Sound absorption and acoustic surface impedance CHRISTER HEED SD2165 Stockholm October 2008
Marcus Wallenberg Laboratoriet för Ljud- och Vibrationsforskning Sound absorption and acoustic surface impedance Christer Heed Approved Date: Signature: SD2165 Acoustical measurements
ABSTRACT The normal sound absorption coefficients of two different test objects in an impedance tube are measured with two different methods: The standing wave ratio method (ISO 10534-1:1996) and the transfer function method (ISO 10534-2:1998). From the latter is the specific acoustic impedance ratio calculated. The test objects are two cellular plastics: AC 30 S and FIRE 30 VF (both are 30 mm thick), the latter has a thin film on the front side. The effect on absorption coefficient when using a cavity behind the samples is investigated. The cavity results in a displacement towards lower frequencies. The difference when using the FIRE 30 VF is that the absorption coefficient has a wide peak at 350 Hz (which is displaced to 300 Hz with cavity behind). Over 600 Hz is the absorption coefficient reduced when using FIRE 30 VF. There are discrepancies between the two methods except when comparing AC 30 S over 1200 Hz where the methods agree. The transfer function method is in general more accurate.
TABLE OF CONTENTS 1 INTRODUCTION... 1 1-1 Task... 1 1-2 Test objects... 1 1-3 Sound pressure reflection factor... 1 2 STANDING WAVE RATIO METHOD (ISO 10534-1:1996)... 2 2-1 Theory... 2 2-2 Measurement environment... 2 2-3 Instrumentation... 3 2-4 Measurement procedure... 3 2-5 Results... 3 3 TRANSFER FUNCTION METHOD (ISO 10534-2:1998)... 4 3-1 Theory... 4 3-2 Measurement environment... 4 3-3 Instrumentation... 4 3-4 Measurement procedure... 5 3-5 Results... 6 4 DISCUSSION... 7 5 REFERENCES... 8
1 INTRODUCTION 1-1 Task The task is to measure normal sound absorption coefficients of two different test objects in an impedance tube with two different methods: The standing wave ratio method (ISO 10534-1:1996) and the transfer function method (ISO 10534-2:1998). From the latter is the specific acoustic impedance ratio calculated. The effect of using a cavity behind the test objects is being investigated. 1-2 Test objects The test objects used are the cellular plastics AC 30 S (grey) and FIRE 30 VF (black) shown from left in figure 1. To the right one can see the specimen holder of the impedance tube containing a distance ring used to create a cavity behind the specimen. The effect on absorption using the cavity compared to no cavity is measured for both specimens with the transfer function method, and AC 30 S only with the standing wave ratio method where the absorption without cavity is measured for FIRE 30 VF. Figure 1: The test objects are two cellular plastics: To the left, the grey AC 30 S and in the middle the black FIRE 30 VF. Note that FIRE 30 VF has a thin film on the front side. To the right, the specimen holder of the impedance tube containing a distance ring with which a cavity behind the specimen is created. The properties of the test objects are as follows: AC 30 S: Grey cellular plastic FIRE 30 VF: Black cellular plastic with a thin film on one side (The film is always pointed towards the sound source during the measurements) Both specimens are 10 cm of diameter and 30 mm of length. The black sample is estimated to have a little higher density. The distance ring used to create a cavity behind the specimen is 10 cm of diameter and 20 mm of length. 1-3 Sound pressure reflection factor The sound absorption coefficient of a material is by definition the ratio of the sound power entering the surface of the test object to the incident sound power. For a plane wave at normal incidence, the sound absorption coefficient has the relation: 1, where r is the sound pressure reflection factor. For a sample with the specific impedance 1
