CN114545331A - Method for reconstructing sound source direct radiation sound field in semi-open space - Google Patents

Method for reconstructing sound source direct radiation sound field in semi-open space Download PDF

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CN114545331A
CN114545331A CN202210145733.9A CN202210145733A CN114545331A CN 114545331 A CN114545331 A CN 114545331A CN 202210145733 A CN202210145733 A CN 202210145733A CN 114545331 A CN114545331 A CN 114545331A
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sound
sound pressure
sound source
basis function
field
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周达仁
卢奂采
程相乐
汝泽宇
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Zhejiang University of Technology ZJUT
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
    • G01S5/18—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using ultrasonic, sonic, or infrasonic waves
    • G01S5/20—Position of source determined by a plurality of spaced direction-finders

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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 source direct radiation sound field in semi-open space
The technical field is as follows:
the invention relates to a method for measuring the total acoustic quantity distribution superposed by the direct radiation of a sound source and the reflection of a plane boundary by using a sensor array in a semi-open space containing the plane reflection boundary to obtain the direct radiation acoustic quantity distribution of the sound source. Belongs to the technical field of sound source identification and positioning, sound field imaging, near-field acoustic holography, sound wave separation and noise control.
Background art:
the measurement and evaluation of the radiated acoustic quantity of a sound source generally needs to be performed in a fully muffled or semi-muffled standard environment. Such an ideal acoustic measurement environment is generally not available for large-size, complex equipment. For example, acoustic radiation testing of large ships can often only be performed in docks or piers where reflective boundaries exist. In this case, the acoustic measurement values contaminated by the boundary reflections can neither faithfully reflect the radiation level of the acoustic source at the survey point nor be combined with the near-field acoustic holography method established in free space to reconstruct the external radiation acoustic quantity distribution of the acoustic source in three-dimensional space.
The half-space sound field reconstruction method based on the Fourier acoustic method is only suitable for the situation that the holographic measurement surface is in a regular geometric shape (a plane, a cylindrical surface or a spherical surface). The reconstruction method of the half-space sound field 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 aiming at the surface of a sound source, so that the requirements of huge measuring point number and inversion calculation amount are introduced.
The invention content is as follows:
the present invention overcomes the above-mentioned shortcomings of the prior art, and provides a method for reconstructing a 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 total sound pressure field contributed by the direct radiation of a sound source and the reflected sound of a plane boundary is expressed as linear superposition of a group of semispherical wave basis functions; arranging a sound pressure holographic measuring surface in a near field of a sound source, measuring a group of total half-space sound pressure values, and matching the total half-space sound pressure values with linear superposition of a half-space spherical wave basis function; solving the semi-space spherical wave basis function coefficient through pseudo-inverse calculation; and reconstructing a direct radiation sound pressure value of the sound source on the holographic measurement surface or any field point by using a group of free space spherical wave basis functions and the obtained basis function coefficients, so as to realize the reconstruction of the sound field.
The planar boundary of the semi-open space is a local Reaction (localization Reaction) boundary whose specific acoustic impedance is a constant and known quantity independent of the incident angle of the acoustic wave and the wavefront of the incident acoustic wave.
The basis for constructing the semi-space spherical wave basis function is an analytic solution of a sound pressure field excited by a multipole sound source in a semi-open space, which is obtained by solving a nonhomogeneous Helmholtz equation and boundary conditions by using a Steepest decision Method.
The expression of the analytic solution comprises three superposed terms which respectively express the direct radiation sound of the multipole, the mirror image multipole radiation sound about the boundary and the boundary sound.
A group of semispherical wave basis functions for expressing the total sound pressure field and a group of free space spherical wave basis functions for expressing the sound source direct radiation sound pressure field share the same group of basis function coefficients.
The invention is suitable for sound sources with any geometric shapes, and the sound pressure holographic measuring surface can be an irregular single-layer measuring surface.
The method for reconstructing the sound source direct radiation sound field in the semi-open space comprises the following contents:
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;
about the geometric center O of the sound source1The projection O on the boundary is used as the origin to establish a global coordinate system O1Is noted as xO1＝(0,0,hs)，hsIs O1Distance to the boundary; o is1The mirror point about the boundary is marked as O2. By translating the global coordinate system by O1And O2Respectively establishing local coordinate systems for the origin, and respectively recording the coordinates of the field point x in the two local coordinate systems as x1≡(r1,θ1,φ1) And x2≡(r2,θ2,φ2) The three satisfy the following relation:
x1＝x-hsez,x2＝x+hsez (1)
wherein e iszIs a z-direction unit vector.
