Method for reconstructing the sound field of direct radiation from sound source in semi-open space

Abstract

The method for reconstructing the sound field of direct radiation of a sound source in a semi-open space is a method for reconstructing the sound field of direct radiation of a sound source in a semi-open space containing a plane reflection boundary, comprising: 1. establishing a mathematical model for expressing the total sound pressure field of direct radiation of a sound source and the contribution of boundary reflection sound by linear superposition of half-space spherical wave basis functions; 2. arranging a holographic measurement surface in the near field of the sound source to perform holographic measurement of sound pressure; 3. using the holographic measurement values ​​of some measuring points as input, reconstructing the sound pressure values ​​of the remaining measuring points, and determining the optimal number of basis function expansion terms based on the principle of minimizing the sound pressure reconstruction error; 4. solving the coefficients of the half-space spherical wave basis function under the condition of the optimal number of expansion terms, obtaining the coefficients of the free space spherical wave basis function that express the sound field of direct radiation of a sound source, and realizing the reconstruction of the sound field of direct radiation of a sound source. The present invention provides a mathematical basis for the measurement and evaluation of radiation of large-scale structural sound sources, and can realize sound field imaging and identification and positioning of structural sound sources in semi-open spaces.

Description

Technical field:

[0001] The present invention relates to a method for obtaining the direct radiation acoustic quantity distribution of a sound source by measuring the total acoustic quantity distribution superimposed by the direct radiation of a sound source and the reflection of a plane boundary in a semi-open space containing a plane reflection boundary using a sensor array. The method belongs to the technical fields of sound source identification and positioning, sound field imaging, near-field acoustic holography, sound wave separation and noise control. Background technology:

[0002] The measurement and evaluation of acoustic radiation from sound sources generally need to be carried out in a fully anechoic or semi-anechoic standard environment. For large-scale, complex equipment, there is usually no such ideal acoustic measurement environment. For example, the acoustic radiation test of a large ship can usually only be carried out in a dock or pier with a reflective boundary. In this case, the acoustic measurement value contaminated by boundary reflections can neither truly reflect the radiation level of the sound source at the measurement point, nor can it be combined with the near-field acoustic holography method established in free space to reconstruct the distribution of the externally radiated acoustic quantity of the sound source in three-dimensional space.

[0003] The half-space sound field reconstruction method based on Fourier acoustics is only applicable to 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 requires setting a large number of discrete nodes and equivalent source points for the sound source surface, which leads to the need for a huge number of measurement points and inverse calculation. Summary of the invention:

[0004] The present invention aims to overcome the above-mentioned shortcomings of the prior art and provides a method for reconstructing the sound field directly radiated by a sound source in a semi-open space.

[0005] The present invention can realize the reconstruction of the sound field directly radiated by the sound source in a semi-open space with a reflective boundary.

[0006] The present invention expresses the total sound pressure field contributed by the direct radiation of a sound source and the reflected sound of a plane boundary as a linear superposition of a group of half-space spherical wave basis functions; arranges a sound pressure holographic measurement surface in the near field of the sound source, measures a group of half-space total sound pressure values, and matches them with the linear superposition of the half-space spherical wave basis functions; solves the coefficients of the half-space spherical wave basis functions by obtaining a pseudo-inverse; uses a group of free-space spherical wave basis functions and the obtained basis function coefficients to reconstruct the direct radiation sound pressure value of the sound source at the holographic measurement surface or at any field point, thereby realizing the reconstruction of the sound field.

[0007] The plane boundary of the semi-open space is a locally reactive boundary, and its boundary acoustic impedance is a constant that is independent of the incident angle of the sound wave and the incident sound wave front, and is a known quantity.

[0008] The basis for constructing the half-space spherical wave basis function is to use the steepest descent method to solve the non-homogeneous Helmholtz equation and boundary conditions to obtain the analytical solution of the sound pressure field excited by the multipole sound source in the semi-open space.

[0009] The expression of the analytical solution contains three superposition terms, which represent the direct radiation sound of the multipole, the radiation sound of the mirror multipole about the boundary, and the boundary sound.

[0010] A set of half-space spherical wave basis functions describing the total sound pressure field and a set of free-space spherical wave basis functions describing the sound pressure field directly radiated by the sound source share the same set of basis function coefficients.

