Absorption Coefficient

Measuring the Absorption Coefficient for Some Common Materials Used in Functional Rooms through Standing Wave Ratio
Adonis Cabigon1, Alaiza Tangaha2
Department of Physics, University of San Carlos, Nasipit, Talamban, Cebu City 6000
1adoniscc@yahoo.com
2mayalaiza92@yahoo.com

Abstract In this paper, we present the measurement of the absorption coefficient α of MDF (Medium Density Fiberboard) and Fiber Cement through an improvised standing wave apparatus consist of an enclosed tube with a speaker on one end and the sample material on the other end. The maximum and minimum sound pressure developed inside the tube is measured by a moving microphone probe and the Standing Wave Ratio SWR and the absorption coefficient of the samples is calculated. The absorption behavior of each sample was also investigated in response to varying frequency. It was found out that the fiber cement was more absorbing than MDF at our chosen frequency rage (100 Hz to700 Hz). 1. Introduction
Explaining the acoustical behavior of large and small rooms requires knowledge of the absorbing properties of materials which cover the surfaces of the room under analysis. The sound absorption coefficient of these materials is ideally defined as the fraction of the randomly incident sound power absorbed by the surface which also depends on the frequency of the emitted sound [1-2]. Measuring the sound absorption of a room is part of the procedure for acoustical measurement, such as determining the sound power level of noise source or the sound transmission loss of a partition. It is also used in certain calculations such as predicting the sound pressure level in a room when the sound power level of a noise source in the room is known [2].
The standing wave method for measuring the acoustic absorption coefficient is by means of a standing wave formed in a tube. This method has contributed greatly to the development of effective acoustic absorbents, which are important in minimizing noise, thus play a significant role in architectural acoustics such as the design of recording studios and listening rooms, automobile interiors and many more [2-3].
In this experiment, we demonstrate the method by using an improvised standing wave apparatus to measure the absorption coefficient of some acoustic samples commonly used in rooms and evaluate its response to varying frequencies.

2. The Experiment
The experimental set-up of the improvised standing wave apparatus is schematically shown in Figure 1. Sound of pressure amplitude A is directed through the length of the tube by means of a loudspeaker at one end, to strike the sample placed at the other end. When the sound waves encounter the sample, part of the sound energy is absorbed

Figure 1.Experimental set-up consist of (A) computer with Waveform Analysis LabView program, (B) frequency generator, (B) loud speaker, (D) improvised standing wave tube, (E) movable microphone probe and (F) absorbing material. and another part is reflected back through the tube at an amplitude B. As a result of interference with the incident wave a standing wave pattern is formed in the tube, whose pressure maximum and minimum can be measured by means of a movable microphone probe. The microphone is connected to a computer with a LabView program that is used to read the frequency and the amplitude of the signal inside the tube.
In Figure 2, it is shown how the sound pressure inside the tube develops as a function of position for both complete reflection and for partly absorbing material. On the left side, shown is the pressure amplitude in the tube with a rigid a rigid termination at x = L, where L is the length of the tube. The entire sound energy incident upon the termination is reflected with the same amplitude. However, there may be some absorption along the walls as the waves travel back and forth along the tube. The figure on the right side represents the case when the pipe is terminated with some acoustic absorbing material. Now some of the incident sound energy is absorbed by the material so that the reflected waves do not have the same amplitude as incident waves. In addition the absorbing material introduces a phase shift upon reflection. The amplitude at a pressure antinode (maximum pressure) is A+B, and the amplitude at a pressure node (minimum pressure) is A-B [1, 3].

Figure 2.Amplitude of the sound pressure inside the tube as a function of position.

The measuring apparatus is used to determine the relation between the sound pressure node and antinode in the tube known as the standing wave ratio:
SWR=PmaxPmin=(A+B)(A-B) . (1)

The absorption coefficient α of the sample is defined as the ratio between the energy absorbed by the sample and the total energy striking the sample. As the energy is proportional to the square of the sound pressure, we have α=1-B2A2 . (2)
However, we cannot experimentally measure A and B but rather the combination of these two terms as expressed in equation (1). Consequently, in terms of measurable quantities, α can be expressed by α=1-(SWR-1)2(SWR+1)2 . (3)
Inside the tube, the location of minima and maxima can easily be located by noting that the distance between them is always one-fourth of the resonant wave length λ of the tube which is given by λ=Vfn , (4) where fn is the resonant frequencies of the tube and can be computed by the equation fn=nV4L , (5) where v is the speed of sound and n is the odd harmonics (only odd harmonics occur in a pipe closed at one end) [3,5-6]. By means of the improvised standing wave apparatus, the standing wave ratio can be computed from the amplitude of pressure maxima and minima. Through this, we can find the absorption coefficient of a particular sample and analyze its response when the frequency of the incident sound is varied.

