EXPERIMENTAL STUDY OF THE ABSORPTION COEFFICIENT OF ACOUSTIC INSULATORS FOR UNDERGRADUATE UNIVERSITY STUDENTS

Objective: This study investigates the techniques for measuring the acoustic absorption coefficient that use an impedance tube, to evaluate their use by students of scientific-technical degrees. Theoretical Framework: This topic presents the concept of absorption coefficient and how it is obtained using impedance tubes. Method: The absorption coefficient of an acoustic insulator is measured as a function of frequency, thickness and density, with an experimental system that uses the transfer function method in an impedance tube. Results and Discussion: The results reveal that the absorption coefficient varies with frequency, being lower at low frequencies. An increase in thickness and/or density produces better acoustic absorption performance. The results are compared with others obtained with an experimental system that uses the standing wave method in an impedance tube. The agreement between the two methods is quite good. Research Implications: The advantages and disadvantages of the two methods are discussed, and their use by students of scientific-technical degrees is assessed. It is recommended to use the transfer function method when students work more autonomously, for example, in final degree projects. Originality/Value: This work offers teachers information related to the experimental study of the acoustic absorption coefficient, which allows them to choose the most appropriate method for each student based on their ability to work autonomously and their knowledge.


INTRODUCTION
Noise is becoming an increasingly serious environmental problem.Acoustic absorbers play an important role in its control.The composition of these materials has evolved over time.
Asbestos, which provides good thermal and acoustic insulation, was an element widely used in construction, until the health risks posed by inhalation of its fibers became known, which is why its use was prohibited in many countries.Subsequently, glass and mineral fibers have been used, which are materials that are difficult to recycle and whose disposal can cause 3 environmental problems.More recently, the development of more environmentally friendly materials has increased, using recycled products and/or less polluting manufacturing processes (Urdanpilleta et al., 2022).
The experimental study of acoustic absorption facilitates the development of new materials and their adaptation to specific requirements or designs.Some works show the behavior of absorption with frequency for different compositions, thicknesses or combinations with air layers (Fatima and Mohanty, 2011;Muhammad et al., 2012).The parameter that measures acoustic absorption is called absorption coefficient.There are three standard methods to measure it in a laboratory: reverberant chamber, standing wave in an impedance tube, and transfer function in an impedance tube.
It seems convenient to include training related to the fundamentals and management of these techniques for university students of scientific-technical degrees, to improve their knowledge and understanding of sound waves and the properties of materials.The two methods using an impedance tube (also called a Kundt tube) are the best alternative for undergraduate students to make absorption coefficient measurements, since the reverberant chamber method requires facilities that are not usually available in a teaching laboratory.Furthermore, the amount of sample required in an impedance tube is much smaller than in a reverberant chamber, which also makes it easier to handle during experiments.
An impedance tube is a rigid cylinder in which the sound produced by a speaker located at one end propagates.If the diameter of the tube is small compared to the wavelength, a plane sound wave is produced that propagates along the axis of the tube.When the wave reaches the other end of the tube, it can be reflected.Successive reflections at both ends of the tube give rise to standing waves and resonance phenomena (Möser, 2009).If a sample of absorbing material is placed at the end of the tube where the wave is reflected, some of the incident energy is absorbed by the sample, so the incident and reflected waves have different amplitudes.
Standards ISO-10534-1 (1996) and ISO-10534-2 (1998) provide two standardized experimental methods that allow the absorption coefficient of the sample to be calculated using an impedance tube.The first describes the standing wave method and the second the transfer function method.
In this work, the behavior of the absorption coefficient of commercial porous insulators is studied based on some easily measurable parameters: frequency, thickness and density.The experimental study has been carried out in a commercial impedance tube that uses the transfer function method and complies with the ISO-10534-2 standard.It is the continuation of a previous work (Macho-Stadler and Elejalde-García, 2017).In it, the same measurements were carried out and on the same samples with a PASCO brand impedance tube available in the 4 teaching laboratory and a procedure based on the ISO 10534-1 standard was followed.The objective is to obtain information on which of the two methods is more appropriate for the experimental study of the acoustic absorption coefficient carried out by students of scientifictechnical degrees.An attempt is made to determine in which cases a basic system that can be found in any Physics teaching laboratory is sufficient, and when it is better to use a more sophisticated system.