[0011] The present invention is applicable to a sound source with any geometric shape, and the sound pressure holographic measurement surface can be an irregular single-layer measurement surface.

[0012] The method for reconstructing the sound field directly radiated by a sound source in a semi-open space of the present invention comprises the following contents:

[0013] 1. Establish a mathematical model that describes the total sound pressure field contributed by the direct radiation of the sound source and the boundary reflected sound by linear superposition of the half-space spherical wave basis functions;

[0014] The global coordinate system is established with the projection O of the sound source geometric center O1 on the boundary as the origin. The global coordinate of O1 is marked as x O1 =(0,0,h s ), h s is the distance from O1 to the boundary; the mirror point of O1 about the boundary is recorded as O2. By translating the global coordinate system, local coordinate systems are established with O1 and O2 as the origins respectively. The coordinates of the field point x in the two local coordinate systems are recorded as x1≡(r1,θ1,φ1) and x2≡(r2,θ2,φ2), and the following relationship is satisfied between the three:

[0015] x1=xh s e z ,x2=x+h s e z (1)

[0016] Among them, e z is the z-unit vector.

[0017] For a steady-state sound field, the total sound pressure p in the half space at the field point x is half (x; ω) can be expressed as a linear superposition of finite half-space spherical wave basis functions:

[0018]

[0019] Where 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 coefficient of the basis function expansion term; j is the expansion term ordinal number, and J is the number of expansion terms. Half-space spherical wave basis function ψ jhalf The expression of (x; ω) is:

[0020] 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)

[0021] Among them, ψ j (x|xh s e z ;ω) and ψ j (x|x+h s e z ;ω) are the jth free space spherical wave basis functions that represent the direct radiation sound of the sound source and its mirror image virtual source. In the spherical coordinate system, ψ j The expression is:

[0022]

[0023] in, is the first kind of spherical Hankel function, k = ω / c is the acoustic wave number, c is the speed of sound; is a spherical harmonic function. In equations (2) to (4), the integers n, l and j satisfy the relationship j = n 2 +n+l+1, where -n≤l≤n, 0≤n≤N, and N is the cutoff value of n. When calculating equation (3), the first two terms on the right side are substituted into the local coordinates x1 and x2 for calculation. ξ j (x|x+h s e z ; ω) describes the boundary sound, and its expression is:

[0024]

[0025] in,

[0026]

[0027]

[0028] as well as

[0029]

[0030] In formulas (5) to (8), R p (θ2; ω), F(w) and w are the sound pressure reflection coefficient, boundary loss factor and numerical spacing respectively; local coordinates r1 and r2 are the distances from the geometric center of the sound source and the geometric center of the mirror virtual source to the field point respectively; θ2 is the incident angle of the sound wave, which is the angle between the line connecting the field point and the geometric center of the virtual source and the positive direction of the z-axis; Complex angle μ p for:

[0031]

[0032] Where β is the normalized boundary acoustic admittance,

[0033]

[0034] Wherein, Z is the boundary acoustic impedance, Z0 is the normalized boundary acoustic impedance, and ρ0 is the density of the fluid medium. The implementation of this method assumes that the acoustic impedance Z0 is a known quantity, and Z0 can be obtained according to the in-situ measurement method of acoustic impedance.

[0035] 2. Arrange a holographic measurement surface in the near field of the sound source to perform holographic measurement of sound pressure; arrange a group of sound pressure sensors in the near field of the sound source to form a sound pressure holographic measurement surface to measure the total sound pressure distribution contributed by the direct radiation sound of the sound source and the boundary reflected sound. According to the different fluid media where the sound source is located, the sound pressure sensor used in this method can be a microphone, a hydrophone or other types of sensors.

[0036] 3. 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 basis function expansion terms based on the principle of minimizing the sound pressure reconstruction error; mark the coordinates of the measuring points on the holographic measurement surface as M is the number of sound pressure measurement points. According to the method of selecting points at intervals, the sound pressure measurement points are divided into two groups. The coordinates of the first group of measurement points are marked as The coordinates of the second set of measurement points are marked as in, and Indicates rounding up and rounding down respectively.