Z, the sound pressure reflection factor is:, where ρ is the air density and c the sound speed. The subscripts i and r denotes incident and reflected waves respectively. The specific acoustic impedance ratio can be solved:, which means that all problems are reduced to measure the complex sound pressure reflection coefficient r, refer to [1]. 2 STANDING WAVE RATIO METHOD (ISO 10534-1:1996) 2-1 Theory The impedance tube used in this method is illustrated in figure 2. When the frequency is below the cut-on frequency of the impedance tube there will be only plane wave travels inside the tube. The incidence sound absorption coefficient is given by the formula: 1 1, where. This relation comes from the fact that when the loudspeaker in the tube is excited by a sinusoidal signal with the frequency ω the sound pressure waves at position x could be expressed as (time variation factor is omitted), where φ is the phase of the pressure,, and then. This means that in order to get r, maximum and minimum sound pressure levels for a given frequency has to be measured. Specimen holder,,, Probe,, Figure 2: Schematic view of the standing wave ratio method. 2-2 Measurement environment The environmental data and date are as follows: Measure location: Room MWL 75, KTH in Stockholm Sweden o Dimensions (L W H): 6.7 m 6 m 4.5 m Date: 14 October 2008 Air temperature: 21 C Relative humidity: 47 % 2
2-3 Instrumentation The following equipment is used for the measurement with the standing wave method: Standing wave tube: Brüel & Kjaer type 4002 o Serial No.: 632422 o Dimensions: Inner diameter: 0.1 m and length: 1 m Beat frequency oscillator: Brüel & Kjaer type 1022 o Serial No.: 290374 Measurement amplifier: Brüel & Kjaer type 2607 o Serial No.: 308989 Band pass filter set: Brüel & Kjaer type 1615 o Serial No.: 424121 The Beat frequency oscillator is connected to the loudspeaker in the standing wave tube (figure 2), generating pure (sinusoidal) tones at the centre frequencies of the one third octave band that will be measured. The microphone probe inside the tube is connected via the band pass filter set to the measurement amplifier where the voltage is read off. Further details of the instrumentation refer to the lecture notes [1]. 2-4 Measurement procedure The loudspeaker inside the tube, figure 2, is designed with a middle hole where a microphone probe can be moved to detect maximum and minimum sound pressure levels. The probe is supported by a small car at the end so it can be kept in the middle line of the tube. The signal generator is tuned to the frequency of interest and the sound pressure level is adjusted to an appropriate level. The band pass filter set is set to the same frequency. The probe is moved from the specimen and out towards the loudspeaker, and the first sound pressure level minimum is detected and the corresponding voltage is read off the measurement amplifier. Then the following maximum is detected. Normally the sound pressure level will first experience a maximum when moving the probe from the test sample towards the speaker, as illustrated in figure 2, but in this case the second maximum is used. This procedure is repeated for all frequencies of interest. The measurements are performed at centre frequencies of the one third octave band in the useful frequency range 360 Hz up to 2 khz. That is, the useful upper and lower frequencies are determined by the formulas: 200 and 3 respectively, where d is the inner diameter and l is the length of the tube. The centre frequencies 250 Hz and 315 Hz are measured for reference although not fulfil the requirement [1]. The measurements are performed twice for the test sample AC 30 S, that is, with and without cavity behind. The test sample FIRE 30 VF is measured without cavity for reference. The results are presented in the next section. 2-5 Results The ratio of the maximum and the minimum voltages corresponding to the measurements of the maximum to minimum sound pressure levels are used to calculate the incidence sound absorption coefficient according to the formula in section 2-1. Figure 3 shows the results in the centre frequencies of the one third octave band in the frequency range from 250 Hz to 2 khz. 3
1 AC 30 S AC 30 S + Cavity FIRE 30 VF Normal incident sound absorption coefficient in one third ocave band 0.9 0.8 0.7 0.6 Absorption coefficient 0.5 0.4 0.3 0.2 0.1 0 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Figure 3: The measured absorption coefficient in the centre frequencies of the one third octave band from 250 Hz to 2 khz for the specimen AC 30 S with and without a cavity behind. The second specimen, FIRE 30 VF is measured without a cavity. 3 TRANSFER FUNCTION METHOD (ISO 10534-2:1998) 3-1 Theory The impedance tube used for this method is illustrated in figure 4. This method uses two microphone positions at points x 1 and x 2. The sound pressure at these points is expressed as: ;, where the time variation factor is omitted. The complex acoustic transfer function between these two microphone signals is then, which could be solved for, where and, where is the distance between the measurement points. The normal sound absorption coefficient is calculated from: 1 and the specific acoustic impedance ratio is obtained as:. 3-2 Measurement environment See section 2-1. 3-3 Instrumentation The following equipment is used for the measurement with the transfer function method: Impedance tube: Brüel & Kjaer type 4002 o Serial No.: 491939 o Dimensions: Inner diameter: 0.1 m and length: 1 m Charge amplifier: Brüel & Kjaer type 2635 o Serial No.: 669742 Power amplifier: Zachry D250 o Serial No.: 614142 4
