1 Introduction

An impedance tube is primarily used to measure the acoustic properties of materials. This apparatus consists of a tube, a sample holder, a sound source and one or more microphones. The primary use of this device is to determine the frequency-dependent normal incident sound absorption coefficient of materials, denoted here as \(\alpha \). This property is essential for understanding how materials can be used for sound insulation or noise control in various applications, such as building acoustics [1, 8, 14], 3D printed metallic material [2, 3], metamaterial [2, 3] and automotive design [19]. Impedance Tube test results also provide insight into the physical properties of porous materials such as tortuosity and porosity [7, 11, 16, 18].

A standard impedance tube can be made using one microphone that travels along the tube while the sample is exposed to a monotone sound [6] or uses two [4, 12] or more microphones [5] while the sample is exposed to a broadband noise. The advantage of using an impedance tube with multiple microphones is that sound absorption of the sample can be measured in a wide range of frequencies in a single measurement, and this method is widely accepted as the standard method [13]

Figure 1 shows the schematic of a two-microphone impedance tube. The speaker generates broadband white noise, and acoustic pressure is collected simultaneously by the two microphones, M1 and M2. The transfer function \(H\) is the ratio of the cross-power spectrum (\({G}_{12}\)) to the auto-power spectrum (\({G}_{11}\)) of M1 about M2:

Fig. 1
figure 1

A two-microphone impedance tube to measure the sound absorption coefficient in a medium

Full size image
$$\overline{H }=\frac{{G}_{12}}{{G}_{11}}$$
(1)

Using the transfer method introduced by [9], the Complex Reflection coefficient (\(R\)) is calculated as follows:

$$R=\frac{H-{e}^{-\text{jks}}}{{e}^{\text{jks}}-H}{e}^{j2k\left(l+s\right)}$$
(2)

where \(s\) is the microphone spacing, \(k=2\pi f/c\) and is the sound wavenumber in the air, \(c\) is the speed of sound, \(l\) is the distance between the sample surface and the nearest microphone, \(j=\sqrt{-1}\), and \(H\) is the calibrated complex transfer function between the two microphones and is calculated as follows:

$$H=\frac{\overline{H} }{{\overline{H} }_{\text{c}}}$$
(3)

where \({\overline{H} }_{\text{c}}\) is the complex calibration factor.

The frequency-dependant normal incidence sound absorption coefficient \(\alpha \) is calculated as follows:

$$\alpha =1-{\left|R\right|}^{2}$$
(4)

A standard impedance tube can be obtained commercially from a limited number of manufacturers. The average cost of a commercially available two-microphone impedance tube is 15,000–20,000 USD, which may not be affordable for many research institutes. Da Silva et al. [10] suggested the design of a low-cost impedance tube for less than 300 USD, which was affordable. However, the impedance tube described by Da Silva et al. required a pair of pressure field microphones and associated signal conditioning/collecting equipment, which consists of a significant portion of the costs, estimated to be more than 10,000 USD.

The microphones used in a commercial two-microphone impedance tube are typically pressure field microphones, which are costly measurement devices supplied by a limited number of acoustic specialist suppliers. A pair of these microphones, including signal conditioning equipment, cost 5–10 thousand USD.

The gap in the available literature is the evaluation of the use of consumer-grade acoustical equipment in an impedance tube design and the assessment of how such a modification impacts the results obtained using an impedance tube compared to when dedicated equipment is utilized. To do this, the results from both types of acoustical equipment, the consumer-grade microphones and signal conditioners, need to be compared with pressure field microphones and signal conditioners on the same tube. With this direct comparison available, researchers and practitioners can have a cost-friendly alternative to commercial impedance tube machines and gain prior knowledge of what to expect regarding the reliability of results from their budget-friendly impedance tube design.

This study presents a vertical impedance tube design using commercially available omnidirectional condenser microphones costing less than 1 USD; suggestions to improve reliability are also made. The results are compared with those obtained using a dedicated pressure field microphone kit for an impedance tube to suggest the practicality of using the economic design for acoustic research facilities with a limited budget.

