Sound Power Measurement A sound source will radiate different sound powers in different environments, especially at low frequencies when the wavelength is comparable to the size of the room 1. Fortunately this difference is not very large for most sound sources, in particular mechanical sources, and we can always use the sound power radiated in the free field as an approximation. 1 Principle The sound power is defined as W = I ds = I n ds, (1) S where I = pu is the sound intensity vector and I n is the sound intensity normal to the surface S. The surface should completely enclose the sound source. Sound power level is then L w = 10 log W ( ) = W L S I + 10 log, (2) 0 S 0 where L I is the averaged normal sound intensity level and S is the area of surface enclosing the source. S 0 = 1 m 2 is the reference surface, and the reference sound power W 0 is defined through the reference intensity I 0 as W 0 = I 0 S 0. 2 Sound pressure method In practical situations, the relation between sound pressure level and sound power level or normal sound intensity level is very complicated, and one cannot get accurate estimation of sound power level by using sound pressure measurements. However, in two ideal cases free field and a diffuse field sound power is explicitly related to the sound pressure. Free field and reverberation room methods have been developed which can be used to estimate sound power level rather accurately. Engineering and survey methods with lower accuracy have also been developed for other environments and for in situ measurements. 1 More accurately, the mean free path ; the average distance a sound ray will travel between successive reflections. S 1
Free field In a free field there is only direct sound and no reflection exists. For a progressive wave, there is a unique relation between the mean-squared sound pressure and the intensity in the direction of the wave propagation, I = p2 ρc. (3) Hence the normal sound intensity level over the surface can be expressed as L I = 10 log I I 0 = 10 log p2 ρci 0 = L p 10 log(c) (4) where C = I 0 ρc / p 2 0 = ρc/400. In room temperature (20 C) with an ambient pressure of 1 atm (1.013 10 5 N/m 2 ), 10 log(c) = 0.146 db and L I = L p 0.146 db. Splitting the surface S into N different pieces S n, the discrete form of equation (1) becomes N free field N p W = I n S n 2 ns n ρc. (5) n=1 Sound power level is then estimated as L w = 10 log W W 0 = 10 log n=1 N n=1 p 2 ns n ρcw 0 db. (6) This can be expressed as by using averaged sound pressure level L w = L p + 10 log( S S 0 ) + 10 log( p2 0S 0 ρcw 0 ) db, (7) where S is the total surface in m 2 and L p = 10 log N p 2 ns n 1 s n=1 p 2 0 db (8) is the averaged sound pressure level. If the reference values of sound pressure and sound power are put into equation (7) the last term vanishes when ρc = 400. Equation (7) is the basic equation for sound pressure method in a free field. Diffuse field In a diffuse field the sound pressure level is essentially independent of the distance to the sound source. Based on the concept of mean free path, the relation between sound energy density E(t) and the sound power W radiated by the source can be obtained as E(t) = 4W cᾱs ( ( 1 exp cᾱs )) 4V t, (9) 2
where V and S are the volume and wall surface of the room respectively, and ᾱ is the mean absorption coefficient. The steady-state sound energy density can be obtained by letting t as E 0 = 4W cᾱs. (10) The steady-state sound energy density in above equation can also be expressed by the sound pressure as E 0 = p2 ρc 2. (11) Combining above two equations and rearranging them we can obtain the relation between sound power and sound pressure in a reverberation field W = p2 ρc ᾱs 4 = p2 ρc A 4 (12) where A = ᾱs is the equivalent absorption. The formula may also be expressed by using the reverberation time to replace the equivalent absorption as W = p2 ρc 13.8V ct, (13) since A = 55.26V. (14) ct Based on above formula one can calculate sound power levels from measured sound pressure levels if the parameters of the reverberation room are known. The sound power level of a sound source will then have the form ( ) ( ) W L w = 20 log = L A p + 10 log 6 + C. (15) W 0 The first three terms are obtained from (12), A 0 is a reference absorption of 1 m, and C is a term related to small correction factors for other influences such as air absorption, ambient pressure, temperature, and interface patterns formed near the room surfaces. ISO has suggested a few formulas for the correction term in ISO 3740 series of standards depending on the accuracy of the measurement methods. The details can be found in the corresponding standards. Field (in situ) In this case both direct and reverberation fields exist and the mean-squared sound pressure can be written as ( p 2 Qθ = W ρc 4πr 2 + 4 ), (16) R where W is the sound power, R = Sᾱ/(1 ᾱ) is the room constant and Q θ is the directivity of the sound source. In this situation it is difficult to measure sound power accurately by measuring sound pressure level. It is quite common to use comparison methods with reference sound sources or to introduce an environment correction K 2 to get more reliable measurement results. A 0 3
