CN114545331A - Method for Reconstructing Sound Field Directly Radiated by Sound Source in Semi-Open Space - Google Patents

Abstract

A method for reconstructing a sound source direct radiation sound field in a semi-open space, which is a method for reconstructing the sound source direct radiation sound field in the semi-open space containing a plane reflection boundary, comprises the following steps: 1. establishing a mathematical model for expressing the total sound pressure field of the direct radiation and the boundary reflected sound contribution of the sound source by linear superposition of a semispherical wave basis function; 2. arranging a holographic measuring surface at the near field of a sound source to perform sound pressure holographic measurement; 3. taking the holographic measured values of part of the measuring points as input, reconstructing the sound pressure values of the other measuring points, and determining the optimal basis function expansion term number by taking the sound pressure reconstruction error as the minimum; 4. and solving the spherical wave basis function coefficient of the semispace under the condition of the optimal expansion term number, acquiring the spherical wave basis function coefficient of the free space for expressing the sound source direct radiation sound field, and realizing the reconstruction of the sound source direct radiation sound field. The invention provides a mathematical basis for measurement and evaluation of large-size structure sound source radiation, and can realize sound field imaging and identification and positioning of the structure sound source in a semi-open space.

Description

Method for Reconstructing Sound Field Directly Radiated by Sound Source in Semi-Open Space

Technical field:

The invention relates to a method for obtaining the direct radiated acoustic quantity distribution of the sound source by measuring the total acoustic quantity distribution superimposed by the direct radiation of the sound source and the reflection from the planar boundary by using a sensor array in a semi-open space containing a plane reflection boundary. It belongs to the technical fields of sound source identification and positioning, sound field imaging, near-field sound holography, sound wave separation and noise control.

Background technique:

The measurement and evaluation of the acoustic quantity radiated by a sound source generally needs to be carried out in a fully anechoic or semi-anechoic standard environment. For large-scale, complex equipment, there is often no such ideal acoustic measurement environment. For example, acoustic radiation testing of large ships can usually only be performed in docks or piers with reflective boundaries. In this case, the acoustic measurements polluted by boundary reflection can neither faithfully reflect the radiation level of the sound source at the measuring point, nor can it be combined with the near-field acoustic holography method established in free space to reconstruct the sound source at the measuring point. External radiated acoustic volume distribution in three-dimensional space.

The half-space sound field reconstruction method based on the Fourier acoustic method is only suitable for the case where the holographic measurement surface is a regular geometric shape (plane, cylinder or sphere). The half-space sound field reconstruction method based on the inverse boundary element method and the equivalent source method needs to set a large number of discrete nodes and equivalent source points on the surface of the sound source, which leads to the requirement of a huge number of measuring points and the amount of inversion calculation.

Invention content:

The present invention aims to overcome the above shortcomings of the prior art, and provides a method for reconstructing the direct radiation sound field of a sound source in a semi-open space.

The invention can realize the reconstruction of the sound field directly radiated by the sound source in the semi-open space containing the reflection boundary.

The invention expresses the total sound pressure field contributed by the direct radiation of the sound source and the reflected sound from the plane boundary as the linear superposition of a set of half-space spherical wave base functions; The total sound pressure value is matched with the linear superposition of the half-space spherical wave basis function; the half-space spherical wave basis function coefficients are solved by calculating the pseudo-inverse; a set of free-space spherical wave basis functions and the obtained basis function coefficients are repeated. The direct radiation sound pressure value of the sound source on the holographic measurement surface or any field point is constructed to realize the reconstruction of the sound field.

The plane boundary of the semi-open space is the Locally Reaction boundary, and its boundary acoustic impedance is a constant irrelevant to the incident angle of the acoustic wave and the incident acoustic wave front, and is a known quantity.

The basis of constructing the half-space spherical wave basis function is the analytical solution of the sound pressure field excited by the multipole sound source in the half-open space, which is obtained by solving the inhomogeneous Helmholtz equation and boundary conditions by the Steepest Descent Method.

The expression of the analytical solution contains three superposition terms, which represent the direct radiation of the multipole, the mirrored multipole radiation about the boundary, and the boundary sound.

A set of half-space spherical wave basis functions that express the total sound pressure field and a set of free-space spherical wave basis functions that express the sound pressure field directly radiated by the sound source share the same set of basis function coefficients.