2 Materials and Methods

2.1 Impedance Tube

A two-microphone vertical impedance tube was designed according to the American Society for Testing and Materials (ASTM) E1050 standard [4]. Here, this design is referred to as “Fandoq”. Figure 2 displays the design details, and the complete 3D design of the compartments can be found via the link provided in the Appendix. In Fig. 2, the letter R stands for radius, and φ stands for diameter.

Fig. 2
figure 2

The impedance tube referred to as “Fandoq”. Dimensions are in mm

Full size image

The top compartments, labelled (A) and (B), house a 4″ speaker driver (F1034 R, Pioneer, Japan). These components were machined from a cylindrical block of recycled high-density polyethylene. Compartment A was filled with sound-absorptive glass wool (Earthwool Glasswool R3.6, Knauf Insulation Ltd, New Zealand) to prevent resonance in the speaker driver. Compartment B featured a funnel shape to smoothly reduce the acoustic transfer channel diameter from 102 to 70 mm, preventing strong channel reflections. It was also filled with glass wool material, in accordance with ASTM 1050, to prevent tube resonance. The speaker driver installation ring was mechanically isolated from compartments (A) and (B) using five layers of melt-blown material (extracted from blue surgical masks) on each side to dampen mechanical vibrations. A layer of grease was applied between the speaker enclosure and the impedance tube to seal the connection acoustically.

The impedance tube (Compartment C) was machined from a mild steel pipe with an outer diameter of 90 mm and a wall thickness of 5.69 mm. Two pieces of steel were welded to one side of the tube to accommodate a bench fixture housing using a bolt (Fig. 2, Section C). On the other side, three 10 mm housings were made (Part D) for microphone fitting. O-ring beds were machined inside microphone housings to seal the microphones from the outside of the tube. The machined tube was coated using hard chromium to avoid corrosion.

Fandoq was designed with three microphone holes to work in two modes: High Frequency (110–2800 Hz) and low frequency (55–2200 Hz). The high-frequency mode is for when the microphones are 30 mm apart, and the low-frequency mode is for when the microphones are placed 60 mm apart. The operating frequency range of the impedance tube for the given dimensions was calculated using the criteria expressed in ASTM 1050. Compartment (E) is a piston made from an aluminium block to adjust sample thickness from 0 to 70 mm. Compartment (F) is a stainless steel sample holder that accommodates the piston and the sample. An O-ring sealed the gap between the sample holder and the tube. Figure 2 (3D View) shows that three adjustable clamps spaced 120° apart radially secure the sample holder to the tube.

The total cost of the raw material for building the impedance tube (not including the electronics) was approximately 140 USD. The workshop costs were approximately 200 USD.

2.2 Electronics

Figure 3 shows the electronic diagram of Fandoq. The speaker driver was a 4-inch Pioneer (TS-F1034R, pioneer, Japan) speaker advertised for car stereo system. The amplifier (Amp) used to drive the speaker was a 100-W mono-channel amplifier (XH-M542, China, price: ~ 1 USD). A computer sound card (no part number, branded Kebidu, China, price: ~ 15 USD) with dual-channel line-in input was used as the data acquisition hardware. This sound card uses a CM6206 (Cmedia, Taiwan) audio chipset. Initially, two separate sound cards with mono input channels were tested, and it was observed that such a setup resulted in an inconsistency in signal phase between the audio channels and a discontinuity in audio sampling which rendered the results useless.

Fig. 3
figure 3

Electronical instrumentation

Full size image

The only variable for comparison was the microphone setup. The setup called “$1 microphone” consists of two omnidirectional condenser microphone amplifier modules for student robotic projects (Part Number KY-037, unbranded, China). This module utilizes a two-stage signal amplification using an op-amp with adjustable gain. The microphones used in this module are omnidirectional electret microphones (EM-B9760Ul, S.G., China). According to the manufacturer’s datasheet, the frequency response is 70–10,000 Hz, with a highly nonlinear sensitivity for up to a 20 dB variation in its response range. The bias voltage of the amplifier’s output signal was removed using a high-pass filter. The total cost of this microphone setup was ~ 1 USD.

