Boundary impedance in-situ measurement method based on sound admittance optimization

Abstract

The invention provides a boundary impedance in-situ measurement method based on sound admittance optimization, which comprises the following steps: arranging a tested sound source in a semi-open space containing a reflection boundary, arranging a microphone array near the tested sound source to form a holographic sound pressure measurement surface, and arranging a reference microphone near the boundary; starting a tested sound source, and collecting sound pressure superposed by sound source radiation and boundary reflection through the microphone array and the reference microphone; describing a half-space sound pressure field by linear superposition of a half-space spherical wave basis function, and establishing a mathematical model of the half-space sound pressure field by taking boundary sound admittance as a to-be-optimized parameter; traversing all values in a boundary sound admittance value range, solving a primary function coefficient, reconstructing the sound pressure at the reference microphone, and determining the optimal boundary sound admittance by taking the minimum relative error between the reconstructed sound pressure and the measured sound pressure as a criterion; and calculating the boundary acoustic impedance rate according to the optimal boundary acoustic admittance. Compared with a measurement method based on reflection coefficient optimization, higher measurement precision can be obtained.

Description

Technical Field

[0001] This invention relates to the field of acoustic impedance measurement technology, and specifically to an in-situ measurement method for boundary acoustic impedance based on acoustic admittance optimization. Background Technology

[0002] Acoustic impedance and acoustic admittance are important physical quantities characterizing the acoustic properties of material surfaces, with wide applications in architectural acoustics, noise control, and underwater acoustic engineering. Methods for measuring the acoustic impedance of material surfaces mainly include three categories: impedance tube method, reverberation chamber method, and in-situ measurement method. The impedance tube method and reverberation chamber method rely on standard acoustic equipment or specific measurement environments to measure material samples. Although they offer high measurement accuracy, they require the material to be prepared into samples of specific sizes and sent to a laboratory, making it difficult to reflect the acoustic performance of the material under actual installation conditions. The in-situ measurement method, on the other hand, typically involves setting up standard sound sources and specifically distributed sound pressure measurement points on-site at the material being tested. The acoustic impedance of the material surface is calculated using the sound pressure measurements. Compared to the impedance tube method or reverberation chamber method, this method is more efficient and easier to implement in specific engineering applications, especially in existing semi-open space acoustic testing environments.

[0003] In semi-open acoustic test fields with reflective boundaries, the implementation of direct radiation field reconstruction algorithms for sound sources typically requires prior acquisition of the boundary acoustic impedance as a known parameter in the sound field mathematical model. This makes accurate measurement of the boundary acoustic impedance a prerequisite for related testing techniques. Existing in-situ acoustic impedance measurement methods based on optimizing the reflection coefficient at the field point of a holographic sound pressure measurement surface utilize the existing test sound source in the test field as the excitation source. Combined with in-situ measurement of the sound pressure field using a holographic sound pressure measurement surface, the acoustic impedance ratio of the spatial boundary can be obtained by optimizing the sound pressure reflection coefficient in the mathematical model. However, when the reflection coefficient is used as an optimization parameter, the range of acoustic impedance ratios mapped by its value range is relatively wide. Under the same bounded constraints and mesh density, the accuracy of the optimization calculation is limited to a certain extent. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides an in-situ boundary impedance measurement method based on acoustic admittance optimization, comprising the following steps: Step S1: Arrange the test sound source in a semi-open space containing a reflective boundary, arrange a microphone array near the test sound source to form a holographic sound pressure measurement surface, and arrange a reference microphone near the boundary. Step S2: Activate the test sound source and collect the sound pressure of the superposition of sound source radiation and boundary reflection through the microphone array and the reference microphone; Step S3: Describe the sound pressure field of the half-space using the linear superposition of the basis functions of the half-space spherical waves, and establish a mathematical model of the sound pressure field of the half-space using the boundary acoustic admittance as the parameter to be optimized. Step S4: Traverse all values ​​within the range of the boundary acoustic admittance, solve the basis function coefficients, reconstruct the sound pressure at the reference microphone, and determine the optimal boundary acoustic admittance based on minimizing the relative error between the reconstructed sound pressure and the measured sound pressure. Step S5: Calculate the boundary acoustic impedance ratio based on the optimal boundary acoustic admittance.

