1. The Decibel Scale Senior Mathematics B Exponential and Logarithmic Functions Senior Mathematics B Exponential and Logarithmic Functions

2. The Decibel Scale2 Human Hearing

3. The Decibel Scale3 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=RavdHOj-BA8

4. The Decibel Scale4 The range of human hearing is impressive. If the intensity of sound is measured in watts per square metre, you can just hear a noise of 10 -12 W/m 2, i.e. 0.000 000 000 001 W/m 2. You would suffer permanent ear damage if subjected to a noise of 100 W/m 2. A logarithmic scale helps to make sense of this wide range of numbers. A tenfold increase in the intensity of the sound increases the perceived loudness by one bel, a unit named in honour of Alexander Graham Bell. Our formula is where n is the number on the loudness scale in bels, S 0 is the strength of the reference sound: the faintest audible sound, and S is the strength of the measured sound.

5. The Decibel Scale5 It is thought that the smallest change in sound that a human can detect is one tenth of a bel, that is, a decibel. This gives us the unit that is actually used. One bel would equal 10 decibels. The relevant formula is then: In this case D is measured in decibels. You can see that the result is the same as assigning the value of 0 decibels to the barely audible sound of intensity S 0 and then each time the intensity of the sound is increased by multiplying by a factor of ten, the loudness of the sound as measured in decibel units is increased by adding ten decibels. It is thought that the smallest change in sound that a human can detect is one tenth of a bel, that is, a decibel. This gives us the unit that is actually used. One bel would equal 10 decibels. The relevant formula is then: In this case D is measured in decibels. You can see that the result is the same as assigning the value of 0 decibels to the barely audible sound of intensity S 0 and then each time the intensity of the sound is increased by multiplying by a factor of ten, the loudness of the sound as measured in decibel units is increased by adding ten decibels.

6. The Decibel Scale6

7. 7

8. 8 Two electric guitars

9. The Decibel Scale9 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=6653_PHmf3o

10. The Decibel Scale10 The same website includes this example: “Two electric guitars create SIL values of 45 dB and 50 dB respectively. What dB reading would you expect when both are played simultaneously.” “The answer is 51.2 dB.” Justify both fully, showing any mathematical rules or laws used. The same website includes this example: “Two electric guitars create SIL values of 45 dB and 50 dB respectively. What dB reading would you expect when both are played simultaneously.” “The answer is 51.2 dB.” Justify both fully, showing any mathematical rules or laws used. According to the website http://physics.mtsu.edu/~wmr/log_4.htm: http://physics.mtsu.edu/~wmr/log_4.htm “if I have two uncorrelated sources each creating sounds of 30 dB then when played together they emit 33 dB! What kind of crazy maths is that you say?”

11. The Decibel Scale11 “if I have two uncorrelated sources each creating sounds of 30 dB then when played together they emit 33 dB! What kind of crazy maths is that you say?” From Question 1, 30 dB corresponds to a Sound Intensity of 10 -9 W/m 2. Therefore, two uncorrelated sources each creating sounds of 30 dB will produce a Total Sound Intensity of 2 x 10 -9 W/m 2. Substituting S = 2 x 10 -9 into the given formula : D = 10 log 10 ( 2 x 10 -9 / 10 -12 ) i.e. D = 10 log 10 ( 2 x 10 3 ) D = 10 (log 10 ( 2 ) + log 10 (10 3 )) D = 10 (0.3010 + 3) Giving D = 33 dB (approx.)

12. The Decibel Scale12 “Two electric guitars create SIL values of 45 dB and 50 dB respectively. What dB reading would you expect when both are played simultaneously.” Rearranging : S = 10^( D / 10 ) x 10 -12 For D = 45, S = 10^( 45 / 10 ) x 10 -12 = 10 -7.5 W/m 2 (or 3.162 x 10 -8 W/m 2 ) For D = 50, S = 10^( 50 / 10 ) x 10 -12 = 10 -7 W/m 2 Total Sound Intensity = 3.162 x 10 -8 + 10 -7 = 0.3162 x10 -7 + 1 x 10 -7 = 1.3162 x 10 -7 W/m 2 For this S-value, D = 10 log 10 (1.3162 x 10 -7 / 10 -12 ) = 10 log 10 (1.3162 x 10 5 ) = 51.2 dB