3 Direct method vs. comparison method Before intensity methods became widely used, it was common to find the sound power of a sound source using sound pressure measurements. Even nowadays, when sound intensity probes are commonly found and corresponding standards are published, sound pressure methods are still popular in practice. There are two main types of sound pressure methods used to evaluate sound power: Direct methods and comparison methods. Direct method In the direct methods, the sound power is calculated using the measured averaged sound pressure level with equation (7) for a free field and equation (15) for a diffuse field. When the diffuse field is a concern, the equivalent sound absorption should also be measured with the sound source in question through reverberation time measurements. For other non-ideal situations, the relation between sound power and sound pressure is more complicated as indicated by (16). ISO suggests two parameters to describe environment corrections: K 1 for background noise correction and K 2 for environment correction. Those corrections should be included when evaluating the sound power level to take into account the effect of the environment and of background noise. It is the environment that determines which type of measurements should be performed and which accuracy level the measurement can reach. Comparison method As stated above, in order to calculate sound power from sound pressure level measured in a diffuse field, one needs to measure reverberation time when the sound source under test is mounted in the room. This may take longer time than the measurement of the sound pressure level radiated by the sound source under test. Furthermore, when measuring sound power level in situ, the environmental conditions may be so severe that the environment correction K 2 exceeds 7 db. In that case, no direct method with sound pressure level may be performed. In these situations, comparison methods can be an alternative. In comparison methods, the radiated sound pressure level of the sound source in question is compared with that radiated by a standard sound source. The sound power radiated from this reference sound source is precisely calibrated at the exactly the same environmental conditions. The sound power difference between the reference source and the source in question should then be the same as the difference between the measured sound pressure levels. The basic assumption of this method is that the environment influence (absorption, reflection, etc.) on the sound source under test should be the same as the reference sound source. This is usually a mechanical sound source with a stable output of sound power which is calibrated with high accuracy in a laboratory. It is used to find the influence of the environment which contributes in relating the sound power of the source to the resulting sound pressure level. This method is obviously suitable for mechanical sound sources with high internal impedance. Electrodynamic sound sources tend to have low internal 4
impedance and are thus more easily affected by the environment. Therefore, this method may not very suitable for this type of sound source. In the comparison method, one avoids measuring reverberation time, making sound power measurements much easier. The method is also commonly used to get the environment correction K 2 for special test environments. Since the comparison method compares the reflected or reverberant sound fields of a reference source and the source in question, it is clearly not suitable for a free field measurement. The more reverberant the sound field is, the higher accuracy this method can reach. We can get precision accuracy (grade 1) when this method is applied to a reverberation room and engineering accuracy (grade 2) when this method is applied to a field with the environment correction K 2 bigger than 7 db. For the case of K 2 smaller than 7 db, only survey accuracy (grade 3) can be reached. When the dimension of the sound source under test is much larger than the reference sound source or when the sound source has a strong directivity, the environment influence may not be same. Care must be taken in this situation. One solution is to measure the sound pressure level of the reference sound source when it is located at different positions and then the average is taken. The procedure is described in detail in corresponding ISO standards. 4 Sound intensity method This method directly measures the sound intensity instead of sound pressure level and hence it can in principle be used in free field measurements. The total intensity will be I = I direct + I diffuse. (17) In a diffuse field, waves propagate in random directions. Therefore, the intensity vectors of different waves have random directions and tend to cancel each other out, so that I diffuse 0. Thus, we are left with approximately only the intensity vector directly from the sound source, I I direct. Measuring intensity will thus give us the same intensity as if there were no reverberation. There are generally two methods to get averaged sound intensity level: Scanning method to get averaged sound intensity level by scanning over a hypothetical surface which completely encloses the noise source under test. Measuring at discrete points to get averaged sound intensity level by measuring at discrete points on the measurement surface which completely encloses the noise source under test. In principle, intensity methods can be applied to any environment but in practice they are restricted by instruments, background noise and the situation of the acoustic field. Measuring the normal intensity I n on each piece S i of the enclosing surface, the sound power component W i on that piece is W i = I n S i (18) 5