[0005] Preferably, the geometry of the holographic sound pressure measurement surface is conformal to the geometry of the test sound source surface.

[0006] Preferably, the measuring aperture of the microphone array is not less than 1.2 times the surface area of ​​the tested sound source.

[0007] Preferably, the spacing between adjacent microphones in the microphone array is less than one-sixth of the sound wave wavelength.

[0008] Preferably, the distance between the reference microphone and the boundary is one-quarter of the sound wave wavelength.

[0009] Preferably, in step S3, the origin of the coordinate system is the projection of the geometric center of the tested sound source onto the boundary, and the coordinate system is the plane containing the boundary. x - y Establish a global coordinate system in the coordinate plane; denote the mirror point of the geometric center of the test sound source about the boundary as the mirror virtual source position, and translate the global coordinate system to the geometric center of the test sound source and the mirror virtual source position respectively to establish two local coordinate systems.

[0010] Preferably, in step S3, the mathematical relationship between the sound pressure reflection coefficient and the boundary acoustic admittance is expressed as follows: ; in, The sound pressure reflection coefficient is... The angle of incidence of the sound wave. For normalized boundary acoustic admittance, ω is the angular frequency of the sound wave.

[0011] Preferably, in step S4, the boundary acoustic admittance is expressed as: ; in, Represents the real part of a complex number; Represent the imaginary part of a complex number; The imaginary unit; with The horizontal axis is... Establish a rectangular coordinate system for the vertical axis, divide the grid into a circle with the origin as the center and a radius of 1, and solve for the optimal boundary acoustic admittance by traversing the grid nodes.

[0012] Preferably, in step S5, the boundary acoustic impedance is calculated according to the following formula: ; in, This is the calculated value of the boundary acoustic impedance. β The optimal boundary acoustic admittance is given.

[0013] The beneficial effects of the present invention include at least the following: First, this invention addresses the need to pre-obtain the boundary acoustic impedance as a parameter for reconstructing the direct radiation field of a sound source in a semi-open space with a reflective boundary. It directly utilizes the sound pressure distribution collected by the existing sound source under test and microphone array in the test field, and solves for the boundary acoustic admittance and obtains the boundary acoustic impedance through an optimization algorithm. Compared to measurement methods based on reflection coefficient optimization, this achieves higher measurement accuracy.

[0014] Secondly, this invention enables in-situ measurement of the acoustic impedance of a material surface without the need for specialized measuring instruments or special measurement conditions. Traditional methods rely on specialized equipment such as impedance tubes or reverberation chambers, while this invention performs measurements directly in the actual engineering application environment, eliminating the need to send material samples to a laboratory, and thus accurately reflecting the acoustic performance of the material under actual installation conditions.

[0015] Third, this invention fully utilizes the sound pressure information obtained from the existing test sound source and microphone array in the semi-open space sound source radiation test field to calculate the boundary acoustic admittance and acoustic impedance, without the need to arrange standard sound sources and specifically distributed microphone arrays to obtain the boundary acoustic impedance, which significantly improves the implementation efficiency of related testing technologies in practical engineering application scenarios.

[0016] Fourth, this invention uses boundary acoustic admittance as the optimization parameter, making full use of the boundedness of acoustic admittance values. For common finite acoustic impedance boundaries, since the modulus of the normalized boundary acoustic admittance is no greater than 1, the range of acoustic impedance rate mapped by its value range is narrower than that of the reflection coefficient. Therefore, under the same bounded constraint conditions and mesh density, the method based on acoustic admittance optimization can achieve higher measurement accuracy.