13. The Decibel Scale13 Loudspeaker / Amplifier

14. The Decibel Scale14 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=tiBZC_nXnw w&feature=PlayList&p=A214ED9F6E5E925E& playnext=1&playnext_from=PL&index=82

15. The Decibel Scale15 Answer the following question posed on the website http://www.phys.unsw.edu.au/j w/dB.html#soundfiles: http://www.phys.unsw.edu.au/j w/dB.html#soundfiles “All else equal, how much louder is a loudspeaker driven (in its linear range) by a 100 W amplifier than by a 10 W amplifier?” Answer the following question posed on the website http://www.phys.unsw.edu.au/j w/dB.html#soundfiles: http://www.phys.unsw.edu.au/j w/dB.html#soundfiles “All else equal, how much louder is a loudspeaker driven (in its linear range) by a 100 W amplifier than by a 10 W amplifier?”

16. The Decibel Scale16 “All else equal, how much louder is a loudspeaker driven (in its linear range) by a 100 W amplifier than by a 10 W amplifier?” Substituting in : For 10 W amplifier, D 10 = 10 log 10 (x / 10 -12 ) = (10 log 10 x) + 120. For 100 W amplifier, D 100 = 10 log 10 (10x / 10 -12 ) = (10 log 10 x) + 130. D 100 - D 10 = [(10 log 10 x) + 130] – [(10 log 10 x) + 120] = 10 dB. All else being equal, a loudspeaker driven (in its linear range) by a 100 W amplifier would be expected to be 10 dB louder than a similar one driven by a 10 W amplifier. Let the Sound Intensity produced by the 10 W amplifier be x W/m 2. All else being equal, it can be assumed that the Sound Intensity produced by the 100 W amplifier would be 10x W/m 2.

17. The Decibel Scale17 Deaf Sentence

18. The Decibel Scale18 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=mGgRdmEutmI

19. The Decibel Scale19 A letter to the editor of “The Age” newspaper asks for further clarification. Respond in layman’s terms in no more than 300 words. The following is taken from http://www.theage.com.au/news/reviews/deaf- sentence/2006/03/01/1141095746578.html: http://www.theage.com.au/news/reviews/deaf- sentence/2006/03/01/1141095746578.html “Part of the problem is that few people understand sound measurement. Decibels increase in a logarithmic scale. Maybe you don't hear much difference between 100dB and 103dB, but at 103dB you're getting twice the sound energy of 100dB. At 106dB, twice that. An 80dB sound is 100 times more intense than one at 60dB.”

20. The Decibel Scale20 Dear Editor, Let me please clear up some confusion regarding the measurement of sound in decibels. Sound intensities take inconvenient numbers such as 0.000 000 000 001 Watts per square metre for the slightest sound audible to the average human (this is our threshold of hearing), 0.000 001 Watts per square metre for ordinary conversation to 1 Watt per square metre for a Jet aircraft take off at 60 metres. For convenience, these values have been coded into decibel readings, on what is called a “logarithmic” scale. This scale helps to make sense of this wide range of numbers. A tenfold increase in the intensity of the sound increases the perceived loudness by one bel, a unit named in honour of Alexander Graham Bell, the inventor of the telephone. It is thought that the smallest change in sound that a human can detect is one tenth of a bel, that is, a decibel. This gives us the unit that is actually used. One bel would equal 10 decibels. (continued next slide)

21. The Decibel Scale21 Thus the threshold of hearing (0.000 000 000 001 W/m 2 ) is assigned the value 0 decibels (dB), a very very faint whisper (with a Sound Intensity of 0.000 000 000 01 W/m 2 ) is designated as 10 dB, ordinary conversation (at 0.000 001 W/m 2, one million times the intensity of the threshold of human hearing) comes in at 60 dB, and the jet aircraft taking off at 60 metres (a trillion times the sound intensity of the threshold of hearing) is measured as 120 dB. An additive increase of one on the decibel scale is thus equivalent to multiplying the Sound Intensity by a factor of approximately 1.26 (10 to the power 0.1). An increase of 3 dB is equivalent to multiplying the Sound Intensity by (1.26 x 1.26 x 1.26), i.e. 2. Whereas an increase of 10 dB is equivalent to multiplying the Sound Intensity by a factor of 10, an increase of 20 DB is equivalent to multiplying the Sound Intensity by (10 x 10) or 100. I trust this provides some clarification. I. M. Smart

