Measurement and evaluation uncertainties of impact sound insulation An investigation of light weight structures Master of Science Thesis in the Master Degree Sound and Vibrations ERIK BACKMAN HENRIK LUNDGREN Department of Civil and Environment Engineering Division of Applied Acoustics Vibroacoustics Group CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden, 2010 Master´s Thesis 2010: 136 Master´s Thesis 2010: 136 Measurement and evaluation uncertainties of impact sound insulation An investigation of light weight structures ERIK BACKMAN & HENRIK LUNDGREN Department of Civil and Environment Engineering Division of Applied Acoustics CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden, 2010 Measurement and evaluation uncertainties of impact sound insulation An investigation of light weight structures © ERIK BACKMAN & HENRIK LUNDGREN, 2010 Master’s Thesis 2010: 136 Department of Civil Engineering Division of Applied Acoustics Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone + 46 (0)31-772 1000 Cover: Examples of sources of impact sound Furrer W., Room and Building Acoustics and Noise Abatement, BUTTERWORTH & CO (1964) Reproservice / Department of Civil and Environmental Engineering Gothenburg, Sweden 2010 i Measurement and evaluation uncertainties of impact sound insulation An investigation of light weight structures ERIK BACKMAN & HENRIK LUNDGREN Department of Civil and Environmental Engineering Division of Applied Acoustics Chalmers University of Technology Abstract Light weight building technique for multi-storey residential houses become more and more common, approximately 15 – 20 % of all new multi storey apartment buildings in Sweden are built with this technique [1]. However, it is well known among engineers and scientists in the field of acoustics that the methods of measurement and evaluation of impact sound according to ISO 140-7 [7] and ISO 717-2 [3] suffer from shortcomings considering light weight floor structures. These methods do not manage to create an objective single number quantity of the impact sound insulation which sufficiently correlates to the habitant’s subjective judgment regarding the sound climate in the building. This is a large disadvantage which might prevent a positive future development of the light weight technique in multi storey residential buildings. This Master thesis aims to extend the knowledge in this field by identifying reasons for measurement and evaluation uncertainties of impact sound insulation measurements in light weight buildings. The results in this thesis indicate significant uncertainties in current measurement and evaluation methods. According to experiences from engineers, the uncertainties are most crucial in the low frequency range where the largest vagueness of the impact sounds arise. Light weight structures generate highest sound levels in low frequencies where the human ear is most sensitive to level differences. This means in practice, when the impact sound is perceived small changes in noise levels might cause large changes in the subjective experience of the impact sound insulation in a light weight building. The results emphasize the need of further knowledge, especially at low frequencies to be able to revise the standards (ISO 140-7 and ISO 717-2) successfully which would result in measurement results giving objective single number quantities which correlate well to the subjective perception. Keywords: Impact sound pressure level, Reverberation time, Distribution, Confidence interval, Reference curve, Low frequencies, Receiving and Source positions. ii Acknowledgements First of all we would like to thank our principal mentor Klas Hagberg at ÅF Ingemansson for giving us the opportunity to work with this project and provide us with necessary data and advice during our work. Thanks for a great commitment. We would like to thank our second mentor Pontus Thorsson at Akustikverkstan for providing us with advice, data and equipment for our own measurements. We would also like to thank following persons for helping us: staff at ÅF Ingemansson and WSP which were providing us with necessary data, Bo Gärdhagen for advice and important data, Aila Särkkä for essential help in hard times with statistics and, of course, Bo Daniel Söderström Wahrolén and Olof Olsson for continuous help and support during our work! iii Table of contents 1. INTRODUCTION........................................................................................................................................ 1 1.1. GOAL ....................................................................................................................................................... 3 1.2. METHOD .................................................................................................................................................. 4 2. THEORY .................................................................................................................................................... 6 2.1. BUILDING ACOUSTICS .................................................................................................................................. 6 2.2. STATISTICS .............................................................................................................................................. 12 3. UNCERTAINTIES OF MEASUREMENTS .................................................................................................... 16 3.1. DISTRIBUTION .......................................................................................................................................... 16 3.2. EFFECT OF RECEIVING ROOM VOLUME ........................................................................................................... 23 4. IMPACT SOUND IN LOW FREQUENCIES .................................................................................................. 32 5. EXTENDED MEASUREMENTS .................................................................................................................. 34 5.1. REVERBERATION TIME MEASUREMENT .......................................................................................................... 34 5.2. IMPACT SOUND PRESSURE MEASUREMENT ..................................................................................................... 35 6. EVALUATION OF REFERENCE CURVE ...................................................................................................... 42 6.1. EXTENDED INVESTIGATION IN THIS THESIS ...................................................................................................... 44 6.2. ALTERATION OF REFERENCE CURVE ............................................................................................................... 46 6.3. APPLYING FOUR DIFFERENT REFERENCE CURVES .............................................................................................. 48 7. EVALUATION OF MEASUREMENT REPORTS ........................................................................................... 52 8. SUMMARY OF EVALUATED RESULTS ...................................................................................................... 53 8.1. DISTRIBUTION WITH REGARD TO SPATIAL AVERAGING PROCEDURE ...................................................................... 53 8.1. EFFECT OF RECEIVING ROOM VOLUME ........................................................................................................... 53 8.2. IMPACT SOUND IN LOW FREQUENCIES ........................................................................................................... 53 8.3. EXTENDED MEASUREMENTS ........................................................................................................................ 54 8.1. EVALUATION OF REFERENCE CURVE .............................................................................................................. 54 8.2. EVALUATION OF MEASUREMENT REPORTS ..................................................................................................... 54 9. DISCUSSION ........................................................................................................................................... 55 9.1. DISTRIBUTION .......................................................................................................................................... 55 9.2. EFFECT OF RECEIVING ROOM VOLUME ........................................................................................................... 55 9.3. IMPACT SOUND IN LOW FREQUENCIES ........................................................................................................... 56 9.4. EXTENDED MEASUREMENTS ........................................................................................................................ 57 9.1. EVALUATION OF REFERENCE CURVE .............................................................................................................. 57 9.2. EVALUATION OF MEASUREMENT REPORTS ..................................................................................................... 60 10. CONCLUSIONS ................................................................................................................................... 61 11. REFERENCES....................................................................................................................................... 62 11.1. LITERATURE ............................................................................................................................................. 62 11.2. INTERNET ................................................................................................................................................ 63 11.3. INTERVIEWS ............................................................................................................................................. 63 12. APPENDIX I – ELEMENTS OF CONFIDENCE INTERVAL CALCULATIONS ................................................... I 13. APPENDIX II – ADDITIONAL DISTRIBUTION PLOTS ............................................................................... II iv 14. APPENDIX III – EQUATIONS OF LINEAR ESTIMATIONS......................................................................... IV 15. APPENDIX IV – ADDITIONAL LINEAR ESTIMATIONS ............................................................................ IX 16. APPENDIX V –LOW FREQUENCY IMPACT SOUND ............................................................................ XVIII 18. APPENDIX VI – REFERENCE CURVE COORDINATES ........................................................................... XXII 19. APPENDIX VII - EVALUATION OF MEASUREMENT REPORTS ............................................................ XXIV 19.1. PROJECT 1............................................................................................................................................. XXV 19.2. PROJECT 2............................................................................................................................................ XXVI 19.3. PROJECT 3............................................................................................................................................ XXVI 19.4. PROJECT 4............................................................................................................................................ XXVI 19.5. PROJECT 5............................................................................................................................................ XXVI 19.6. PROJECT 6........................................................................................................................................... XXVII 19.7. PROJECT 7........................................................................................................................................... XXVII 20. APPENDIX VIII – CONTRIBUTE TO ICA 2010. .................................................................................... XXIX 1 1. Introduction It is well known among engineers and scientists in the field of acoustics that the methods of impact sound measurements and in particular the evaluation of impact sound insulation according to ISO 140-7 [7] and ISO 717-2 [3] respectively suffer from shortcomings considering light weight floor structures. These shortcomings get clear when it comes to create an objective single number quantity of the impact sound insulation, correlating sufficiently to the inhabitant’s subjective judgment regarding the sound climate in multi storey light weight buildings. Bodlund [6] claims that there are three different ways to find a solution to the problem: 1 By introducing a new or modified impact sound source which effectively simulates normal impact sources and footsteps (by changing ISO 140-7 [7]). 