and the sound power level can thus be determined as ( N ) W i L w = 10 log. (19) W 0 5 Standards The above sections describe basic principles and procedures of measurements in different conditions. In practice many factors have to be taken into account in order to reduce errors and to make results more reliable and repeatable. The international standards for determination of sound power levels of machines and equipments are outlined in the ISO standard 3740:2000 where ISO 3741 ISO 3747 are sound pressure methods while ISO 9614-1, ISO 9614-2 and ISO 9614-3 (not listed in ISO 3740) are sound intensity methods. They can be divided into three groups according to the measurement accuracy: Precision methods (grade 1), engineering methods (grade 2) and survey methods (grade 3). Consequently, they also have different requirements on environment and background noise level. In the followingm details on these standards are discussed with focus on precision laboratory methods (ISO 3745 and ISO 3741). Other in situ methods (ISO 3747 and 3746) will be discussed in the lab exercise. 5.1 Free field methods (ISO 3744, 3745 and 3746) For all three methods hypothetical surfaces over a reflecting plane are used with equation (7) to calculate the sound power. (The exception is ISO 3745, which uses no reflecting plane.) Direct measurements are always are used in these methods. The main differences among them are: 1. Accuracy 2. Required test environment 3. Background noise 4. Obtainable sound powers i=1 Here only ISO 3745:2003 will be discussed in detail. relevant standards. For others, see the Room The standards specify different kinds of requirements for the rooms to achieve different grades of accuracy. Background noise At least 10 db lower than the sound pressure level measured from the source under test in each frequency band. 6
Temperature Should be within the range 10 C to 30 C. Within this range the influence of humidity can be neglected. Instrumentation The accuracy of instruments should be consistent with the accuracy of the method. Installation of source Whenever a typical condition of mounting exists for the source, it should be used or simulated, if practicable. Radius of measurement sphere (or hemisphere) The radius of the test hemisphere should be equal to or larger than all of following: Twice the largest source dimension or three times the distance of the acoustic center of the source from the reflecting plane, whichever is larger (for a semi-anechoic room) The wavelength of the lowest frequency of interest 1 m Microphone positions One of the following four methods is used to obtain the average value of the mean squared pressure on the test sphere (or hemisphere). Any measurement point should be at least 1 m away from the absorptive surfaces of the room. Note that the averages must be performed on mean squared values instead of on db levels. a. Fixed microphone positions The standard recommends an array of microphone positions associated with equal areas on the measurement surface. In general, the number of measurement points is sufficient if the difference in decibels between the highest and lowest sound pressure levels measured in any frequency band of interest is numerically less than half of the number of measurement points. If this requirement is not satisfied using 20-point array, an additional 20-point array must be used. If the requirement is still not satisfied by the 40-point, one might need to use microphone positions associated with unequal areas. In this case it should be taken into account when making average of sound pressure level. See ISO 3745 for details. b. Coaxial circular paths in parallel planes c. Meridional arc traverses d. Spiral path 7
Measurement time For the frequency bands centered on or below 160 Hz, the measurement time must be at least 30 seconds. For A-weighted sound pressure levels and for the frequency bands centered on or above 200 Hz, the measurement time must be at least 10 seconds. Correction for background sound pressure levels Background noise correction is often denoted by K 1. If the difference of the sound pressure level when the source is in operation and the background noise level is between 10 db and 20 db for each frequency band, the influence of the background noise must be corrected. For frequency band i, the background noise correction is given by where K 1i = 10 log(1 10 0.1 Li ) db, (20) L i = L pi L pi is the difference in sound pressure level at the same measurement