Abstract
This paper focuses on the deduction of surface impedance and absorption coefficient of locally and non-locally reactive materials in free field. A sound wave model based on the complex image theory is introduced and used for deducing the characteristics of both locally and non-locally reactive materials. For the locally reactive material, the method based on the complex image model is compared with the other two existing deduction methods based on the Q-term model and the F-term model. The comparisons are made through numerical experiment in which the sound field above an impedance plane is simulated by the boundary element method. The results show that, considering both the accuracy and the efficiency, the complex image model-based method would be an optimal choice. For the non-locally reactive material, the method based on the complex image model is then compared with the existing deduction method based on the sound wave model proposed by Allard. The numerical experiment shows that, the method based on the complex image model can obtain almost the same result as that of the method based on the Allard’s model but cost less time.
1 Introduction
Numerous techniques have been established for the measurement of surface impedance and sound absorption coefficients of materials in free-field or in situ environments. These methodologies can be systematically divided into two principal categories. The first category includes methods that are independent of any specific sound wave model. A representative method within this category is the approach introduced by Tamura [1, 2], which applies the spatial Fourier transform and requires measuring complex sound pressure fields on two parallel planes located close to the surface of the test specimen. Theoretically, this method is compatible with sound sources of arbitrary configuration; however, in practical applications, sources that effectively reduce errors arising from the finite size of the measurement area are preferred. A significant advantage of Tamura’s method is its capability to determine reflection coefficients across a wide range of incidence angles from a single measurement setup, thereby improving measurement efficiency. Furthermore, its applicability to both locally and non-locally reacting surfaces enhances its versatility in characterizing diverse material types. Despite these benefits, the technique demands extensive measurement time and a large sample size, making it less efficient in practice.
The second category of methods is based on specific models of sound wave, three primary models have been widely recognized: (a) the plane wave model [3, 4]; (b) the mirror image model [5, 6], which assumes that incident spherical wavefronts are reflected as plane waves; and (c) the spherical wave model [4, 7]. Among techniques relying on the plane wave assumption, the transfer function technique introduced by Allard and Sieben [8, 9] remains the most widely employed. This method uses two closely spaced microphones to measure sound pressure at two adjacent positions near the surface of the test material. Extensive validation has demonstrated excellent agreement between results obtained with the transfer function method and those from the Kundt’s tube method [10], particularly at frequencies exceeding 500 Hz, thereby confirming its accuracy under laboratory conditions. Shortly after the introduction of the PU-probe [4, 11], this sensor was incorporated into the transfer function methodology [12] significantly enhancing measurement flexibility. However, due to the reliance on plane wave propagation, the method requires a considerable distance between the sound source and the material surface—a condition that is typically not met in situ measurement setups.
The mirror image model delivers a more physically accurate depiction of the sound field above an impedance plane by explicitly integrating the effects of geometric spreading, thereby surpassing the plane wave model in predictive precision. This modeling framework has been successfully applied to two key measurement techniques: the transfer function method and the pulse-echo method. When employing the mirror image model within the transfer function method, the sound source may be placed in close proximity to the material surface, as the theoretical formulation inherently corrects for geometric spreading, which significantly improves its practical utility in experimental applications. And both two closely-spaced microphones [13] and the PU-probe [14] have been used in this method. However, the correction would fail at low frequencies. For the pulse-echo method [15, 16], the direct wave and reflected wave are separated by the window or the subtraction technique in the time domain. The pulse-echo method is applicable to a common measurement condition like in an ordinary room. But, the surface of the material under test is required to be large to make the unwanted reflected waves from ambient objects separated in the time domain, which limits its application in some realistic condition.