2 By changing the procedure for evaluation of the single number characterizing the impact sound insulation (by changing ISO 717-2 [3]). 3 By changing 1 and 2. Many attempts have been made to replace the ISO impact machine or to combine it with a heavier sound source which would produce a sound corresponding to more typical footstep impact on the floor structure [8]. However, if a new standardized impact sound generator would be constructed, consequently the evaluation stated in ISO 717-2 [3] needs to be modified as well. An example of alternative impact sound sources are the heavy rubber wheel or the rubber ball which are used in the Japanese national standard, JIS A 1418 [6]. These sources are soft, heavy and able to generate low frequency sound of another characteristic than the tapping machine to better simulate the sound of a walking person [8]. However, the wheel generates much higher forces on the floor structure, which increase the risk of structural damage as a result of the testing. Regarding uncertainties in measurements of impact sound insulation, the measurement procedure is performed as stated in ISO 140-7. The results are evaluated according to ISO 717-2. Initially, the frequency range considered in these standards was adapted to measurement of impact sound insulation on concrete structures and concrete elements. Then the methods were acceptable in general since it was not allowed to build light weight (primarily wooden) multi storey buildings at that time. In the early 1990’s this changed and it became acceptable to use wood as building material for multi storey residential buildings [8]. Based on this, it is not surprising that the standards ISO 140- 7 and ISO 717-2 are not working optimally for light weight floor structures since light weight structures differ from traditional heavy structures from an acoustical point of view. One weakness with impact sound pressure level measurements on light weight structures is that the measured and evaluated single number impact sound pressure level is not consistent in general with the subjective impression for those who are living in buildings with light weight floor structures. Furthermore, the methods are not adapted to measure as low in frequency as necessary when they are applied to light weight structures. The main problem is the impact sound pressure level in the low frequency domain where high levels are common for light weight structures [8], while for heavy structures the levels are normally low and furthermore decreasing with decreasing frequency, hence not affecting the final result. Figure 1-1 shows typical difference in frequency content from an impact sound pressure level measurement on one heavy and one light floor structure respectively. The measurements presented in Figure 1-1 are made in a laboratory with the same receiving room. 2 Remarkable is the weighted single number L’n,w + CI,50-2500 which is calculated to 52 dB for both structures even if significant differences exists for the measured levels, especially in the low frequency region. Figure 1-1: Impact sound pressure level measurements performed on a concrete and a wooden floor structure during laboratory conditions [9]. Also, the human ear is more sensitive to level differences in the low frequency region. Once the signal appears, a sound pressure level difference of 3-5 dB is perceived as a doubling of the sound level in the lowest frequencies, compared to 1000 Hz where a 10 dB difference is perceived as a sound level doubling [10], Figure 1-2. The impact sound pressure level characteristics may not be the only reason why the subjective perception differs from the measured results. Concrete floor structures are homogeneous while wooden floor structures are more complex constructed of joists and beams connected in different manner due to different systems. This may create high uncertainty of the impact sound pressure measurements with respect to the system and where the tapping machine is placed. If it is placed direct over a beam the vibrations can be lead straight to the receiving room compared to when it is placed between beams since no strong path from the tapping machine to the receiving room exists. Frequency (Hz) 3 Figure 1-2: Phon curves extended to very low frequencies, describing how the human ears perceive the sound pressure level in different frequencies [27]. Regarding the evaluation method stated in ISO 717-2, a reference curve is used to estimate a weighted single value of the measured impact sound pressure level [3]. Since the result from the existing evaluation method does not sufficiently correlate to the subjective perception, many attempts have been made to define another shape of the evaluation curve designed to take lower frequencies into account, i.e. optimally shaped when still using the tapping machine as sound source. Similar to earlier research of e.g. Hagberg [8] and Bodlund [6], an analysis regarding reference curve shape was included also in this work, based on existing material using a linear regression analysis. The mean weighted values from objective impact sound pressure level measurements are compared with mean values from subjective judgements. The aim with the reference curve investigation is to state whether results from earlier research still hold when objective and subjective data are extended. 1.1. Goal The main purpose of this investigation is to point out and study uncertainties in the ISO measurement and evaluation method of impact sound insulation (sound pressure level) on light weight structures. Further on, the design of the reference curve will be investigated with the aim to verify a reference curve from earlier investigations used for single numbers adapted to the subjective evaluation of impact sound pressure level. This is made in order to confirm whether the suggested reference curve from Hagberg [8] will alter if the data set is further extended. Finally, a description of existing measurements based on current ISO 140-7, available at different consultant companies, on impact sound insulation in light weight floor structures and their usefulness for further analyzes in the Akulite research project should be made. Hence, if necessary, a description on existing measurements and their need for supplementary information for the Akulite project should be included. 4 1.2. Method Input data has been achieved by contact with various consultants in the acoustical field. The measurements have been assigned a project number to keep them anonymous. For each project there have been measurements on different number of objects. The objects are referred to the appendix number in each measurement report or similar. The measurements of impact sound pressure level has been divided into concrete and light weight (mainly wooden) floor structures for both vertical and horizontal measurements, Table 1-1. All the input data has been treated and analyzed in Matlab R2009b. Measurements according to ISO 140-7 are divided into several parts. There is a need for level measurements to state the impact sound level in the room. However, there is also a need for some corrections of the measured sound level if the room is not fully furnished or sparsely furnished for example, i.e. not corresponding to a normally furnished room. To make reasonable corrections there are thus a need for reverberation time measurements which will be compared to a “normal” reverberation time or equivalent sound absorption area. Furthermore, the background level has to be measured in order to establish that it is not affecting the final results. This investigation has been focused on mainly impact sound pressure level and reverberation time measurements and their influence on the final results. Table 1-1: Table presenting the measurement data which have been used for the analysis. The projects have gained a number to keep them anonymous. If it has been both horizontal and vertical measurements in a project they have been separated in the table. Project Direction Floor structure Measurement 1 Vertical Light weight A05 A07 A10 A11 A12 1 Horizontal Light weight A06 A08 A09 2 Vertical Light weight 1 3 Vertical Light weight 2008 (1) 2008 (2) 2009 (1) 2009 (2) 2007 4 Vertical Light weight A01 A02 A03 A04 A05 A06 5 Vertical Light weight A01 A02 A03 A04 A05 A06 A07 A08 A09 6 Vertical Light weight A10 A11 6 Horizontal Light weight A12 7 Vertical Light weight A15 A16 A17 A18 A19 A20 A21 A22 A23 A24 A25 7 Horizontal Light weight A26 8 Vertical Concrete 8 9 9 Vertical Concrete A04 10 Vertical Light weight 1 2 3 4 5 6 7 8 11 Vertical Concrete 1 2 12 Vertical Concrete 1 13 Vertical Concrete 1 2 3 4 14 Vertical Light weight 1 15 Vertical Concrete 1 16 Vertical Light weight 1 5 1.2.1. Uncertainties of measurements The data have been analyzed with regard to different measurement positions in the receiving room. The distribution of data from impact sound pressure level and reverberation time measurements were investigated with a Weibull distribution plot. The distribution of measured reverberation times in various measurements positions have been more detailed investigated by using quaintile-quantile plots. Also, different receiving room volumes and their influence on the results have been studied, in order to illustrate how the uncertainties of measurements depend of the receiving room volume. 1.2.2. Impact sound in low frequencies Some measured impact sound pressure levels down to very low frequencies have been investigated in 1/3 octave bands. The investigation was based on projects with measured impact sound pressure level down to either 6.3 Hz or 25 Hz. The frequency range varies between the investigated measurements depending on the operator’s choice at the moment for performing the measurements due to certain needs in the specific case. 1.2.3. Extended measurements The authors have made own measurements regarding impact sound insulation, performed vertically between two rooms similar in size and volume, the measurement is referred as project 16 in Table 1-1. The aim of these measurements has been to investigate how different positions of the tapping machine in the sending room influence the final weighted impact sound pressure level. The deviation of the impact sound pressure level and reverberation time data from the measurement has been analyzed with a Weibull distribution. The number of tapping machine positions was totally 35, limited by the room boundaries. 1.2.4. Evaluation of reference curve The correlation analysis to investigate different reference curves has, similar to Hagbergs work [8], been based on a linear regression analyze. Mean values from objective impact sound pressure level measurements have been compared with mean values from subjective judgments using current data from two independent investigations. 