3. Results and Discussion
To be able to have a reliable and efficient evaluation of the absorbing properties of a material, we are required with a greatest possible frequency range at which its response is analyzed. In our measurement, we fail to meet this requirement for we are limited as far as the lower frequency limit is concerned by reason that the apparatus shall be a little more than a quarter wavelength long. That is to say, that if the apparatus shall be held within a reasonable length, one can hardly measure less than 90 to100 Hz. For higher frequency, we are limited by the necessity that the diameter of the tube shall be less than 0.586 A, in that there is a possibility for a first transverse resonance at this wavelength. As the tube method presupposes a flat sound field in the measuring tube 's cross section, transverse resonances can naturally not be allowed, as these would give rise to a varying sound pressure in the tube 's transverse direction [3]. In addition, we are also limited of the time in performing the experiment and it takes a quit long duration just to complete a single measurement. As a result, we get only ten data point for every sample (see Appendix) which is insufficient to make judgment about their absorbing properties since we fail to investigate its response to frequency region higher than 700 Hz.
Nevertheless, we are still able to demonstrate the principles behind the standing wave method for finding the absorption coefficient of a certain material. Here, we use MDF (medium density fiber board) and fiber cement as our absorbing sample. These materials are commonly used as walls and ceilings for functional rooms. Illustrated in Figure 3 is the calculated absorption coefficient profile of the two acoustic samples. We can see from the graph that MDF has lower absorbing coefficients for frequencies ranging from 100 to 700 Hz compared to that of fiber cement. However, its absorption coefficient peaks remarkably at about 533 Hz. Around this frequency, its absorption coefficient is up to three times greater than that of the fiber cement. The graph also shows that the absorbing coefficient of the fiber cement has decreasing values at higher frequency region within the upper mentioned rage. For frequencies below 300 Hz there is a noticeable range of difference between the trends of the absorption coefficients of MDF and fiber cement. Above this frequency, similarities of these trends can be observed not including the region where that of the MDF peaks. It is also evident from the graph that the absorption coefficient values for both materials rage only from 0 to 0.05 within the investigated range of frequencies. These values are quit small to say that the two samples are good absorbing at the chosen frequency region. The accumulated data are also superficial to predict the materials’ absorbing behavior beyond this range.

Figure 3 Graph of absorption coefficients of MDF (black) and Fiber Cement (red) as functions of frequency.

In our experiment, we cannot expect for an absolute accuracy of the measured values as we consider the uncertainties in measuring the speed of sound and length of the tube for the computation of the fundamental resonant frequency to produce a standing wave inside the tube. Furthermore, through the frequency generator, we were not able to produce a signal whose frequency is exactly the same as that of our computation (see Appendix). These differences introduce an initial phase shift of the reflected sound added to that of what is caused by the absorbing material and yields a different value for the standing wave ratio. The rapid fluctuation of the sound pressure amplitude inside the tube as read by the LabView program is also a great contributor to the inaccuracy since we are not able to read all the fluctuating values in order to get a more reliable average. The uncertainties in finding the maxima and minima because of the former also have to be taken into account. In the sample, a circular opening with minimal radius is made as channel for the microphone probe introduces a leak that can change the pressure inside the tube, thus; also have a contribution to the discrepancy.

4. Conclusion Using the improvised standing wave apparatus, we were able to demonstrate the principles in measuring the absorption coefficient of a material through standing wave ratio. On the other hand, we failed in making a reliable evaluation for the absorbing properties of the sample material in response to varying frequencies since we only have chosen a very short frequency rage not sufficient for the evaluation. The inaccuracy of the measured values is brought about by inadequate calibration of the experimental set-up as a whole. Nevertheless, we can still conclude from our data that fiber cement is more absorbing than MDF at the chosen frequency rage (100 Hz to 700 Hz). In experiments like ours, a highly calibrated standing wave apparatus that can provide a wide range of frequencies for measurement is well recommended so that accurate results and responsible evaluation can be realized. References 1. D. Russel, “Absorption Coefficient and Impedance” GMI Engineering and Management Institute, flint, MI, 48504. 2. A. Reynolds, Engineering Principles of Acoustics: Noise and Vibration control, Allyn & Bacon, 1981. 3. P. Brüel, D. Kjær, “Standing Wave Apparatus”, Technical, Acoustical and Vibration Research, No.1, January 1995. 4. W. Dong, T. Faltens, M. Pantoll, D. Simon, T. Thomson, “Acoustic Properties of Organic/Inorganic Composite Aerogels”, Matter, Res. Soc. Synap. Proc. Vol1188, 2009. 5. H. Young, R. Freedman, University Physics, 11th Ed, Pearson Education South Asia PTE LTP, 2004. 6. M. Baclayon, “Locally-assembled Taylor Tube Apparatus for Sound Absorption and Reflection measurements” National Institute of Physics, College of Science, UP Diliman, 2005.

Appendix
Data Accumulation

The temperature inside the tube is measured to be 27 Celsius degree, so the speed of sound is calculated as 347.6 m/s, the length of the tube is (3.08 ± 0.003) m these measurements yields to the following computations:

Computed Frequency and Wavelength

Computation for the Absorption Coefficients

Fiber Cement

MDF (Medium Density Fiberboard)

References: 1. D. Russel, “Absorption Coefficient and Impedance” GMI Engineering and Management Institute, flint, MI, 48504. 2. A. Reynolds, Engineering Principles of Acoustics: Noise and Vibration control, Allyn & Bacon, 1981. 5. H. Young, R. Freedman, University Physics, 11th Ed, Pearson Education South Asia PTE LTP, 2004. 6. M. Baclayon, “Locally-assembled Taylor Tube Apparatus for Sound Absorption and Reflection measurements” National Institute of Physics, College of Science, UP Diliman, 2005.