Among the three models, the spherical wave model provides the greatest accuracy in computing the sound field produced by a monopole source above an impedance surface, especially when the distance between the source and the test surface is small—a condition frequently encountered in situ measurement scenarios. There are two typical spherical wave models for the locally reactive material: the F-term model [17], and the Q-term model [18, 19]. The F-term model yields an asymptotic solution valid under high-frequency conditions or when the source-receiver distance is large. In contrast, the Q-term model enables precise computation of the sound field through a numerically stable image integral formulation. For non-locally reacting materials, Allard [20] proposed a spherical wave-based approach that has been employed to characterize their acoustic behavior. However, in the application of the spherical wave model, surface impedance cannot be directly obtained from experimental acoustic measurements, requiring a supplementary analytical process. Hess [21] determined the impedance values by examining the difference in sound pressure levels recorded using two microphones positioned in a vertical alignment. A similar approach was adopted by Nocke [22, 23], who instead utilized excess attenuation data rather than level differences. More Studies [24–29] have employed the specific impedance—readily measurable using a PU-probe—to infer the surface impedance of materials. In the measurement setup using the PU-probe, two physical quantities are directly measured: the sound pressure, obtained from the miniature microphone integrated in the probe, and the particle velocity, obtained from a Microflown sensor. The specific impedance is then derived from the ratio of these two directly measured quantities. Notably, the PU-probe [24] exhibits strong resilience to variations in surface area effects. Moreover, impedance measurement systems incorporating PU-probes can be designed with greater compactness. Given these advantages, this study concentrates on methodologies grounded in specific impedance measurements.
This study further explores the derivation of surface impedance and absorption coefficient using the spherical wave model. Leveraging the advantages of the PU-probe highlighted previously, specific impedance—instead of excess attenuation or sound pressure level difference—is chosen as the input for the deduction process. The primary contributions of this work are as follows. First, the complex image model, a spherical wave-based approach introduced by Li and White [31], is introduced and validated as a novel inverse algorithm for material characterization. This establishes a complete measurement and deduction pipeline for both locally and non-locally reacting materials using PU-probe data. Second, it is demonstrated that for locally reacting surfaces, the complex image method provides a computationally efficient alternative to the F-term and Q-term methods. It achieves results nearly identical to the Q-term method but requires approximately one-third of the computation time by avoiding numerical integration. Third, for non-locally reacting materials, the complex image method is shown to be an order of magnitude faster (one-tenth the time) than Allard’s approach, while maintaining comparable accuracy. This gain in efficiency highlights its potential for in situ or real-time measurements.
2 Theoretical background
Once the specific impedance (), defined as the ratio of sound pressure to particle velocity at a given measurement point, is determined, the surface impedance () of the material under test can be directly computed from it when employing either the plane wave model or the mirror image model. But for the in situ measurement, the measurement equipment would better be portable. Therefore, the sound source needs to be placed near the surface of the material under test. In this case, it is more appropriate to use the spherical wave model, especially at low frequencies. However, the surface impedance of the material cannot be estimated directly because the calculation of the spherical wave field also needs the value of the surface impedance which is to be identified. Therefore, a deductive procedure is required to determine the surface impedance of the material. Two main factors are involved in the deduction procedure: One is the spherical wave model describing the sound field above the material and the other is the deduction algorithm. These two factors are to be described in the following.
2.1 Spherical wave models
A time-harmonic point source with angular frequency is situated at point above an infinite surface characterized by a reflection coefficient , as illustrated in Figure 1. The real wavenumber in air is given by , where represents the speed of sound in air. The time factor is assumed throughout and suppressed for notational simplicity. Applying the spatial Fourier transform, the sound pressure at receiver location can be expressed aswhere denotes the zeroth-order Bessel function of the first kind, , , and represent the receiver height, the source height, and the horizontal distance between the source and receiver, respectively, is the horizontal wave number, , the vertical wavenumber in air, is expressed as , and is the distance between the real source and the receiver. The reflection coefficient depends on the sample’s characteristics (locally reactive or not) and configuration of the sample (with infinite thickness or finite thickness backed by a rigid wall). In the present paper, the sample with a thickness backed by a rigid wall is considered. The reflection coefficient in Equation 1 is given as follows:where is the normalized surface impedance of the material, with being the characteristic impedance of the air and the surface impedance of the material, , , and are the complex wave number, characteristic density and characteristic impedance of the material under test, respectively, and is the vertical wave number in the material.