Microphone: MK224 PCP233/15 o Serial No.: 951246 Preamplifier: MWL-UNO 06 Frequency Analyzer: Tektronix type 2630 o Serial No.: B010319 PC with analyzing software o Serial No.: 001472 Data post processing software: Matlab The output of the frequency analyzer, which is a FFT based narrow band analyzer, is connected via the power amplifier to the loudspeaker in the impedance tube (figure 4), and generate white noise. The microphone with preamplifier is connected to the input of the frequency analyser. The analyzer is controlled from the Personal Computer. 3-4 Measurement procedure The frequency analyzer is set to random noise, further settings refer to page 136 and 137 in the reference [1]. The test sample, AC 30 S, is mounted at the end of the impedance tube in the specimen holder. Plane waves are generated in the tube by the loudspeaker and the sound pressures are measured at two points, 1 and 2 in figure 4. The complex transfer function of the two measured points is determined. The upper frequency limit is determined by the size of the tube as in the standing wave ratio method, section 2-4. For this method the distance between the microphone positions,, must fulfil the requirements 0.45 and 0.05. This is because the two microphones have to be placed within a half of the shortest wavelength and to ensure that there is a certain phase difference between them. In this experiment three microphone positions are used, that is two pairs of microphone distance, 0.04 and 0.25. The reason is to cover a wider frequency range. In this case position 1 will cover the frequency range from 450 Hz to 3800 Hz and position 1 will cover the frequency range from 70 Hz to 600 Hz. But the size of the tube limits the upper frequency range to 2 khz. The distance between the microphone position 2 and the opening of the specimen holder is 0.375 m. Since the distance ring used to get a cavity behind the specimen is 20 mm of length, the distances used to calculate the sound pressure reflection factor from the transfer functions (see section 3-1) below will be: 0.375 0.020 and, 0.375 respectively for position 1 (high frequencies). For position 1 (low frequencies) the distances are: 0.375 0.020 and, 0.375. Specimen holder s 2 1 1 X 1 Figure 4: Schematic view of the impedance tube used with two microphones. The microphone positions are 2, 1 and 1. 5
Usually the transfer function between the microphone positions are measured twice with the microphones interchanged. The reason is to overcome possible microphone mismatch. In this experiment however one microphone technique is used due to the fact that Matlab has a certain problem in making complex square root since the big range of phase involved. Instead the output of the signal generator will be the reference signal. The level of the amplifier is set to an appropriate level and is kept during the measurements. The microphone is placed in position 1 as indicated in figure 4. The transfer function between the microphone and reference signal is measured and saved. This is repeated for microphone positions 2 and 1. Then the three transfer functions are measured with cavity behind the test sample AC 30 S. This procedure is then repeated for the other test sample, FIRE 30 VF, without and with a cavity behind. A total of 12 transfer functions are measured. The measured data is converted from binary format to Matlab readable format using the program pcmatlnk.exe. The 8 transfer functions used in the final calculation are for each of the four test sample configuration estimated from and, where s is the reference signal. The normal sound absorption coefficient of the test sample and the specific acoustic impedance is calculated according to section 3-1. The results are presented in the next section. 3-5 Results The normal incident absorption coefficient and the magnitude of the specific acoustic impedance ratio are presented in figure 5 and 6 respectively. The figures are divided in the low frequency range to the left and the high frequency range to the right as measured. Figure 5: The measured absorption coefficient for the test samples AC 30 S and FIRE 30 VF with and without a cavity behind. To the left is the low frequency range (from 70 Hz to 600 Hz) and to the right is the high frequency range (from 450 Hz to 2 khz). 6