For a steady state sound field, the total sound pressure p of the half space at the field point xhalf(x; ω) can be expressed as a linear superposition of finite term semispherical basis functions:
Figure BDA0003508829930000021
wherein, ω is the acoustic angular frequency; psijhalf(x; ω) is a semispherical wave basis function; c. Cj(ω) is the coefficient of the basis function expansion term; j is the expansion term ordinal number and J is the expansion term item number. Semi-space spherical wave basis function psijhalfThe expression of (x; ω) is:
ψjhalf(x；ω)＝ψj(x|x-hsez；ω)+ψj(x|x+hsez；ω)+ξj(x|x+hsez；ω) (3)
wherein psij(x|x-hsez(ii) a ω) and ψj(x|x+hsez(ii) a ω) are the j-th free-space spherical wave basis functions of directly radiated sound respectively representing the sound source and the mirror image virtual source thereof. In a spherical coordinate system, #jThe expression of (a) is:
Figure BDA0003508829930000031
wherein,
Figure BDA0003508829930000032
the method is characterized in that the method is a first ball Hankel function, k is omega/c is the wave number of sound waves, and c is the sound velocity;
Figure BDA0003508829930000033
is a spherical harmonic function. In the formulae (2) to (4), the integers n, l and j satisfy the relationship that j is n2+ N + l +1, wherein-N is not less than l and not more than N, N is not less than 0 and not more than N, and N is a truncation value of N. When calculating the formula (3), the first two terms on the right side are substituted into the local coordinate x1And x2And (6) performing calculation. Xij(x|x+hsez(ii) a ω) represents the boundary sound, which is expressed as:
Figure BDA0003508829930000034
wherein,
Figure BDA0003508829930000035
Figure BDA0003508829930000036
and
Figure BDA0003508829930000037
in the formulae (5) to (8), Rp(θ2(ii) a Omega), F (w) and w are respectively sound pressure reflection coefficient, boundary loss factor and numerical spacing; local coordinate r1And r2Respectively the distances from the geometric center of the sound source and the geometric center of the mirror image virtual source to the field point; theta2The sound wave incident angle is the included angle between the connecting line of the field point and the virtual source geometric center and the positive direction of the z-axis;
Figure BDA0003508829930000038
complex angle mupComprises the following steps:
Figure BDA0003508829930000039
wherein beta is a normalized boundary acoustic admittance,
Figure BDA0003508829930000041
wherein Z is the boundary specific acoustic impedance, Z0For normalized boundary specific acoustic impedance, p0Is the fluid medium density. The implementation of the method assumes specific acoustic impedance Z0Is a known amount, Z0Can be obtained according to an in-situ measurement method of acoustic impedance.
2. Arranging a holographic measuring surface at the near field of a sound source to perform sound pressure holographic measurement; a group of sound pressure sensors are arranged in a near field of a sound source to form a sound pressure holographic measuring surface, and total sound pressure distribution contributed by direct radiation sound and boundary reflection sound of the sound source is measured, wherein the sound pressure sensors adopted by the method can be microphones, hydrophones or other types of sensors according to different fluid media of the sound source.
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; the coordinates of the measuring points on the holographic measuring surface are recorded as
Figure BDA0003508829930000042
And M is the number of sound pressure measuring points. According to the mode of taking points at intervals, the sound pressure measuring points are divided into two groups. The first set of measured point coordinates is recorded as
Figure BDA0003508829930000043
Figure BDA0003508829930000044
The coordinates of the second set of measuring points are recorded as
Figure BDA0003508829930000045
Figure BDA0003508829930000046
Wherein,
Figure BDA0003508829930000047
and
Figure BDA0003508829930000048
respectively representing a round-up and a round-down.