FIGURE 1
2.1.1 Typical sound wave models for locally reactive materials
2.1.1.1 F-term model
Chien and Soroka [17] used steepest decent technique to evaluate the integral in Equation 1. The sound pressure at the receiver is then given bywhere is the plane-wave reflection coefficient andis defined as the boundary loss factor. The parameter is designated as the numerical distance,and is called the complementary error function, which can be calculated as
Note that Equation 4 is valid under the condition of [17].
2.1.1.2 Q-term model
Di and Gilbert [18] proposed an exact solution for the sound field of a point source above an impedance plane by representing the plane-wave reflection coefficient as the Laplace transform of an image source distribution. For a locally reactive plane, the sound pressure at the receiver can be obtained fromwhere .
In the situation that the value of () and the imaginary part of are both very small, the amplitude of the integrand in Equation 8 shows a distinct peak at where a steep descent of the amplitude of the integrand occurs [30]. When evaluating the integral numerically, enough integration points should be used to track this steep descent. For example, Ochmann applied an adaptive multigrid quadrature by which the integration points accumulate automatically in the vicinity of . In fact, if the sound source and receiver are at some typical measurement heights, like = 0.1 m and = 0.03 m, the amplitude of the integrand will decrease monotonically with the increase of , and the peak no longer arise. Another thing that should be noted in the Q-term model is the upper limit of the integral in Equation 8. In some references the upper limit was determined by an empirical formula [18], , where is the wave length. But as observed in the simulations the value of obtained by this formula sometimes is overlarge so that the amplitude of the integrand at is too small, which will lead to the floating point overflow in the numerical integration. In this paper the upper limit is obtained by another way, i.e., truncating the integral at the position where the amplitude of the integrand decreases to a sufficiently small value .
2.1.2 Typical sound wave model for non-locally reactive materials
Allard [20] proposed a model for calculating the sound field generated by a point source above a non-locally reactive material with finite thickness backed by a rigid wall. The pressure at the receiver is given by,where , , , , is the effective complex wavenumber and is the effective density in the material. Note that numerical integration is needed to calculate the third term in Equation 9. Equation 9 has been used for the measurement of characteristics of non-locally reactive sample [27].
2.1.3 Complex image model for non-locally and locally reactive materials
Li and White [31] proposed an efficient method for computing the sound field above an infinite surface, and this method is very accurate for the computation in near field where usually the in situ impedance measurement is performed. In the complex image model, the slowly varying factor in the integrand of Equation 1 is approximated by a short series in terms of complex exponential function. Then, by applying the Sommerfeld identity, Equation 1 is transformed into,where is the complex distance. for locally reactive material, and for non-locally reactive material. is the number of complex images, which was advised to be chosen from 3 to 5 [31], and and are complex numbers that satisfy
The details of derivation from Equations 1–10 and calculation of and can be found [31, 32], and are not shown here for the sake of brevity. Note that Equation 10 does not involve numerical integration, and thus the time needed for the computation of sound field is short. Therefore, it is expected that deducing the characteristics of a material based on this model is very efficient, because the deduction procedure involves repeated calculation of sound field generated by a point source above the material under test. This is essential for the non-locally reactive material, because the number of iteration required when deducing the characteristics of non-locally reactive material is usually far bigger than that required when deducing the characteristics of locally reactive material. Note that the particle velocity at the receiver point can be obtained through Euler’s equation for all above spherical wave models. Thus the specific impedance at the receiver can be calculated by using these models.
2.2 Deduction algorithms
2.2.1 Deduction algorithm for locally reactive materials
Actually, deducing the surface impedance of locally reactive material is equivalent to solving an implicit non-linear equation,where is the frequency. is the specific impedance predicted by a kind of spherical wave model. If is the analytical function of , and therefore has a continuous derivative, the Newton-Raphson method [33] can be used to solve this non-linear equation. The Newton-Raphson method approaches the root of Equation 12 by the following iteration procedure,where the prime denotes the differentiation with respect to the argument. If is not the analytical function of or it is hard to obtain the derivative of , the Secant method [34] has to be used. In the Secant method, the differentiation in Equation 13 is estimated by the finite difference. Therefore, two initial approximations are needed in the Secant method rather than one in the Newton-Raphson method. An alternative method for solving Equation 12 is the minimization method. The value of normalized surface impedance that best fits the specific impedance, , can be found by minimizing the objective function .
