CN114935399A - Method for in-situ measurement of half-space boundary surface acoustic impedance by single-layer microphone array - Google Patents
Method for in-situ measurement of half-space boundary surface acoustic impedance by single-layer microphone array Download PDFInfo
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Abstract
The method for measuring the acoustic impedance of the boundary surface of the half space in situ by the single-layer microphone array comprises the following steps: 1. arranging a sound source and a microphone array on one side of a half space of a boundary, and carrying out sound pressure holographic measurement; 2. establishing a mathematical model of a half-space sound field on one side of a boundary; 3. determining the optimal expansion term number of the basis function; 4. establishing a mathematical relation between the holographically measured sound pressure and the boundary surface sound pressure, and reconstructing boundary surface sound pressure distribution; 5. establishing a mathematical relationship between holographic measurement sound pressure and the normal vibration velocity of boundary surface medium particles, and reconstructing the distribution of the normal vibration velocity of the boundary particles; 6. and calculating the acoustic impedance rate of the boundary surface by using the reconstructed sound pressure and the normal vibration velocity of the mass point. The method is based on near-field acoustic holography and microphone array in-situ acquisition of physical boundary near-field sound pressure information, and realizes in-situ measurement and calculation of boundary surface acoustic impedance, so that the measurement of boundary material sample acoustic impedance is implemented without depending on standard impedance measurement equipment and measurement environment.
Description
The technical field is as follows:
the invention relates to a method for acquiring holographic sound pressure of a semi-space boundary near field in situ by using a single-layer microphone array, reconstructing and calculating boundary surface acoustic quantity based on near field acoustic holography and a virtual source principle, and further obtaining boundary surface acoustic impedance, belonging to the technical field of acoustic in-situ measurement, holographic measurement, sound field imaging and near field acoustic holography.
The background art comprises the following steps:
accurate acquisition of acoustic characteristic parameters such as surface acoustic impedance of a physical boundary of a sound field is a precondition for correct control and utilization of the sound field and a sound source nearby the sound field. The impedance tube method and the reverberation chamber method are standard methods for measuring the acoustic impedance of small-size and large-size material samples, a special acoustic measurement device and a standard acoustic environment are required in the implementation process, and the measurement result cannot accurately reflect the acoustic characteristics and acoustic parameters of the measured material in the actual application environment.
The method is an effective method for measuring the acoustic impedance of the boundary surface in situ in the actual engineering environment.
The boundary surface acoustic impedance measurement method based on the Fourier acoustic method, the statistical optimal near-field acoustic holography and the equivalent source method needs to acquire holographic sound pressure by using a double-layer plane microphone array or a spherical microphone array; the measurement method based on the inverse boundary element method requires two microphone arrays respectively arranged to surround the sound source and to surround the boundary surface. In other words, these methods either require more microphone arrays, such as a dual layer microphone array; or require a microphone array of a special geometry, such as a spherical microphone array.
The invention content is as follows:
the invention overcomes the limitation of the boundary surface acoustic impedance in-situ measurement method and provides a method for in-situ measurement of the boundary surface acoustic impedance only by a single-layer microphone array.
In the invention, at the center of a sound source and a mirror image point of the sound source relative to a boundary surface, a group of spherical wave basis functions are superposed to respectively express a sound source radiation and a boundary reflection sound field, a mathematical model of a half-space sound field of a boundary near field is established, a transfer function between the sound pressure of a half-space field point and the sound pressure of the boundary surface and the normal vibration velocity of a boundary fluid medium particle is constructed, and the sound pressure distribution acquired by a single-layer microphone array is utilized to reconstruct the sound pressure of the boundary surface and the normal vibration velocity distribution.
According to the method, the acoustic impedance of the boundary surface is calculated according to the reconstructed sound pressure and normal vibration velocity distribution of the boundary surface, and other parameters representing the acoustic characteristics of the boundary, such as a reflection coefficient, a sound absorption coefficient and the like, can be further calculated.
The boundary surface inspected by the invention is a plane, and the acoustic angle of the sound wave is changed by adjusting the direction of the sound source, so that the acoustic impedance of the boundary surface under the condition of different incident angles is calculated.
