Measurement of the Sound Absorption Coefficient of Materials with a New Sound Intensity Technique

The paper presents a completely new measurement technique of the sound absorption properties of materials, based on the measurements of active intensity and sound energy density. It allows one to measure the absorption coefficient with a wide band excitation, to use frequency bands of any width and to make measurements both inside a tube or in situ. The intensity technique is compared to Transfer Function Method (as defined in the ASTM E-1050 standard), by means of a complete theoretical study and of a large experimental validation. The results suggest that the new method is at least as accurate and reliable as the ASTM E1050 standard. It also has many advantages: it is faster, easier, it directly produces results in /3 or 1 octave bands, it can be implemented with portable, low cost instrumentation. 1. Absorption coefficient measurement techniques Absorption coefficient measurements are not easy, especially if one needs to do them in situ, i.e. where materials are placed and having no regard of their shape or extension. Usually the measurement of absorption coefficient is made in a reverberant room, according to the ISO 354 standard [1], or using the traditional standing wave tube technique, described in the ISO/DIS 10534 standard [2]. Both of these methods do not allow one to make measurements in situ, and the second one requires single frequency measurements, so that a complete test takes a long time. Moreover with the second one small samples have to be used and the tube/sample diameter has a strong influence on the frequency range and measurement limits. Anyway, in a standing wave tube it is also possible to make just a single, fast measurement, using a wide band signal instead of repeating measurements for each single pure tone. The use of a wide band signal speeds up the test and also allows one to be sure that collected values at each frequency were obtained in the same environmental conditions, for instance at the same room temperature and wave propagation velocity. These improvements can be achieved if the Chung & Blaser Transfer Function Method [3,4], described in the ASTM E-1050 standard [5], is chosen. This method requires a two-channel FFT analyzer and two closely spaced microphones, which have to be previously calibrated for phase and gain matching. As it is well known, the ASTM standard method involves the separation of a stationary, random, broad band signal into its incident and reflected components. It is based on the transfer function between the two sound pressure signals measured by two microphones placed along the tube wall. These two sound pressure signals are considered ergodic processes. This assumption has to be made in order to use time averages instead of statistical functions. In fact time averages allow one to describe the stochastic process on the base of just one of its realizations, so that it is not necessary to collect many of them to characterize the process in terms of statistical parameters [6]. The more the averaging time is long, the more the averages approach the theoretical value, so that a good accuracy requires a long averaging time. This standard is applied using a FFT analyzer, which works with narrow band filters and thus constant-percentage bands analysis can not be done directly: furthermore, due to the indetermination principle, the narrower the frequency bands are, the longer the averaging time must be, in order to reduce the error below a fixed threshold [7]. When it is necessary to characterize a material for technical or trading purposes, a few values of the absorption coefficient are required, at the IEC octave or /3 octave center-band frequencies. It is a good practice to get these values from the average absorption coefficient of all the frequencies included in each octave or /3 octave band, and not simply from the “local” value of the absorption coefficient at the center-band frequency: thus the results of a narrow-band analysis require a proper post processing to yield the required average absorption coefficients. In some previous works of one of the authors [8,9] several implementations of the Transfer Function Method were analyzed, in comparison with the ISO/DIS 10354 standard. These works pointed out that a strict implementation of the Transfer Function Method causes many problems, which can be solved only by using a properly designed experimental apparatus, and limiting the analysis to certain kinds of absorbing materials, which do not exhibit strong variations of the absorption coefficient with frequency. Similar results were also obtained by Chu [10]. An alternative extended method to measure the absorption coefficient, that uses a single microphone, subsequently placed in many different positions along the tube, and that computes the transfer function between each pair of positions, has been developed by one of the authors, as reported in the above mentioned works [8,9]. This measurement technique is slower than the original Transfer Function Method, it requires an accurate manual placement of the probe, and it still suffers of the same limitations due to narrow-band analysis and averaging time of the original Transfer Function Method. In the second work [9] an implementation of the new Intensity Method was tested for the first time, but the results were wrong because of an error in the computing formulas. 2. The new sound intensity technique The new intensity technique was created to make available an easy and quick solution to the above mentioned problems. With the new method it is possible to make measurements in situ because this method is robust and its basic requirements are very simple. It must be clearly stated that the proposed method is completely new, and it is not at all similar to other proposed “intensity methods” for absorption coefficient measurements, as those reported by Fahy [11] which relies on the use of the intensity meter as an “impedance meter”and in wichthe impedance is obtained by the sound pressure and particle velocity ratio. This new Intensity method uses only of the concepts of Sound Intensity and Energy Density, and thus is much more robust than the other methods in which ratios of complex quantities are involved. Furthermore, this new method does not rely on the previous knowledge of the incident intensity, obtained by a reference measurement of the same sound source in a free field: a single measurement near the absorbing surface is all that is required to extract the results. Something is worth to be underlined is that in the intensity technique no hypothesis about stochastic processes need in principle to be made, as the Sound Intensity definitions yield for any kind of sound field, both with deterministic and random signals. Obviously the ergodic assumption should be made if the Intensity Technique has to be compared with the Transfer Function Method, but just because this is required from the latter, or in general from any measurement method which is based on spectral analysis [12]. There are still limits in frequency range, according to sample extensions and placement, being necessary to have plane waves on the sample surface and to avoid border effects, but these limits are significantly broader than those required for the validity of the Transfer Function Method. With the new technique, the absorption coefficient can be calculated from the measurement of the Sound Intensity I and the sound field Energy Density D, on the base of simple mathematical relationship existing among them. These relationships allow one to know what the reflected and incident sound intensities are, so that the absorption coefficient can be calculated exactly from its definition. The Sound Intensity I is a vector quantity and it can be measured using a 3-D Sound Intensity probe (which uses 3 phase-matched microphones pairs) or a B-format Soundfield microphone, which is a special probe using a 4 microphones tetrahedral array and a proper circuitry to recover the pressure gradients along the three Cartesian axes. As initially suggested by Chu [13], it is even possible to use a single, omni-directional microphone properly placed in several closely spaced positions, provided that the sound field is excited with a deterministic broad-band signal, as, for instance, the MLS signal. All these techniques involve the computation of pressure gradient components along the three Cartesian axes (X,Y,Z), then the Cartesian components of the particle velocity u can be derived from these sound pressure gradients using Euler’s equation (usually written with finitedifference approximation, in which the pressure gradient is simply obtained from the difference of the pressure signals measured by the two microphones placed along the measurement axis, divided by the microphone spacing), as clearly explained by Fahy [10] and Rasmussen [14]. The result of particle velocity vector and sound pressure product is the active intensity vector I; the knowledge of sound pressure and particle velocity also allows one to know the sound field energy density D, being D t u t p t c ( ) ( ) ( ) = ⋅ ⋅ + ⋅       1 2 2 2