2.2.2 Deduction algorithm for non-locally reactive material
For a locally reactive material, is a function of only one unknown quantity: the normalized surface impedance. But for a non-locally reactive material, depends on two parameters, and . Actually, deducing the surface impedance of non-locally reactive material is equivalent to solving a system of implicit non-linear equations,
The least-squares method can be employed to find and that yield the optimal fit to the experimentally obtained impedance with respect to the source height . The similar method has been used by [27]. Then, the normalized surface impedance at an angle can be obtained by the following equations
2.2.3 Acquirement of the initial approximation of sensitive parameter
It should be noted that the acquirement of the initial approximation of sensitive parameter(s), for locally reactive material and and for non-locally reactive material, is essential for the deduction procedure. For locally reactive material, the initiative approximation of can be obtained by the PWA method [26] which utilizes the mirror model. For non-locally reactive material, the initiative approximations of and can be obtained by the following strategy proposed by Brandao. The normal absorption coefficient is calculated as if the material is locally reactive. Then a least square fit is performed to find the flow resistivity that, when inserted into the model proposed by Delany and Bazley [35], would result in the best fit of the normal absorption coefficient obtained above. A flow resistivity which is 75% of the flow resistivity obtained above is re-inserted into the Delany and Bazley model to find the initiative approximation of and .
When the surface impedance of the material is obtained, it is easy to calculate the normal-incidence absorption coefficient from the following formulation.
3 Numerical experiments
3.1 Locally reactive material
To take the effect of edges of the material into consideration, the BEM was used to calculate the specific impedance , like what Hirosawa [25] and Brandao [26] have done. The schematic illustration of BEM calculation is shown in Figure 2. Using the external direct BEM, one can obtain [36]where is the free-field Green’s function in three dimensional space, is an arbitrary point on the surface, is if the receiver is located on the surface and is if is located at any point above the surface. The surface is discretized into elementary elements, and with the receiver positioned on it, the surface sound pressure is obtained by solving a system of linear equations. After computing the surface pressure distribution, this result is substituted into Equation 18 to calculate the sound pressure at any point above the surface. Subsequently, the particle velocity is derived from Euler’s equation. In this paper, the quadrilateral constant element was used, the integral in Equation 18 was evaluated by standard Gauss-Legendre quadrature using 6 points.
FIGURE 2
In the numerical experiment a squared material with dimensions of 1 m × 1 m × 0.025 m was under test and was assumed to be of the Allard and Champoux type [37], the flow resistivity, porosity and tortuosity of the material were 25 cgs, 0.96 and 1.1, respectively, the source height () was 0.3 m, the receiver height () was 0.03 m, the number of complex images was 5, and the highest frequency was 4 kHz. The reason to choose 4 kHz as the highest frequency is that as the highest frequency increases, the size of basic elements needs to decrease, but it would be difficult to simulate with BEM because of the computational cost. Regarding the performance at higher frequencies (up to 6–7 kHz), it is expected that the proposed method will maintain or even improve its accuracy. This is because at higher frequencies, the wavelength becomes shorter relative to the material dimensions, and the specific impedance exhibits greater sensitivity to parameter variations, which generally benefits the deduction process. Therefore, while direct validation beyond 4 kHz is not presented, the trend observed in the following results suggests reliable performance in the higher frequency range. The specific impedance at the same receiver was also calculated using Equation 8 under the assumption of an infinitely large material. By comparing the specific impedance used for deduction with the BEM simulation result, the SNR was found to vary with frequency, ranging from approximately 22 dB–42 dB across the analyzed bandwidth. The integral in Equation 8 was evaluated by the adaptive Gauss-Kronrod quadrature (using the function “quadgk” of MATLAB).