[0017] Fifth, this invention utilizes the precise mathematical relationship between boundary acoustic admittance and acoustic pressure reflection coefficient to directly calculate boundary acoustic impedance ratio through acoustic admittance. The calculation process is simple and efficient, relaxes the constraints on the coordinates of the reference microphone, and expands the engineering applicability of the in-situ measurement method. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the method flow according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the arrangement of the microphone array and the reference microphone in an embodiment of the present invention. Figure 3 This is a schematic diagram illustrating the geometric relationship between two local coordinate systems relative to the global coordinate system in an embodiment of the present invention. Figure 4 This is a schematic diagram of the arrangement of the microphone array, reference microphone, and board sound source according to an embodiment of the present invention. Figure 5 This is a schematic diagram of the unit circle mesh division for boundary acoustic admittance optimization according to an embodiment of the present invention. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of the present invention.

[0020] like Figure 1 As shown, this embodiment of the invention provides an in-situ boundary impedance measurement method based on acoustic admittance optimization, which includes the following steps: Step S1: Arrange the test sound source in a semi-open space containing a reflective boundary, arrange a microphone array near the test sound source to form a holographic sound pressure measurement surface, and arrange a reference microphone near the boundary.

[0021] The sound source under test is placed in a semi-open space sound source radiation test field containing a reflective boundary. The boundary acoustic impedance of the semi-open space is unknown, which is the target quantity to be measured in this invention. Figure 2 As shown, a microphone array is arranged near the sound source to form a holographic sound pressure measurement surface, used to collect spatial distribution information of the sound source's radiation field. Simultaneously, a reference microphone is placed near the boundary to provide an independent sound pressure measurement point for calculating reconstruction errors in subsequent optimization algorithms.

[0022] The geometry of the holographic sound pressure measurement surface should be conformal to the geometry of the test sound source surface in order to collect spatial distribution information of the direct radiation field of the test sound source more uniformly.

[0023] To ensure the integrity and validity of the measurement data, the arrangement of the microphone array also needs to consider the spatial sampling problem related to the sound wave wavelength. Let the wavelength of the sound wave under consideration be... The measuring aperture of the microphone array should not be less than 1.2 times the surface area of ​​the sound source under test. This setting ensures that the array can fully acquire the spatial distribution of the direct radiation field of the sound source under test. The spacing between adjacent microphones in the microphone array should be less than [missing information]. This is to meet the requirements of the spatial sampling theorem and improve the spatial resolution of the acquired sound field. The distance between the reference microphone and the boundary should be set to... This position can effectively distinguish the contributions of incident and reflected waves, improving the sensitivity of acoustic ductility optimization.

[0024] Step S2: Start the test sound source and collect the sound pressure of the superposition of sound source radiation and boundary reflection through the microphone array and reference microphone.

[0025] After the measurement system setup is complete, the test sound source is activated to emit sound waves. The sound waves emitted by the source propagate in the semi-open space; part of them directly reach the microphone array, forming a direct sound field, while the other part is reflected off the boundary and reaches the microphone array, forming a reflected sound field. The sound pressure signals collected by the microphone array and the reference microphone are the result of the superposition of the direct radiation field of the sound source and the boundary reflection field. During data acquisition, the position coordinates of each microphone and its corresponding complex sound pressure value need to be recorded. The sound pressure data at each measurement point of the microphone array will be used to establish a mathematical model of the sound pressure field in the half-space and solve for the basis function coefficients, while the sound pressure data of the reference microphone will serve as a reference benchmark for the optimization algorithm.

[0026] Step S3: Describe the sound pressure field of the half-space using the linear superposition of the basis functions of the half-space spherical wave, and establish a mathematical model of the sound pressure field of the half-space using the boundary acoustic admittance as the parameter to be optimized.

[0027] To describe the sound pressure field distribution in a semi-open space with reflecting boundaries, this invention employs the linear superposition of the basis functions of the spherical waves in the half-space to establish a mathematical model, and uses the boundary acoustic admittance as the parameter to be optimized. Figure 3 As shown, with the geometric center of the sound source The projection of the boundary of the surface of the material under test is taken as the origin of the coordinate system. Taking the plane containing the boundary as xy Establish a global coordinate system in the coordinate plane. Denote the mirror image of the geometric center of the sound source about the boundary as... And translate the global coordinate system to respectively and Establish two local coordinate systems. For any point in half-space... The coordinates in the two local coordinate systems are denoted as follows: and They are related to global coordinates The relationship between them can be represented as: ; ; In the formula, From the geometric center of the sound source to xy Distance between planes, for z A unit vector in direction.