Setting the upper limit of the possible values of the expansion term number J of the basis function as JmaxI.e. J is 1. ltoreq. J.ltoreq.Jmax. To this endAny J within the range, according to equation (2), the sound pressure value collected by the first set of measuring points of the holographic measuring surface can be expressed in the form of a matrix as follows:
Figure BDA0003508829930000049
wherein,
Figure BDA00035088299300000410
column vector consisting of half-space total sound pressure measurements:
Figure BDA00035088299300000411
wherein, the superscript T is the vector transposition. { C (ω) }J×1Column vectors consisting of semispherical wave basis function coefficients:
Figure BDA00035088299300000412
Figure BDA0003508829930000051
the matrix is composed of expansion terms of the semispherical wave basis function at each measuring point:
Figure BDA0003508829930000052
solving equation (11) can obtain a coefficient column vector:
Figure BDA0003508829930000053
wherein, the upper label
Figure BDA0003508829930000054
It is shown that the pseudo-inverse of the matrix,
Figure BDA0003508829930000055
wherein, the superscript H is the conjugate transpose of the matrix.
When coefficient column vector { C (ω) }J×1After the determination, the sound pressures of the second set of measurement points can be further reconstructed:
Figure BDA0003508829930000056
and calculating the relative error between the sound pressure reconstruction value and the measured value of the second group of measuring points:
Figure BDA0003508829930000057
wherein | · | purple sweet2Is the 2-norm of the vector.
From 1 to JmaxTraversing all J, calculating a relative error epsilon by using the formulas (11) to (18), and determining the expansion term number corresponding to the minimum value of epsilon as an optimal expansion term number Jopt。
4. Solving the spherical wave basis function coefficient of the semispace under the condition of the optimal expansion term number, obtaining 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; setting the number of expansion terms of the basis function to JoptAccording to equation (2), the sound pressure value collected by the hologram measuring plane can be expressed in a matrix form as follows:
Figure BDA0003508829930000061
pseudo-inverting equation (19) to solve the coefficient column vector
Figure BDA0003508829930000062
Figure BDA0003508829930000063
Thus, the reconstruction value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:
Figure BDA0003508829930000064
wherein,
Figure BDA0003508829930000065
the sound pressure reconstruction point coordinates are shown, S is 1,2, …, and S are the number of reconstruction points;
Figure BDA0003508829930000066
for free space spherical wave basis function at reconstruction point
Figure BDA0003508829930000067
The matrix of expansion terms of (c):
Figure BDA0003508829930000068
on the basis of a mathematical model for expressing a sound source radiation sound field based on free space spherical wave basis function superposition, a semi-space spherical wave basis function meeting Helmholtz equation and boundary conditions is constructed by taking boundary acoustic impedance as a parameter, and the mathematical model for expressing a semi-space total sound field based on semi-space spherical wave basis function superposition is established. The method comprises the steps of performing holographic measurement on a sound field of a structural sound source with any geometric shape in a semi-open space containing a plane boundary, and obtaining a basis function coefficient of a sound source direct radiation sound field by reversely solving the basis function coefficient to realize reconstruction of the sound source direct radiation sound field.
The invention has the beneficial effects that:
1. the mathematical model based on the superposition of the spherical wave basis functions in the semispace can express a semi-open space sound field with a limited impedance boundary, and provides a mathematical basis for implementing measurement and evaluation of radiation of a structural sound source, particularly a large-size structural sound source, under the condition that an ideal acoustic measurement condition is not met.
2. The half-space total sound pressure distribution measured by the array is substituted into a mathematical model of the sound field to be solved, so that the purpose that a near-field acoustic holography method can be realized, namely the sound field imaging and the identification and positioning of a structural sound source in a half-open space can be realized.
3. The invention is suitable for structural sound sources with any geometric shapes, and the sound pressure holographic measuring surface can be an irregular single-layer holographic measuring surface.
Description of the drawings:
FIG. 1 is a schematic diagram of a semi-open space sound field formed by a sound source and a planar boundary;
FIG. 2 is a geometric relationship between the geometric center, field point and planar boundary of a sound source and its mirrored virtual source;
FIG. 3 is a diagram illustrating grouping of measurement points when the optimal number of expansion terms is selected;
FIG. 4 is a schematic diagram of a simulated sound field consisting of a pulsating spherical sound source, a planar boundary and a hydrophone array;
FIG. 5 is a schematic illustration of the distribution and numbering rules of hydrophones on an array;
FIG. 6 is a distribution curve of total semi-space sound pressure values, sound source direct radiation sound pressure reconstruction values and sound source direct radiation sound pressure true values on holographic measuring points.