1.2.5. Evaluation of performed measurements The measurement reports of project 1 - 7 in Table 1-1 have been analyzed based on requirements in standard SS-EN ISO 140-7 [7], ISO 717-2 [3] and SS 25267 [2]. The arrangement of the reports has been compared to see if there are deficiencies in the description of measurement procedure. The work has been limited to only evaluate measurements on light weight (mainly wooden) floor structures for frequencies from 50 to 3150 Hz; since this is the current standard limits (can be extended to 5000 Hz, however not interesting in this project). The input data evaluation has been limited to focus on the following aspects; • Are there any differences between measurement reports regarding acoustical results? • Are the measurement procedures sufficiently clear? • Are the building structures sufficiently described in the reports? • Are there any existing risks to use existing measurements in the AkuLite project? 6 2. Theory 2.1. Building acoustics A short description and explanation in theory of building acoustics concerning standards, concepts, parameters, procedures and evaluation methods in this thesis follows in this chapter. 2.1.1. Measurement standards SS-EN ISO 140-7: Swedish, European and International standard describing how to perform impact sound pressure level measurements in the field. SS-EN ISO 717-2: second edition: Swedish, European and International standard describing how to evaluate single number levels from the results of measurements, for example performed according to SS- EN ISO 140-7. SS 25267:2004, third edition: Swedish sound classification standard, containing a scheme with four sound classes stating requirement levels, where sound class C corresponds to the Swedish national requirements in the building code, BBR. It also contains additional requirements to the SS EN ISO 140-7 standard. 2.1.2. Acoustic concepts Impact sound Sound generated, through mechanical contact to a floor structure, in one room transmitted to another room. Impact sound can have different origins such as walking persons, falling objects or chairs being moved. Impact sound pressure level: The sound pressure level in dB which is perceived in a receiving room when a tapping machine is running on the floor in a sending room. Impact sound level is measured between two rooms at different floors where the floor is separating the rooms or between two rooms at the same floor with a separating wall. Consequently, the impact sound insulation can be measured both vertically and horizontally. The Impact sound insulation in the field situation is presented by the normalized or standardized weighted single number parameters, L’n,w and/or L’n,w+CI,50-2500 or L’nT,w and/or L’nT,w+CI,50-2500. The impact sound insulation is also commonly presented as a function of frequency in 1/3 octave bands between 50- 3150 Hz. Schroeder frequency: The frequency where the modal spacing changes from having less than three modes to having at least three modes within a given mode´s half-power bandwidth [32]. 7 Sound classification: A floor structure can be classified using different sound classes. In Sweden and other Nordic countries four classes exist, A, B, C and D where A is the highest and D is the lowest sound class. Sound class C corresponds to minimum national requirements. In special cases sound class D is used. A structure might be rated with sound class D if it is not possible to achieve the minimum sound class C, e.g. due to the cultural or historical reasons of a building. The limits for impact sound of the sound classes are defined by Table 2-1. Table 2-1: Impact sound pressure level requirements for classification according to SS 25267 [2]. Sound class A Sound class B Sound class C Sound class D Highest level2) 48 dB1) 52 dB1) 56 dB1) 60 dB 1) Both L’n,w and L’n,w+C I,50-2500 has to be fulfilled. 2) Receiving room volume is limited to 31 m3 Tapping machine ISO standard impact sound generator which is used in the measurement procedure according to ISO 140-7. The machine includes five steel-faced hammers, each with a weight of 0.5 kg which strikes the floor structure 10 times per second from a height of 40 mm. 2.1.3. Acoustic parameters The following description of parameters is due to the measurement standards SS-EN ISO 140-7, SS- EN ISO 717-2 and SS 25267:2004. �: Equivalent absorption area in the receiving room, given in [m2]. ��: Reference equivalent absorption area, set to 10 m2. ��,�������: Spectrum adaptation term to be used to extend the frequency range down to 50 Hz, evaluated according to ISO 717-2 [dB]. ′: Equivalent impact sound pressure level in the field for each 1/3 octave band in the frequency range 50-3150 Hz in the receiving room. The index ‘ on L indicates that flank transmission is included (i.e. field situation), given for each 1/3 octave band with one digit [dB]. �: Averaged measured equivalent sound pressure level for each 1/3 octave band in the frequency range 50-3150 Hz of the background noise in the receiving room, given for each 1/3 octave band with one digit [dB]. ′�: Averaged normalized impact sound pressure level measured in field. L’n is normalized to the absorption 8 area A0 = 10 m2. In Sweden, this parameter is used for rooms with a volume less than 31 m3, given for each 1/3 octave band with one digit [dB]. ′� : Standardized impact sound pressure level measured in field. L’nT is standardized to the reverberation time, T = 0.50 s. In Sweden, this parameter is used for rooms with a volume equal to or greater than 31 m3, given for each 1/3 octave band with one digit [dB]. ′�,�: Weighted and normalized impact sound pressure level, given in one single number value [dB]. ��,� + ��,������� Weighted and normalized impact sound pressure level considering the frequency range 50 – 2500 Hz, given in one single number value [dB]. ′� ,�: Weighted and standardized impact sound pressure level, given in one single number value [dB]. �� ,� + ��,������� Weighted and standardized impact sound pressure level considering the frequency range 50 – 2500 Hz, given in one single number value [dB]. �: Reverberation time in the receiving room for each 1/3 octave band, given for each 1/3 octave band with two digits [s]. ��: Reference reverberation, set to 0.50 s for all 1/3 octave bands. The reverberation time corresponds approximately to a furnished room in a “normal dwelling” independent of frequency. �: Volume of the receiving room (in Sweden limited to 31 m3) [3]. 2.1.4. Measurement procedure The measurement procedure is described in the international standard SS-EN ISO 140-7. In general, the measurements are performed with a tapping machine in at least four randomly distributed positions in the source room. If the measurement is performed on a light weight floor structure, the tapping machine shall be orientated 45° to the direction of the beams and ribs in the floor structure. There are two possible approaches to perform the spatial averaging procedure of the impact sound pressure level in receiving rooms, either by fixed microphone positions or by sweeping microphone positions. Either if the measurements are performed with fixed microphone positions or by a sweeping microphone, there shall be at least four sending positions. If the approach with fixed microphone positions is used, the positions shall be at least four. Further on, at least six measurements shall be done. If the measurement is performed with sweeping microphone, the 9 sweep shall be performed with a minimum radius and during a minimum measurement time. At least four measurements shall be done. Further on, some demands regarding distance to boundaries and distances between microphone positions have to be considered for both procedures. Regarding reverberation time measurements, at least six measurements shall be performed with at least one source position and three microphone positions with two readings in each position. As a complement to SS-EN ISO 140-7, there are some demands by the Swedish standard SS 25267:2004 in appendix H. This text describes further demands regarding microphone and source positions and also instructions to handle equipment, small rooms and measurement uncertainties. 2.1.5. Evaluation of measurements The spatial averaged impact sound pressure level is calculated for each 1/3 octave band according to Equation 2-1. Equation 2-1 = 10log�� �1� � 10�� ��⁄ !"� # where Li are the sound pressure levels L1 to Ln at n different positions in the room. If needed, the averaged impact sound pressure level is adjusted due to the background noise. If the level difference between the impact sound pressure level and background noise is greater than 6 dB but less than 10 dB the correction shall be made according to Equation 2-2. If the level difference is less than 6 dB the correction shall be made according to Equation 2-3. However, if the difference is greater than 10, no correction shall be made. Equation 2-2 = 10log��$10�%&,� ��⁄ − 10�&,� ��⁄ ( Equation 2-3 = ! − 1,3 where L is the adjusted signal in dB Lsb is the level of the signal and the background noise combined in dB Lb is the background noise The properties (i.e. the furnishing) of the receiving room affect the measured impact sound pressure level, hence the level will differ in a room depending on if the room is furnished or not. Therefore, the impact sound pressure level is adjusted to the level of a normally furnished room. This is done either by normalization (i.e. normalize to 10 m2 absorption area) or by standardization to a normally furnished room (i.e. T = 0.50 s). The most proper way is to standardize to 0.50 s since this makes the level independent of room volume. If normalize to 10 m2 (i.e. using L’n,w) this imply big errors if the receiving room volumes are big. In Sweden L’n,w is used, however there is a limit not to exceed receiving room volumes of 31 m3 in the evaluation which means that in reality L’nT,w is valid when 10 room volumes exceed 31 m2. Hence, in Sweden the value is normalized according to Equation 2-4 and 2-5 However, the impact sound pressure level is standardized according to Equation 2-6 if the receiving room volume is equal or greater than 31 m3. Equation 2-4 ′�,! = ! + 10log�� *�!��+ where Equation 2-5 �! = 0,16��! and �� = 10 .� The standardized impact sound pressure level is calculated according to Equation 2-6. Equation 2-6 ′� ,! = ! − 10log�� *�!��+ where �� = 0,50 0 The 1/3 octave band values are then compiled to one single number through a weighting procedure. The weighting procedure to calculate the single number levels L’n,w and L’nT,w is performed in the frequency range 100 – 3150 Hz by comparing L’n or L’nT with a reference curve, Figure 2-1, which is defined in ISO 717-2 [3]. ISO 717-2 states that the single number quantity called L’n,w and L’nT,w equals the impact sound pressure level at 500 Hz, after the reference curve has been shifted in steps of 1 dB until the sum of the unfavorable deviations from the measured curve is as large as possible but not larger than 32.0 dB. An unfavorable deviation is present in a specific frequency band when the measured level is higher than the value of the reference curve. 11 Figure 2-1: Reference curve used for weighting of measured impact sound pressure level, according to ISO 717-2 [3]. To take frequencies lower than 100 Hz into account, the adaptation term CI,50-2500, might be added to L’n,w or L’nT,w, calculated according to Equation 2-7 and Equation 2-8. Equation 2-7 ��,������� = ′�,123 − 15 − ′�,� Equation 2-8 ��,������� = ′� ,123 − 15 − ′� ,� The sum of the impact sound pressure levels are calculated according to Equation 2-9. Note that only the measured values for the 1/3 octave bands 50 – 2500 Hz are considered in Equation 2-9 and Equation 2-10. Equation 2-9 ′�,123 = 10log�� 4� 10�5,� ��⁄6 !"� 7 Equation 2-10 ′� ,123 = 10log�� 4� 10�58,� ��⁄6 !"� 7 The “low frequency” single number rating is then specified simply by adding the adaptation term to the weighted value i.e. L’n,w+CI,50-2500 and L’nT,w+CI,50-2500. 