Figures 3, 4 show the results of deduced normalized surface impedance and normal absorption coefficient based on the three spherical wave models, respectively. The legend “Allard-Champoux” describes the theoretical reference value of normalized surface impedance and absorption coefficient calculated by physical model [37]. It can be figured out that the results of the method based on the locally complex image model (referred to hereafter as “locally complex image method”) are almost the same as those of the method based on the Q-term model (referred to hereafter as “Q-term method”). The oscillation around the theoretical normal absorption coefficient, observed in Figure 4, is due to the edge effect of finite surface area. The same oscillation was also found by Hirosawa [25] and Brandao [26]. They showed that the amplitude of the oscillations can be reduced by moving the receiver off-center. Further work has been done by Brandao and an empirical formulation calculating the optimal position of receiver has been given.
FIGURE 3
FIGURE 4
Figures 5, 6 show the results when the receiver was put at the optimal position (0.3, 0.1, 0.03) m, which is recommended by Brandao [28]. It can be seen that the locally complex image method still leads to almost the same results as those of Q-term method. For the surface impedance, the oscillation disappears, but large discrepancies between real parts of surface impedance obtained by the complex image method or Q-term method and the theoretical value are observed below 300 Hz. These deviations might be due to the insensitivity of the specific impedance to the change of the surface impedance. However, it should be emphasized that the imaginary part of the surface impedance shows good consistency with theoretical values throughout the entire frequency range (from 100 Hz to 4,000 Hz). Moreover, since the imaginary part is considerably larger than the real part at low frequencies, the normal absorption coefficient—which depends on both parts—remains accurate across the full frequency band analyzed. Therefore, for practical applications where the absorption coefficient is the primary concern, the proposed method can be reliably used over the entire frequency range, including below 300 Hz. In contrast, for applications dedicated to accurately estimating the real part below 300 Hz, additional caution or supplementary measurement strategies are needed.
FIGURE 5
FIGURE 6
It should be noted that, under the passive material assumption, the absorption coefficient is theoretically bounded between 0 and 1. However, a few instances where the values become slightly below 0 are observed, and these occurrences are also confined to frequencies below 300 Hz. This phenomenon is attributed to the large errors in the real part of the surface impedance in the low-frequency range, as discussed earlier.
In Figures 3–6, the results of the method based on the F-term model (referred to hereafter as “F-term method”) have large deviation compared with the theoretical value at low frequencies, at which the condition of is not satisfied.
Based on the analysis above, the locally complex image method leads to almost the same results as the Q-term method does, and has better performance than the F-term method. But, the time for the whole deduction procedure (totally 49 frequencies) when using the complex image method is nearly one-third of that when using the Q-term method. If more frequencies are considered and less powerful equipment are used, which is the fact for the in situ measurement, the complex image model would be the better choice than the Q-term model. The same conclusions have been obtained when the flow resistivity of the material under test was changed. Note that the results given in this paper were all obtained by using the Secant method. In the secant method, the iteration termination criterion is set such that the difference between two consecutive iteration results is less than 10–5. It was observed that approximately 5–9 iterations were required for each frequency. All calculations were performed on a computer equipped with an Intel Core(TM) i5-2380P processor and 4 GB of RAM. Spending more time on the procedure of deduction, the minimization method (using the function “fminsearch” of MATLAB) leads to the same results as those obtained by the Secant method, which are not given here for the sake of brevity.
3.2 Non-locally reactive material
Since BEM is not suitable for calculating the sound field generated by a point source above a non-locally reactive material, the specific impedance at the receiver is calculated by Equation 9 instead of BEM. Five source heights ( = 0.1m, 0.15m, 0.2m, 0.25m, 0.3 m) are chosen. The flow resistivity is set to 10 cgs so that the material under test can be treated as a non-locally reactive material. In this case, the sample under test is assumed to be infinite, and there is no error caused by the edge effects, so both the source and receiver are located at the -axis. The other parameters used in this section are the same as used in Section 3.1. The locally complex image method is used to find the initiative approximations of and .