The present invention is suitable for sound sources with arbitrary geometric shapes, and the sound pressure holographic measurement surface can be an irregular single-layer measurement surface.

The method for reconstructing the direct radiation sound field of a sound source in a semi-open space of the present invention includes the following contents:

1. Establish a mathematical model that expresses the total sound pressure field contributed by the direct radiation of the sound source and the boundary reflection sound by the linear superposition of the half-space spherical wave basis functions;

A global coordinate system is established with the projection O of the sound source geometric center O ₁ on the boundary as the origin. The global coordinate of O ₁ is marked as x _O1 =(0,0,h _s ), and h _s is the distance from O ₁ to the boundary; O ₁ The mirror point about the boundary is denoted as O ₂ . By translating the global coordinate system, a local coordinate system is established with O ₁ and O ₂ as the origins, respectively, and the coordinates of the field point x in the two local coordinate systems are denoted as x ₁ ≡(r ₁ , θ ₁ , φ ₁ ) and x ₂ ≡(r ₂ , θ ₂ , φ ₂ ), the following relationship is satisfied between the three:

x ₁ =xh _s e _z , x ₂ =x+h _s e _z (1)

where ez is the _z -direction unit vector.

For a steady-state sound field, the half-space total sound pressure p _half (x; ω) at field point x can be expressed as a linear superposition of finite-term half-space spherical wave basis functions:

Among them, ω is the angular frequency of the acoustic wave; ψ _jhalf (x; ω) is the half-space spherical wave basis function; c _j (ω) is the coefficient of the expansion term of the basis function; j is the ordinal number of the expansion term, and J is the number of the expansion term. The expression of the half-space spherical wave basis function ψ _jhalf (x; ω) is:

ψ _jhalf (x; ω)=ψ _j (x|xh _s e _z ; ω)+ψ _j (x|x+h _s e _z ; ω)+ξ _j (x|x+h _s e _z ;ω) (3)

Among them, ψ _j (x|xh _s e _z ; ω) and ψ _j (x|x+h _s e _z ; ω) are the j-th free-space spherical waves representing the direct radiation sound of the sound source and its mirror image virtual source, respectively basis function. In the spherical coordinate system, the expression of ψ _j is:

为第一类球汉克尔函数，k＝ω/c为声波波数，c为声速；

为球谐函数。在式(2)～式(4)中，整数n，l和j满足关系式j＝n²+n+l+1，其中，-n≤l≤n，0≤n≤N，N为n的截断值。在计算式(3)时，其右边的前两项分别代入局部坐标x₁和x₂进行计算。ξ_j(x|x+h_se_z；ω)表述边界声，其表达式为：in,

is the first kind of spherical Hankel function, k=ω/c is the wave number of the sound wave, and c is the speed of sound;

is a spherical harmonic function. In equations (2) to (4), the integers n, l and j satisfy the relational expression j=n ² +n+l+1, where -n≤l≤n, 0≤n≤N, and N is n cutoff value. When calculating formula (3), the first two items on the right side are respectively substituted into the local coordinates x ₁ and x ₂ for calculation. ξ _j (x|x+h _s e _z ; ω) represents the boundary sound, and its expression is:

in,

as well as

复角μ_p为：In equations (5) to (8), R _p (θ ₂ ; ω), F(w) and w are the sound pressure reflection coefficient, boundary loss factor and numerical spacing, respectively; the local coordinates r ₁ and r ₂ are respectively The distance from the geometric center of the sound source and the geometric center of the image virtual source to the field point; θ ₂ is the incident angle of the sound wave, which is the angle between the line connecting the field point and the virtual source geometric center and the positive direction of the z-axis;

The complex angle μ _p is:

where β is the normalized boundary acoustic admittance,

Among them, Z is the boundary acoustic impedance ratio, Z ₀ is the normalized boundary acoustic impedance ratio, and ρ ₀ is the density of the fluid medium. The implementation of this method assumes that the acoustic impedance rate Z ₀ is a known quantity, and Z ₀ can be obtained according to the in-situ measurement method of acoustic impedance.

2. Arrange a holographic measurement surface in the near field of the sound source for sound pressure holographic measurement; arrange a group of sound pressure sensors in the near field of the sound source to form a sound pressure holographic measurement surface, and measure the total contribution of the direct radiation sound of the sound source and the boundary reflection sound. Sound pressure distribution, wherein, according to the difference of the fluid medium where the sound source is located, the sound pressure sensor used in this method may be a microphone, a hydrophone or other types of sensors.