[0028] In the global coordinate system, sound source radiation and boundary reflection occur at a half-space field point. sound pressure response It can be described by a linear superposition of a set of half-space spherical wave basis functions: ; In the formula, For the basis functions of the half-space spherical wave, These are the corresponding basis function coefficients. The angular frequency of the sound wave; The ordinal number of the expansion term of the basis function; is the number of terms in the basis function expansion.

[0029] The basis function of a half-space spherical wave consists of two parts: centered on the sound source... The spherical wave basis function at the origin represents the contribution of the direct field, with the mirror point as the reference point. The product of the spherical wave basis function and the reflection coefficient, centered at the origin, represents the contribution of the reflected field. Its expression is: ; Among them, with and The spherical wave basis functions with the origin as the coordinate point are defined as follows: ; ; In the formula, and These represent the spherical coordinates of the field point in the two local coordinate systems, respectively. For the first kind of spherical Hankel function, For sound wave number, For the speed of sound, It is a spherical harmonic function. (Number) , and Satisfying the relation ,in , , for The cutoff value.

[0030] The core of this invention lies in using boundary acoustic admittance as the parameter to be optimized. Sound pressure reflection coefficient. With boundary acoustic admittance There is a precise mathematical relationship between them: ; In the formula, The angle of incidence of the sound wave. The normalized boundary acoustic admittance is the reciprocal of the boundary acoustic impedance. For finite acoustic impedance boundaries commonly found in engineering applications, the boundary acoustic admittance is... The values ​​of satisfy: ; In the formula, The modulus represents a complex number. This boundedness of acoustic admittance values ​​is key to this invention. Since the modulus of normalized acoustic admittance is no greater than 1, the range of acoustic impedance that its value range maps to is narrower than that of the reflection coefficient. This means that, under the same mesh density, higher computational accuracy can be achieved based on acoustic admittance optimization.

[0031] Step S4: Traverse all values ​​within the range of boundary acoustic admittance, solve the basis function coefficients, reconstruct the sound pressure at the reference microphone, and determine the optimal boundary acoustic admittance based on minimizing the relative error between the reconstructed sound pressure and the measured sound pressure.

[0032] Based on the mathematical model established in step S3, this step determines the optimal boundary acoustic admittance value through a traversal optimization algorithm. The sound pressure collected at each measuring point of the microphone array can be expressed in the following matrix form: ; In the formula, This is a column vector composed of half-space sound pressure measurements. It is a column vector consisting of the basis function coefficients of the half-space spherical wave. The coordinates of the measuring point are... , M The number of measurement points on the array. It is a matrix composed of the expansion terms of the half-space spherical wave basis functions at each measuring point.

[0033] Boundary acoustic admittance For a complex number, it can be represented as: In the formula, Represents the real part of a complex number; Represent the imaginary part of a complex number; It is the imaginary unit; and ;by The horizontal axis is... If a rectangular coordinate system is established with the vertical axis as the coordinate axis, then All possible values ​​should fall within a circle centered at the origin with a radius of 1; the region within the circle is then discretized into a grid according to a specific grid division method, and the corresponding value of any grid node is... This is the possible optimal acoustic admittance value. , This represents the total number of nodes.

[0034] The optimization calculation is performed according to the following process: Set the number of terms in the expansion of the half-space spherical wave basis function. The range of values ​​is ,in This is the maximum number of expanded items. The number of expanded items is then set sequentially within this range. For each expansion term, iterate through all possible values ​​of the acoustic admittance. For a given and First, solve for the basis function coefficients: ; In the formula, The pseudo-inverse of a matrix is ​​expressed by the following formula: ; In the formula, H represents the conjugate transpose of the matrix. After obtaining the basis function coefficients, the reference microphone... The reconstructed sound pressure value of the half-space at that location can be calculated by the following formula: ; Let the number of terms in the expansion of the basis functions of the half-space spherical wave be... ,Will The value is assigned to the basis function of the spherical wave in the half-space, and the process is iterated through all... The corresponding sound pressure reconstruction value is obtained. Then let the number of expanded terms be... Repeat the above reconstruction calculation process; repeat this cycle until the number of expanded terms is completed. Reconstruction calculation at that time. Calculate the relative error between the reconstructed sound pressure value and the measured value: ; The optimal value is determined based on minimizing the reconstruction error, and this value is used as the boundary acoustic admittance. The value of .