Detailed Description
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 sound source direct radiation sound field in the semi-open space is carried out according to the following steps:
step 1, establishing a sound field mathematical model based on half-space spherical wave basis function superposition.
As shown in fig. 1 and 2, with the geometric center O of the sound source1The projection O on the boundary is used as the origin to establish a global coordinate system O1Is noted as a global coordinate
Figure BDA0003508829930000071
hsIs O1Distance to the boundary; o is1The mirror point about the boundary is marked as O2. By translating the global coordinate system with O1And O2Respectively establishing local coordinate systems for the origin, and respectively recording the coordinates of the field point x in the two local coordinate systems as x1≡(r1,θ1,φ1) And x2≡(r2,θ2,φ2) The three satisfy the following relation:
x1＝x-hsez,x2＝x+hsez (1)
wherein e iszIs a z-direction unit vector.
For a steady state sound field, the total sound pressure p of the half space at the field point xhalf(x; ω) can be expressed as a linear superposition of finite term semispherical basis functions:
Figure BDA0003508829930000081
wherein, ω is the acoustic angular frequency; psijhalf(x; ω) is a semispherical wave basis function; c. Cj(ω) is the coefficient of the basis function expansion term; j is the expansion term ordinal number and J is the expansion term item number. Semi-space spherical wave basis function psijhalfThe expression of (x; ω) is:
ψjhalf(x；ω)＝ψj(x|x-hsez；ω)+ψj(x|x+hsez；ω)+ξj(x|x+hsez；ω) (3)
wherein psij(x|x-hsez(ii) a ω) and ψj(x|x+hsez(ii) a ω) are the j-th free-space spherical wave basis functions of directly radiated sound respectively representing the sound source and the mirror image virtual source thereof. In a spherical coordinate system, #jThe expression of (a) is:
Figure BDA0003508829930000082
wherein,
Figure BDA0003508829930000083
the method is characterized in that the method is a first ball Hankel function, k is omega/c is the wave number of sound waves, and c is the sound velocity;
Figure BDA0003508829930000084
is a spherical harmonic function. In the formulae (2) to (4), the integers n, l and j satisfy the relationship of j-n2+ N + l +1, where-N is not less than l and not more than N, 0 is not less than N and not more than N, and N is the cut-off value of N. When calculating the formula (3), the first two terms on the right side are substituted into the local coordinate x1And x2And (6) performing calculation. Xij(x|x+hsez(ii) a ω) represents the boundary sound, which is expressed as:
Figure BDA0003508829930000085
wherein,
Figure BDA0003508829930000086
Figure BDA0003508829930000087
and
Figure BDA0003508829930000088
in the formulae (5) to (8), Rp(θ2(ii) a Omega), F (w) and w are the sound pressure reflection coefficient, the boundary loss factor and the numerical distance respectively; local coordinate r1And r2Respectively the distances from the geometric center of the sound source and the geometric center of the mirror image virtual source to the field point; theta2The sound wave incident angle is the included 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 FIG. 2;
Figure BDA0003508829930000091
complex angle mupComprises the following steps:
Figure BDA0003508829930000092
wherein beta is a normalized boundary acoustic admittance,
Figure BDA0003508829930000093
wherein Z is the boundary specific acoustic impedance, Z0For normalized boundary specific acoustic impedance, p0Is the fluid medium density. The implementation of the method assumes specific acoustic impedance Z0Is a known amount, Z0Can be obtained according to an in-situ measurement method of acoustic impedance.
And 2, acquiring a holographic measurement value.
As shown in fig. 1, a group of sound pressure sensors are arranged in a near field of a sound source to form a sound pressure holographic measurement surface, and a total sound pressure distribution contributed by the directly radiated sound and the boundary reflected sound of the sound source together is measured, wherein the sound pressure sensors adopted in the method can be microphones, hydrophones or other types of sensors according to different fluid media in which the sound source is located.
And 3, selecting the optimal expansion term number of the semi-space spherical wave basis function.
