# Simplified Three-Microphone Acoustic Test Method

^{*}

## Abstract

**:**

^{2}= 0.994. A simulation study highlighted several unique accuracy advantages of this new proposed method in comparison to the existing standard methods. The proposed new method represents an easy-to-use technique that requires little in the way of equipment and can be set up with minimal training and expense.

## 1. Introduction

^{−3}), 42.5 mm thick glass wool (density of 40 kg·m

^{−3}), and 76 mm acoustical/thermal insulator wood (density of 38.5 kg·m

^{−3}). The techniques they reported on were:

- The two-microphone dual cavity-backed method [13].
- A modified variant of the Iwase method [14] that utilized microphones on both sides of a sample backed by a hard reflector, with no air space between the sample and the reflector. The modification was to utilize a two-microphone method up-stream, versus the original Iwase method that used a microphone directly on the surface of the sample. They noted that their proposed modification enhanced the speed and utility of the original Iwase method. Of note however, is that this modification was only designed to streamline the test method for speed. It was not designed to improve the accuracy of the method.

## 2. Materials and Methods

#### 2.1. Theoretical Derivation

_{i}(x,t) = |P

_{i}| e

^{(−jk}°

^{x + θi)}e

^{(jωt)}

_{r}(x,t) = |P

_{r}|e

^{(jk}°

^{x + θr)}e

^{(jωt)}

_{i}(x,t) is the incident wave as a function of x and t, P

_{r}(x,t) is the reflected wave as a function of x and t, |P

_{i}| is the magnitude of the incident wave, |P

_{r}| is the magnitude of the reflected wave, t is time (s), j is complex value where j = sqrt(−1), and x is position with respect to the sample surface (m). The propagation wave number of the acoustic wave in air (rad·m

^{−1}) is given by k

_{o}, and ω is the frequency of the sound wave (rad·s

^{−1}).

_{i}(x) + P

_{r}(x) = |P

_{i}| e

^{(−jkox)}+ |P

_{r}| e

^{(jkox)}

^{−1}).

_{i}and P

_{r}, respectively):

_{1}= P

_{i}e

^{(jko|x1|)}+ P

_{r}e

^{(−jko|x1|)}

_{2}= P

_{i}e

^{(jko|x2|)}+ P

_{r}e

^{(−j ko|x2|)}

_{f}= P

_{M0}= P

_{i}+ P

_{r}

_{M0}is the pressure at the location corresponding to microphone 0.

_{r}in terms of P

_{i}, as modified by transport through the material twice (forward and back). Of importance to note here is that for many materials, reports have been provided for which α is not constant across the frequency spectrum and is, for some materials, a function of the frequency [15,16,17]. Prime examples of materials that exhibit this behavior are water, metals and other crystalline objects [18]. The literature reports, noted above, report success in utilizing a simple power-law model that describes how α changes with frequency, and is shown here for convenience in Equation (7):

_{o}(2πf)

^{η}

_{o}is a very low frequency value for the propagation attenuation constant α, and f is the frequency (Hz).

_{r}= P

_{i}e

^{(−2d}

^{γ)}= P

_{i}R

^{(−2d}

^{γ)}= e

^{(−2d[}

^{α + jβ])}

= e

^{(−2d}

^{α)}e

^{(−2 jβd)}

^{(−2d}

^{α)}

_{i}and P

_{r}yields the pressure P

_{f}at the sample surface (location of microphone 0), which is shown in Equation (9); noting P

_{r}= P

_{i}e

^{(−2γd)}provides

_{f}= P

_{M0}= P

_{i}+ P

_{r}

P

_{f}= P

_{i}+ P

_{i}e

^{(−2}

^{γd)}

_{bi}= P

_{i}e

^{(−}

^{γd)}

_{bi}, there is also the reflected pressure wave, P

_{br}, at the microphone 3 location and shown as Equation (11):

_{br}= P

_{i}e

^{(−}

^{γd)}

_{M3}= P

_{bi}+ P

_{br}= 2 Pi e

^{(−}

^{γd)}

_{30}= P

_{M0}/P

_{M3}= [Pi + Pi e

^{(−2}

^{γd)}]/(2 Pi e

^{(−}

^{γd)})

= ½ (e

^{(}

^{γd)}+ e

^{(−}

^{γd)})

_{30}= cosh(γd)

_{30})/d

^{(−2}

^{γd)}

_{30}; as such, the measurement must measure the phase as well as the magnitude.