63 125 250 500 1k 2k 4k 42 44 46 48 50 52 54 56 58 60 62 Reference curve L e v e l [d B ] Frequency [Hz] ISO 717 12 According to the description above and guidelines in SS 25267, L’n,w and L’n,w+CI,50-2500 are used even if the real value should be stated as the standardized level (L’nT,w) when room volumes exceed 31 m3. 2.2. Statistics Two commonly used statistic methods to investigate the measurement uncertainties are standard deviation and confidence interval. These measures can be used in the field of acoustics for instance in evaluation of uncertainties between different microphone positions regarding impact sound pressure level and reverberation time measurements. 2.2.1. Standard deviation Standard deviation, σ, is a single value describing to what extent the different values in a statistical population deviate from the mean value, which is illustrated in Figure 2-2. A low value on standard deviation indicates that the data tend to be close to the mean value, whereas a high value indicates that the data are widely spread out over a large range from the mean value. The standard deviation is expressed in the same units as the data and is defined as the square root of the sample variance s2, Equation 2-11. The variance can also be defined as theoretical as σ2. Since the theoretical variance σ2 cannot be calculated, the variance is estimated by the sample variance s2. Equation 2-11 9 = :0� where Equation 2-12 0� = 1� − 1 �$;< − ;̅(� <"� and Equation 2-13 ;̅ = 1� � ;< <"� In the field of acoustics, standard deviation is commonly used to measure the uncertainty of measurements, for instance the uncertainty which different microphone positions in a receiving room gives rise to within each 1/3 octave band. A typical indication regarding uncertainties in a measurement could be if the standard deviation for a certain 1/3 octave band is significantly higher than other 1/3 octave bands, then there is likely some error caused by poor circumstances like loud time-varying background noise. Figure 2-2: A normal distribution standard deviation diagram. Each colored band has a 2.2.2. Confidence interval for an unknown parameter Confidence intervals are in mathematical statistics a interval which is estimated for a given confidence level means in this work that for each set of measurement series, measured mean value lies somewhere within the confidence interval confidence levels are 90 and 99 %. A confidence level of 90 % ends up with a narrower interval than a 95 % confidence level. In the opposite way, a 99 % confidence level ends up with a wider interval. The confidence interval covers the true parameter value with a certain probabilit A confidence interval can be based on different distributions as; normal distribution, binomial distribution and Poisson distribution. The confidence interval for the expected value of a normal distribution µ, is defined as Equation be estimated from the data by using the sample mean value, and s2 is the estimated variance, calculated as n is number of observations. c is the coverage probability from table in Appendix I suit to when estimated as s2, this operation is known as Student’s t Appendix I from calculated values from 13 : A normal distribution standard deviation diagram. Each colored band has a width of one standard deviation. [4]. Confidence interval for an unknown parameter re in mathematical statistics a measure of the uncertainty expressed in an interval which is estimated for a given confidence level >, often 95 %. A confidence level of 95 % means in this work that for each set of measurement series, there is a 95 % probability that lies somewhere within the confidence interval. Examples of other common nd 99 %. A confidence level of 90 % ends up with a narrower interval than a 95 % confidence level. In the opposite way, a 99 % confidence level ends up with a wider interval. The confidence interval covers the true parameter value with a certain probabilit A confidence interval can be based on different distributions as; normal distribution, binomial distribution and Poisson distribution. The confidence interval for the expected value of a normal Equation 2-14. µ is the theoretical mean value of the distribution and can be estimated from the data by using the sample mean value, ;̅, Equation 2-13. Equation 2-14 CONFCD;̅ − E F G F ;̅ + EH where Equation 2-15 E = I√0� √� is the estimated variance, calculated as Equation 2-12. is number of observations. is the coverage probability from table in Appendix I suit to when , this operation is known as Student’s t-Distribution[ from calculated values from Equation 2-16 and Equation width of one standard deviation. of the uncertainty expressed in an , often 95 %. A confidence level of 95 % is a 95 % probability that the . Examples of other common nd 99 %. A confidence level of 90 % ends up with a narrower interval than a 95 % confidence level. In the opposite way, a 99 % confidence level ends up with a wider interval. The confidence interval covers the true parameter value with a certain probability. A confidence interval can be based on different distributions as; normal distribution, binomial distribution and Poisson distribution. The confidence interval for the expected value of a normal is the theoretical mean value of the distribution and can is the coverage probability from table in Appendix I suit to when σ2 is unknown and Distribution[11]. c is chosen in Equation 2-17. 14 Equation 2-16 KLIM = 12 L1 + >M where > is the confidence level. Equation 2-17 OP = L� − 1M where df is the number of degrees of freedom The degree of freedom is defined as the number of values in the final calculation of a statistic that are free to vary. A low value of degrees of freedom will give rise to a high value of the coverage probability c which results in a wide confidence interval. 2.2.3. Normal distribution The normal distribution or Gaussian distribution is in statistics and probability theory a distribution which is providing a compatible description of data that are aggregated around the mean. The probability density function is defined as Equation 2-18. Equation 2-18 QL;M = 1 σ√2π e �Lx�μM2 2R2 where x is a random variable with mean µ and variance σ2 in the domain ; ∈ (−∞, ∞) [5]. A graph over the probability density function is called a Normal curve or a Bell curve and is denoted with a characteristic peak located at the mean which has clustered symmetric data around. A normal distribution has different normal density curves depending on values of mean µ and standard deviation σ, Figure 2-3. The so-called standard normal distribution is given if the normal distribution has a mean of 0 and a variance of 1. 15 Figure 2-3: An example of a normal distribution plot. 2.2.4. Weibull distribution A Weibull distribution is a flexible and adjustable distribution, “The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter, β”, [30]. A variable x has a Weibull distribution with parameters α and β is the density function of x is as Equation 2-19 [28]. Equation 2-19 PL;M S TUV ;LV��MW�*XY+Z, X[� 0, ; F 0 \ A normal distribution is not a special case of a Weibull distribution, but the shape of the Weibull distribution can still be compared to a normal distribution to establish if the data can be seen as approximately normally distributed [26]. 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normal distribution plot P ro b a b ili ty D e n s it y Mean value = 3, standard deviation = 0.5 16 3. Uncertainties of measurements 3.1. Distribution The spatial distribution of data points within a receiving room from an impact sound pressure level and a reverberation time measurements performed in one light weight and one concrete floor structure have been investigated using a Weibull distribution. The data on the estimated Weibull distribution has been calculated by a direct estimation of the maximum likelihood. The Weibull distribution has appeared to be the best matching distribution after practical comparisons to other distributions, e g normal or Poisson. When studying the distributions for individual 1/3 octave bands it has turned out that the spatial distribution of the reverberation time at various data points differs from a normal distribution. This can clearly be seen in a quantile-quantile plot, which displays the sample quantiles versus the theoretical quantiles for a normal distribution. If the distribution of X is normal, the plot will be close to linear [23]. In the quantile-quantile plots presented in this report, the standard deviation of the measured values is presented on the y-axis and the standard deviation of the normal distribution of the measurements is presented on the x-axis. The quantile-quantile plots show how each measurement data from each measurement point, marked as a “+” sign, are distributed compared to an extrapolated theoretical normal distribution, a straight line. The measurements can be considered to have a normal distribution if the “+” signs are close to the straight line. Studying measurements from different objects included in this thesis, project 3 (see Table 1-1), was analyzed in particular since this project was a light weight project including most number of positions in the receiving room (12 discrete positions). An object with high number of positions has been of great interest since the reliability of the distribution increases with number of data. Project 15 has been analyzed as a reference object since project 15 has homogenous concrete floor structure including a lot of measurement data. The available data on impact sound pressure level has been picked from sweep measurements in project 15 where each equivalent level of 1 second segments works as a discrete position. 3.1.1. Impact sound pressure level The shape of the spatial distribution curve regarding impact sound pressure data points from one measurement in a light weight structure divided into the 1/3 octave bands 50 - 400 Hz is illustrated in Figure 3-1. The Weibull distributions are calculated from the measurement positions using maximum likelihood estimation. The calculations rely on the data from the 2007 years measurement in project 3, which consists of impact sound pressure level measurements in 12 discrete positions. The deviation for higher 1/3 octave bands can be seen in Appendix II. The tails of the distributions of impact sound pressure data are not, compared to an ideal normal distribution, equal in shape to each other. It turns out that the left tail is wider than the right tail in all investigated 1/3 octave bands. The distributions are judged to be sufficiently close to the shape of a normal distribution to use the simpler distribution. If the distribution plots are modified to have equal mean impact sound pressure level, is it possible to see how the width of the distributions changes due to different 1/3 octave bands, Figure 3-1. 