22. The Decibel Scale22 Twisted Sister

23. The Decibel Scale23 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=Z6GzD92P3X 8&feature=related

24. The Decibel Scale24 http://www2.glenbrook.k12.il.us/gbssci/phys /Class/sound/u11l2b.html http://www2.glenbrook.k12.il.us/gbssci/phys /Class/sound/u11l2b.html poses the question: “On a good night, the front row of the Twisted Sister concert would surely result in a 120 dB sound level. A Walkman produces 100 dB. How many Walkmen would be needed to produce the same intensity as the front row of the Twisted Sister concert?” http://www2.glenbrook.k12.il.us/gbssci/phys /Class/sound/u11l2b.html http://www2.glenbrook.k12.il.us/gbssci/phys /Class/sound/u11l2b.html poses the question: “On a good night, the front row of the Twisted Sister concert would surely result in a 120 dB sound level. A Walkman produces 100 dB. How many Walkmen would be needed to produce the same intensity as the front row of the Twisted Sister concert?”

25. The Decibel Scale25 How many Walkmen would be needed to produce the same intensity as the front row of the Twisted Sister concert? From Question 1, a 120 dB sound level results from a Sound Intensity of 1 W/m 2, while a 100 dB sound level results from 10 -2 (or 0.01) W/m 2. 0.01 W/m 2 must be multiplied by 100 to equate to 1 W/m 2. 100 Walkmen would be needed to produce the same intensity as the front row of the Twisted Sister concert.

26. The Decibel Scale26 Jack Russell Terriers

27. The Decibel Scale27 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=AX0ODTqWp1Y

28. The Decibel Scale28 According to http://www.ncbi.nlm.nih.gov/entrez/query. fcgi?cmd=Retrieve&db=PubMed&list_ui ds=9097241&dopt=Abstract, researchers at the University of Southampton have measured the mean threshold of hearing of a sample of 20 Jack Russell Terriers at –5 decibels. How can this be? Interpret the meaning of negative sound levels. http://www.ncbi.nlm.nih.gov/entrez/query. fcgi?cmd=Retrieve&db=PubMed&list_ui ds=9097241&dopt=Abstract According to http://www.ncbi.nlm.nih.gov/entrez/query. fcgi?cmd=Retrieve&db=PubMed&list_ui ds=9097241&dopt=Abstract, researchers at the University of Southampton have measured the mean threshold of hearing of a sample of 20 Jack Russell Terriers at –5 decibels. How can this be? Interpret the meaning of negative sound levels. http://www.ncbi.nlm.nih.gov/entrez/query. fcgi?cmd=Retrieve&db=PubMed&list_ui ds=9097241&dopt=Abstract

29. The Decibel Scale29 The mean threshold of hearing of a sample of 20 Jack Russell Terriers has been measured at –5 decibels. How can this be? Interpret the meaning of negative sound levels. Substituting in S = 10^( D / 10 ) x 10 -12 (see Question 2) to find the Sound Intensity corresponding to -5 dB: S = 10^( -5 / 10 ) x 10 -12 = 10 -12.5 = 3 x 10 -13 W/m 2. Sound Levels in negative decibels correspond to Sound Intensities below the Human Threshold of Hearing (10 -12 W/m 2 ). Jack Russell Terriers have, on average, keener hearing than Humans, as they can hear sounds of less intensity.

30. The Decibel Scale30 Happy Hour

31. The Decibel Scale31 For a video clip to introduce this section, please go to: http://www.youtube.com/watch?v=GpYd2DLe7 XM&NR=1

32. The Decibel Scale32 According to http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf: http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf “The current noise restrictions for licensed premises in Queensland are 75 decibels if no acoustic report is submitted to indicate the premises can contain a higher noise level.” The noise level at a local tavern at “happy hour” has been measured as 65 decibels. The next door neighbour has learnt that the licensee is planning to install a juke box (70 decibels), 6 “standard” video gaming machines (45 decibels each), and 4 “Super Jackpot” gaming machines (60 decibels each). The licensee, eagerly anticipating increased revenue, and the neighbour, upset about the noise pollution which he claims will total 645 decibels, are about to come to blows. Your job is to mediate a solution (or suggest several possible solutions) which conform to Queensland regulations. According to http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf: http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf “The current noise restrictions for licensed premises in Queensland are 75 decibels if no acoustic report is submitted to indicate the premises can contain a higher noise level.” The noise level at a local tavern at “happy hour” has been measured as 65 decibels. The next door neighbour has learnt that the licensee is planning to install a juke box (70 decibels), 6 “standard” video gaming machines (45 decibels each), and 4 “Super Jackpot” gaming machines (60 decibels each). The licensee, eagerly anticipating increased revenue, and the neighbour, upset about the noise pollution which he claims will total 645 decibels, are about to come to blows. Your job is to mediate a solution (or suggest several possible solutions) which conform to Queensland regulations.