[0037] 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 sound pressure values ​​collected by the first group of measurement points on the holographic measurement surface can be expressed in the following matrix form:

[0038]

[0039] in, is a column vector of the total sound pressure measurements in the half space:

[0040]

[0041] Where superscript T is the vector transpose. {C(ω)} J×1 is a column vector of the coefficients of the half-space spherical wave basis functions:

[0042]

[0043] is the matrix composed of the expansion items of the half-space spherical wave basis function at each measuring point:

[0044]

[0045] Solving equation (11), we can get the coefficient column vector:

[0046]

[0047] Among them, the superscript It means to find the pseudo-inverse of the matrix.

[0048]

[0049] The superscript H is the conjugate transpose of the matrix.

[0050] When the coefficient column vector {C(ω)} J×1 After this is determined, the sound pressure of the second set of measurement points can be further reconstructed:

[0051]

[0052] And calculate the relative error between the reconstructed sound pressure value and the measured value of the second set of measurement points:

[0053]

[0054] Here, ||·||2 is the 2-norm of the vector.

[0055] From 1 to J max Traverse all J, use equations (11) to (18) to calculate the relative error ε, and determine the number of expansion items corresponding to the minimum value of ε as the optimal number of expansion items J opt .

[0056] 4. Under the condition of the optimal number of expansion terms, solve the coefficients of the half-space spherical wave basis function, obtain the coefficients of the free-space spherical wave basis function that describes 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 number of basis function expansion terms to J opt According to formula (2), the sound pressure value collected by the holographic measurement surface can be expressed in the following matrix form:

[0057]

[0058] Find the pseudo-inverse of equation (19) to solve the coefficient column vector

[0059]

[0060] Thus, the reconstructed value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:

[0061]

[0062] in, is the coordinate of the sound pressure reconstruction point, s=1,2,…,S, S is the number of reconstruction points;

[0063] is the free space spherical wave basis function at the reconstruction point The matrix composed of the expanded items of :

[0064]

[0065] The present invention is based on a mathematical model for describing the sound field radiated by a sound source by superposition of spherical wave basis functions in free space, takes boundary acoustic impedance as a parameter, constructs a half-space spherical wave basis function that satisfies the Helmholtz equation and boundary conditions, and establishes a mathematical model for describing the total sound field of a half-space by superposition of spherical wave basis functions in a half-space. Holographic measurement is performed on the sound field of a structural sound source with an arbitrary geometric shape in a semi-open space with a plane boundary, and the basis function coefficients of the sound field directly radiated by the sound source are obtained by inversely solving the basis function coefficients, thereby realizing reconstruction of the sound field directly radiated by the sound source.

[0066] Beneficial effects of the present invention:

[0067] 1. The mathematical model based on the superposition of half-space spherical wave basis functions proposed in the present invention can describe the sound field of a semi-open space with a finite impedance boundary, and provides a mathematical basis for the measurement and evaluation of structural sound sources, especially the radiation of large-size structural sound sources, in the absence of ideal acoustic measurement conditions.

[0068] 2. The total sound pressure distribution in the half space measured by the array is substituted into the mathematical model of the sound field for solution, which can achieve the purpose of the near-field acoustic holography method, that is, it can realize the sound field imaging and identification and positioning of the structural sound source in the semi-open space.

[0069] 3. The present invention is applicable to structural sound sources with arbitrary geometric shapes, and the sound pressure holographic measurement surface can be an irregular single-layer holographic measurement surface. Description of the drawings:

[0070] Figure 1 It is a schematic diagram of the sound field in a semi-open space formed by the sound source and the plane boundary;

[0071] Figure 2 It is the geometric relationship between the geometric center, field point and plane boundary of the sound source and its mirror image virtual source;

[0072] Figure 3 This is a schematic diagram of measuring point grouping when selecting the optimal number of expansion items;

[0073] Figure 4 It is a schematic diagram of the simulated sound field composed of the pulsating spherical sound source, the plane boundary and the hydrophone array;

[0074] Figure 5 It is a schematic diagram of the distribution and numbering rules of the hydrophones on the array;

[0075] Figure 6 It 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 true value of the sound pressure directly radiated by the sound source at the holographic measurement point. Specific implementation method:

[0076] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0077] The method for reconstructing the sound field directly radiated by a sound source in a semi-open space of the present invention is implemented by the following steps:

[0078] Step 1: Establish a mathematical model of the sound field based on the superposition of half-space spherical wave basis functions.