17 Figure 3-1: A: Weibull distributions of 12 impact sound pressure level measurements performed on a light weight floor structure presented in 1/3 octave bands 50 - 400 Hz. The result is based on the data from project 3 measurement 2007, where the weighted mean impact sound pressure level was evaluated from these 12 measurements. B: Weibull distributions of the impact sound pressure level measurements normalized to the highest probability density, project 3 measurement 2007. The distance between 2 ticks is 10 dB The shape of the distribution curve regarding impact sound pressure levels from a measurement on a concrete structure is illustrated in Figure 3-2 for the 1/3 octave bands between 50 – 400 Hz. The distributions for higher 1/3 octave bands can be seen in Appendix II. The wideness of the distribution varies between different 1/3 octave bands. Furthermore, the left tail is wider than the right one, in nearly all 1/3 octave bands, see Figure 3-2. Figure 3-2: A: Weibull distributions of the time sequences of impact sound pressure level measurements performed on a concrete floor structure presented in 1/3 octave bands 50 - 400 Hz. The result is based on the data from project 15 where a long time measurement of impact sound pressure level have been divided into 1 second time sequences. B: Weibull distributions of the impact sound pressure level measurements normalized to the highest probability density, project 15. The Distance between 2 ticks is 10 dB 3.1.2. Reverberation time In this section the reverberation time is analyzed, primarily due to its variation depending on measurement positions, room volumes etc. The shape of the distribution curves, representing typical reverberation time data from measurements in various positions on a light weight structure, changes 40 50 60 70 80 90 100 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Impact sound presssure level [dB] P ro b a b ili ty D e n s it y Distribution plot 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 200 Hz 315 Hz 400 Hz 30 40 50 60 70 80 90 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Impact sound presssure level [dB] P ro b a b ili ty D e n s it y Distribution plot 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 200 Hz 315 Hz 400 Hz A B A B 18 with frequency. The distributions for the 1/3 octave bands 50 - 400 Hz are illustrated in Figure 3-3 while higher 1/3 octave bands are found in Appendix II. One typical measurement representing 24 reverberation time measurements in discrete positions have been used from the project from year 2007 (project 3). It is rather clear that the distribution for the lowest frequencies is wider and more spread than for the higher frequencies. Furthermore, the left tail of the distribution is wider than the right tail in nearly all 1/3 octave bands apart from the lowest 1/3 octave bands, see Figure 3-3. The distribution in the two lowest 1/3 octave bands (50 and 63 Hz) shows opposite behavior, with a right tail wider than the left. Figure 3-3: A: Weibull distributions of 24 reverberation time measurements performed in a light weight floor structure presented in 1/3 octave bands 50 - 400 Hz. The result is based on the data from project 3, measurement 2007 where the mean reverberation time was evaluated from these 24 measurements. B: Weibull distributions of the reverberation times normalized to the highest probability density, project 3 measurement 2007. The distance between 2 ticks is 0.5 s The skew behavior of the spatial distribution for the two lowest 1/3 octave bands, 50 and 63 Hz was further investigated with quantile-quantile plots, see Figure 3-4 and Figure 3-5. It turns out that the measured reverberation time for these two 1/3 octave bands has two dominant outliers which significantly deviate from the other data and the normal distribution. It is notable that the deviating outliers in the 1/3 octave bands 50 and 63 Hz are not arising from only two measurement points, i.e. the deviating values do not arise from same loudspeaker and measurement positions. The deviations can thus not be judged to be deterministic characteristics for the loudspeaker and microphone positions; instead they seem to be random. This conclusion is reasonable since the measurement was performed using three repetitions for each loudspeaker - receiver combination. 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Reverberation time [s] P ro b a b ili ty D e n s it y Distribution plot 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 200 Hz 315 Hz 400 Hz A B 19 Figure 3-4: Quantile-quantile plot of 24 reverberation time measurements in a light weight structure presented in 1/3 octave band 50 Hz. The figure shows how the measured data deviate from a normal distribution. Each “+”-sign indicates a measurement and the line the estimated normal distribution of the measurements. The result is based on data from project 3, measurement 2007 where the mean reverberation time was evaluated from these 24 measurements. Figure 3-5: Quantile-quantile plot of 24 reverberation time measurements in a light weight structure presented in 1/3 octave band 63 Hz. The figure shows how the measured data deviate from a normal distribution. Each “+”-sign indicates a measurement and the line the estimated normal distribution of the measurements. The result is based on data from project 3, measurement 2007 where the mean reverberation time was evaluated from these 24 measurements. If the reverberation time measurement data points, containing the outliers in the 1/3 octave bands 50 and 63 Hz were erased, the distribution plots would show a distribution that almost equals a normal distribution, see Figure 3-6. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Standard Normal Quantiles R e v e rb e ra ti o n t im e [ s ] Quantile-quantile plot -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Standard Normal Quantiles R e v e rb e ra ti o n t im e [ s ] Quantile-quantile plot 20 Figure 3-6: A: Weibull Distribution plot for a 50 Hz reverberation time measurements in a light weight structure presented with the two strong deviating measurements erased, project 3 measurement 2007. B: Weibull Distribution plot for a 63 Hz reverberation time measurements in a light weight structure presented with the two strong deviating measurements erased, project 3 measurement 2007. The fact that few outliers can give strange values might cause unpredictable errors, for instance when the instruments are calculating the standardized value (i.e. normalization to the reverberation time 0.5 s). If comparing the results between 1. the “full” reverberation time measurement series and 2. the “reduced” reverberation time measurement series (the two outliers in each of the 1/3-octave bands 50 and 63 Hz excluded) the standardization of the impact sound pressure level was influenced (see Equation 2-6). Figure 3-7 shows the reduction of the impact sound pressure level when the outliers are excluded. It is clear that the four deviating reverberation time measurement positions in this specific case cause an error that approximately equals 1 dB for the two lowest 1/3 octave bands 50 and 63 Hz. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5 Reverberation time [s] P ro b a b ili ty D e n s it y Distribution plot 50 Hz 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Reverberation time [s] P ro b a b ili ty D e n s it y Distribution plot 63 HzA B 21 Figure 3-7: Standardization of the impact sound pressure level on a reverberation time measurement in 1/3 octave bands 50 – 3150 Hz with and without the deviating measurements at 50 and 63 Hz, project 3, measurement 2007. As a comparison similar reverberation time data from measurements on a heavy structure is shown in Figure 3-8. The shape of the distribution curves in the figure are evaluated based on measurement results from 24 discrete reverberation time measurement positions in one room in a building with solid concrete structure, in 1/3 octave bands 50 – 400 Hz. Frequencies above 400 Hz are presented in Appendix II. In general, the distribution of the lowest frequencies is wider and more evenly distributed than the distribution of the higher, however still close to a normal distribution. Furthermore, the left tail of the distribution is wider than the right in all 1/3 octave bands apart from the lowest 1/3 octave bands, see Figure 3-8. 31.5 63 125 250 500 1k 2k 4k -5 -4 -3 -2 -1 0 1 Standardization correction Frequency [Hz] R e d u c tio n [ d B ] All measured reverberation times Deviating reverberation times excluded 22 Figure 3-8: A: Weibull distributions of 24 reverberation time measurements performed in a concrete floor structure presented in the 1/3 octave bands 50 - 400 Hz. The result is based on the data from project 15 where the mean reverberation time was evaluated from these 24 measurements. B: Weibull distributions normalized to the highest probability density, project 15. The distance between 2 ticks is 10 dB Hence, measurements from heavy structures appear to exhibit more normal distribution in all frequencies considered in the standard measurement procedure. The two widest and most spread spatial distributions regarding reverberation time data points from measurements in this specific case in a building with concrete structure were found in the 1/3 octave bands 50 and 100 Hz. These 1/3 octaves were further investigated with quantile-quantile plots, see Figure 3-9 and Figure 3-10. No typical outlier is noticed in 1/3 octave band 50 Hz, just a wide spectra of the measured reverberation times. However, the measured reverberation time in the 100 Hz 1/3 octave band has some significant outliers, indicating that similar problems might appear for heavy structures as for light weight structures. However, the final result from measurements in heavy structures is less affected by single errors in the low frequencies since the single number normally is determined by higher frequencies. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 Reverberation time [s] P ro b a b ili ty D e n s it y Distribution plot 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 200 Hz 315 Hz 400 Hz 23 Figure 3-9: Quantile-quantile plot of 24 reverberation time measurements in a concrete structure presented in 1/3 octave band 50 Hz. The figure shows how the measured data deviates from normal distribution. Each “+”-sign indicates a measurement and the line the estimated normal distribution of the measurements. The result is based on the data from project 15 where the mean reverberation time was evaluated from these 24 measurements. Figure 3-10: Quantile-quantile plot of 24 reverberation time measurements in a light weight structure presented in 1/3 octave band 100 Hz. The figure shows how the measured data deviates from normal distribution. Each “+”-sign indicates a measurement and the line the estimated normal distribution of the measurements. The result is based on the data from project 15 where the mean reverberation time was evaluated from these 24 measurements. 3.2. Effect of receiving room volume During a measurement, the receiving room volume has to be stated. The receiving room volume is one parameter that might affect the final results, in addition to all other possible details. Therefore, -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -1 0 1 2 3 4 5 6 Standard Normal Quantiles R e v e rb e ra ti o n t im e [ s ] Quantile-quantile plot -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 3.5 Standard Normal Quantiles R e v e rb e ra ti o n t im e [ s ] Quantile-quantile plot 24 the receiving room volume dependence of the confidence interval for the measured mean impact sound pressure level and mean reverberation time were evaluated for each 1/3 octave band between 50 - 3150 Hz. All investigated impact sound pressure measurements have been performed vertically. Concerning the confidence interval calculations, the spatial distribution of the different positions regarding impact sound pressure level and reverberation time were assumed to have a normal distribution of data for all 1/3 octave bands, even if earlier investigation indicates that the measurements not perfectly fit to a normal distribution. The confidence interval of a measurement explains how much the mean value theoretically can vary between different measurement series. If the confidence interval is narrow, it indicates high accuracy while a wide interval indicates high uncertainty. A confidence interval has been chosen instead of standard deviation since it reflects more clearly the probability of variation concerning measurement averaged value. As Erwin Kreyszig writes in “Advanced Engineering Mathematics” - “Most important methods of statistical interference are estimation of parameters, determination of confidence intervals and hypothesis testing” [11]. Standard deviation is more proper when investigating individual measurements to see if any 1/3 octave band or bands deviate from other 1/3 octave bands, which in that case indicates on an uncertain measurement. 