The other setup, “Dedicated microphone”, consisted of a pair of dedicated pressure microphones, typically used and advertised for impedance tubes. The microphone part number was 377A10 (PCB Piezotronics, USA). The signal was powered using a dedicated preamp (426B03, PCB Piezotronics, USA) and amplified using a dedicated signal conditioner, model 482C05 (PCB Piezotronics, USA). The total cost of this dedicated setup was > 8,000 USD. These microphones provide a highly linear (± 2 dB) output in a 4–70 kHz frequency range.

3 Method

To keep this paper short, we avoid presenting the equations used in impedance tube measurements. All the calculations for obtaining acoustical properties, including Normal Incidence Sound Absorption Coeffıcient \(\left(\alpha \right),\) were performed strictly according to definitions, methods and equations covered in Sect. 8.4.6 of ASTM E1050 [4] with the equations derived from the transfer function method proposed by Chung & Blaser [9]. The total signal sampling duration was 60 s at the sampling rate of 44,100 Hz. The sampled signal underwent linear RMS averaging with 60 iterations, each iteration 1 s long. A key point in the design is that both input signals from the microphones underwent Flat-Top windowing to avoid spectral leakage. The standard method [4] suggests using the Hanning window on input signals which we found to result in coherences < 0.8 and noisy datsa in the $1 microphone group. The Computer software to handle all the processes from data collection, analysis and logging was coded using LabVIEW [17]. For each microphone setup, the complex calibration factor \({\overline{H} }_{\text{c}}\) was determined (According to ASTM E1050) to offset channel phase and amplitude mismatches and used for microphone calibration. To correct for the amplitude and phase mismatch of the microphones and the sound card channels, the complex calibration factor \({\overline{H} }_{\text{c}}\) was determined by calculating the transfer function for two-microphone positions: standard position \(({\overline{H} }^{I})\) and the swapped position \(({\overline{H} }^{II})\) and using the following equation:

$${\overline{H} }_{\text{c}}={({\overline{H} }^{I}\times {\overline{H} }^{II})}^{1/2}$$
(5)

For calculating \({\overline{H} }_{\text{c}}\), the sample holder was adjusted to fit a 70 mm thick bed of glass wool (Earthwool Glasswool R3.6, Knauf Insulation Ltd, New Zealand). \({\overline{H} }^{I}\) was calculated using Eq. 1, for the standard position of the microphones where M1 and M2 are in their corresponding positions, as shown in Fig. 1. Both microphones, with the same signal wires attached to them, were physically swapped and carefully penetrated the tube, keeping the same radial angle and penetration into the tube to exactly occupy the same space as for calculating \({\overline{H} }^{\text{I}}\), and the transfer function was calculated again using Eq. 1 to obtain \({\overline{H} }^{\text{II}}\). This process was repeated for each microphone setup. With \({\overline{H} }_{\text{c}}\) obtained, the calibrated complex transfer function \(H\) was calculated using Eq. 3. The sound absorption coefficient \(\alpha \) was calculated using Eq. 4.

The samples tested in acoustic examinations included a 70 mm thick bed of glass wool and a 70 mm bed of stainless steel ball bearings with a particle diameter of 3.175 mm (1/8 inch). The ball bearings were poured into the sample holder, and their surface was levelled using a metal ruler. For each sample, sound absorption was measured using both microphone setups. For each microphone setup, the calibration file containing \({\overline{H} }_{\text{c}}\) was loaded into the software, and the acoustic properties were evaluated.

A GitHub repository [15] in the Appendix contains all the source codes LabView graphical user interface and supplementary information needed to implement the impedance method and replicate the test results.