_{30}. Starting with the definition of H

_{30},

_{30}= P

_{M0}/P

_{M3}=[Pi + Pi e

^{(−2}

^{γd)}]/(2 Pi e

^{(−}

^{γd)})

= ½ (e

^{(}

^{γd)}+ e

^{(−}

^{γd)})

= ½ [e

^{(d[}

^{α + jβ])}+ e

^{(−d[}

^{α + jβ])}]

_{30}= ½ [e

^{(d}

^{α)}e

^{( jβd)}+ e

^{(−d}

^{α)}e

^{(−jβd)}]

_{30}= ½ [e

^{(d}

^{α)}cos(ωt + βd) + e

^{(−d}

^{α)}cos(ωt − βd)]

H

_{30}= ½ [e

^{(d}

^{α)}{cos(ωt)cos(βd) − sin(ωt)sin(βd)} + e

^{(−d}

^{α)}cos(ωt − βd)]

_{33}as shown in Equation (15), using Equation (12) as the expression for P

_{M3}:

_{33}= P

_{M3}/P

_{M3air}= [2 Pi e

^{(−γd)}]/[2 Pi e

^{(−γaird)}]

= e

^{[−d (}

^{α}

^{+ jβ)]}/e

^{[−d (}

^{αair + jβair)]}

_{air}= 0, then converting into the time domain leads to Equation (16):

_{33}= [e

^{(−}

^{αd)}cos(ωt − βd)]/[cos(ωt − β

_{air}d)]

_{33}= e

^{(−}

^{αd)}cos(ωt − θ)

= e

^{(−}

^{αd)}cos(ωt − d(β−β

_{air}))

_{air}) is the relative phase between the sampled signal to the air reference reading.

^{(−αd)}. Hence the attenuation of the signal, as induced by the presence of the sample, is readily obtained by measuring the time-averaged signal strength with and again without a sample in the test cell, and then taking the ratio of the two measurements. In practice, the measurement of the sound pressure level utilizes a measurement of the root-mean-squared (rms) value of the signal, which in the discrete form of a time-averaged version of the magnitude of H

_{33}is provided by Equation (18):

_{33}|

_{RMS}= [(1/n) (∑ (H

_{33})

^{2})]

^{0.5}

_{33}|

_{RMS}is the rms estimate of the magnitude of H

_{33}, and n is the number of readings to perform the sum of squares over.

_{33}|

_{RMS}that provides Equation (19).

_{33}|

_{RMS}= e

^{(−}

^{αd)}

_{M3}|

_{RMS}= V

_{pp_signal}e

^{(−}

^{αd)}/(2)

^{0.5}

_{M3 air}|

_{RMS}= V

_{pp_signal}/(2)

^{0.5}

_{33}|)/d

^{2}

^{(−2dα)}(Equations (8) and (14) re-stated for clarity), and ρ is the acoustic attenuation constant (dimensionless; range of 0–1).

#### 2.2. Accuracy

_{30}is a mathematical identity to the other standard methods; as such, theoretically they should all provide exactly the same value. Hence, of primary interest is how the standard methods compare to:

- The simplified variant of the SMM method, SSMM (|H
_{33}|; pressure magnitudes only). - The complex SMM variant the CSMM (H
_{30}; pressure and phase). - The simplified variant of CSMM (|H
_{30}|; pressure magnitudes only).

_{30}|), was not derived from first principles; as such, the simulation study did not include noise, but instead examined how large the error was when the full complex values with phase information were not captured in the measurements. Of potential interest for this variant is in the application for online systems, where it is inconvenient or impossible to obtain an air-reference, periodically throughout the day, that would preclude the use of the SSMM (|H

_{33}|) method, while still preserving the ease and simplicity of a magnitude-only version.

#### 2.2.1. Accuracy Comparison in the Presence of Noise

#### 2.2.2. Accuracy Comparison for the Online Pressure Only

_{33}. This yields the lowest-cost approach that can be performed with a simple tube, a sample holder and an inexpensive sound meter. This raises the question as to the accuracy of such a simple and low-cost system. To examine this, a simulation comparison was performed, whereby the material properties of the sample were allowed to vary across the full range expected, for typical materials such as air, water, metals, wood, rubber cork and others, as reported in Beis and Hansen [19]. For most of these materials, the propagation velocity is higher than that of air, yielding a wave-number of β < 1; in the exception to these typical materials are a few unique materials such as cork and rubber, for which the maximum β is at most equal to 5. Of note is that the authors were unable to find materials in the literature for which the velocity was slower than one-fifth that of air (β > 5). Hence, the presented range should be representative of nearly all materials of interest, with the bulk of the materials of interest having β < 1 [19]. Also of note is that light, fibrous materials that are full of air can be expected to exhibit β’s near 1. Noting all these factors then provided the bounds for the simulation study, whereby the values of α were allowed to range from very small, α = 5, to large, α > 200. Across this two-parameter variational boundary, the error was tracked and is presented in the results section as the deviation for the measured α and its associated reflectance, R. Again, the study utilized a sample thickness of 19.05 mm at a temperature of 23 °C.