3.2.1. Impact sound pressure level The volume effect of the receiving room was investigated for impact sound pressure measurements performed both on homogenous concrete structures and light weight floor structures. The confidence interval was calculated according to Equation 2-14 which requires that the investigated measurements have the same number of measurement positions. Hence, it was preferable to find many measurements with equal numbers of measurement positions to gain estimation with high accuracy. For light weight structures 22 measurements with five measurement positions in each measurement were found, see Table 3-1. The table also presents the volume of the receiving room, the degrees of freedom df, the coverage probability c and finally, the confidence level > for each project included. See Equation 2-14 in theory chapter 2.2.2 where also the mathematical elements are described. Notice that five positions are one more position than required according to ISO 140-7. 25 Table 3-1: Measurements of impact sound pressure level performed on light weight structures with five measurement positions, used for confidence interval calculations. Project Measurement Volume [m 3 ] df c(γ,df) ] 1 A05 56.60 4 2.78 95% A10 56.60 4 2.78 95% A11 46.30 4 2.78 95% A12 46.30 4 2.78 95% 3 2008 (1) 60 4 2.78 95% 2008 (2) 38,9 4 2.78 95% 2009 (1) 60 4 2.78 95% 2009 (2) 38,9 4 2.78 95% 5 A01 169.25 4 2.78 95% A02 169.25 4 2.78 95% A03 169.25 4 2.78 95% A04 169.25 4 2.78 95% A05 169.25 4 2.78 95% A06 169.25 4 2.78 95% A07 68.24 4 2.78 95% A08 68.24 4 2.78 95% A09 68.24 4 2.78 95% 6 A10 50.00 4 2.78 95% A11 50.00 4 2.78 95% 7 A15 34.50 4 2.78 95% A18 78.60 4 2.78 95% A22 32.10 4 2.78 95% Concerning the measurements performed on homogenous concrete floor structures, four measurements with four measurement positions were available, see Table 3-2. In the table, also necessary input data for the confidence interval estimations are stated. Table 3-2: Measurements of impact sound pressure level performed on homogenous concrete floor structures with four measurement positions, used for confidence interval calculations. Project Measurement Volume [m 3 ] df c(γ,df) ] 8 8 80.3 3 3.18 95% 9 107.5 3 3.18 95% 9 A04 32 3 3.18 95% 14 1 90 3 3.18 95% The confidence interval was calculated for each measurement and 1/3 octave band, indicated as circles in Figure 3-11. Figure 3-11 shows the linear least squares estimation of the volume dependence regarding impact sound pressure level measurements performed on light weight floor structures for the 1/3-octave band 50 Hz. To these 22 calculated confidence intervals of the mean value of the five impact sound pressure level measurement positions, a straight line was fit, marked 26 with squares. Each square of the line indicates a receiving room volume; in this case there were ten different volumes of the receiving rooms for the investigated 22 measurements. Figure 3-11: Confidence interval of impact sound pressure level measurements as a function of volume for 1/3 octave band 50 Hz. The calculations are based on a 95% confidence level and 22 measurements, each with five measurement positions. The “o”-signs show the calculated confidence interval for each measurement and the square marked line present the linear estimation of the calculated intervals. This calculation has been performed for each 1/3 octave band 50 - 3150 Hz, all linear estimations can be seen in Appendix IV. Some 1/3 octave bands have been selected for a more detailed analysis since all 1/3 octave bands are not of interest. The six lowest 1/3 octave bands 50 - 160 Hz were selected since discrepancies were mainly found at 1/3 octave bands below 160 Hz. To show the relation between volume and high frequencies, the 1/3 octave bands 250, 500, 1000 and 2000 Hz has also been selected, see Figure 3-12. It has been found irrelevant to show all 1/3 octave bands above 160 Hz since they are of a similar nature. The confidence intervals for the chosen 1/3 octave bands based on 22 impact sound pressure level measurements in a light weight structures are shown in Figure 3-12. The correlations and equations for all 1/3 octave bands between 50 and 3150 Hz are stated in Appendix III. The differences of the confidence interval for the biggest and smallest room for each 1/3 octave band can be seen in Figure 3-13. The figures indicate that the receiving room volume has less influence on the lower 1/3 octave bands 50 - 160 Hz than on the higher 1/3 octave bands. For the higher 1/3 octave bands, 200 - 3150 Hz, all measurements result in a decreasing confidence interval with an increased volume, i.e. less measurement uncertainty with big receiving room volumes for “high” 1/3 octave bands. 20 40 60 80 100 120 140 160 180 0 1 2 3 4 5 6 7 8 9 10 Volume [m3] C o n fi d e n c e i n te rv a l [d B ] Confidence interval of impact sound pressure level measurements 27 Figure 3-12: Confidence interval of impact sound pressure level measurements performed on light weight structures as a function of volume for selected 1/3 octave bands. The calculations are based on a 95% confidence level and 22 measurements, each with 5 measurement positions. Figure 3-13: Difference between confidence interval over impact sound pressure level in light weight structures of biggest and smallest receiving room for each 1/3 octave band. Black bars presents the difference between the linear estimations of the 1/3 octave bands in Figure 3-12 while the light bars are the remaining 1/3 octave bands. The calculations are based on a 95% confidence level and 18 measurements, each with 5 measurement positions. The dependence of receiving room volume for each 1/3 octave band for impact sound pressure level measurements performed on homogenous concrete floor structures is presented in Figure 3-14. The figure shows the linear dependency of the (with percentage) confidence interval as a function of receiving room volume for the selected 1/3 octave bands. Figure 3-15 shows the difference between 20 40 60 80 100 120 140 160 180 0 1 2 3 4 5 6 7 8 9 10 Confidence interval of impact sound pressure level measurements Volume [m3] C o n fid e n c e in te rv a l [ d B ] 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 50 63 80 100 125 160 250 500 1k 2k -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Impact sound pressure measurements Dif ference of confidence interval D iff e re n c e [ d B ] Frequency [Hz] 28 the biggest and smallest receiving room volume of the linear estimation. The result which these two figures present indicates that the biggest volume dependency of impact sound pressure level measurement uncertainty are found at the lowest and highest 1/3 octave bands. In general, the figures indicate that the uncertainty of the low 1/3 octave bands increases with receiving room volume (quite contradictory to what would be expected due to normal diffuse field theory since the Schroeder frequency decreases with increasing room volume. The Schroeder frequency can be used to indicate the lower frequency limit where the sound field in the room can be considered as statistical, i.e. there are a large number of room modes within each 1/3 octave band. At lower frequencies individual room modes can be significant which theoretically would increase the confidence interval.) The uncertainty decreases for higher 1/3 octave bands, which follows the common theory. The correlation and equations of each linear estimation for all 1/3 octave bands between 50 and 3150 Hz are stated in Appendix III. Figure 3-14: Confidence interval of impact sound pressure level measurements performed on homogenous concrete floor structures as a function of volume for selected 1/3 octave bands. The calculations are based on a 95% confidence level and 4 measurements, each with four measurement positions. 20 40 60 80 100 120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Confidence interval of impact sound pressure level measurements Volume [m3] C o n fid e n c e in te rv a l [ d B ] 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 29 Figure 3-15: Difference between confidence interval over impact sound pressure level in concrete structures in biggest and smallest receiving room for each 1/3 octave band. Black bars presents the difference between the linear estimations of the 1/3 octave bands in Figure 3-14 while the light bars are the remaining 1/3 octave bands. The calculations are based on a 95% confidence level and 10 measurements, each with 4 measurement positions. 3.2.2. Reverberation time For the reverberation time measurements, the variation of the confidence interval due to receiving room volume was estimated by using ten different measurements (receiving rooms). These measurements were all performed in light weight structure (wooden) buildings. Each measurement has been performed using three microphone positions with two readings in each position, Table 3-3. The measurement equipment has made an average of the two readings in each of the three positions. Hence, this ended up in three measurement data points for each 1/3 octave band. This had consequences for the confidence interval calculations since only the variance for three measurement positions could be calculated (instead of six). It also affected the number of degrees of freedom since only two degrees of freedom could be used instead of five, Equation 2-17. This has only influenced the width of the confidence interval, not the characteristic of the linear regression, shown in Figure 3-16. 50 63 80 100 125 160 250 500 1k 2k -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Impact sound pressure measurements performed on homogenous concrete f loor structures Dif ference of confidence interval D iff e re n c e [ d B ] Frequency [Hz] 30 Table 3-3: Measurements of reverberation time with three measurement positions with two readings in each position, used for confidence interval calculations. Project Measurement Volume [m 3 ] df c(γ,df) ] 1 A10 56.60 2 4.3 95% A12 46.30 2 4.3 95% 4 A02 109.00 2 4.3 95% A04 109.00 2 4.3 95% 6 A10 50.00 2 4.3 95% A11 50.00 2 4.3 95% 7 A15 34.50 2 4.3 95% A16 61.00 2 4.3 95% A18 78.60 2 4.3 95% A22 32.10 2 4.3 95% The measured mean reverberation time and its dependence on the receiving room volume can be seen in Figure 3-16. The figure includes the same low frequency 1/3 octave bands as for the level measurements (Figure 3-12). The correlations and equations of each estimated line for all 1/3 octave bands within the frequency range 50 - 3150 Hz can be found in Appendix III. The differences of the confidence interval for the biggest and smallest room for each 1/3 octave are shown in Figure 3-17. The uncertainty of the reverberation time measurements increases with receiving room volume for the three lowest 1/3 octave bands 50, 63 and 80 Hz, quite contradictory to what could be expected since an increased volume should result in more diffuse field. However, it could be due to the shape of the room (extended in only two dimensions – still same height) and its effect on certain mode shapes. The higher the frequency the less the volume affect the final results, i.e. the confidence interval of the remaining 1/3 octave bands are relative constant over receiving room volume. 31 Figure 3-16: Confidence interval of reverberation time measurements performed in light weight structures as a function of volume for selected 1/3 octave bands. The calculations are based on a 95% confidence level and 10 measurements, each with 3 measurement positions with 2 readings in each position. Figure 3-17: Difference between confidence interval over reverberation time in light weight structures in biggest and smallest receiving room for each 1/3 octave band. Black bars presents the difference between the linear estimations of the 1/3 octave bands in Figure 3-16 while the light bars are the remaining 1/3 octave bands. The calculations are based on a 95% confidence level and 10 measurements, each with 3 measurement positions with 2 readings in each position. 