[0079] like Figure 1 and Figure 2 As shown, the global coordinate system is established with the projection O of the sound source geometric center O1 on the boundary as the origin. The global coordinate of O1 is h s is the distance from O1 to the boundary; the mirror point of O1 about the boundary is recorded as O2. By translating the global coordinate system, local coordinate systems are established with O1 and O2 as the origins respectively. The coordinates of the field point x in the two local coordinate systems are recorded as x1≡(r1,θ1,φ1) and x2≡(r2,θ2,φ2), and the following relationship is satisfied between the three:

[0080] x1=xh s e z ,x2=x+h s e z (1)

[0081] Among them, e z is the z-unit vector.

[0082] For a steady-state sound field, the total sound pressure p in the half space at the field point x is half(x; ω) can be expressed as a linear superposition of finite half-space spherical wave basis functions:

[0083]

[0084] Where 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 coefficient of the basis function expansion term; j is the expansion term ordinal number, and J is the number of expansion terms. Half-space spherical wave basis function ψ jhalf The expression of (x; ω) is:

[0085] 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)

[0086] Among them, ψ j (x|xh s e z ;ω) and ψ j (x|x+h s e z ;ω) are the jth free space spherical wave basis functions that represent the direct radiation sound of the sound source and its mirror image virtual source. In the spherical coordinate system, ψ j The expression is:

[0087]

[0088] in, is the first kind of spherical Hankel function, k = ω / c is the acoustic wave number, c is the speed of sound; is a spherical harmonic function. In equations (2) to (4), the integers n, l and j satisfy the relationship j = n 2 +n+l+1, where -n≤l≤n, 0≤n≤N, and N is the cutoff value of n. When calculating equation (3), the first two terms on the right side are substituted into the local coordinates x1 and x2 for calculation. ξ j (x|x+h s e z ; ω) describes the boundary sound, and its expression is:

[0089]

[0090] in,

[0091]

[0092]

[0093] as well as

[0094]

[0095] In formulas (5) to (8), R p (θ2; ω), F(w) and w are the sound pressure reflection coefficient, boundary loss factor and numerical spacing respectively; the local coordinates r1 and r2 are the distances from the geometric center of the sound source and the geometric center of the mirror virtual source to the field point respectively; θ2 is the sound wave incident angle, which is the angle between the line connecting the field point and the geometric center of the virtual source and the positive direction of the z-axis, such as Figure 2 As shown; Complex angle μ p for:

[0096]

[0097] Where β is the normalized boundary acoustic admittance,

[0098]

[0099] Wherein, Z is the boundary acoustic impedance, Z0 is the normalized boundary acoustic impedance, and ρ0 is the density of the fluid medium. The implementation of this method assumes that the acoustic impedance Z0 is a known quantity, and Z0 can be obtained according to the in-situ measurement method of acoustic impedance.

[0100] Step 2, obtain holographic measurement values.

[0101] like Figure 1 As shown, a group of sound pressure sensors are arranged in the near field of the sound source to form a sound pressure holographic measurement surface to measure the total sound pressure distribution contributed by the direct radiation sound of the sound source and the boundary reflected sound. According to the different fluid media where the sound source is located, the sound pressure sensor used in this method can be a microphone, a hydrophone or other types of sensors.

[0102] Step 3, select the optimal number of expansion terms of the half-space spherical wave basis function.

[0103] The coordinates of the measuring points on the holographic measuring surface are marked as M is the number of sound pressure measurement points. According to the method of selecting points at intervals, the sound pressure measurement points are divided into two groups. The coordinates of the first group of measurement points are marked as The coordinates of the second set of measurement points are marked as in, and Indicates rounding up and rounding down respectively. Taking the plane array composed of 6 rows and 6 columns of evenly distributed sound pressure measurement points as an example, the measurement points are grouped, as shown in the schematic diagram Figure 3 shown.

[0104] 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 sound pressure values ​​collected by the first group of measurement points on the holographic measurement surface can be expressed in the following matrix form:

[0105]

[0106] in, is a column vector of the total sound pressure measurements in the half space:

[0107]

[0108] Where superscript T is the vector transpose. {C(ω)} J×1 is a column vector of the coefficients of the half-space spherical wave basis functions:

[0109]

[0110] is the matrix composed of the expansion items of the half-space spherical wave basis function at each measuring point:

[0111]

[0112] Solving equation (11), we can get the coefficient column vector:

[0113]

[0114] Among them, the superscript It means to find the pseudo-inverse of the matrix.