[0035] Step S5: Calculate the boundary acoustic impedance ratio based on the optimal boundary acoustic admittance.

[0036] The optimal boundary acoustic admittance obtained using step S4 Boundary acoustic impedance Calculate using the following formula: ; This calculation formula demonstrates the superiority of the present invention compared to the optimization method based on the reflection coefficient: the acoustic admittance and acoustic impedance are in a simple reciprocal relationship, the calculation process is concise and direct, it does not involve geometric parameters such as the incident angle, and it relaxes the constraints on the coordinates of the reference microphone.

[0037] The effectiveness of the method of the present invention will be verified through a specific numerical simulation example below.

[0038] like Figure 4 As shown, in Figure 2 Measurements are performed in the global coordinate system shown, and the coordinates of the material surface boundary are... The test sound source is assumed to be a square plate with a side length of 0.10m, with the source surface parallel to the material surface and 0.25m from the boundary. The frequency of the sound wave under investigation is 1700Hz, and the density of the air medium is 1.20kg / m³. 3 When the speed of sound is 343 m / s, the corresponding wavelength of the sound wave is 0.20 m.

[0039] A square array of 36 microphones (6×6=36) is arranged on one side of the sound source surface. The array's measuring aperture is 0.15m×0.15m, which meets the requirement of being no less than 1.2 times the area of ​​the sound source surface. The spacing between adjacent microphones is approximately 0.03m, less than one-sixth of the wavelength (approximately 0.033m). The array is 0.05m away from the sound source surface. The geometric center coordinates of the square holographic sound pressure measurement surface formed by the array are... Place a reference microphone near the boundary, at coordinates [coordinates missing]. The distance from the boundary is approximately one-quarter of the wavelength.

[0040] The test sound source was activated, and the sound pressure level, calculated from the superposition of source radiation and boundary reflection, was collected via microphones. The half-space sound pressure field measured by the microphone array was described using the linear superposition of basis functions of a half-space spherical wave, and a mathematical model was established using boundary acoustic admittance as the parameter to be optimized. The maximum number of expansion terms for the basis functions was set to be... .like Figure 5 As shown, respectively with and Establish a rectangular coordinate system for the horizontal and vertical axes. For the region within a circle centered at the origin with a radius of 1, calculate the coordinates according to a step size of... Divide the grid.

[0041] Let the number of terms to expand be We iterate through all values ​​within the range of acoustic admittance, solve for the basis function coefficients, reconstruct the sound pressure at the reference microphone, and calculate the relative error. Then, let the number of expansion terms be... Repeat the above solution process until the number of expanded terms is [number missing]. The optimal calculation is performed at that time. The optimal value of the boundary acoustic admittance is determined based on the criterion of minimizing the reconstruction error, and the acoustic impedance ratio is calculated using the obtained optimal acoustic admittance.

[0042] To verify the accuracy of the calculated acoustic impedance, numerical simulation was performed. Assuming a frequency of 1700 Hz, the true acoustic impedance of the material surface is... The Young's modulus of the plate structure is Poisson's ratio is 0.3, and the thickness is The density is A simple harmonic excitation force of 0.1N is applied to the geometric center of the plate. The sound pressure field radiated by the sound source on the plate in half-space is calculated using relevant analytical formulas, and the sound pressure measurement value of the microphone is obtained. To simulate the influence of microphone measurement error, Gaussian white noise with a signal-to-noise ratio of 30dB is added to the measured sound pressure.

[0043] To quantify the accuracy of acoustic impedance calculation, the calculated acoustic impedance value is defined. Compared with the true value relative error between ε for: ; Simulation results show that when the number of expanded terms is When, relative error E Get the minimum value At this point, the boundary acoustic admittance is obtained through optimization. The calculated acoustic impedance value is The relative error between it and its true value is .