M为声压测点数目。按照隔点取点的方式，将声压测点分为两组。第一组测点坐标记为

第二组测点坐标记为

其中，

和

分别表示向上取整和向下取整。3. Using the holographic measurement values of some measuring points as input, reconstruct the sound pressure values of the remaining measuring points, and determine the optimal basis function expansion terms based on the principle of the sound pressure reconstruction error being the smallest; Coordinates are marked as

M is the number of sound pressure measurement points. The sound pressure measurement points are divided into two groups according to the method of taking points at intervals. The first group of measuring point coordinates is marked as

The second group of measuring point coordinates is marked as

in,

and

Represent round up and round down, respectively.

The upper limit of the possible value of the basis function expansion term J is set to be J _max , that is, 1≤J≤J _max . For any J within this range, according to formula (2), the sound pressure values collected by the first group of measuring points on the holographic measurement surface can be expressed as the following matrix form:

为半空间总声压测量值组成的列向量：in,

Column vector of half-space total sound pressure measurements:

Among them, the superscript T is the vector transpose. {C(ω)} _J×1 is the column vector composed of the half-space spherical wave basis function coefficients:

为半空间球面波基函数在各测点的展开项组成的矩阵：

is the matrix composed of the expansion terms of the half-space spherical wave basis function at each measuring point:

Solving equation (11), the coefficient column vector can be obtained:

表示对矩阵求伪逆，Among them, the superscript

represents the pseudo-inverse of the matrix,

where the superscript H is the conjugate transpose of the matrix.

After the coefficient column vector {C(ω)} _J×1 is determined, the sound pressure of the second group of measuring points can be further reconstructed:

And calculate the relative error between the sound pressure reconstruction value and the measured value of the second set of measuring points:

where ||·|| ₂ is the 2-norm of the vector.

Traverse all J from 1 to J _max , use equations (11) to (18) to calculate the relative error ε, and determine the number of expansion terms corresponding to the minimum value of ε as the optimal number of expansion terms J _opt .

4. Solve the half-space spherical wave basis function coefficients under the condition of the optimal expansion term, obtain the free-space spherical wave basis function coefficients representing the sound field directly radiated by the sound source, and realize the reconstruction of the sound field directly radiated by the sound source; set the basis function expansion The number of terms is J _opt . According to equation (2), the sound pressure value collected by the holographic measurement surface can be expressed in the following matrix form:

Pseudo-inverse equation (19) to solve for the column vector of coefficients

Therefore, the reconstructed value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:

为声压重构点坐标，s＝1,2,…,S，S为重构点数目；in,

is the sound pressure reconstruction point coordinates, s=1,2,...,S, S is the number of reconstruction points;

为自由空间球面波基函数在重构点

的展开项组成的矩阵：

for the free-space spherical wave basis function at the reconstruction point

A matrix of expanded terms of :

The invention constructs a half-space spherical wave basis function satisfying the Helmholtz equation and boundary conditions on the basis of a mathematical model expressing the sound source radiated sound field based on the superposition of the free-space spherical wave basis functions, and establishes a half-space spherical wave based on the boundary acoustic impedance as a parameter. Basis function superposition expresses a mathematical model of the total sound field in the half-space. Holographic measurement of the sound field of a structural sound source with any geometric shape in a semi-open space with a planar boundary, and by inversely solving the basis function coefficients, the basis function coefficients of the sound source's direct radiation sound field are obtained, and the reconstruction of the sound source's direct radiation sound field is realized. .

Beneficial effects of the present invention:

1. The mathematical model based on the superposition of the semi-space spherical wave basis functions proposed by the present invention can express the semi-open space sound field with finite impedance boundary. Measurement and evaluation of radiation from dimensionally structured sound sources, providing a mathematical basis.

2. Substitute the total sound pressure distribution in the half space measured by the array into the mathematical model of the sound field to solve, which can achieve the goal that the near-field acoustic holography method can achieve, that is, it can realize the identification and localization of sound field imaging and structural sound sources in the semi-open space .

3. The present invention is applicable to structured sound sources with arbitrary geometric shapes, and the sound pressure holographic measurement surface can be an irregular single-layer holographic measurement surface.