The coordinates of the measuring points on the holographic measuring surface are recorded as
Figure BDA0003508829930000094
And M is the number of sound pressure measuring points. According to the mode of taking points at intervals, the sound pressure measuring points are divided into two groups. The first set of measured point coordinates is recorded as
Figure BDA0003508829930000095
Figure BDA0003508829930000096
Figure BDA0003508829930000097
Second group of measuring points seatsMarking as
Figure BDA0003508829930000098
Figure BDA0003508829930000099
Wherein,
Figure BDA00035088299300000910
and
Figure BDA00035088299300000911
respectively representing a round-up and a round-down. Taking a planar array composed of 6 rows and 6 columns of sound pressure measurement points which are uniformly distributed as an example, the measurement points are grouped, and a schematic diagram of the measurement points is shown in fig. 3.
Setting the upper limit of possible values of the expansion term J of the basis function as JmaxI.e. J is 1. ltoreq. J.ltoreq.Jmax. For any J in this range, the sound pressure values collected by the first set of measurement points of the holographic measurement surface can be expressed in the form of a matrix according to equation (2):
Figure BDA0003508829930000101
wherein,
Figure BDA0003508829930000102
column vector consisting of half-space total sound pressure measurements:
Figure BDA0003508829930000103
wherein, the superscript T is the vector transposition. { C (ω) }J×1Column vectors consisting of semispherical wave basis function coefficients:
Figure BDA0003508829930000104
Figure BDA0003508829930000105
the method is a matrix formed by expansion terms of the semispherical wave basis function at each measuring point:
Figure BDA0003508829930000106
solving equation (11) can obtain a coefficient column vector:
Figure BDA0003508829930000107
wherein, the upper label
Figure BDA0003508829930000108
It is shown that the pseudo-inverse of the matrix,
Figure BDA0003508829930000109
wherein, the superscript H is the conjugate transpose of the matrix.
When coefficient column vector { C (ω) }J×1After the determination, the sound pressures of the second set of measurement points can be further reconstructed:
Figure BDA00035088299300001010
and calculating the relative error between the sound pressure reconstruction value and the measured value of the second group of measuring points:
Figure BDA00035088299300001011
wherein | · | purple sweet2Is the 2-norm of the vector.
From 1 to JmaxTraversing all J, calculating a relative error epsilon by using the formulas (11) to (18), and determining the expansion term number corresponding to the minimum value of epsilon as an optimal expansion term number Jopt。
And 4, reconstructing a direct radiation sound field of the sound source.
Setting the number of expansion terms of the basis function to JoptAccording to equation (2), the sound pressure value collected by the hologram measuring plane can be expressed in a matrix form as follows:
Figure BDA0003508829930000111
solving the coefficient column vector by pseudo-inverting equation (19)
Figure BDA0003508829930000112
Figure BDA0003508829930000113
Thus, the reconstruction value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:
Figure BDA0003508829930000114
wherein,
Figure BDA0003508829930000115
the sound pressure reconstruction point coordinates are shown, S is 1,2, …, and S are the number of reconstruction points;
Figure BDA0003508829930000116
for free space spherical wave basis function at reconstruction point
Figure BDA0003508829930000117
The matrix of expansion terms of (c):
Figure BDA0003508829930000118
example (b): the placement of the source of the pulsating spherical sound and the planar boundary is shown in FIG. 4, where the boundary lies in the plane where Z is 0 and the specific acoustic impedance Z is02+3 i; radius of the pulsating ball a ═0.05m, its geometric center O1Of (2)
Figure BDA0003508829930000121
Surface particle radial vibration velocity V00.01m/s, frequency f 3000 Hz; the sound pressure holographic measurement is carried out by using a plane hydrophone array, the array plane is vertical to an x-axis, and the geometric center of the array plane is vertical to the spherical center O1Is perpendicular to the array plane and is d away from the sphere centers0.15 m; the array aperture is 0.15m multiplied by 0.15m, and the array aperture is composed of 6 multiplied by 6 measuring points, and the distance between adjacent measuring points is 0.03 m. For convenience of description and analysis of the calculation results, the 36 measuring points are numbered in sequence, and as shown in FIG. 5, the coordinates of the measuring point No. 1 are (0.150m, -0.075m and 0.375m), and the coordinates of the measuring point No. 36 are (0.150m,0.075m and 0.225 m). Density of aqueous medium rho0＝1000kg/m3The sound speed is c 1500 m/s. Simulating the influence of measurement errors of the hydrophone, and adding white Gaussian noise with the signal-to-noise ratio of 30dB to the measured value.