[0115]

[0116] The superscript H is the conjugate transpose of the matrix.

[0117] When the coefficient column vector {C(ω)} J×1 After this is determined, the sound pressure of the second set of measurement points can be further reconstructed:

[0118]

[0119] And calculate the relative error between the reconstructed sound pressure value and the measured value of the second set of measurement points:

[0120]

[0121] Here, ||·||2 is the 2-norm of the vector.

[0122] From 1 to J max Traverse all J, use equations (11) to (18) to calculate the relative error ε, and determine the number of expansion items corresponding to the minimum value of ε as the optimal number of expansion items J opt .

[0123] Step 4: reconstruct the direct radiation sound field of the sound source.

[0124] Set the number of basis function expansion terms to J opt According to formula (2), the sound pressure value collected by the holographic measurement surface can be expressed in the following matrix form:

[0125]

[0126] Find the pseudo-inverse of equation (19) to solve the coefficient column vector

[0127]

[0128] Thus, the reconstructed value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:

[0129]

[0130] in, is the coordinate of the sound pressure reconstruction point, s=1,2,…,S, S is the number of reconstruction points; is the free space spherical wave basis function at the reconstruction point The matrix composed of the expanded items of :

[0131]

[0132] Example: The arrangement of the pulsating ball sound source and the plane boundary is as follows Figure 4 As shown, the boundary is located in the z = 0 plane, the acoustic impedance Z0 = 2 + 3i; the radius of the pulsating ball a = 0.05m, and the coordinates of its geometric center O1 The radial vibration velocity of the surface particle V0 = 0.01 m / s, the frequency f = 3000 Hz; the sound pressure holographic measurement is performed using a planar hydrophone array, the array surface is perpendicular to the x-axis, the line connecting its geometric center and the sphere center O1 is perpendicular to the array surface, and the distance from the sphere center 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, such as Figure 5As shown, 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 ρ0 = 1000kg / m 3 , the speed of sound is c = 1500m / s. To simulate the influence of hydrophone measurement error, Gaussian white noise with a signal-to-noise ratio of 30dB is added to the measurement value.

[0133] Figure 6 The figure shows the distribution of the dimensionless sound pressure amplitude |p / ρ0cV0| at each measuring point, including the total sound pressure value in the half space, the reconstructed value of the sound pressure directly radiated by the sound source, and the true value of the sound pressure directly radiated by the sound source. Figure 6 It is found that the reconstructed value of the sound pressure directly radiated by the sound source is consistent with the real value. The results show that the present invention can achieve the reconstruction of the sound field directly radiated by the sound source in the semi-open space.

[0134] The contents described in the embodiments of this specification are only one of the examples of the implementation of the invention concept. The protection scope of the present invention includes but is not limited to the specific forms and parameters described in the embodiments, and also includes equivalent technical means that can be thought of by those skilled in the art based on the invention concept.