Description of drawings:

Figure 1 is a schematic diagram of a semi-open space sound field composed of a sound source and a plane boundary;

Figure 2 is the geometric relationship between the geometric center, field point and plane boundary of the sound source and its mirror image virtual source;

Fig. 3 is a schematic diagram of measuring point grouping when selecting the optimal number of expansion items;

Figure 4 is a schematic diagram of a simulated sound field composed of a pulsating ball sound source, a plane boundary and a hydrophone array;

Figure 5 is a schematic diagram of the distribution and numbering rules of the hydrophones on the array;

Figure 6 is the distribution curve of the total sound pressure value in the half space, the reconstructed value of the sound pressure directly radiated by the sound source and the real value of the sound pressure directly radiated by the sound source on the holographic measuring point.

Detailed ways

Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

The implementation of the method for reconstructing the direct radiation sound field of a sound source in a semi-open space of the present invention is carried out according to the following steps:

Step 1, establish a sound field mathematical model based on the superposition of the half-space spherical wave basis functions.

h_s为O₁到边界的距离；O₁关于边界的镜像点记为O₂。通过对全局坐标系平移，以O₁和O₂分别为原点建立局部坐标系，场点x在两局部坐标系中的坐标分别记为x₁≡(r₁,θ₁,φ₁)和x₂≡(r₂,θ₂,φ₂)，三者之间满足如下关系：As shown in Figure 1 and Figure 2, a global coordinate system is established with the projection O of the sound source geometric center O ₁ on the boundary as the origin, and the global coordinate of O ₁ is marked as

h _s is the distance from O ₁ to the boundary; the mirror point of O ₁ about the boundary is denoted as O ₂ . By translating the global coordinate system, a local coordinate system is established with O ₁ and O ₂ as the origins, respectively, and the coordinates of the field point x in the two local coordinate systems are denoted as x ₁ ≡(r ₁ , θ ₁ , φ ₁ ) and x ₂ ≡(r ₂ , θ ₂ , φ ₂ ), the following relationship is satisfied between the three:

x ₁ =xh _s e _z , x ₂ =x+h _s e _z (1)

where ez is the _z -direction unit vector.

For a steady-state sound field, the half-space total sound pressure p _half (x; ω) at field point x can be expressed as a linear superposition of finite-term half-space spherical wave basis functions:

Among them, ω is the angular frequency of the acoustic wave; ψ _jhalf (x; ω) is the half-space spherical wave basis function; c _j (ω) is the coefficient of the expansion term of the basis function; j is the ordinal number of the expansion term, and J is the number of the expansion term. The expression of the half-space spherical wave basis function ψ _jhalf (x; ω) is:

ψ _jhalf (x; ω)=ψ _j (x|xh _s e _z ; ω)+ψ _j (x|x+h _s e _z ; ω)+ξ _j (x|x+h _s e _z ;ω) (3)

Among them, ψ _j (x|xh _s e _z ; ω) and ψ _j (x|x+h _s e _z ; ω) are the j-th free-space spherical waves representing the direct radiation sound of the sound source and its mirror image virtual source, respectively basis function. In the spherical coordinate system, the expression of ψ _j is:

为第一类球汉克尔函数，k＝ω/c为声波波数，c为声速；

为球谐函数。在式(2)～式(4)中，整数n，l和j满足关系式j＝n²+n+l+1，其中，-n≤l≤n，0≤n≤N，N为n的截断值。在计算式(3)时，其右边的前两项分别代入局部坐标x₁和x₂进行计算。ξ_j(x|x+h_se_z；ω)表述边界声，其表达式为：in,

is the first kind of spherical Hankel function, k=ω/c is the wave number of the sound wave, and c is the speed of sound;

is a spherical harmonic function. In equations (2) to (4), the integers n, l and j satisfy the relational expression j=n ² +n+l+1, where -n≤l≤n, 0≤n≤N, and N is n cutoff value. When calculating formula (3), the first two items on the right side are respectively substituted into the local coordinates x ₁ and x ₂ for calculation. ξ _j (x|x+h _s e _z ; ω) represents the boundary sound, and its expression is:

in,

as well as

复角μ_p为：In equations (5) to (8), R _p (θ ₂ ; ω), F(w) and w are the sound pressure reflection coefficient, boundary loss factor and numerical spacing, respectively; the local coordinates r ₁ and r ₂ are respectively The distance from the geometric center of the sound source and the geometric center of the mirror virtual source to the field point; θ ₂ is the incident angle of the sound wave, which is the angle between the line connecting the field point and the virtual source geometric center and the positive direction of the z-axis, as shown in Figure 2;

The complex angle μ _p is:

where β is the normalized boundary acoustic admittance,

Among them, Z is the boundary acoustic impedance ratio, Z ₀ is the normalized boundary acoustic impedance ratio, and ρ ₀ is the density of the fluid medium. The implementation of this method assumes that the acoustic impedance rate Z ₀ is a known quantity, and Z ₀ can be obtained according to the in-situ measurement method of acoustic impedance.

