i,Q)}{{\&PartialD;\mathrm{no}}_Q}+\mathrm{\&beta;}\frac{\&PartialD;G(P_i,Q)}{\&PartialD;{\mathrm{no}}_{P_i}{\&PartialD;\mathrm{no}}_Q})\mathrm{d\&Gamma;},$ m=1,...,n

$f_i={\mathrm{j\&rho;\&omega;q}}_{\mathrm{\&omega;}}\left(r_0\right)[G(P_i,r_0)+\frac{j}{k}\frac{\&PartialD;G(P_i,r_0)}{{\&PartialD;\mathrm{no}}_{P_i}}],$ i=1,...,n

表示P_i与声场中任意一点Q之间的距离，且Q点与声源点和接受点不重合；为周期结构板节点模型或参考板节点模型的形函数中对应P_i及Q的元素，n_Q表示周期结构板节点模型或参考板节点模型上经过点Q的法向向量，

表示周期结构板节点模型或参考板节点模型上经过点P_i的法向向量；Γ表示周期结构板节点模型或参考板节点模型的表面边界；ρ为空气密度，ω为圆频率，k为波数，j为虚数单位；r₀表示声源点，q_ω(r₀)表示声源点r₀的声源强度，

表示P_i与r₀之间的距离；Both G(P _i ,Q) and G(P _i ,r ₀ ) are Green's functions, specifically expressed as

Indicates the distance between P _i and any point Q in the sound field, and the Q point does not coincide with the sound source point and the receiving point; is the element corresponding to P _i and Q in the shape function of the periodic structural slab joint model or the reference slab joint model, n _Q represents the normal vector passing through point Q on the periodic structural slab joint model or the reference slab joint model,

Indicates the normal vector passing through the point P _i on the periodic structural plate joint model or the reference plate joint model; Γ indicates the surface boundary of the periodic structural plate joint model or the reference plate joint model; ρ is the air density, ω is the circular frequency, and k is the wave number , j is the imaginary unit; r ₀ represents the sound source point, q _ω (r ₀ ) represents the sound source intensity of the sound source point r ₀ ,

Indicates the distance between P _i and r ₀ ;

According to the public literature, using the finite element theory, the establishment process of the system equation is:

In the space where the periodic structural plate is located, there is a sound source that provides the medium mass of ρ ₀ q (q is the volume velocity of the sound source), then the active wave equation of the sound field at this time is:

${<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}\frac{{<span\; class="notranslate">\; \&PartialD;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; p</span>}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; q</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; t</span>}}$

When the sound source is in simple harmonic vibration, the sound pressure at any point in the space has the same frequency as the sound source, and the function of sound source and sound pressure can be written as:

$\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j\&omega;t</span>}}\\ \mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{\mathrm{<span\; class="notranslate">\; j\&omega;t</span>}}\end{array}\right.$

where q _ω and p _ω are the sound source volume velocity and sound pressure in the frequency domain, respectively. According to the above two formulas, the active Helmholtz equation can be obtained:

${<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j</span>}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; \&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}$

Where k=ω/c is the wave number.

In three-dimensional problems, the Helmholtz equation has a basic solution G(P,Q)=e ^-jkr /4πr, where P and Q are any two points in the sound field, and r represents the distance between the two points. It represents the influence on another point when there is a "source" of unit intensity at a certain point in the sound field, which satisfies the following equation:

${<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}$

Where δ(P,Q) is a Dirac δ function, when P and Q coincide, δ is infinite, otherwise it is 0.

According to Green's second formula:

${{<span\; class="notranslate">\; \&Integral;</span>}_{}}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; v</span>}{<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}{<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&Omega;</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; v</span>}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; u</span>}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; v</span>}}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}$

表示边界表面上f点处的外法向导数，将上面三式联立可得到：In the formula

Represents the external normal derivative at point f on the boundary surface, and the above three equations can be combined to get:

${<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&Omega;</span>}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}{\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&Omega;</span>}$

Combined with an important property of the Dirac delta function:

${<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&Omega;</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; \&Element;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; \&NotElement;</span>\mathrm{<span\; class="notranslate">\; \&Omega;</span>}\end{array}\right.$

And assuming that the sound source is a point sound source located at r ₀ (x ₀ ,y ₀ ,z ₀ ), then

${\&Integral;}_{\mathrm{\&Omega;}}\mathrm{\&delta;}(P,Q)p_{\mathrm{\&omega;}}\left(Q\right)\mathrm{d\&Omega;}={\&Integral;}_{\mathrm{\&Gamma;}}G(P,Q)\frac{\&PartialD;p_{\mathrm{\&omega;}}\left(Q\right)}{{\&PartialD;\mathrm{no}}_Q}\mathrm{d\&Gamma;}-{\&Integral;}_{\mathrm{\&Gamma;}}{p}_{\mathrm{\&omega;}}\left(Q\right)\frac{\&PartialD;G(P,Q)}{{\&PartialD;\mathrm{no}}_Q}\mathrm{d\&Gamma;}+{\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{j\&rho;\&omega;q}}_{\mathrm{\&omega;}}G(P,Q)\mathrm{d\&Omega;}$ can be written as:

$\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>$

那么上式中右边第一项即为0。In the formula α is the solid angle of the surface at point P. Since the surface of the periodic structure is a rigid surface, the related boundary conditions are

Then the first term on the right side of the above formula is 0.