The invention discloses a method for measuring the acoustic impedance of a semi-space boundary surface in situ by a single-layer microphone array, which comprises the following steps:
s1, arranging a sound source and a single-layer microphone array on one side of a half space of a boundary, and carrying out sound pressure holographic measurement;
arranging the sound sources in half-space, with the center of the sound source being denoted as O 1 (ii) a Arranging a single-layer planar microphone array parallel to the boundary surface at the near field of the boundary surface, the distance between the array surface and the boundary being h a (ii) a Recording the projection of the center of the array surface on the boundary as a point O, and recording the point O and the center O of the sound source 1 Is taken as the angle between the connecting line of (A) and the boundary normal at the point O as the sound wave incident angle theta inc (ii) a After the arrangement is finished, starting a sound source to emit sound waves, and collecting holographic sound pressure by the single-layer microphone array; after the collection is finished, the sound source is turned off, and the position of the sound source is adjusted to change the sound wave incidence angle theta inc (ii) a The holographic sound pressure collection is repeatedly carried out to obtain different incidence angles theta inc Holographic sound pressure under the condition;
when the fluid medium is an aqueous medium, the sound source and the microphone array are replaced by a sound source and a hydrophone array in the aqueous medium;
s2, establishing a mathematical model of a half-space sound field on one side of the boundary;
establishing a global coordinate system by taking the point O as a coordinate origin and the plane where the boundary is located as an x-y coordinate plane; mixing O with 1 The mirror point about the boundary is marked as O 2 Respectively with O 1 And O 2 Establishing a local coordinate system for the coordinate origin, and respectively recording the coordinates of the half-space field point x in the two local coordinate systems as x 1 ≡(r 1 ,θ 1 ,φ 1 ) And x 2 ≡(r 2 ,θ 2 ,φ 2 ) (ii) a For a steady sound field, in two local coordinate systems, sound pressure responses of sound source radiation and boundary reflection at the field point are expressed by linear superposition of a set of spherical wave functions, and the expressions are respectively:
and
wherein psi 1,j (x 1 (ii) a ω) and ψ 2,j (x 2 (ii) a ω) are each O 1 And O 2 A spherical wave basis function as the origin of coordinates; c. C 1,j (omega) and c 2,j (ω) are two sets of basis function expansion term coefficients, respectively; omega is the angular frequency of sound wave; j is the ordinal number of the expansion term of the basis function; j is the number of expansion term terms of the basis function; in a spherical coordinate system, # j The expression of (a) is:
wherein the content of the first and second substances,
the first kind of spherical Hankel function, where k is omega/c is the wave number of sound wave, and c is the sound velocity;
is a spherical harmonic function; in the formulae (1) to (3), the integers n, l and j satisfy the relationship j ═ n 2 + 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;
half-space sound pressure p at field point x half (x; ω) is a linear superposition of the source radiation and the boundary reflection, which can be expressed as:
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;
record the acoustic pressure measurement point coordinates of the array as
M is the number of sound pressure measuring points; array-acquired set of sound pressure values according to equation (4)
Can be expressed in the form of a matrix as follows:
wherein, the spherical wave function expansion term matrix
Comprises the following steps:
coefficient vector C (ω) } 2J×1 Comprises the following steps:
wherein, superscript T represents the transpose of the vector;
dividing the sound pressure measuring points into two groups according to the way of taking points at intervals, and recording the coordinates of the first group of measuring points as
The coordinates of the second set of measuring points are recorded as
Wherein, the first and the second end of the pipe are connected with each other,
and
respectively representing rounding-up and rounding-down; according to the formula (5), establishing a mathematical relationship between the sound pressure reconstruction value of the second group of measuring points and the sound pressure measurement value of the first group of measuring points:
wherein, the first and the second end of the pipe are connected with each other,
a vector of sound pressure reconstruction values for a second set of points,
a transfer matrix of sound pressure measurements from the first set of transducers to sound pressure reconstruction values from the second set of transducers:
wherein, the upper label
Represents the pseudo-inverse of the matrix:
wherein, the superscript H represents the conjugate transpose of the matrix;