Step 2, obtaining holographic measurement values.

As shown in Figure 1, a group of sound pressure sensors are arranged in the near field of the sound source to form a sound pressure holographic measurement surface, and the total sound pressure distribution contributed by the direct radiation sound of the sound source and the boundary reflected sound is measured. Depending on the medium, the sound pressure sensor used in this method may be a microphone, a hydrophone or other types of sensors.

Step 3, select the optimal expansion term of the half-space spherical wave basis function.

M为声压测点数目。按照隔点取点的方式，将声压测点分为两组。第一组测点坐标记为

第二组测点坐标记为

其中，

和

分别表示向上取整和向下取整。以6行6列均匀分布的声压测点组成的平面阵列为例，对测点进行分组，其示意图如图3所示。Mark the coordinates of the measuring point on the holographic measuring surface as

M is the number of sound pressure measurement points. The sound pressure measurement points are divided into two groups according to the method of taking points at intervals. The first group of measuring point coordinates is marked as

The second group of measuring point coordinates is marked as

in,

and

Represent round up and round down, respectively. Taking a plane array composed of sound pressure measurement points with 6 rows and 6 columns evenly distributed as an example, the measurement points are grouped, and the schematic diagram is shown in Figure 3.

The upper limit of the possible value of the basis function expansion term J is set to be J _max , that is, 1≤J≤J _max . For any J within this range, according to formula (2), the sound pressure values collected by the first group of measuring points on the holographic measurement surface can be expressed as the following matrix form:

为半空间总声压测量值组成的列向量：in,

Column vector of half-space total sound pressure measurements:

Among them, the superscript T is the vector transpose. {C(ω)} _J×1 is the column vector composed of the half-space spherical wave basis function coefficients:

为半空间球面波基函数在各测点的展开项组成的矩阵：

is the matrix composed of the expansion terms of the half-space spherical wave basis function at each measuring point:

Solving equation (11), the coefficient column vector can be obtained:

表示对矩阵求伪逆，Among them, the superscript

represents the pseudo-inverse of the matrix,

where the superscript H is the conjugate transpose of the matrix.

After the coefficient column vector {C(ω)} _J×1 is determined, the sound pressure of the second group of measuring points can be further reconstructed:

And calculate the relative error between the sound pressure reconstruction value and the measured value of the second set of measuring points:

where ||·|| ₂ is the 2-norm of the vector.

Traverse all J from 1 to J _max , use equations (11) to (18) to calculate the relative error ε, and determine the number of expansion terms corresponding to the minimum value of ε as the optimal number of expansion terms J _opt .

Step 4, reconstruct the direct radiation sound field of the sound source.

Set the number of expansion terms of the basis function as J _opt , according to formula (2), the sound pressure value collected by the holographic measurement surface can be expressed as the following matrix form:

Pseudo-inverse equation (19) to solve for the column vector of coefficients

Therefore, the reconstructed value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:

为声压重构点坐标，s＝1,2,…,S，S为重构点数目；

为自由空间球面波基函数在重构点

的展开项组成的矩阵：in,

is the sound pressure reconstruction point coordinates, s=1,2,...,S, S is the number of reconstruction points;

for the free-space spherical wave basis function at the reconstruction point

A matrix of expanded terms of :

表面质点径向振动速度V₀＝0.01m/s，频率f＝3000Hz；使用平面水听器阵列进行声压全息测量，阵列面与x-轴垂直，其几何中心与球心O₁的连线垂直于阵列面，距离球心d_s＝0.15m；阵列孔径0.15m×0.15m，由6×6个测点组成，相邻测点间距为0.03m。为了便于描述和分析计算结果，对36个测点依次编号，如图5所示，其中，第1号测点坐标为(0.150m,-0.075m,0.375m)，第36号测点坐标为(0.150m,0.075m,0.225m)。水介质密度为ρ₀＝1000kg/m³，声速为c＝1500m/s。模拟水听器测量误差的影响，对测量值附加信噪比为30dB的高斯白噪声。Example: The arrangement of the pulsating sphere sound source and the plane boundary is shown in Figure 4, where the boundary is located on the z=0 plane, the acoustic impedance rate Z ₀ =2+3i; the radius of the pulsating sphere a=0.05m, its geometric center O ₁ 's coordinates