THEORETICAL FRAMEWORK
When sound waves reach a surface, they are reflected and/or absorbed.The absorbed sound can be transmitted or dissipated.Sound energy is dissipated by the simultaneous action of viscous and thermal mechanisms.The amount of energy reflected or absorbed depends on the acoustic properties of the surface.The absorption coefficient is defined as: where:

Ir = reflected intensity
This coefficient takes values between 0 (total reflection) and 1 (total absorption).It is known that it does not depend on the incident intensity, but rather on the frequency and angle of incidence.For normal incidence, it is usually denoted n.The simplest way to estimate n is to use an impedance tube.

STANDING WAVE METHOD IN AN IMPEDANCE TUBE
This method is described in the ISO-10534-1 standard.Figure 1 shows a diagram of the experimental setup.

Scheme of the experimental setup used in the standing wave method
The length of the tube is L. If the speaker is placed at x=0, emitting a harmonic sound, and the sample is placed at x=L, the resulting sound field in the tube is the superposition of the incident wave of amplitude p0 that moves in the positive direction of the tube axis and the reflected amplitude rp0 moving in the negative direction.In general, reflection can produce attenuation of the amplitude of the incident wave and a phase difference between the incident and reflected waves.The reflection coefficient r can be represented by a complex number of module R and phase .The resulting sound field can be written as: p = p 0 (e −jkx + re +jkx ) = 2p 0 r cos kx + p 0 (1 − r)e −jkx (2) Being the root mean square (rms) value of p 2 : The quotient between the maximum value and the minimum value of the pressure amplitude of the standing wave obtained from equation (3) allows calculating the value of R and that of n: To determine the value of n using equation ( 4), it is sufficient to measure the values of pmax and pmin near the sample, to avoid errors due to sound attenuation inside the tube.To do this, the speaker is fed with a harmonic signal, and the microphone is moved until a 6 maximum and minimum pressure is found.This method requires a fairly long measurement time, because in each experiment only one frequency is measured.

TRANSFER FUNCTION METHOD IN A TUBE OF IMPEDANCES
The method is described in the ISO-10534-2 standard.Figure 2 shows a diagram of the experimental setup.

Scheme of the experimental setup used in the transfer function method
The speaker is placed at one end of the tube and the sample at the other.The speaker emits noise and plane waves are generated.The sound field is measured with the help of two microphones mounted on the wall of the tube and located at x1 and x2 with respect to the sample: p 1 = p + e −jkx 1 + rp + e +jkx 1 = p + e −jk(ℓ+s) + rp + e +jk (ℓ+s)  (5) where: p+= amplitude of the incident pressure wave ℓ= distance between sample and microphone 2 s= distance between the two microphones.
The transfer function between the microphones, H12, allows calculating the reflection coefficient and absorption coefficient of the sample for normal incidence: To determine the value of n using equation ( 7), it is sufficient to measure the transfer function H12.To do this, the speaker is fed with white noise.A frequency analyzer converts the signal to the frequency domain.This method allows us to obtain n as a function of frequency in each of the measurements, which greatly reduces the measurement time compared to the standing wave method.

METHODOLOGY
n has been measured in several samples of a commercial absorbent material based on various parameters.For this, the ACUPRO measurement system from Spectronics has been used, which uses the transfer function method and complies with the ISO 10534-2 standard.
The inner diameter of the tube is 34.9 mm and its total length is 1.20 m.The sound source is placed at one end of the tube and the sample at the other.The pressure value is measured at two points on the tube.The distance between the sample and the nearest microphone is ℓ=50.8mm.
The distance between the two microphones is s=29.21mm.The frequency range of the system varies between 50 and 5700 Hz.The data acquisition module is a signal analyzer integrated into the instrument software.At the end of the measurement, the software calculates the absorption coefficient.Figure 3 shows the calibration and measurement window of the ACUPRO system software., 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000 and 5000 Hz.

SAMPLES
The samples have been cut from sheets of a commercial porous insulator (Copopren) composed of cohesive polyurethane particles of different properties with an acoustic additive.
Two different plates are available: (i) the first (RF80) with a thickness of 40 mm and apparent density of 80 kgm-3 and (ii) the second (RF150) with a thickness of 20 mm and apparent density of 150 kgm-3.The apparent density of a porous solid is determined by dividing its mass by its total volume, including that of the pores (apparent volume).The cut samples are cylinders with both faces flat and parallel, as shown in Figure 4.It is important to note that there are no reference materials with which to compare the results obtained, so performing a sufficient number of measurements carefully is important.