FIG. 6 shows the dimensionless sound pressure amplitude | p/ρ at each measurement point0cV0And the distribution of the I comprises a total semi-space sound pressure value, a reconstruction value of the sound source direct radiation sound pressure and a true value of the sound source direct radiation sound pressure. Observing the figure 6, the reconstructed value of the sound source direct radiation sound pressure can be well matched with the real value. The result shows that the method 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 the present specification is only one of the cases of the implementation forms of the inventive concept. The scope of the present invention includes, but is not limited to, the specific forms and parameters set forth in the examples, as well as equivalent technical means that may occur to those skilled in the art upon consideration of the present inventive concept.

Claims (2)

1. The method for reconstructing the sound source direct radiation sound field in the semi-open space is characterized in that: comprises the following steps:
s1, establishing a mathematical model for expressing the total sound pressure field of the direct radiation and boundary reflected sound contribution of a sound source by linear superposition of a semi-space spherical wave basis function; about the geometric center O of the sound source1Projection O on the boundary establishes global for originCoordinate system, O1Is noted as a global coordinate
Figure FDA0003508829920000015
hsIs O1Distance to the boundary; o is1The mirror point about the boundary is marked as O2(ii) a By translating the global coordinate system by O1And O2Respectively establishing local coordinate systems for the origin, and respectively recording the coordinates of the field point x in the two local coordinate systems as x1≡(r1,θ1,φ1) And x2≡(r2,θ2,φ2) The three satisfy the following relation:
x1＝x-hsez,x2＝x+hsez (1)
wherein e iszIs a z-direction unit vector;
for a steady state sound field, the total sound pressure p in half space at field point xhalf(x; ω) can be expressed as a linear superposition of finite term semispherical basis functions:
Figure FDA0003508829920000011
wherein, ω is the acoustic angular frequency; psijhalf(x; ω) is a semispherical wave basis function; c. Cj(ω) is the coefficient of the basis function expansion term; j is the expansion term ordinal number, and J is the expansion term item number; semi-space spherical wave basis function psijhalfThe expression of (x; ω) is:
ψjhalf(x；ω)＝ψj(x|x-hsez；ω)+ψj(x|x+hsez；ω)+ξj(x|x+hsez；ω) (3)
wherein psij(x|x-hsez(ii) a ω) and ψj(x|x+hsez(ii) a Omega) are respectively the jth free space spherical wave basis function for expressing the direct radiation sound of the sound source and the mirror image virtual source thereof; in a spherical coordinate system, #jThe expression of (a) is:
Figure FDA0003508829920000012
wherein,
Figure FDA0003508829920000013
the method is characterized in that the method is a first ball Hankel function, k is omega/c is the wave number of sound waves, and c is the sound velocity;
Figure FDA0003508829920000014
is a spherical harmonic function; in the formulae (2) to (4), the integers n, l and j satisfy the relationship that j is n2+ N + l +1, wherein l is more than or equal to-N and less than or equal to N, N is more than or equal to 0 and less than or equal to N, and N is a cutoff value of N; when calculating the formula (3), the first two terms on the right side are substituted into the local coordinate x1And x2Calculating; xij(x|x+hsez(ii) a ω) represents the boundary sound, which is expressed as:
Figure FDA0003508829920000021
wherein,
Figure FDA0003508829920000022
Figure FDA0003508829920000023
and
Figure FDA0003508829920000024
in the formulae (5) to (8), Rp(θ2(ii) a ω), F (w) and w are the sound pressure reflection coefficient and the boundary loss factor, respectivelyDaughter (boundry Loss Factor) and Numerical Distance (Numerical Distance); local coordinate r1And r2Respectively the distances from the geometric center of the sound source and the geometric center of the mirror image virtual source to the field point; theta2The sound wave incident angle is the included angle between the connecting line of the field point and the virtual source geometric center and the positive direction of the z-axis;
Figure FDA0003508829920000025
complex angle mupComprises the following steps:
Figure FDA0003508829920000026
wherein beta is a normalized boundary acoustic admittance,
Figure FDA0003508829920000027
wherein Z is the boundary specific acoustic impedance, Z0For normalized boundary specific acoustic impedance, p0Is the fluid medium density; the implementation of the method assumes specific acoustic impedance Z0Is a known amount, Z0The acoustic impedance can be obtained according to an in-situ measurement method of acoustic impedance;