Surface particle radial vibration velocity V ₀ =0.01m/s, frequency f = 3000Hz; sound pressure holographic measurement using a planar hydrophone array, the array surface is perpendicular to the x-axis, and the line connecting its geometric center and the center of the sphere O ₁ Vertical to the array surface, the distance from the center of the sphere is d _s =0.15m; the array aperture is 0.15m×0.15m, consisting of 6×6 measuring points, and the distance between adjacent measuring points is 0.03m. In order to facilitate the description and analysis of the calculation results, the 36 measuring points are numbered in sequence, as shown in Figure 5, where the coordinates of the No. 1 measuring point are (0.150m, -0.075m, 0.375m), and the coordinates of the No. 36 measuring point are (0.150m, 0.075m, 0.225m). The density of the water medium is ρ ₀ =1000kg/m ³ , and the speed of sound is c=1500m/s. The influence of the measurement error of the hydrophone is simulated, and the Gaussian white noise with a signal-to-noise ratio of 30dB is added to the measurement value.

Figure 6 shows the distribution of the dimensionless sound pressure amplitude |p/ρ ₀ cV ₀ | at each measuring point, including the half-space total sound pressure value, the reconstructed value of the sound pressure directly radiated by the sound source, and the direct sound source radiation True value of sound pressure. Observing Figure 6, it is found that the reconstructed value of the sound pressure directly radiated by the sound source is in good agreement with the real value. The results show that the invention can realize the reconstruction of the sound field directly radiated by the sound source in the semi-open space.

The content described in the embodiments of this specification is only one example of the implementation form of the inventive concept. The protection scope of the present invention includes, but is not limited to, the specific forms and parameters stated in the embodiments, and also includes equivalent technical means that can be conceived by those skilled in the art according to the inventive concept.

Claims (2)

1. the method for reconstructing the direct radiation sound field of a sound source in a semi-open space, is characterized in that: comprise the following steps: S1. Establish a mathematical model that expresses the total sound pressure field contributed by the direct radiation of the sound source and the sound reflected by the boundary by the linear superposition of the half-space spherical wave basis functions; establish the global coordinate with the projection O of the geometric center O ₁ of the sound source on the boundary as the origin system, the global coordinates of O ₁ are marked as

h _s is the distance from O ₁ to the boundary; the mirror point of O ₁ about the boundary is denoted as O ₂ ; by translating the global coordinate system, a local coordinate system is established with O ₁ and O ₂ as the origin respectively, and the field point x is at the two local coordinates The coordinates in the system are respectively denoted as x ₁ ≡(r ₁ , θ ₁ , φ ₁ ) and x ₂ ≡(r ₂ , θ ₂ , φ ₂ ), and the following relations are satisfied between the three: x ₁ =xh _s e _z , x ₂ =x+h _s e _z (1) Among them, e _z is the z-direction unit vector; For a steady-state sound field, the half-space total sound pressure p _half (x; ω) at field point x can be expressed as a linear superposition of finite-term half-space spherical wave basis functions:

Among them, ω is the angular frequency of the sound wave; ψ _jhalf (x; ω) is the half-space spherical wave basis function; c _j (ω) is the basis function expansion term coefficient; j is the expansion term ordinal number, J is the expansion term number; The expression of the spherical wave basis function ψ _jhalf (x; ω) is: ψ _jhalf (x; ω)=ψ _j (x|xh _s e _z ; ω)+ψ _j (x|x+h _s e _z ; ω)+ξ _j (x|x+h _s e _z ;ω) (3) Among them, ψ _j (x|xh _s e _z ; ω) and ψ _j (x|x+h _s e _z ; ω) are the j-th free-space spherical waves representing the direct radiation sound of the sound source and its mirror image virtual source, respectively Basis function; in spherical coordinate system, the expression of ψ _j is:

in,

is the first kind of spherical Hankel function, k=ω/c is the wave number of the sound wave, and c is the speed of sound;