#### 2.2.3. External Accuracy Factors

#### 2.3. Experimental Confirmation Testing

- Hewlett Packard, Santa Clara, CA, USA, 33120A signal generator
- Agilent, Santa Clara, CA, USA, DSO1024 digital oscilloscope
- Peavey, Meridian, MS, USA, Power Amplifier IPR-1600 DSP
- Crowne Audio, Elkhart, IN, USA, 15 cm diameter speaker
- Behringer, Behringer City, China, two cascaded equalizers to provide a full range of attenuation of +/−24 dBu
- Earth instrument microphones
- Extech Instruments, Nashua, NH, USA, NIST Traceable Sound Level Meter 407,732
- Extech Instruments, Nashua, NH, USA, NIST Traceable Piston Sound Level Calibrator 407,722
- TC20 Earthworks measurement microphone (mounted in the end-reflection plate for through-transmission characterization)
- FMR Audio, Austin, TX, USA, RNP 8380, microphone pre-amplifier and phantom voltage source (two units, one for each microphone)
- PCB Piezotronics, Depew, NY, USA, 426B03 Pre-polarized Condenser Microphone (used on the traveling trolley, for inside the tube standing-wave ratio, for SWR measurements, and via the ISO 10534-1 standard method)
- PCB Piezotronics, Model 482C15 ICP sensor signal conditioner

#### 2.4. Experimental Design

## 3. Results

#### 3.1. Accuracy Comparison in the Presence of Noise

_{33}|) and the CSMM method (which utilizes measurements of magnitude and phase at microphone locations 0 and 3 to obtain the complex transfer function H

_{30}) as a comparison to the standard methods: the SWR (ISO 10534-1)[2] and the two-microphone methods (ASTM E1050 and ISO 10534-2) [5,6]. The metric selected for the performance comparison, as impacted by noise, was the percent error of the estimate of the true value of the reflectance coefficient R, which was tracked as the reflection coefficient varied from near 0 to 0.99 with a Gaussian white noise level of 0.005. As the value for α is built into R, the only other variable to be accounted for was β. To examine the influence of β on the results, β was allowed to vary from 0.1 to 5.25, and the results are detailed in Figure 5, Figure 6 and Figure 7.

_{30}, exhibited much better performance for highly reflective samples, but then faltered as the absorption decreased. It was especially poor for low values of β. Of particular interest, however, is the uniformly strong performance provided by the SSMM method that utilizes the |H

_{33}| transfer function by which to measure R. Out of all the methods, the SSMM technique proved to be the most robust estimator when presented with a real signal that included noise.

#### 3.2. Accuracy Comparison for Online System

_{30}| transfer function. As this method was not derived from first principles, it was expected that errors would be present in comparison to the first-principle methods of the CSMM methodology. To examine the potential for this alternative simplified method, a simulation comparison was performed, whereby the material properties of the sample were allowed to vary across the full range expected from typical materials such as air, water, metals, wood, rubber cork and others, as reported in Beis and Hansen [19]. For most of these materials, the propagation velocity is higher than that of air, yielding a wave-number of β < 1, and in exception to these typical materials are a few unique materials such as cork and rubber, for which the maximum β is at most equal to 5. This then provided the bounds for the simulation study, whereby the values of α were allowed to range from very small, α = 5, to large, α > 200. Across this two-parameter variational boundary, the error was tracked and is presented in Figure 8 as the deviation for the measured α, and in Figure 9 as the deviation of measured R.

#### 3.3. Experimental Confirmation

_{33}| method and the original Iwase method produced a very strong correlation, resulting in a coefficient of determination of r

^{2}= 0.994. There was a minor small slope and offset bias. As such, in practice, for comparative studies utilizing only the new SSMM method, the near-perfect correlation provides strong supporting evidence for this methodology. Figure 10 details the regression analysis for how well the new formulation tracked the original Iwase [14] three-microphone method.

## 4. Discussion

_{33}| transfer function, is similar to running spectroscopy samples whereby a blank reference sample is periodically run. As such, it provides the accuracy of the one-microphone method utilized for the two- and four-microphone methods, where the microphone is physically moved between readings, while avoiding the burden of physically moving the microphone, which is tedious and time-consuming. Hence, if this method is automated utilizing the mono-frequency approach, it provides enhanced accuracies over the afore-mentioned methods, and can obtain a scan in a relatively short duration of 1–5 s, depending upon the integration times selected to reduce noise levels in the measurements.

_{33}transfer function, with relative-phase capture, is expected to outperform the standard methods for samples with low reflectance properties, and should be the method of choice for these materials, circumstances of the application permitting.