setting the value upper limit of the expansion term number J of the basis function as J max I.e. J is 1. ltoreq. J.ltoreq.J max (ii) a For any J in the range, reconstructing the sound pressure value of the second set of measuring points by using the expressions (8) to (10), and calculating the relative error between the sound pressure reconstruction value of the second set of measuring points and the measured value:
wherein | · | purple sweet 2 Is a vector 2-norm; from 1 to J max Traversing all J, and determining the number of expansion terms corresponding to the epsilon minimum value as the optimal number of expansion terms J opt ；
S4, establishing a mathematical relation between the holographically measured sound pressure and the boundary surface sound pressure, and reconstructing the boundary surface sound pressure;
setting the number of expansion terms of the basis function to J opt Then, equation (5) is in the form:
according to equation (12), a mathematical relationship between the holographically measured sound pressure and the sound pressure of the boundary surface is established, and the sound pressure of the boundary surface can be reconstructed and calculated by the following equation:
wherein, the first and the second end of the pipe are connected with each other,
is the reconstructed point coordinates of the boundary surface, S is 1,2, …, S is the number of reconstructed points,
for holographic measurement of the transfer matrix of sound pressure to boundary surface sound pressure:
wherein the content of the first and second substances,
at the s-th reconstruction point for two sets of spherical wave functions
The matrix is formed by the expansion terms, and the arrangement form of matrix elements refers to the formula (6);
s5, establishing a mathematical relation between the holographic measurement sound pressure and the boundary surface fluid medium particle normal vibration velocity, and reconstructing the boundary particle normal vibration velocity;
sound pressure p at field point x half (x; omega) and particle vibration velocity v half (x; ω) satisfies the Euler equation:
where ρ is 0 In order to be the density of the fluid medium,
in order to be a gradient operator, the method comprises the following steps,
according to the formula (12) and the formula (15), a mathematical relation between the holographic measurement sound pressure and the normal vibration velocity of the particle of the fluid medium on the boundary surface is established, and the normal vibration velocity of the particle on the boundary surface can be reconstructed and calculated by the following formula:
wherein the content of the first and second substances,
for holographic measurement of the transmission matrix from sound pressure to the normal vibration velocity of the boundary surface particles:
wherein n is the boundary surface at the reconstruction point
The unit normal vector of (2);
s6, calculating the acoustic impedance rate of the boundary surface;
the specific acoustic impedance of the boundary surface is defined as the boundary surface sound pressure p half (x; omega) and the particle normal vibration velocity v nhalf (x; ω); calculating the surface acoustic impedance rate of each reconstruction point by using a group of reconstructed sound pressure and normal vibration velocity, and averaging to obtain the boundary surface acoustic impedance rate as follows:
the invention has the beneficial effects that:
1. the method is based on near-field acoustic holography and microphone array in-situ acquisition of physical boundary near-field sound pressure information, and realizes in-situ measurement and calculation of boundary surface acoustic impedance, so that the measurement of the boundary material sample acoustic impedance is implemented without depending on standard impedance measurement equipment and measurement environment, and meanwhile, the boundary surface acoustic impedance obtained by the method better accords with the acoustic characteristics of the boundary material in the practical engineering application environment.
2. Compared with other measuring methods using double-layer microphone arrays or spherical microphone arrays, the method provided by the invention has the advantages that the holographic sound pressure information is acquired by using the single-layer microphone array, and the difficulty of field arrangement and installation when the microphone array is designed, processed and applied to in-situ holographic sound pressure measurement is reduced.
Description of the drawings:
FIG. 1 is a schematic view of the arrangement of a sound source and microphone array in half-space;
FIG. 2 is a schematic diagram of grouping of measuring points when determining the number of expansion terms of the optimal spherical wave basis function;
FIG. 3 is a comparison of calculated values of specific acoustic impedance of the boundary surface with theoretical values.
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 measuring the acoustic impedance of the boundary surface of the half space in situ by the single-layer microphone array is carried out according to the following steps:
step 1, arranging a sound source and a single-layer microphone array on one side of a half space of a boundary, and carrying out sound pressure holographic measurement.