得到下式：Due to the mathematical derivation in the form of boundary integrals, the above formula cannot obtain a unique solution at certain frequencies, and these corresponding frequencies are called "pseudo frequencies". In order to overcome the problems caused by non-unique solutions, one can first obtain a partial derivative of the above formula and apply boundary conditions

Get the following formula:

$\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}$

Then linearly combine the above two equations to obtain an equation that can obtain a unique solution at any frequency:

$\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}$

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}$

In the formula, β is a non-zero coupling constant, and the imaginary part is generally required to be non-zero, usually taken as β=j/k.

A further assumption is made on the periodic structure model here, that is, the periodic structure model is considered to have no thickness, and only the reflection of its surface acoustic wave is considered. This is feasible in practical situations, because in the actual sound field, the size of the periodic structure is usually large, and its thickness is negligible compared with the overall size. Here, such a model is called a periodic structure thin plate. According to this assumption, and considering the point P located on the surface of the periodic structure, the formula

$c\left(P\right)p_{\mathrm{\&omega;}}\left(P\right)=-{\&Integral;}_{\mathrm{\&Gamma;}}{p}_{\mathrm{\&omega;}}\left(Q\right)\frac{\&PartialD;G(P,Q)}{\&PartialD;{\mathrm{no}}_Q}\mathrm{d\&Gamma;}-\mathrm{\&beta;}{\&Integral;}_{\mathrm{\&Gamma;}}{p}_{\mathrm{\&omega;}}\left(Q\right)\frac{\&PartialD;G(P,Q)}{{\&PartialD;\mathrm{no}}_Q{\&PartialD;\mathrm{no}}_P}\mathrm{d\&Gamma;}$ can be transformed into:

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}$

$c\left(P\right)p_{\mathrm{\&omega;d}}\left(P\right)=-{\&Integral;}_{\mathrm{\&Gamma;}}{p}_{\mathrm{\&omega;d}}\left(Q\right)\frac{\&PartialD;G(P,Q)}{\&PartialD;{\mathrm{no}}_Q}\mathrm{d\&Gamma;}-\mathrm{\&beta;}{\&Integral;}_{\mathrm{\&Gamma;}}{p}_{\mathrm{\&omega;d}}\left(Q\right)\frac{\&PartialD;G(P,Q)}{{\&PartialD;\mathrm{no}}_Q{\&PartialD;\mathrm{no}}_P}\mathrm{d\&Gamma;}$ P∈Γ

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}$

In the formula, p _Qd (P)＝p _ω (P ₁ )-p _ω (P ₂ ), p _Qd (Q)＝p _ω (Q ₁ )-p _ω (Q ₂ ); At , the sound pressure difference between P ₁ , P ₂ and Q ₁ , Q ₂ on both sides of the periodic structure. Similarly, the sound pressure at a certain point on a thin plate with a periodic structure can also be represented by the sound pressure difference on both sides of this point.

In the meshless method, the sound pressure difference at any point (such as Q) on the periodic structure thin plate can be expressed as:

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="N\; Q"\; filenames="HDA00002273005700021.png"\; state="\{\{state\}\}">N</figure-callout></span><figure-callout\; id="1"\; label="N\; Q"\; filenames="HDA00002273005700021.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}<span\; class="notranslate">\; ]</span>\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; )</span>\end{array}\right\}$

In the formula, P ₁ , P ₂ ,..., P _n represent n nodes, N _Q is the shape function vector of each node at Q, and P _d is the vector of sound pressure difference at each node.

Assuming that the periodic structure thin plate model has been divided by P ₁ , P ₂ ,..., P _n n nodes, then the sound pressure difference formula at any point (such as Q) on the periodic structure thin plate is substituted into

$\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{<span\; class="notranslate">\; \&PartialD;</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; Q</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}\mathrm{<span\; class="notranslate">\; d\&Gamma;</span>}$

$<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;j\&rho;\&omega;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; P</span>}}}<span\; class="notranslate">\; ,</span>$

After simplification, the system equation for calculating the node sound pressure difference can be obtained:

(C+D)·p _d =F.