9 Specific acoustic impedance ratio 9 Specific acoustic impedance ratio AC 30 S AC 30 S AC 30 S + Cavity AC 30 S + Cavity FIRE 30 VF FIRE 30 VF FIRE 30 VF + Cavity FIRE 30 VF + Cavity 8 8 7 7 6 6 5 5 Z/(ρ c) Z/(ρ c) 4 4 3 3 2 2 1 1 0 100 150 200 250 300 350 400 450 500 550 600 Frequency (Hz) 500 1000 1500 2000 Frequency (Hz) Figure 6: The magnitude of the specific acoustic impedance ratio is plotted for the test samples AC 30 S and FIRE 30 VF with and without a cavity behind. To the left is the low frequency range (from 70 Hz to 600 Hz) and to the right is the high frequency range (from 450 Hz to 2 khz). 0 4 DISCUSSION When using a cavity behind a material, the material will act as a (frequency dependent) membrane with a certain mass. The air inside the cavity is analogue to a spring. From figure 5 (the transfer function method), it is obvious that, when using a cavity behind the specimen AC 30 S, the absorption coefficient is higher all over the measured frequency range except for the very lowest frequencies around 100 Hz and the highest, 2 khz. The absorption is displaced to lower frequencies due to the fact that there is more air behind the material and the stiffness is reduced. This is more obvious when analyzing the absorption coefficient for the specimen FIRE 30 VF which has a thin film on the front side (membrane). One can see that this results in a wide peak at 350 Hz and with the cavity behind, the stiffness is less which results in that the peak is displaced to 300 Hz. From say 600 Hz and up the thin film results in very low absorption compared to the AC 30 S as expected (FIRE 30 VF is slightly stiffer too). At higher frequencies the cavity behind FIRE 30 VF makes the absorption coefficient swing more. Putting a thin film on the front side of an absorbing material could be a good choice for low frequencies or if there is a certain narrow low frequency range that is of interest. AC 30 S has a much higher absorption coefficient for high frequencies (in this case over 600 Hz). The use of a cavity is in general displacing the frequency range of the absorption coefficient to lower frequencies. Figure 6 shows the magnitude of the specific acoustic impedance ratio and the curves are more or less inverted compared to the absorption, that is, when the impedance is low the absorption is high and vice versa, a little bit more inert though. 7
When comparing methods, the transfer function method is generally better than the standing wave method [1]. The absorption coefficient measured with the standing wave method for AC 30 S with cavity is higher than without a cavity all over the measured frequency range, figure 3, in according to the transfer function method, figure 5. In figure 3, the absorption coefficient for FIRE 30 VF is lower than AC 30 S except for a peak at a low frequency. The discrepancies are that the peak at 450 Hz in figure 3 for FIRE 30 VF is displaced higher in frequency and the absorption coefficient is a lot lower for higher frequencies (over 1600 Hz it is close to zero) compared to figure 5. The former is probably due to the fact that the measurements are made in the centre frequencies of the one third octave band. The absorption coefficient for AC 30 S agrees somewhat between 1200 Hz and 2 khz between the methods. Below 1200 Hz the absorption coefficient decrease too fast for the standing wave method compared to the transfer function method. The standing wave method has certain problems, it is difficult to find the minimum sound pressure levels due to the fact that they have a narrow peak, refer to figure 2 and this could be one reason for the discrepancies. The voltage is read off manually which can introduce errors. The main reason for the discrepancies is believed to be due to the fact that the standing wave method is not as accurate as the transfer function method. One can see in figures 5 and 6 that if the two frequency ranges from the transfer function method would be jointed together there would be a step at 600 Hz, especially when using cavity behind the material. Since the impedance tubes sizes limits the upper frequency to 2 khz, the two frequency ranges could have been chosen more effectively. 5 REFERENCES [1]: Leping Feng, Acoustical measurements, Lecture notes, TRITA-AVE 2007:07, ISSN 1651-7660, 2 nd print (2008) 8