[0044] The above results demonstrate that the in-situ measurement method for material surface acoustic impedance based on acoustic admittance optimization of this invention, by directly utilizing existing sound sources and holographic measurement surface arrangements in the test field, supplementing the sound pressure information of the field point with a single reference microphone, and combining the sound field mathematical model of the superposition of half-space spherical wave basis functions, can accurately obtain the boundary acoustic impedance ratio through the boundary acoustic admittance optimization algorithm. Since the range of acoustic impedance ratios mapped by the range of acoustic admittance values ​​is relatively narrow, it has a higher measurement accuracy advantage compared to methods based on reflection coefficient optimization.

[0045] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described; only preferred embodiments of the present invention are illustrated. The descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the present invention. As long as the combination of these technical features does not contradict each other, it should be considered within the scope of this specification.

[0046] It should be noted that those skilled in the art can make various modifications and improvements without departing from the inventive concept, and these all fall within the scope of protection of this invention. Therefore, the scope of protection of this invention should be determined by the appended claims.

Claims

1. A method for in-situ measurement of boundary impedance based on acoustic admittance optimization, characterized in that, Includes the following steps: Step S1: Arrange the test sound source in a semi-open space containing a reflective boundary, arrange a microphone array near the test sound source to form a holographic sound pressure measurement surface, and arrange a reference microphone near the boundary. Step S2: Activate the test sound source and collect the sound pressure of the superposition of sound source radiation and boundary reflection through the microphone array and the reference microphone; Step S3: Describe the sound pressure field of the half-space using the linear superposition of the basis functions of the half-space spherical waves, and establish a mathematical model of the sound pressure field of the half-space using the boundary acoustic admittance as the parameter to be optimized. Step S4: Traverse all values ​​within the range of the boundary acoustic admittance, solve the basis function coefficients, reconstruct the sound pressure at the reference microphone, and determine the optimal boundary acoustic admittance based on minimizing the relative error between the reconstructed sound pressure and the measured sound pressure. Step S5: Calculate the boundary acoustic impedance ratio based on the optimal boundary acoustic admittance.

2. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 1, characterized in that: The geometry of the holographic sound pressure measurement surface is conformal to the geometry of the test sound source surface.

3. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 1, characterized in that: The measuring aperture of the microphone array is not less than 1.2 times the surface area of ​​the tested sound source.

4. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 1, characterized in that: The spacing between adjacent microphones in the microphone array is less than one-sixth of the wavelength of the sound wave.

5. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 1, characterized in that: The distance between the reference microphone and the boundary is one-quarter of the sound wave wavelength.

6. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 1, characterized in that: In step S3, the projection of the geometric center of the tested sound source onto the boundary is taken as the origin of the coordinate system, and the plane containing the boundary is taken as the coordinate system. x - y Establish a global coordinate system in the coordinate plane; denote the mirror point of the geometric center of the test sound source about the boundary as the mirror virtual source position, and translate the global coordinate system to the geometric center of the test sound source and the mirror virtual source position respectively to establish two local coordinate systems.

7. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 6, characterized in that: In step S3, the mathematical relationship between the sound pressure reflection coefficient and the boundary acoustic admittance is expressed as follows: ； in, The sound pressure reflection coefficient is... The angle of incidence of the sound wave. For normalized boundary acoustic admittance, ω is the angular frequency of the sound wave.

8. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 7, characterized in that: In step S4, the boundary acoustic admittance is expressed as: ； in, Represents the real part of a complex number; Represent the imaginary part of a complex number; The imaginary unit; with The horizontal axis is... Establish a rectangular coordinate system for the vertical axis, divide the grid into a circle with the origin as the center and a radius of 1, and solve for the optimal boundary acoustic admittance by traversing the grid nodes.

9. The in-situ boundary impedance measurement method based on acoustic admittance optimization according to claim 1, characterized in that: In step S5, the boundary acoustic impedance is calculated according to the following formula: ； in, This is the calculated value of the boundary acoustic impedance. β The optimal boundary acoustic admittance is given.

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