Figures 7, 8 show the results of normalized surface impedance and absorption coefficient in the normal incidence deduced by the locally complex image method, by the method based on the non-locally complex image model (referred to hereafter as “non-locally complex image method”), and by the method based on the Allard model (referred to hereafter as “Allard method”). The locally complex image method leads to large deviation, which indicates that it is not appropriate to utilize a sound wave model for locally reactive material when the flow resistivity of the material is small. It can be figured out that the results of the non-locally complex image method are almost the same as those of the Allard method, and both have a good agreement with the theoretical value. But the time cost in the deduction procedure when using the Allard model is ten times of that when using the complex image model.
FIGURE 7
FIGURE 8
3.3 Noise analysis
To assess the robustness of the proposed method under realistic in situ conditions—where factors such as finite sample size and edge scattering introduce deviations from the ideal wave field—a sensitivity analysis was performed by adding controlled levels of Gaussian white noise to the input specific impedance. Signal-to-noise ratios (SNR) were considered to simulate increasing levels of measurement disturbance. The deduction procedure was then repeated on the noisy data to evaluate the stability and accuracy of the deduced surface impedance for both locally and non-locally reacting materials.
The results reveal that for both material types, the method remains stable for SNR greater than 20 dB, producing surface impedance values that closely match the noise-free case. However, when the SNR is smaller than 20 dB, the results become unacceptable. Further analysis indicates that the degradation is primarily caused by excessive errors in the real part of the surface impedance, which is inherently less robust to measurement uncertainties. This finding is consistent with the earlier observation that the real part inversion shows reduced accuracy, particularly at low frequencies.
4 Conclusion
This paper focuses on the deduction of the surface impedance and absorption coefficient of locally and non-locally reactive materials in free field. A deduction method based on the complex image model is introduced. For the locally reactive material, the locally complex image method can present almost the same result as that obtained by the Q-term method, but costs less time than the Q-term method (nearly one-third for 49 frequencies) since the complex image model does not involve the numerical integration. For the non-locally reactive material, the similar results have been obtained. The time cost of the non-locally complex image method is one-10th as that of the Allard method. Therefore, the deduction method based on the complex image model has a potential application to the in situ or real-time measurement, especially for the non-locally reactive material. It should be noted that, although numerical experiments provide a controlled assessment of accuracy and efficiency, serving as a necessary foundation before proceeding to physical measurements. Physical validation on real materials is recognized as an important direction for future work.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
YZ: Data curation, Methodology, Conceptualization, Investigation, Formal Analysis, Writing – review and editing, Writing – original draft. KW: Project administration, Supervision, Resources, Writing – review and editing. WY: Writing – review and editing, Validation, Funding acquisition.
Funding
The author(s) declared that financial support was received for this work and/or its publication. This work was supported by Anhui Provincial Education Department Scientific Research Project (No. 2025AHGXZK20193), Natural Science Foundation Research Program of Anhui Higher Education Institutions (No. KJ2020B13) and Chuzhou University Key Teaching Research Project (No. 2025jyz029, 2023jyz011).
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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Summary
Keywords
absorption coefficient, free field, locally reactive, non-locally reactive, surface impedance
Citation
Zhang Y, Wang K and Yu W (2026) A method for determining the surface impedance and absorption coefficient of locally and non-locally reactive acoustic materials in free field. Front. Phys. 14:1785611. doi: 10.3389/fphy.2026.1785611
Received
12 January 2026
Revised
13 March 2026
Accepted
31 March 2026
Published
29 April 2026
Volume
14 - 2026
Edited by
Glauber T. Silva, Federal University of Alagoas, Brazil
Reviewed by
José Henrique Araújo Lopes, Federal University of Alagoas, Brazil
Asif Israr, Institute of Space Technology, Pakistan
Updates
Copyright
© 2026 Zhang, Wang and Yu.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Kai Wang, wangkai@ujs.edu.cn
Disclaimer
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