As shown in FIG. 1, the sound sources are arranged in a half-space, with the center of the sound source being denoted as O 1 . Arranging a single-layer planar microphone array parallel to the boundary surface at the near field of the boundary surface, the distance between the array surface and the boundary being h a . Recording the projection of the center of the array surface on the boundary as a point O, and recording the point O and the center O of the sound source 1 Is taken as the angle between the connecting line of (A) and the boundary normal at the point O as the sound wave incident angle theta inc . After the arrangement is finished, starting a sound source to emit sound waves, and collecting holographic sound pressure by the single-layer microphone array; after the collection is finished, the sound source is turned off, and the position of the sound source is adjusted to change the sound wave incidence angle theta inc (ii) a The holographic sound pressure collection is repeatedly carried out to obtain different incidence angles theta inc Holographic sound pressure under the conditions.
When the fluid medium is an aqueous medium, the sound source and microphone array is replaced with a sound source and hydrophone array in an aqueous medium.
And 2, establishing a mathematical model of the half-space sound field on one side of the boundary.
As shown in FIG. 1, a global coordinate system is established by taking a point O as a coordinate origin and a plane where a boundary is located as an x-y coordinate plane. Mixing O with 1 The mirror point about the boundary is marked as O 2 Respectively with O 1 And O 2 Establishing a local coordinate system for the coordinate origin, and respectively recording the coordinates of the half-space field point x in the two local coordinate systems as x 1 ≡(r 1 ,θ 1 ,φ 1 ) And x 2 ≡(r 2 ,θ 2 ,φ 2 ). For a steady sound field, in two local coordinate systems, sound pressure responses of sound source radiation and boundary reflection at the field point are expressed by linear superposition of a set of spherical wave functions, and the expressions are respectively:
and
wherein psi 1,j (x 1 (ii) a ω) and ψ 2,j (x 2 (ii) a ω) are each independently O 1 And O 2 A spherical wave basis function as the origin of coordinates; c. C 1,j (omega) and c 2,j (ω) are two sets of basis function expansion term coefficients, respectively; omega is the angular frequency of sound wave; j is the ordinal number of the expansion term of the basis function; j is the number of terms of the expansion term of the basis function. In a spherical coordinate system, # j The expression of (a) is:
wherein the content of the first and second substances,
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;
is a spherical harmonic function. In the formulae (1) to (3), the integers n, l and j satisfy the relationship j ═ n 2 + 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.
Half-space sound pressure p at field point x half (x; ω) is a linear superposition of the source radiation and the boundary reflection, which can be expressed as:
and 3, 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.
Recording the coordinates of the sound pressure measuring points of the array as
And M is the number of sound pressure measuring points. Array-acquired set of sound pressure values according to equation (4)
Can be expressed in the form of a matrix as follows:
wherein, the spherical wave function expansion term matrix
Comprises the following steps:
coefficient vector C (ω) } 2J×1 Comprises the following steps:
where the superscript T represents the transpose of the vector.
As shown in figure 2, the sound pressure measuring points are divided into two groups according to the way of taking points at intervals, and the coordinates of the first group of measuring points are recorded as
The coordinates of the second set of measuring points are recorded as
Wherein the content of the first and second substances,
and
respectively representing a round-up and a round-down. And (5) establishing a mathematical relation between the sound pressure reconstruction value of the second set of measuring points and the sound pressure measurement value of the first set of measuring points according to the formula (5):
wherein the content of the first and second substances,
a vector of sound pressure reconstruction values for a second set of points,
and (3) a transfer matrix of the sound pressure measured value of the first set of measuring points to the sound pressure reconstruction value of the second set of measuring points is as follows:
wherein, the upper label
Represents the pseudo-inverse of the matrix:
wherein, the superscript H represents the conjugate transpose of the matrix;
setting the value upper limit of the expansion term number J of the basis function as J max I.e. J is 1. ltoreq. J.ltoreq.J max . For any within the rangeAnd J, reconstructing the sound pressure values of the second set of measuring points by using the equations (8) to (10), and calculating the relative error between the sound pressure reconstruction values of the second set of measuring points and the measured values:
wherein | · | charging 2 Is the vector 2-norm. From 1 to J max Traversing all J, and determining the expansion item number corresponding to the epsilon minimum value as the optimal expansion item number J opt 。
And 4, establishing a mathematical relation between the holographically measured sound pressure and the boundary surface sound pressure, and reconstructing the boundary surface sound pressure.