RESULTS AND DISCUSSION
Below, the measurements obtained and their behavior with three parameters are presented: (i) the frequency, (ii) the thickness of the material and (iii) the density of the material.

BEHAVIOR WITH FREQUENCY
Figure 5 shows the results obtained with the ACUPRO system for a 40 mm thick RF80 sample.The results show that n grows with frequency until reaching a maximum value between 0.8 and 1.Once the maximum is reached, the variations of n are less important.This behavior is similar in the rest of the samples.10 Table 1 shows the n values (with their corresponding error) obtained for a 40mm thick RF80 sample and a 20mm thick RF150 sample.For each sample, the values obtained in a previous study using a PASCO brand impedance tube available in the teaching laboratory are also included (Macho-Stadler and Elejalde-García, 2017).The results obtained in this study (ACUPRO system, ISO 10534-2) agree quite well with those obtained for the same samples in the previous study (PASCO system, ISO 10534-1).The difference between the values obtained for n with both methods is less than 15%, except in the lower frequency area for the thinnest sample.In these cases, a worsening of the signal-to-noise ratio has been observed in the measurements made with the PASCO system, which decreases their precision.

BEHAVIOR WITH SAMPLE THICKNESS
For RF80 samples, measurements have been made for thicknesses between 20 and 40 mm in intervals of 5 mm.For RF150 samples, measurements have been made for thicknesses between 20 and 80 mm in intervals of 20 mm.
To obtain samples with thicknesses smaller than that of the commercial plate, the cylinders are carefully cut to the desired thickness.To obtain samples of thickness greater than that of the commercial sheet, several samples are combined until the desired thickness is reached.
Figure 6 shows the results obtained in this study.The results previously obtained with the PASCO system are similar.

Figure 6
Variation of n with the thickness of the sample for the two insulators studied.Measurements obtained with the ACUPRO system for third octave bands.Data is represented by points.Lines are included to aid in visualization.
For samples of the same material, thickness seems to be a determining property: when the thickness of the sample increases, so does its absorption coefficient at low and medium frequencies.For porous materials, acoustic absorption occurs due to the loss of energy of acoustic waves due to the vibration of the fibers that make up the material and also due to the friction of the air with the skeleton of the material.Studies on acoustic absorption in porous materials have concluded that low-frequency acoustic absorption is directly related to thickness (Seddeq, 2009).The general rule is that effective sound absorption of a porous absorber is achieved when the thickness of the material is approximately one-tenth the wavelength of the incident sound.If we consider that an "effective acoustic absorption" occurs when the absorption coefficient is greater than 0.6, the experimental results agree with this rule.
A porous material has an open pore structure.When air passes through it, the energy is transformed into heat due to the friction of the air molecules with the walls of the pores.More energy is dissipated when air molecules move at a faster speed, which occurs at /4 and 3 /4 from the surface ( wavelength).Therefore, the material absorbs more at those frequencies for which the value of /4 is less than its thickness.When the sample thickness decreases below /4, the absorption is less effective and n decreases.As can be seen in Figure 5, the absorption peak appears around that frequency.Once the absorption peak is exceeded, the value of n decreases again due to the lower speed of the air molecules.

BEHAVIOR WITH SAMPLE DENSITY
To determine the relationship of n with the density of the material, the results of RF80 and RF150 samples of the same thickness have been compared.Figure 7 shows the results obtained with the ACUPRO system for samples of thicknesses 20 and 40 mm of both materials.

Figure 7
Variation of n with the density of the material: 80 kg/m3 for RF80 and 150 kg/m3 for RF150.
Measurements obtained with the ACUPRO system for third octave bands.The data obtained is represented by points.Lines are included to aid in visualization.
The results show that when the density of the sample increases, the absorption coefficient at medium frequencies also increases.In the rest of the bands the behavior is more dubious.The results previously obtained with the PASCO system were not clear in any area of the spectrum, possibly due to its lower precision in the low frequency area.
The density of a material is usually used as a parameter to define its acoustic efficiency, although there is not as much literature as in the case of sample thickness.McGrory et al. (2012) found an increase in acoustic absorption in the medium/high frequency zone, when the density of the material increases, which coincides with the results of this study.For recycled polyurethane (RPF) samples Asdrubali et al. (2012) show some measurements based on density between 400 and 3150 Hz.The results of the 80 kg/m3 and 150 kg/m3 samples also corroborate an improvement in the medium frequency area, below 1250 Hz.A more complete study based on density seems interesting, but samples with other densities would be needed to complete it.