4 Results

Figure 4 shows the validation results of the Fandoq impedance tube using a calibration sample of polyurethane (PU) foam with a thickness of 38 mm and density of 0.0164 g.cm−3. The grey curve, denoted as “Control” is the test results on the validation sample provided by the University of Auckland Acoustics Research Centre, using a calibrated industrial impedance tube designed based on Brüel & Kjær (Denmark) Type 4026. As shown in Fig. 4, the results obtained using the Dedicated microphone, 1$ microphone and Control impedance tube closely match with an average deviation of < 5%. It is expected that using calibration with a higher absorption coefficient can minimize the appearance of noise and outliers in the data, as it reduces strong in-tube reflections. However, the calibration PU sample was deliberately chosen not to be a perfect sound absorber; therefore, the results of the validation test are expected to fall within the higher error spectrum.

Fig. 4
figure 4

Validation test

Full size image

For impedance tubes designed for higher frequencies, it is recommended to repeat the validation test.

Figure 5 shows the complex microphone calibration factor (Hc) polar coordinates. Graph (a) is magnitude in dB (reference value = 1), and graph (b) shows phase in radians. As shown in Fig. 5, the Dedicated Microphone setup had fewer fluctuations in the calibration factor components, ensuring a more consistent sensitivity in its pressure measurement.

Fig. 5
figure 5

Microphone Calibration

Full size image

Figures 6 and 7 show the test results for the Sound Absorption Coefficient of Glass Wool and the ball bearings samples obtained using Fandoq.

Fig. 6
figure 6

Sound Absorption test for the glass wool sample

Full size image
Fig. 7
figure 7

Sound Absorption test for the ball bearings sample

Full size image

The green curve in Fig. 7 shows the sound absorption tests of the $1 microphone group after applying a 36-point moving average filter (MA36).

As is shown in Figs. 6 and 7, the sound absorption coefficients obtained using the two setups closely track each other. However, the results obtained using the “$1 microphone setup” are relatively noisy, especially for the lower end of the frequency spectrum (< 250 Hz). For frequencies higher than 300 Hz, the average deviation between the two-microphone groups was less than 1% (0.3% for the glass wool and 0.7% for the ball bearings). Based on the error analysis, the sound absorption measurements using the $1 microphone setup show a variation that, with a 95% Confidence Interval, is 0.3% ± 0.1%. This indicates that we can be 95% confident that the true mean variation in sound absorption measurements lies within this range.

The experiments were repeated for two PU foam samples with thicknesses ranging from 24 and 70 mm, and a range of powders with mean particle diameters from 0.06 to 2.5 mm and similar results in terms of agreement between the two-microphone groups were observed. Due to the similarity of the results and the agreement between the two-microphone groups, figures for other samples are not provided in this paper to keep it concise.

5 Conclusion

Spending 2–5% of the cost of a commercially available apparatus, a vertical two-microphone impedance tube was designed and tested according to ASTM E1050. The design used cheap consumer-grade electret microphones, a dual-channel computer sound card, and an automotive stereo system speaker. For calibration, the channel phase and amplitude mismatches between the microphones were corrected by generating a calibration factor \({H}_{\text{c}}\), as recommended by the standard method covered in ASTM 1050 standard. The results obtained using the consumer-grade microphone setup were compared with those obtained from the dedicated pressure field microphone setup, typically used in impedance tubes, costing several thousand USD.

Sound absorption measurements taken with consumer-grade microphones closely matched those obtained using dedicated pressure field microphones, with an average deviation of less than 1% between the two-microphone groups. However, the trade-off is the reduced accuracy within the low-end frequency spectrum, specifically when the incident sound frequency falls below 250 Hz. Subject to calibration and application of a moving average filter, generic microphones can reproduce almost identical results with dedicated pressure microphones.

These findings suggest that constructing an impedance tube to measure the acoustical properties of materials is feasible using affordable consumer-grade microphones and acoustical equipment. This feasibility is contingent upon the careful selection of the windowing method and the calibration of the microphones.

For future studies, the practicality of using low-cost audio microphones in other acoustical tests, including but not limited to noise control, transmission loss (ASTM E2611), and acoustic mapping, can be investigated. This is subject to confirmation that the microphone's frequency response and dynamic range meet the requirements of the experiment. Such studies enable researchers with limited budgets to explore affordable alternatives for conducting their experiments, which is impactful in the field of acoustical engineering.