Setting the number of expansion terms of the basis function to J opt Then, equation (5) is in the form:
according to equation (12), a mathematical relationship between the holographically measured sound pressure and the boundary surface sound pressure is established, and the boundary surface sound pressure can be reconstructed and calculated by the following equation:
wherein the content of the first and second substances,
is the reconstructed point coordinates of the boundary surface, S is 1,2, …, S is the number of reconstructed points,
for holographic measurement of the transfer matrix of sound pressure to boundary surface sound pressure:
wherein, the first and the second end of the pipe are connected with each other,
at the s-th reconstruction point for two sets of spherical wave functions
The matrix is composed of the expansion terms, and the arrangement form of the matrix elements refers to the formula (6).
And 5, establishing a mathematical relation between the holographic measurement sound pressure and the normal vibration velocity of the boundary surface fluid medium particle, and reconstructing the normal vibration velocity of the boundary particle.
Sound pressure p at field point x half (x; omega) and particle vibration velocity v half (x; ω) satisfies the Euler equation:
where ρ is 0 In order to be the density of the fluid medium,
in order to be a gradient operator, the method comprises the following steps of,
according to the formula (12) and the formula (15), a mathematical relation between the holographic measurement sound pressure and the normal vibration velocity of the particle of the fluid medium on the boundary surface is established, and the normal vibration velocity of the particle on the boundary surface can be reconstructed and calculated by the following formula:
wherein, the first and the second end of the pipe are connected with each other,
for holographic measurement of the transmission matrix from sound pressure to the normal vibration velocity of the boundary surface particles:
wherein n is the boundary surface at the reconstruction point
The unit normal vector of (c).
And 6, calculating the acoustic resistivity of the boundary surface.
The specific acoustic impedance of the boundary surface is defined as the boundary surface sound pressure p half (x; omega) and the particle normal vibration velocity v nhalf (x; ω). Calculating the surface acoustic impedance rate of each reconstruction point by using a group of reconstructed sound pressure and normal vibration velocity, and averaging to obtain the boundary surface acoustic impedance rate as follows:
example (b): and (3) taking the pulsating sphere as a sound source, and carrying out numerical simulation calculation on the half-space sound field on one side of the boundary. In the global coordinate system shown in fig. 1, the boundary coordinate z is 0, and the surface specific acoustic impedance satisfies the acoustic impedance model:
Z 0 ＝0.436(1+i)(σ e /f) 0.5 +19.48iα e /f (19)
wherein σ e Taking 38kPasm as the effective flow resistivity of the boundary -2 ；α e For the reduction rate of the boundary porosity with the boundary depth, 15m is taken -1 (ii) a f is the acoustic frequency.
The single layer microphone array plane is parallel to the boundary, the geometric center is positioned on the z-axis, and the distance from the boundary is h a 0.3 m; the aperture of the array is 0.15m multiplied by 0.15m, 6 multiplied by 6 sound pressure measuring points are uniformly distributed on the array, the distance between adjacent measuring points is 0.03m, and the measuring point distribution schematic diagram is shown in figure 2. The coordinates of the center of the pulsating sphere are set as
At this time, the incident angle of the sound wave is θ inc 0 deg.. The radius of the pulsating sphere is 0.05m, and the radial vibration speed V of the surface particle 0 0.01 m/s. Density of air medium is rho 0 ＝1.29kg/m 3 The speed of sound in air is 343 m/s. Simulating measurement errors of microphonesThe effect is to add white gaussian noise with a signal to noise ratio of 30dB to the measurements.