CONCLUSIÓN
In this work, the effect of frequency, material thickness and density on the acoustic absorption coefficient of two commercial porous materials has been analyzed.The measurements have been carried out in an impedance tube that complies with the ISO 10534-2 standard.The results obtained agree with those that appear in the literature for this type of materials.In general, an increase in the thickness of the sample and/or its density produces better performance in acoustic absorption in the mid/low frequency zone.To improve acoustic absorption, combining the effect of thickness and density seems to be a good strategy.
The results have been compared with those obtained in a previous study for the same samples with a non-certified system available in the teaching laboratory following the experimental method described in the ISO 10534-1 standard.The agreement between the two methods between 800 and 5000 Hz is reasonably good, being better for higher values of frequency and sample thickness when the signal-to-noise ratio of the method described in ISO 10534-1 is better.
These results allow us to know the advantages and disadvantages of the two experimental procedures compared.One of the advantages of the standing wave method is that material available in a teaching laboratory can be reused.In a Physics laboratory, it is common to find impedance tubes that are used for the study of standing wave patterns in closed or open tubes (Haeberli, 2009) and the measurement of the speed of sound (Aljalal, 2014).By placing an absorbent sample at one end of the tube and following the method detailed in the ISO-10534-1 standard, the absorption coefficient of the sample can be obtained.The experimental method is simple and the subsequent manipulation of the data can be done with a spreadsheet, so it seems suitable for short practices in the Physics laboratory.It could also be part of a miniresearch when active learning methodologies are used, such as problem-or project-based learning.In both cases you have to think carefully about the number of measurements to be carried out since the great disadvantage of this method is that it requires a fairly long measurement time, because in each experiment only one frequency is measured.
The transfer function method requires a shorter measurement time than the standing wave method, since it allows many frequencies to be measured simultaneously.To do this, the experimental system must include a frequency analyzer.If the equipment is automated, taking measurements and subsequent manipulation of the data is quick, since the software carries out all the stages.This does not seem to be the best option for first-year undergraduate students whose training in experimental skills should include obtaining and directly manipulating data.14 However, this speed in obtaining experimental results is a great advantage for students who want to complete their Final Degree Project (TFG).Generally, a TFG proposal includes the study of some materials based on their properties (thickness, density, composition, etc.).The student must demonstrate that he/she has acquired during his/her studies a series of knowledge and skills important for his/her future work.For this reason, her work is much more autonomous and she can make decisions about the type of cases to study, which could be too complex to analyze or have inconclusive results, as usually happens in scientific research.A measurement system that provides results quickly enough prevents students from giving up when experiments do not lead to clear conclusions.
During these last academic years, a student from the Industrial Technology Engineering Degree and another from the Physics Degree have completed their TFGs studying the absorption coefficient of materials manufactured from recycled textile fibers.After the study based on the thickness and composition of the samples, they found it interesting to include a brief section on the influence on the absorption coefficient when samples of the material alternate with layers of air.Their results show that by using an adequate thickness of air, an increase in sound absorption in the mid/low frequency area can be achieved, similar to that produced by increasing the thickness of the material.The inclusion of air chambers allows the use of a smaller amount of acoustic insulation, reducing the weight of the acoustic treatment, and is already used in real constructions.In both cases, the students showed their satisfaction with the development of the TFG.
A better characterization of the influence of air layers (different thicknesses, location with respect to the insulation, etc.) on the absorption coefficient seems to be a good alternative study for future TFGs.

Figure 3 ACUPRO
Figure 3 ACUPRO System Software: Calibration and Measurement Window

Figure 5
Figure 5Variation of n with frequency.Measurements obtained with the ACUPRO system for a 40 mm thick RF80 sample.Measurements are taken between 0 and 5000 Hz every 15 Hz.

Table 1
Absorption coefficient at normal incidence obtained for two different samples in the central frequencies of the third octave bands between 800 and 5000 Hz.For comparison, the values obtained in a previous study using the standing wave method are included.