point and same frequency band with and without the source in operation. When L = 10 db, the factor K 1i is about 0.5 db. The corrected sound pressure level is then L pi = L pi K 1i db. (21) If the background noise is more than 20 db below the sound pressure level with the source under test, no correction is needed. Calculation of sound power level The sound power is calculated as where L w = L p + 10 log C 1 = 10 log B B 0 313.15 273.15 + θ ( S1 S 0 ) + C 1 + C 2 db, (22) and C 2 = 15 log B B 0 296.15 273.15 + θ (23) are small correction factors involving the ambient pressure B during the measurements, standard atmospheric pressure B 0 = 1.013 10 5 Pa, and temperature θ (in C) during measurements. For an anechoic room, S 1 is equal to 4πr 2 where r is the radius of the test sphere. For a semi-anechoic room, S 1 is equal to 2πr 2. 5.2 Reverberation room methods (ISO 3741 and 3743) ISO 3741 is a precision method (grade 1) while ISO 3743-1 and ISO 3743-2 are engineering methods (grade 2). If frequencies above 300 Hz are included in the frequency range of interest, the volume of the test room should not exceed 300 m 3. 8
1/3 octave band center frequency Upper value of standard deviation of reproducibility σ R (db) (Hz) Anechoic room Semi-anechoic room 50 80 a 2.0 2.0 100 630 1.0 1.5 800 5000 0.5 1.0 6300 10000 1.0 1.5 12500 20000 b 2.0 2.0 A-weighted 0.5 0.5 a: if the sound field is qualified b: if the instruments allows and if correction is made for absorption of sound by the atmosphere Table 1: Estimated upper values of the standard deviations of reproducibility of sound power levels determined in accordance with ISO 3745:2003 Bandwidth Center frequency (Hz) Upper value of standard deviation of reproducibility (db) 100 160 3 1/3 octave 200 315 2 400 5000 1.5 6300 10000 3 125 2.5 octave 250 1.5 500 4000 1 8000 2 A-weighted 0.5 Table 2: Estimated upper values of the standard deviations of reproducibility of sound power levels determined in accordance with ISO 3741:1999 Lowest band of interest Minimum volume of the test room (m 3 ) 125 Hz (octave) or 100 Hz (1/3 octave) 200 125 Hz (1/3 octave) 150 160 Hz (1/3 octave) 100 250 Hz (octave) or 200 Hz (1/3 octave) 70 Table 3: Minimum volume of the test rooms 9
Requirements for absorption of test room The absorption of the room can influence the minimum distance between the noise source and the microphone positions. It also influences the sound radiation of the source and the frequency response characteristics of the test space. Due to these reasons the absorption of the test room should be neither too large nor extremely small, requiring that α < 0.06 for the surface closest to the source, and that T > V/S. Location of the source Should be placed at least 1.5 m away from any wall of the room. Microphones The minimum distance between source and microphone is d min = C 1 V /T, where C 1 = 0.08, or 0.16 for a higher grade of accuracy. If the microphone is being moved during the measurement, it must move at constant speed over a path at least 3λ in length, where λ is the wavelength of the center frequency of the lowest frequency band (about 10 m if the lowest frequency band is 100 Hz). For an array of several microphones, the microphone positions must be spaced at a distance of at least λ/2 from each other and 1 m from the room surface. Number of microphone positions and source locations When fixed microphone positions are used, at least six microphone positions at different heights should be used. If the standard deviation for any frequency band is larger than 1.5 db, additional microphone positions must be used. See ISO 3741:1999 for details. Although a reverberation room represents a diffuse field, it is still far away from a real diffuse field where sound pressure is the same everywhere. That is the reason averaging over many microphone positions or over a long microphone traverse path is required. Radiation on discrete frequencies or narrow bands of noise If a source radiates narrow band or discrete frequency sound, a precision determination of sound power requires greater effort than that of a broadband sound source. The standard gives additional precautions which have to be observed for this type of sound source. The methods are often complex and time consuming for sources that mainly emit on discrete frequencies below 200 Hz. For such sources measurement in a free field described in ISO 3745 are likely to be more appropriate. 5.3 Environment correction K 2 Environment correction K 2 is a correction term to account for the effect of reflected or absorbed sound on the surface sound pressure level measured. It is 10