Step 5: Using the following formula and the sound pressure difference p _d at the upper node of the periodic structural plate node model and the reference plate node model obtained in step 4, respectively calculate the sound pressure at the receiving point corresponding to the periodic structural plate node model and the reference plate node model :

p _ω (R)＝-B·p _d

in p _ω (R) represents the sound pressure at the receiving point R, N _RQ is the element corresponding to R and Q in the shape function of the periodic structural plate node model or the reference plate node model, and G(R,Q) is the Green’s function;

Step 6: Calculate the directional scattering coefficient of each sound source point according to the following formula:

分别代表声源点相对于周期结构板节点模型几何中心的俯仰角与方位角；Where: θ and

Represent the pitch angle and azimuth angle of the sound source point relative to the geometric center of the periodic structural plate node model;

是周期结构板节点模型所对应的第l个接收点的声压；

是参考板节点模型所对应的第l个接收点的声压；*代表复共轭；s is the number of receiving points in the periodic structural plate node model; θ _l and represent the elevation angle and azimuth angle of the lth receiving point respectively;

is the sound pressure of the lth receiving point corresponding to the periodic structural plate node model;

is the sound pressure of the lth receiving point corresponding to the node model of the reference plate; * stands for complex conjugate;

Step 7: After obtaining the directional scattering coefficients of all sound source points, calculate the average scattering coefficient δ _r according to the following formula:

In this example, the average scattering coefficient of this periodic structure is calculated from 50Hz to 2000Hz on the 1/3 octave band. Compared with the measurement experiment, it is found that the two are very close, and the average difference is only 0.027. It is also consistent that the scattering coefficient increases with the increase of frequency. Effectively proved the validity of the present invention.

Claims (1)

1. A method for calculating the acoustic scattering coefficient of a periodic structure plate, characterized in that: comprising the following steps: Step 1: Divide the periodic structural plate to be calculated into a periodic structural plate node model containing n uniformly distributed nodes; establish a flat reference plate, which has the same shape as the planar projection of the periodic structural plate, and place the reference plate It is also divided into a reference plate node model containing n uniformly distributed nodes; Step 2: Evenly set no less than 100 sound source points and receiving points on a hemispherical surface with the geometric center of the periodic structural plate node model as the center of the sphere. Twice the maximum distance from the edge of the structural plate; also set the same position and number of sound source points and receiving points on a hemispherical surface with the geometric center of the reference plate node model as the center of the sphere; Step 3: Using the moving least squares method, establish the shape functions of the periodic structural plate joint model and the reference plate joint model; Step 4: Calculate the system equations corresponding to the periodic structural plate node model and the reference plate node model respectively, and obtain the sound pressure difference p _d at the nodes on the periodic structural plate node model and the reference plate node model; The system equation is obtained by discrete Helmholtz differential equation, and the form of the system equation is: (C+D)·p _d ＝F In the formula, C and D are coefficient matrices of order n×n, p _d is a vector of order n×1, which refers to the sound pressure difference of all nodes on the periodic structural plate node model or reference plate node model, and F is a vector of order n×1 The loading matrix of ; where

c(P _i )=α/4π, i=1,...,n α is the solid angle of the surface at node _Pi ;

m=1,...,n

i=1,...,n Both G(P _i ,Q) and G(P _i ,r ₀ ) are Green's functions, specifically expressed as

Indicates the distance between P _i and any point Q in the sound field, and the Q point does not coincide with the sound source point and the receiving point;

is the element corresponding to P _i and Q in the shape function of the periodic structural slab joint model or the reference slab joint model, n _Q represents the normal vector passing through point Q on the periodic structural slab joint model or the reference slab joint model,

Indicates the normal vector passing through the point P _i on the periodic structural plate joint model or the reference plate joint model; Γ indicates the surface boundary of the periodic structural plate joint model or the reference plate joint model; ρ is the air density, ω is the circular frequency, and k is the wave number , j is the imaginary unit; r ₀ represents the sound source point, q _ω (r ₀ ) represents the sound source intensity of the sound source point r ₀ ; Step 5: Using the following formula and the sound pressure difference p _d at the upper node of the periodic structural plate node model and the reference plate node model obtained in step 4, respectively calculate the sound pressure at the receiving point corresponding to the periodic structural plate node model and the reference plate node model : p _ω (R)＝-B·p _d in

p _ω (R) represents the sound pressure at the receiving point R, N _RQ is the element corresponding to R and Q in the shape function of the periodic structural plate node model or the reference plate node model, and G(R,Q) is the Green’s function; Step 6: Calculate the directional scattering coefficient of each sound source point according to the following formula:

Where: θ and

represent the pitch angle and azimuth angle of the sound source point relative to the geometric center of the periodic structural plate node model; s is the number of receiving points in the periodic structural plate node model; θ _l and

represent the elevation angle and azimuth angle of the lth receiving point respectively;

is the sound pressure of the lth receiving point corresponding to the periodic structural plate node model;

is the sound pressure of the lth receiving point corresponding to the node model of the reference plate; * stands for complex conjugate; Step 7: After obtaining the directional scattering coefficients of all sound source points, calculate the average scattering coefficient δ _r according to the following formula: .

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