Claims

1. A method for reconstructing a sound field directly radiated by a sound source in a semi-open space, characterized in that: The following steps are included: S1. Establish a mathematical model that describes the total sound pressure field contributed by the direct radiation of the sound source and the boundary reflected sound by the linear superposition of the half-space spherical wave basis functions; establish a global coordinate system with the projection O of the geometric center of the sound source O1 on the boundary as the origin, and the global coordinate of O1 is marked as h s is the distance from O1 to the boundary; the mirror point of O1 about the boundary is recorded as O2; by translating the global coordinate system, local coordinate systems are established with O1 and O2 as the origins respectively, and the coordinates of the field point x in the two local coordinate systems are recorded as x1≡(r1,θ1,φ1) and x2≡(r2,θ2,φ2), respectively, and the following relationship is satisfied between the three: x1=xh s And z ,x2=x+h s And z (1) Among them, e z is the z-direction unit vector; For a steady-state sound field, the total sound pressure p in the half space at the field point x is half (x; ω) can be expressed as a linear superposition of finite half-space spherical wave basis functions: Where 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 coefficient of the basis function expansion term; j is the expansion term ordinal number, J is the number of expansion terms; the half-space spherical wave basis function ψ jhalf The expression of (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 (oh) (3) Among them, ψ j (x|xh s e z ;ω) and ψ j (x|x+h s e z ; ω) are the jth free space spherical wave basis functions that represent the direct radiation sound of the sound source and its mirror image virtual source; in the spherical coordinate system, ψ j The expression is: in, is the first kind of spherical Hankel function, k = ω / c is the acoustic wave number, c is the speed of sound; is a spherical harmonic function; in equations (2) to (4), the integers n, l and j satisfy the relationship j = n 2 +n+l+1, where -n≤l≤n, 0≤n≤N, and N is the cutoff value of n; when calculating formula (3), the first two terms on the right side are substituted into the local coordinates x1 and x2 for calculation; ξ j (x|x+h s e z ; ω) describes the boundary sound, and its expression is: in, as well as In formulas (5) to (8), R p (θ2; ω), F(w) and w are the sound pressure reflection coefficient, boundary loss factor and numerical distance respectively; the local coordinates r1 and r2 are the distances from the geometric center of the sound source and the geometric center of the mirror virtual source to the field point respectively; θ2 is the incident angle of the sound wave, which is the angle between the line connecting the field point and the geometric center of the virtual source and the positive direction of the z-axis; Complex angle μ p for: Where β is the normalized boundary acoustic admittance, Wherein, Z is the boundary acoustic impedance, Z0 is the normalized boundary acoustic impedance, and ρ0 is the density of the fluid medium; the implementation of this method assumes that the acoustic impedance Z0 is a known quantity, and Z0 can be obtained according to the in-situ measurement method of acoustic impedance; S2. Arrange a holographic measurement surface in the near field of the sound source to perform holographic sound pressure measurement; Arrange a group of sound pressure sensors in the near field of the sound source to form a sound pressure holographic measurement surface to measure the total sound pressure distribution contributed by the direct radiation sound of the sound source and the boundary reflection sound; S3. 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 number of basis function expansion terms based on the principle of minimizing the sound pressure reconstruction error; The coordinates of the measuring points on the holographic measuring surface are marked as m=1,2,...,M, M is the number of sound pressure measurement points; the sound pressure measurement points are divided into two groups according to the method of selecting points at intervals; the coordinates of the first group of measurement points are marked as m′=1,2,…,M′, The coordinates of the second set of measurement points are marked as m″=1,2,…,M″, in, and Respectively represent rounding up and rounding down; 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 sound pressure values ​​collected by the first group of measurement points on the holographic measurement surface can be expressed in the following matrix form: in, is a column vector of the total sound pressure measurements in the half space: Where superscript T is the vector transpose; {C(ω)} J×1 is a column vector of the coefficients of the half-space spherical wave basis functions: is the matrix composed of the expansion items of the half-space spherical wave basis function at each measuring point: Solving equation (11), we can get the coefficient column vector: Among them, the superscript It means to find the pseudo-inverse of the matrix. Where, the superscript H is the conjugate transpose of the matrix; When the coefficient column vector {C(ω)} J×1 After this is determined, the sound pressure of the second set of measurement points can be further reconstructed: And calculate the relative error between the reconstructed sound pressure value and the measured value of the second set of measurement points: Among them, ||·||2 is the 2-norm of the vector; From 1 to J max Traverse all J, use equations (11) to (18) to calculate the relative error ε, and determine the number of expansion items corresponding to the minimum value of ε as the optimal number of expansion items J opt ; S4. Solve the half-space spherical wave basis function coefficients under the condition of the optimal number of expansion terms, obtain the free-space spherical wave basis function coefficients that describe 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 number of basis function expansion terms to J opt According to formula (2), the sound pressure value collected by the holographic measurement surface can be expressed in the following matrix form: Find the pseudo-inverse of equation (19) to solve the coefficient column vector Thus, 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 coordinate of the sound pressure reconstruction point, s=1,2,…,S, S is the number of reconstruction points; is the free space spherical wave basis function at the reconstruction point The matrix composed of the expanded items of :

2. The method for reconstructing the sound field directly radiated by a sound source in a semi-open space according to claim 1, characterized in that: The sound pressure sensor described in step S2 uses a microphone or a hydrophone according to the different fluid media where the sound source is located.

Images