20 40 60 80 100 120 0 1 2 3 4 5 6 7 8 9 10 Confidence interval of reverberation time measurements Volume [m3] C o n fid e n c e in te rv a l [ s ] 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 250 Hz 500 Hz 1000 Hz 2000 Hz 50 63 80 100 125 160 250 500 1k 2k -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 Reverberation time measurements Dif ference of confidence interval D iff e re n c e [ s ] Frequency [Hz] 4. Impact sound in low frequencies In this thesis impact sound pressure levels from available measurements in low frequencies have been investigated in 1/3 octave bands. The investigation of project 10, octave bands 6.3 – 3150 Hz. For project 3, 11 and included. Notice that project 3, 10 and 14 are performed on light weight structures while project 11, 12, 13 and 15 are performed on concrete floor structures. T pressure level in two frequency ranges make the highest measured level in 1/3 octave bands visible. below 50 Hz and between 50 – 3150 Hz, which can be se been divided into light weight and concrete floor structures. measured impact sound pressure level for the investigated projects often occurs in 1/3 octave bands below 50 Hz, in particular for light weight structures (in 10 also obvious that the levels are difference between the maximum impact sound pressure levels varies between different projects. The largest level difference is however found in project 10, measurement 4 where a level difference of 18.4 dB occurs. The measured impact sound pressure levels frequency spectra for each pr be seen in Appendix V. Figure 4-1: Highest impact sound pressure level in frequency ranges, below 50 Hz and in the interval 50 In Figure 4-2, an overview of the occurrence of the highest measured impact sound pressure level in different 1/3 octave band is shown. in the 1/3 octave bands between 20 highest levels of impact sound for concrete structures are normally f between 50 - 3150 Hz. 32 sound in low frequencies In this thesis impact sound pressure levels from available measurements in low frequencies have been investigated in 1/3 octave bands. The investigation of project 10, 13, 14 and 3150 Hz. For project 3, 11 and 12, the 1/3 octave bands 25 Notice that project 3, 10 and 14 are performed on light weight structures while project 11, 12, 13 and 15 are performed on concrete floor structures. The highest measured i in two frequency ranges in each measurement have been investigated make the highest measured level in 1/3 octave bands visible. The two chosen frequency ranges are 3150 Hz, which can be seen in Figure 4-1 where also the projects been divided into light weight and concrete floor structures. Figure 4-1 shows that the highest measured impact sound pressure level for the investigated projects often occurs in 1/3 octave bands r light weight structures (in 10 of 13 investigated measurements). It is obvious that the levels are higher in the low frequency region for the light structures. difference between the maximum impact sound pressure levels varies between different projects. is however found in project 10, measurement 4 where a level difference The measured impact sound pressure levels frequency spectra for each pr Highest impact sound pressure level in frequency ranges, below 50 Hz and in the interval 50 room in project 3, 10, 11, 12, 13, 14 and 15. an overview of the occurrence of the highest measured impact sound pressure level in different 1/3 octave band is shown. For light weight structures, most of the highest levels are found 1/3 octave bands between 20 – 31.5 Hz. In contrast to the light weight floor structures highest levels of impact sound for concrete structures are normally found at higher 1/3 octave band In this thesis impact sound pressure levels from available measurements in low frequencies have 13, 14 and 15 included 1/3 bands 25 – 3150 Hz are Notice that project 3, 10 and 14 are performed on light weight structures while project 11, he highest measured impact sound investigated in order to The two chosen frequency ranges are where also the projects has shows that the highest measured impact sound pressure level for the investigated projects often occurs in 1/3 octave bands investigated measurements). It is egion for the light structures. The difference between the maximum impact sound pressure levels varies between different projects. is however found in project 10, measurement 4 where a level difference The measured impact sound pressure levels frequency spectra for each project can Highest impact sound pressure level in frequency ranges, below 50 Hz and in the interval 50 - 3150 Hz for each an overview of the occurrence of the highest measured impact sound pressure level in of the highest levels are found In contrast to the light weight floor structures, the ound at higher 1/3 octave band 33 Figure 4-2: Compilation of in which 1/3 octave band the highest measured impact sound pressure level has occurred. The bars illustrate the occurrence for both light weighted structures and concrete structures in project 3, 10, 11, 12, 13, 14 and 15. 6.3 8 10 12.5 16 20 25 31,5 40 50 63 80 100 125 160 200 250 315 400 1 2 3 4 5 Occurrence for highest impact sound pressure level in different 1/3 octave bands Frequency [Hz] O c c u rr e n c e Light w eigth structure Concrete structure 5. Extended measurements In order to confirm statements from current measurements one light weight (wooden) house Dimensions and volumes of receiving and sending room can be seen in rooms can be seen in Figure 5-3 Table 5-1: Dimensions and volume of the receiving and sending room. Room Receiving Sending 5.1. Reverberation time measurement Reverberation time measurements were performed according to measurement standard ISO 140 16 measurement decays were measured with eight set source positions, Figure 5-1 and performed. Figure 5-1: Sketch illustrating the source and S1 S2 34 measurements confirm statements from current measurements, extended measurements were made in ht weight (wooden) house. The measurements were performed vertically between two rooms. Dimensions and volumes of receiving and sending room can be seen in Table and Figure 5-4. Dimensions and volume of the receiving and sending room. Room Dimensions [m] Volume [m 3 ] Receiving 4.42x3.31x2.50 36.58 Sending 4.42x3.30x2.50 36.47 Reverberation time measurement Reverberation time measurements were performed according to measurement standard ISO 140 16 measurement decays were measured with eight set-ups of two microphones with two different and Table 5-2. For each set-up, an average of five readings was : Sketch illustrating the source and receiving positions for the reverberation time measurements in the receiving room. S2 R1 R2 R3 R4 R5 measurements were made in . The measurements were performed vertically between two rooms. Table 5-1, sketches of the Reverberation time measurements were performed according to measurement standard ISO 140-7. ups of two microphones with two different an average of five readings was positions for the reverberation time measurements in the 35 Table 5-2: Measurement procedure for the reverberation time measurements, the positions are shown in Figure 5-1. Measurement Source Receiver 1 S1 R1, R2 2 S1 R2, R3 3 S1 R3, R4 4 S1 R4, R5 5 S2 R4, R5 6 S2 R3, R4 7 S2 R2, R3 8 S2 R1, R2 The Weibull distributions of the reverberation time measurement are shown for 1/3 octave bands 50 - 400 Hz in Figure 5-2. The measured data seems to be reliable since the measurements did not exhibit any outliers causing a distribution deviating significantly to a normal distribution. A distribution plot with the measured reverberation time normalized to the highest probability density is shown in Figure 5-2. All 1/3 octave bands apart from 1/3 octave band 63 Hz exhibit same pattern as the other investigated measurements in chapter 3.1 Distribution, a left tail wider than the right tail of the distribution curve. Figure 5-2: A: Weibull distributions of 16 reverberation time measurements performed in a light weight floor structure presented in the 1/3 octave bands 50 - 400 Hz. The result is based on the data from project 16 where the mean reverberation time was evaluated from these 16 measurements. B: Weibull distributions of the reverberation time measurements normalized to the highest probability density, project 16. 5.2. Impact sound pressure measurement 35 measurement positions for the tapping machine have been used when performing the impact sound pressure level measurements, in order to investigate different positions and their influence on the spatial average value. The spatial average value in the receiving room was collected by using a rotating boom, continuously sweeping near the middle of the receiving room; Figure 5-3. A sweep -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 5 6 Reverberation time [s] P ro b a b ili ty D e n s it y Distribution plot 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 200 Hz 315 Hz 400 Hz A B sequence of 30 seconds was used. This approach was chosen in order to compare the influence of different positions of the tapping machine and its positions influence on the receiving Figure 5-3: Sketch illustrating the receiving A grid which met demands according to ISO 140 between source positions were set up, marking each tapping machine position floor, Figure 5-4. Within the limits set by the boundaries of the room The tapping machine was running measurements covered a full microphone sweep in the receiving room 36 sequence of 30 seconds was used. This approach was chosen in order to compare the influence of different positions of the tapping machine and its positions influence on the receiving receiving position for the impact sound pressure level measurement room. which met demands according to ISO 140-7 regarding distances to boundaries and distances between source positions were set up, marking each tapping machine position on the sending room . Within the limits set by the boundaries of the room, 35 source positions The tapping machine was running during at least 40 seconds in each position to make sure that all microphone sweep in the receiving room. R1 sequence of 30 seconds was used. This approach was chosen in order to compare the influence of different positions of the tapping machine and its positions influence on the receiving room level. ound pressure level measurement in the receiving distances to boundaries and distances on the sending room 35 source positions were used. at least 40 seconds in each position to make sure that all Figure 5-4: Sketch illustrating the source The Weibull distribution of the 35 m 5-5. The measured data seem distribution which deviate significantly to a normal distribution. measured reverberation time normalized to the highest probability density is shown in 1/3 octave bands exhibit same pattern as the other investigated measur Distribution, a left tail wider than the right tail of the distribution curve. A: Weibull distributions of 35 impact sound pressure level measurements performed in a light weight floor structure presented in the 1/3 octave bands 50 sound pressure level was e B: Weibull distributions of the impact sound pressure level measurements normalized to the highest probability density, 30 40 50 60 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Impact sound presssure level [dB] P ro b a b ili ty D e n s it y Distribution plot S1 S1 S10 S11 S20 S9 S2 S21 S30 A 37 source positions for the impact sound pressure level measurements in the room. The Weibull distribution of the 35 measurements is shown for 1/3 octave bands 50 to be reliable since no extreme outliers were observed, causing a distribution which deviate significantly to a normal distribution. A distribution plot with the red reverberation time normalized to the highest probability density is shown in 1/3 octave bands exhibit same pattern as the other investigated measurements in chapter , a left tail wider than the right tail of the distribution curve. Figure 5-5: Weibull distributions of 35 impact sound pressure level measurements performed in a light weight floor structure presented in the 1/3 octave bands 50 - 400 Hz. The result is based on the data from project 16 where the mean impact sound pressure level was evaluated from these 35 measurements. Weibull distributions of the impact sound pressure level measurements normalized to the highest probability density, project 16. 