By using the method, the sound pressure measurement value of the microphone array is used for reconstructing the sound pressure of the boundary surface and the normal vibration velocity distribution of mass points, and then the acoustic impedance rate of the boundary surface is calculated. The frequency range is considered to be between 1000Hz and 5000 Hz. The comparison of the calculated values of the specific acoustic impedance of the boundary surface with the theoretical values at different frequencies is shown in FIG. 3.
Observing fig. 3, it is found that when the incident angle of the sound wave is θ inc When the temperature is equal to 0 degrees, the boundary surface acoustic impedance rate calculated by the method of the invention under different frequency conditions is consistent with the theoretical value. When the sound source position is adjusted, the sound wave incidence angle is theta inc The method of the present invention can also calculate the specific acoustic impedance corresponding to the theoretical value at other values in the range of 0 ° to 90 °. The result shows that the method for in-situ measurement of the acoustic impedance of the semi-space boundary surface of the single-layer microphone array can accurately obtain the acoustic impedance of the boundary surface by acquiring the holographic sound pressure in situ at the boundary near field.
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 measuring the acoustic impedance of the boundary surface of the half space in situ by the single-layer microphone array is characterized by comprising the following steps of: comprises the following steps:
s1, arranging a sound source and a single-layer microphone array on one side of a half space of a boundary, and carrying out sound pressure holographic measurement;
arranging the sound sources in half-space, with the center of the sound source being denoted as O 1 (ii) a Arranging a single-layer planar microphone array parallel to the boundary surface at the near field of the boundary surface, wherein the distance between the array surface and the boundary is h a (ii) a Recording the projection of the center of the array surface on the boundary as a point O, and recording the point O and the center O of the sound source 1 Is taken as the angle between the connecting line of (A) and the boundary normal at the point O as the sound wave incident angle theta inc (ii) a After the arrangement is completed, startA sound source emits sound waves, and a single-layer microphone array collects holographic sound pressure; after the collection is finished, the sound source is turned off, and the position of the sound source is adjusted to change the incident angle theta of the sound wave inc (ii) a The holographic sound pressure collection is repeatedly carried out to obtain different incidence angles theta inc Holographic sound pressure under conditions;
s2, establishing a mathematical model of a half-space sound field on one side of the boundary;
establishing a global coordinate system by taking the point O as a coordinate origin and the plane where the boundary is located as an x-y coordinate plane; mixing O with 1 The mirror point about the boundary is marked as O 2 Respectively with O 1 And O 2 Establishing a local coordinate system for the coordinate origin, and respectively recording the coordinates of the half-space field point x in the two local coordinate systems as x 1 ≡(r 1 ,θ 1 ,φ 1 ) And x 2 ≡(r 2 ,θ 2 ,φ 2 ) (ii) a For a steady sound field, in two local coordinate systems, sound pressure responses of sound source radiation and boundary reflection at the field point are expressed by linear superposition of a set of spherical wave functions, and the expressions are respectively:
and
wherein psi 1,j (x 1 (ii) a ω) and ψ 2,j (x 2 (ii) a ω) are each O 1 And O 2 A spherical wave basis function as the origin of coordinates; c. C 1,j (omega) and c 2,j (ω) are two sets of basis function expansion term coefficients, respectively; omega is the angular frequency of sound wave; j is the ordinal number of the expansion term of the basis function; j is the number of expansion term terms of the basis function; in a spherical coordinate system, # j The expression of (a) is:
wherein the content of the first and second substances,
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;
is a spherical harmonic function; in the formulae (1) to (3), the integers n, l and j satisfy the relationship j ═ n 2 + 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;
half-space sound pressure p at field point x half (x; ω) is a linear superposition of the acoustic source radiation and the boundary reflection, which can be expressed as:
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;
record the acoustic pressure measurement point coordinates of the array as
M is 1,2, wherein M is the number of sound pressure measuring points; array-acquired set of sound pressure values according to equation (4)
Can be expressed in the form of a matrix as follows:
wherein, the spherical wave function expansion term matrix
Comprises the following steps:
coefficient vector C (ω) } 2J×1 Comprises the following steps:
wherein, superscript T represents the transpose of the vector;
dividing the sound pressure measuring points into two groups according to the way of taking points at intervals, and recording the coordinates of the first group of measuring points as
m′＝1,2,...,M′，
The coordinates of the second set of measuring points are recorded as
m″＝1,2,...,M″，
Wherein the content of the first and second substances,
and
respectively representing rounding-up and rounding-down; according to the formula (5), establishing a mathematical relationship between the sound pressure reconstruction value of the second group of measuring points and the sound pressure measurement value of the first group of measuring points:
wherein the content of the first and second substances,
a vector of sound pressure reconstruction values for a second set of points,
and (3) a transfer matrix of the sound pressure measured value of the first set of measuring points to the sound pressure reconstruction value of the second set of measuring points is as follows:
wherein, the upper label
Represents the pseudo-inverse of the matrix:
wherein, the superscript H represents the conjugate transpose of the matrix;
setting the value upper limit of the expansion term number J of the basis function as J max I.e. J is 1. ltoreq. J.ltoreq.J max (ii) a For any J in the range, reconstructing the sound pressure value of the second set of measuring points by using the expressions (8) to (10), and calculating the relative error between the sound pressure reconstruction value of the second set of measuring points and the measured value:
wherein | · | purple sweet 2 Is a vector 2-norm; from 1 to J max Traversing all J, and determining the number of expansion terms corresponding to the epsilon minimum value as the optimal number of expansion terms J opt ；
S4, establishing a mathematical relation between the holographically measured sound pressure and the boundary surface sound pressure, and reconstructing the boundary surface sound pressure;
setting the number of expansion terms of the basis function to J opt Then, equation (5) is in the form:
according to equation (12), a mathematical relationship between the holographically measured sound pressure and the sound pressure of the boundary surface is established, and the sound pressure of the boundary surface can be reconstructed and calculated by the following equation:
wherein the content of the first and second substances,
is the reconstructed point coordinates of the boundary surface, S is 1,2, …, S is the number of reconstructed points,
for holographic measurement of the transfer matrix of sound pressure to boundary surface sound pressure:
wherein the content of the first and second substances,
at the s-th reconstruction point for two sets of spherical wave functions
The matrix is formed by the expansion terms, and the arrangement form of matrix elements refers to the formula (6);
s5, establishing a mathematical relation between the holographic measurement sound pressure and the boundary surface fluid medium particle normal vibration velocity, and reconstructing the boundary particle normal vibration velocity;
sound pressure p at field point x half (x; omega) and particle vibration velocity v half (x; ω) satisfies the Euler equation:
where ρ is 0 In order to be the density of the fluid medium,
in order to be a gradient operator, the method comprises the following steps of,
according to the formula (12) and the formula (15), a mathematical relationship between the holographic measurement sound pressure and the normal vibration velocity of the particle of the fluid medium on the boundary surface is established, and the normal vibration velocity of the particle on the boundary surface can be reconstructed and calculated by the following formula:
wherein the content of the first and second substances,
for holographic measurement of the transmission matrix of sound pressure to the normal vibration velocity of the boundary surface particles:
where n is the boundary surface at the reconstruction point
A unit normal vector of (d);
s6, calculating the acoustic impedance rate of the boundary surface;
the specific acoustic impedance of the boundary surface is defined as the boundary surface sound pressure p half (x; omega) and the normal particle velocity v nhalf (x; ω); calculating the surface acoustic impedance rate of each reconstruction point by using a group of reconstructed sound pressure and normal vibration speed, and averaging to obtain the boundary surface acoustic impedance rate of：
2. The method for in-situ measurement of acoustic impedance of a boundary surface of a half space of a single layer microphone array as claimed in claim 1 wherein: the sound source and microphone array described in step S1 is suitable for sound wave generation and sound pressure acquisition in an air medium, and when the half-space fluid medium is an aqueous medium, it is replaced with a sound source and hydrophone array in an aqueous medium correspondingly for sound wave generation and sound pressure acquisition in an aqueous medium, and measurement and calculation of acoustic impedance of a boundary surface.
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