frequency dependent and is used in standards using an enveloping measurement surface such as ISO 3744 and 3746. Two methods can be used to determine the environment correction factor: An absolute comparison method or a method based on room absorption. For the absolute comparison method we need a calibrated reference sound source with characteristics which meet the requirements of ISO 6926. If we mount the reference sound source in the test room and then measure and calculate the sound power level according to the procedure described in 5.1 without any environment correction term, the environment correction term of the test environment can then be found as K 2 = L W L W r. (24) Here, L W is the uncorrected sound power level measured according to the procedure described previously for a free field and calculated by using (22), and L W r is the calibrated sound power level of the reference sound source. This procedure is actually exactly the same as the comparison method discussed before to use a calibrated reference sound source to estimate the influence of the environment. The other method to determine the environment correction is based on room absorption measured. In that case the correction term can be calculated as ( K 2 = 10 log 1 + 4 S ), (25) A where A is the equivalent sound absorption area of the room and S is the area of the measurement surface. The problem in this method is that the sound absorption of the room has to be measured, which might be very time consuming. If only A-weighted sound power is required, one can use an approximate method to estimate the environment correction K 2A by using approximate values of the mean sound absorption coefficient of the room. In the lab exercise, we will use this method. 5.4 Intensity methods (ISO 9614-1, 9614-2 and 9614-3) The relationship between sound intensity level and sound pressure level at any point depends on the characteristics of the source, the characteristics of the environment and distance of the measurement positions and the source. For the previously described sound pressure methods, many restrictions therefore have to be made for source characteristics and measurement environments in order to make the uncertainty of the sound power determination within acceptable limits. The procedures specified in the sound pressure methods are not always appropriate since it is not often possible to install and operate large equipment in costly facilities such as anechoic or reverberation chambers to make high accuracy measurements. These methods also can not be used in the presence of high levels of extraneous noise generated by sources other than the source under investigation. The intensity method is a complement of the sound pressure methods and can be used under less restricted test conditions. There are three standards of intensity methods differences in sampling methods and accuracies. ISO 9614-1 is based on discrete-point sampling of the intensity field normal to the measurement surface and can reach precision accuracy 11
(grade 1), engineering accuracy (grade 2) and survey accuracy (grade 3). Both ISO 9614-2 and ISO 9614-3 are based on scanning of the normal intensity over the measurement surface with the engineering and survey accuracy and precision accuracy, respectively. The uncertainties when determining the sound power using intensity methods stem from the characteristics of the sound sources, extraneous sound fields, the absorption of the source under test, the intensity-field sampling and the measurement procedure used. The standards also supply procedures to reach a desired grade of accuracy. When needed, one can consult the corresponding standards for details. Following terms with respect to Field indicators F 1 F 4 in the three standards are often used when describing an intensity field and the way to determine the sound power. In the discussion below we follow the symbols of ISO 9614-1. Temporal variability indicator This is defined as F 1 = 1 1 M ( ) 2, Ink Ī n M 1 Īn (26) where Īn is the mean value for M short-time-average sample I nk. This is an indicator of the temporal variability of the sound field at the selected measurement position or surface. If the indicator is larger than a certain value, say 0.6, the sound field at the measurement surface is not stationary. It is always due to the influence of extraneous noise intensity. In order to increase the grade of accuracy one has to reduce the temporal variability or increase the measurement period. Surface pressure-intensity indicator k=1 This is defined as F 2 = L p L In, (27) where L p is the surface averaged sound pressure level and L In is the surface averaged normal unsigned sound level, i.e., average of the magnitude of the sound intensity only, no matter what is the direction of the intensity. Negative partial power indicator This is defined as F 3 = L p L In, (28) The difference compared with F 2 is that here signed intensity is used. This is called negative partial power indicator. Field non-uniformity indicator This is defined as F 4 = 1 1 N ( ) 2, Iik Ī n N 1 Īn (29) i=1 12
This indicator specifies the field non-uniformity from segment to segment. In order to get a certain level of accuracy the condition L d > F 2 (30) must be fulfilled, where L d is a parameter of the measurement system, called the dynamic capability index. If a chosen measurement surface does not satisfy this criterion, some steps described in the standard have to be taken to improve the measurement accuracy. Otherwise the measurement accuracy grade has to be reduced. In the laboratory exercise, we use sound intensity method with scanning and follow standard ISO 9614-2. 13