70 80 90 Impact sound presssure level [dB] 50 Hz 63 Hz 80 Hz 100 Hz 125 Hz 160 Hz 200 Hz 315 Hz 400 Hz S3 S12 S14 S13 S15 S19 S17 S18 S16 S9 S8 S7 S6 S5 S4 S22 S24 S23 S25 30 S29 S28 S27 S26 S31 S32 S33 S34 S35 B positions for the impact sound pressure level measurements in the sending easurements is shown for 1/3 octave bands 50 - 400 Hz in Figure to be reliable since no extreme outliers were observed, causing a A distribution plot with the red reverberation time normalized to the highest probability density is shown in Figure 5-5. All ements in chapter 3.1 Weibull distributions of 35 impact sound pressure level measurements performed in a light weight floor structure on the data from project 16 where the mean impact Weibull distributions of the impact sound pressure level measurements normalized to the highest probability density, 38 Measurement standard ISO 140-7 states that a proper measurement shall include at least four randomly placed positions of the tapping machine. By randomly use 4 of the 35 source positions, a single weighted values of L’n,w and L’n,w + CI,50-2500 has been calculated for 100 000 combinations. The standardization of the measurements has been based on mean values of the 16 reverberation time measurements. The highest and lowest calculated single number quantities are stated in Table 5-3. Table 5-3: Highest and lowest values on single number quantities L’n,w and L’n,w +CI,50-2500 from different source positions in sending room, project 16. L’n,w [dB] L’n,w +CI,50-2500 [dB] Highest 63 59 Lowest 57 56 The pairs of four source positions of the 35 available that contribute to the minimum value of L’n,w and L’n,w + CI,50-2500 of the 100 000 random source position combinations is illustrated in Figure 5-6 and Figure 5-7. The result show that it is mainly the source positions 1-15, situated deepest in the building which give rise to the minimum level on L’n,w, Figure 5-6. Figure 5-7 shows that the minimum level on L’n,w + CI,50-2500 arises just from one combination of source positions (position 4-7). The source positions can be seen in Figure 5-4. Figure 5-6: Bar plot over how frequently each source position contributes to a minimum value of L’n,w. 39 Figure 5-7: Bar plot over how frequently each source position contributes to a minimum value of L’n,w + CI,50-2500. The source position and its contribution to the highest levels of L’n,w and L’n,w + CI,50-2500 is illustrated in Figure 5-8 and Figure 5-9. The source positions contributing most frequent to the calculated maximum levels on L’n,w are mainly the positions 21 – 35, situated nearest the facade in the building. Source positions 34 and 35 turns out to be very dominant which also was stated subjectively during the measurement and appears to be due to two heat pipes passing vertically between the sending and the receiving room, close to the corner where highest levels were detected. These pipes short circuited the floating floor in the sending room and thus resulted in a clear flanking transmission. The crucial source positions are further confirmed by Figure 5-9. The highest level on L’n,w + CI,50-2500 arises just from one combination of source positions (position 27-28, 34-35). The source positions are illustrated in Figure 5-4. 40 Figure 5-8: Bar plot over how frequently each source position contributes to a maximum value of L’n,w. Figure 5-9: Bar plot over how frequently each source position contributes to a maximum value of L’n,w + CI,50-2500. Carrying out an averaging procedure, based on 100 000 random combinations of four source positions and a sweeping microphone in the receiving room, and then use these different combinations to evaluate L’n,w and L’n,w + CI,50-2500 give a distribution of the single numbers equal to those shown in Figure 5-10. Both distributions have similarities to a normal distribution but, similar to earlier distributions, the left tail is wider than the right tail of the distribution. 41 Figure 5-10: A: Weibull distribution for L’n,w from 100 000 combinations of 4 randomly chosen source positions of 35 available, project 16. B: Weibull distribution for L’n,w + CI,50-2500 from 100 000 combinations of 4 randomly chosen source positions of 35 available, project 16. The measurements were performed in the frequency range 10 - 5000 Hz. Similar to earlier measurements, the highest impact sound pressure levels arise in 1/3 octave bands below 50 Hz. The impact sound pressure levels calculated as a spatial mean value for each 1/3 octave band in the frequency range 10 - 5000 Hz based on all the 35 source positions is presented in Figure 5-11. Figure 5-11: The mean impact sound pressure level from the 35 source positions in the receiving room in the frequency range 10 – 5000 Hz. 50 52 54 56 58 60 62 64 66 68 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 L´ n,w [dB] P ro b a b ili ty D e n s it y Distribution plot 50 55 60 65 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L´ n,w + C I,50-2500 [dB] P ro b a b ili ty D e n s it y Distribution plot 8 16 31.5 63 125 250 500 1k 2k 4k 8k 40 45 50 55 60 65 70 75 Frequency [Hz] [d B ] Impact sound pressure level A B 6. Evaluation of reference curve The ISO 717-2 reference curve has shortcomings since it has a typical shape adapted to heavy structures. When applied to light weight structures large errors might appear and the final single number does not correlate to the experie the housing units [6, 8]. Many attempts have been made during the years to define a reference curve which creates a weighted single value that might be used for any buildi better adaption to the subjective evaluation of impact sound pressure level for light weighted floor structures. The several proposed reference curves differ and covered frequency range, Figure can be seen in Appendix VI. Figure 6-1: Reference curve ISO 717-2 [3] used for single value evaluation of impact sound pressure level together with proposed alternatives; Hagberg [8], Bodlund [6] and Fasold [13]. The reference curves are evaluated based on measurements and intervi this, a correlation analysis might be done. This approach was performed in the work made by Hagberg [8] and when applying linear regression correlation analysis, the reference curve from ISO 717-2 [3] i.e. L’n,w, exhibit a linear subjective results. This might be improved by adding the adaptation term However, still further improvement was proved by applying an alternative reference curve which has a shift from a positive slope of 5.5 dB per 1/3 octave band for the 1/3 octave bands betwee 100 Hz to a straight line. This curve gives a linear regression curve by Bodlund [6] has a linear regre ′�,� � � ′�,�,^_`�ab` 42 Evaluation of reference curve 2 reference curve has shortcomings since it has a typical shape adapted to heavy structures. When applied to light weight structures large errors might appear and the final single correlate to the experienced level of real impact sound from those persons living in the housing units [6, 8]. Many attempts have been made during the years to define a reference curve which creates a weighted single value that might be used for any building structure, i.e. involving better adaption to the subjective evaluation of impact sound pressure level for light weighted floor roposed reference curves differ from each other, both regarding curve shape Figure 6-1. The level in each 1/3 octave band for each weighting curve 2 [3] used for single value evaluation of impact sound pressure level together with proposed alternatives; Hagberg [8], Bodlund [6] and Fasold [13]. The reference curves are evaluated based on measurements and interviews with the tenants. From this, a correlation analysis might be done. This approach was performed in the work made by Hagberg [8] and when applying linear regression correlation analysis, the reference curve from ISO , exhibit a linear regression fit r of 74 % between the measured values and the mean subjective results. This might be improved by adding the adaptation term CI,50-2500 However, still further improvement was proved by applying an alternative reference curve which has a shift from a positive slope of 5.5 dB per 1/3 octave band for the 1/3 octave bands betwee 100 Hz to a straight line. This curve gives a linear regression r of 87 %, Equation curve by Bodlund [6] has a linear regression r of 83 %, Equation 6-3. Equation 6-1 ��,������� � 74.40 ' 4.17f gh � 84%,� � 22k Equation 6-2 ^_`�ab` = 79.28 ' 4.09f gh � 87%,� � 22k 2 reference curve has shortcomings since it has a typical shape adapted to heavy structures. When applied to light weight structures large errors might appear and the final single from those persons living in the housing units [6, 8]. Many attempts have been made during the years to define a reference curve ng structure, i.e. involving better adaption to the subjective evaluation of impact sound pressure level for light weighted floor from each other, both regarding curve shape . The level in each 1/3 octave band for each weighting curve 2 [3] used for single value evaluation of impact sound pressure level together with ews with the tenants. From this, a correlation analysis might be done. This approach was performed in the work made by Hagberg [8] and when applying linear regression correlation analysis, the reference curve from ISO of 74 % between the measured values and the mean 2500, see Equation 6-1. However, still further improvement was proved by applying an alternative reference curve which has a shift from a positive slope of 5.5 dB per 1/3 octave band for the 1/3 octave bands between 50 and Equation 6-2. The proposed k k mn The linear regression for Hagbergs investigated average objective score of the subjective data is plotted in regression using L’n,w + CI,50-2500 using L’n,w,Hagberg proposed by Hagberg. The vertical error bars show the maximum and minimum measured values of objective data within each housing unit. Figure 6-2: Linear regression for whole data sample, wooden floor structures, Δ = hollow concrete structures, 43 Equation 6-3 n = 80.27 ' 3.98f gh � 83%,� � 22k The linear regression for Hagbergs investigated average objective data versus the average mean score of the subjective data is plotted in Figure 6-2 and Figure 6-3. Figure according to ISO 717-2 [3] and Figure 6-3 illustrates the regression proposed by Hagberg. The vertical error bars show the maximum and minimum measured values of objective data within each housing unit. ear regression for whole data sample, L’n,w +CI,50-2500 versus subjective grading; □ = concrete structure, wooden floor structures, Δ = hollow concrete structures, × = light weighted steel structures [8]. data versus the average mean Figure 6-2 illustrates the illustrates the regression proposed by Hagberg. The vertical error bars show the maximum and minimum □ = concrete structure, ◊ = × = light weighted steel structures [8]. Figure 6-3: Linear regression for whole data sample, wooden floor structures, Δ = hollow concrete structures, 6.1. Extended invest In order to further investigate current proposals of reference curve contours from earlier investigations some additional calculations are made in this thesis. Earlier [8] were extended by some current meas different structures, amongst those two additional light weight structures. Hence, the investigation of reference curve contours is based on impact sound pressure level measurements were compared with mean values from subjective judgments. A linear regression model could be applied since