is a spherical harmonic function; in equations (2) to (4), the integers n, l and j satisfy the relational expression j=n ² +n+l+1, where -n≤l≤n, 0≤n≤ N, N is the truncation value of n; when calculating formula (3), the first two items on the right side are respectively substituted into the local coordinates x ₁ and x ₂ for calculation; ξ _j (x|x+h _s e _z ; ω) expression Boundary sound, its expression is:

in,

as well as

In equations (5) to (8), R _p (θ ₂ ; ω), F(w) and w are the sound pressure reflection coefficient, the boundary loss factor (Boundary Loss Factor) and the numerical distance (Numerical Distance), respectively; The coordinates r ₁ and r ₂ are the distances from the geometric center of the sound source and the geometric center of the image virtual source to the field point, respectively; θ ₂ is the incident angle of the sound wave, which is the angle between the line connecting the field point and the virtual source geometric center and the positive direction of the z-axis ;

The complex angle μ _p is:

where β is the normalized boundary acoustic admittance,

Among them, Z is the boundary acoustic impedance ratio, Z ₀ is the normalized boundary acoustic impedance ratio, and ρ ₀ is the density of the fluid medium; the implementation of this method assumes that the acoustic impedance ratio Z ₀ is a known quantity, and Z ₀ can be based on the acoustic impedance. In situ measurement method acquisition; S2. Arrange the holographic measurement surface in the near field of the sound source for sound pressure holographic measurement; Arrange a group of sound pressure sensors in the near field of the sound source to form a sound pressure holographic measurement surface, and measure the total sound pressure distribution contributed by the direct radiation sound of the sound source and the boundary reflection sound; S3. Take the holographic measurement values of some measuring points as input, reconstruct the sound pressure values of the remaining measuring points, and determine the optimal number of expansion terms of the basis function based on the principle of the sound pressure reconstruction error being the smallest; Mark the coordinates of the measuring point on the holographic measuring surface as

M is the number of sound pressure measurement points; the sound pressure measurement points are divided into two groups according to the method of taking points at intervals; the coordinates of the first group of measurement points are marked as

The second set of measuring point coordinates are marked as

in,

and

Represent round up and round down respectively; Set the upper limit of the possible value of the basis function expansion term J to J _max , that is, 1≤J≤J _max ; for any J within this range, according to formula (2), the first set of measuring points on the holographic measurement surface The sound pressure value can be expressed in the following matrix form:

in,

Column vector of half-space total sound pressure measurements:

Among them, the superscript T is the vector transpose; {C(ω)} _J×1 is the column vector composed of the half-space spherical wave basis function coefficients:

is the matrix composed of the expansion terms of the half-space spherical wave basis function at each measuring point:

Solving equation (11), the coefficient column vector can be obtained:

Among them, the superscript

represents the pseudo-inverse of the matrix,

Among them, the superscript H is the conjugate transpose of the matrix; After the coefficient column vector {C(ω)} _J×1 is determined, the sound pressure of the second group of measuring points can be further reconstructed:

And calculate the relative error between the sound pressure reconstruction value and the measured value of the second set of measuring points:

Among them, ||·|| ₂ is the 2-norm of the vector; Traverse all J from 1 to J _max , use equations (11) to (18) to calculate the relative error ε, and determine the number of expansion terms corresponding to the minimum value of ε as the optimal number of expansion terms J _opt ; S4. Solve the half-space spherical wave basis function coefficients under the condition of the optimal expansion term, obtain the free-space spherical wave basis function coefficients representing the sound field directly radiated by the sound source, and realize the reconstruction of the sound field directly radiated by the sound source; The number of expansion terms of the basis function is set as J _opt , according to formula (2), the sound pressure value collected by the holographic measurement surface can be expressed as the following matrix form:

Pseudo-inverse equation (19) to solve for the column vector of coefficients

Therefore, the reconstructed value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:

in,

is the sound pressure reconstruction point coordinates, s=1,2,...,S, S is the number of reconstruction points;

for the free-space spherical wave basis function at the reconstruction point

A matrix of expanded terms of :

2. The method for reconstructing the direct radiation sound field of a sound source in a semi-open space as claimed in claim 1, wherein the sound pressure sensor in step S2 adopts a microphone, a hydrophone according to the difference of the fluid medium where the sound source is located. device or other type of sensor.

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