s2, arranging a holographic measuring surface in a near field of a sound source to perform sound pressure holographic measurement;
arranging a group of sound pressure sensors in a near field of a sound source to form a sound pressure holographic measuring surface, and measuring total sound pressure distribution contributed by direct radiation sound and boundary reflection sound of the sound source together;
s3, taking the holographic measured values of part of the measuring points as input, reconstructing the sound pressure values of the rest measuring points, and determining the optimal basis function expansion term number by taking the sound pressure reconstruction error as the minimum;
the coordinates of the measuring points on the holographic measuring surface are recorded as
Figure FDA0003508829920000031
M is the number of sound pressure measuring points; dividing the sound pressure measuring points into two groups according to a point separating and taking mode; the first set of measured point coordinates is recorded as
Figure FDA0003508829920000032
Figure FDA0003508829920000033
The coordinates of the second set of measuring points are recorded as
Figure FDA0003508829920000034
Figure FDA0003508829920000035
Wherein,
Figure FDA0003508829920000036
and
Figure FDA0003508829920000037
respectively representing rounding-up and rounding-down;
setting the upper limit of possible values of the expansion term J of the basis function as JmaxI.e. J is 1. ltoreq. J.ltoreq.Jmax(ii) a For any J in this range, the sound pressure values collected by the first set of measurement points of the holographic measurement surface can be expressed in the form of a matrix according to equation (2):
Figure FDA0003508829920000038
wherein,
Figure FDA0003508829920000039
column vector consisting of half-space total sound pressure measurements:
Figure FDA00035088299200000310
wherein, superscript T is vector transposition; { C (ω) }J×1Column vectors consisting of semispherical wave basis function coefficients:
Figure FDA00035088299200000311
Figure FDA00035088299200000312
the matrix is composed of expansion terms of the semispherical wave basis function at each measuring point:
Figure FDA00035088299200000313
solving equation (11) can obtain a coefficient column vector:
Figure FDA00035088299200000314
wherein, the upper label
Figure FDA0003508829920000041
It is shown that the pseudo-inverse of the matrix,
Figure FDA0003508829920000042
wherein, the superscript H is the conjugate transpose of the matrix;
when coefficient column vector { C (ω) }J×1After the determination, the sound pressures of the second set of measurement points can be further reconstructed:
Figure FDA0003508829920000043
and calculating the relative error between the sound pressure reconstruction value and the measured value of the second group of measuring points:
Figure FDA0003508829920000044
wherein | · | purple sweet2Is the 2-norm of the vector;
from 1 to JmaxTraversing all J, calculating a relative error epsilon by using the formulas (11) to (18), and determining the expansion term number corresponding to the minimum value of epsilon as an optimal expansion term number Jopt；
S4, solving a half-space spherical wave basis function coefficient under the condition of the optimal expansion term number, obtaining a free-space spherical wave basis function coefficient for expressing a sound field directly radiated by the sound source, and realizing the reconstruction of the sound field directly radiated by the sound source;
setting the number of expansion terms of the basis function to JoptAccording to equation (2), the sound pressure value collected by the hologram measuring plane can be expressed in a matrix form as follows:
Figure FDA0003508829920000045
pseudo-inverting equation (19) to solve the coefficient column vector
Figure FDA0003508829920000046
Figure FDA0003508829920000047
Thus, the reconstruction value of the sound pressure directly radiated by the sound source on the sound pressure reconstruction surface can be obtained:
Figure FDA0003508829920000048
wherein,
Figure FDA0003508829920000049
the sound pressure reconstruction point coordinates are shown, S is 1,2, …, and S are the number of reconstruction points;
Figure FDA00035088299200000410
for free space spherical wave basis function at reconstruction point
Figure FDA00035088299200000411
The matrix of expansion terms of (c):
Figure FDA0003508829920000051
2. the method of reconstructing a sound source direct radiation sound field in a semi-open space as claimed in claim 1, wherein: the sound pressure sensor in step S2 uses a microphone, a hydrophone or other types of sensors according to the fluid medium in which the sound source is located.
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