## 5. Conclusions

^{2}= 0.994 (Figure 10 and Figure 11).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Disclaimer

## References

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**Figure 1.**Schematic diagram of the experimental apparatus detailing the testing configuration for performing the standard two-microphone method, the 1998 three-microphone method of Iwase et al., to obtain complex material propagation coefficients.

**Figure 2.**Picture showing the test apparatus utilized in the study. The apparatus has multiple upstream microphones for performing ISO 10534-2 as well as an insertion traveling microphone, attached to the cable seen entering the tube just to the right of the speaker on the far left of the tube. Seen on the far right of the tube is the silver aluminum sample holder that has a microphone embedded into the side wall to perform the sample surface pressure measurement: Iwase [12] microphone 0. At the far right is Iwase [12] transmission microphone 3 that is embedded in the end reflection-plate.

**Figure 3.**Picture showing the various instrumented end plates for the experimental test apparatus utilized in the study. Each of these end plates provided a different means to obtain the Iwase [12] microphone 3 measurements. The image (

**a**) on the left shows a microphone embedded into the end reflection plate. In the middle image (

**b**) is the sample holder as well as an instrumented movable piston-style reflection plate. The piston configuration allowed for easy adjustment of the reflection plate location. Seen on the far right in image (

**c**) is a reflection plate with an NIST traceable A-weighted sound meter mounted in a pressure-tight sealed reflection plate. Option (

**c**) provided the easiest method of the three, and provided the lowest-cost entry into acoustic testing.

**Figure 4.**Comparison chart depicting pressure inside the tube with and without noise for two different reflection coefficients R. The large amplitude signal is the response when the sample’s R was 0.99. The low-amplitude signal is the response for a sample with a much lower R of 0.05. The frequency for this test case was 1000 Hz with a material-relative phase delay of β = 1.20.

**Figure 5.**Comparison of accuracy from the computation of R, with a normalized noise level of 0.005 and a β value of 0.1. Comparison is between the two standard methods, SWR [2] and the two-microphone method [5,6], and the technique utilized in the three-microphone method [14], versus the new proposed methods utilizing the simplified |H

_{33}| method (amplitude only measurements) and the technique that uses a measurement of H

_{30}(complex measurements that include phase relationships).

**Figure 6.**Comparison of accuracy from the computation of R, with a normalized noise level of 0.005 and a β value of 2.0. Comparison is between the two standard methods, SWR [2] and the two-microphone method [5,6], and the technique utilized in the three-microphone method [14], versus the new proposed methods utilizing the simplified |H

_{33}| method (amplitude only measurements) and the technique that uses a measurement of H

_{30}(complex measurements that include phase relationships).

**Figure 7.**Comparison of accuracy from the computation of R, with a normalized noise level of 0.005 and a β value of 5.25. Comparison is between the two standard methods, SWR [2] and the two-microphone method [5,6], and the technique utilized in the three-microphone method [14], versus the new proposed methods utilizing the simplified |H

_{33}| method (amplitude only measurements) and the technique that uses a measurement of H

_{30}(complex measurements that include phase relationships).

**Figure 8.**Error comparison for the CSMM simplified new method utilizing only pressure-magnitude readings |H

_{30}|. Error is reported as the magnitude of the absolute value of the error of the predicted value of α. Error magnitude was computed as material propagation changes occurred for both α and β (typical materials have a propagation delay with relative β < 1).

**Figure 9.**Error comparison for the CSMM simplified new method utilizing only pressure-magnitude readings |H

_{30}|. Error is reported as magnitude of the absolute value of the reflection coefficient R.

**Figure 10.**Correlation analysis for attenuation between A-weighted response, as measured utilizing the Iwase [12] three-microphone method, versus the new proposed SIM method.

**Figure 11.**Shown above is a comparison graph for the Iwase [14] method versus the new proposed simplified SSMM method (after correction for bias and slope offset). The comparison utilizes the A-weighted spectral reduction metric to provide a single-valued comparison, from multi-frequency spectral data.

**Figure 12.**Simplified acoustic testing equipment setup that is enabled through the use of the SIM method.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pelletier, M.G.; Holt, G.A.; Wanjura, J.D.
Simplified Three-Microphone Acoustic Test Method. *Instruments* **2017**, *1*, 4.
https://doi.org/10.3390/instruments1010004

**AMA Style**

Pelletier MG, Holt GA, Wanjura JD.
Simplified Three-Microphone Acoustic Test Method. *Instruments*. 2017; 1(1):4.
https://doi.org/10.3390/instruments1010004

**Chicago/Turabian Style**

Pelletier, Mathew G., Greg A. Holt, and John D. Wanjura.
2017. "Simplified Three-Microphone Acoustic Test Method" *Instruments* 1, no. 1: 4.
https://doi.org/10.3390/instruments1010004