33. The Decibel Scale33 Again, substituting in S = 10^( D / 10 ) x 10 -12 : For “happy hour” noise of 65 dB: S 1 = 10^( 65 / 10 ) x 10 -12 = 10 -5.5 = 3.162 x 10 -6 W/m 2. For the Jukebox (70 dB): S 2 = 10^( 70 / 10 ) x 10 -12 = 10 -5 W/m 2. For 1 “standard” video gaming machine: S 3 = 10^( 45 / 10 ) x 10 -12 = 10 -7.5 = 3.162 x 10 -8 W/m 2. For 1 “Super Jackpot” machine: S 4 = 10^( 60 / 10 ) x 10 -12 = 10 -6 W/m 2. Total Sound Intensity from all sources listed = S 1 + S 2 + 6S 3 + 4S 4 = (0.3162 + 1 + 6 x 0.003162 + 4 x 0.1) x 10 -5 W/m 2. = 1.735 x 10 -5 W/m 2. Again, substituting in S = 10^( D / 10 ) x 10 -12 : For “happy hour” noise of 65 dB: S 1 = 10^( 65 / 10 ) x 10 -12 = 10 -5.5 = 3.162 x 10 -6 W/m 2. For the Jukebox (70 dB): S 2 = 10^( 70 / 10 ) x 10 -12 = 10 -5 W/m 2. For 1 “standard” video gaming machine: S 3 = 10^( 45 / 10 ) x 10 -12 = 10 -7.5 = 3.162 x 10 -8 W/m 2. For 1 “Super Jackpot” machine: S 4 = 10^( 60 / 10 ) x 10 -12 = 10 -6 W/m 2. Total Sound Intensity from all sources listed = S 1 + S 2 + 6S 3 + 4S 4 = (0.3162 + 1 + 6 x 0.003162 + 4 x 0.1) x 10 -5 W/m 2. = 1.735 x 10 -5 W/m 2.

34. The Decibel Scale34 1.735 x 10 -5 W/m 2 corresponds to a decibel reading of D = 10 log 10 (1.735 x 10 -5 / 10 -12 ) = 72 dB. 72 dB is half the regulated limit of 75 dB, so the Tavern would appear to comply with Queensland Regulations, and the neighbour to have no grounds for complaint. This model, however, assumes that the “happy hour” noise will remain at 65 dB, despite the new attractions possibly attracting increased custom and perhaps customers raising their noise level to be heard above the sound of the jukebox and gaming machines.

35. The Decibel Scale35 Let’s investigate what sound intensity would be required from “happy hour” noise for the total noise to reach the regulated limit. ? + (1 + 6 x 0.003162 + 4 x 0.1) x 10 -5 = 10^( 75 / 10 ) x 10 -12 ? = 3.162 x 10 -5 - 1.418972 x 10 -5 = 1.743 x 10 -5 W/m 2 The decibel level of this Sound Intensity = 10 log 10 (1.743 x 10 -5 / 10 -12 )= 72 dB As this represents approximately 5 times the measured “happy hour” sound intensity of 65 dB, the tavern would appear to easily comply with Queensland Regulations, even if “people noise” doubled, tripled or quadrupled. The mediator should explain the calculations to the neighbour, reassuring him that noise levels were not expected to exceed the current noise restrictions for licensed premises in Queensland.

36. The Decibel Scale36 Advice should be given to the tavern licensee that, even though he was complying with Government regulations, it might be good public relations to do what he could to limit noise by considering the following “helpful hints”: (continued next slide)

37. The Decibel Scale37 http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf

38. The Decibel Scale38 References / Sources: http://www.mansw.nsw.edu.au/members/reflections/vol21no1coupland.htm http://physics.mtsu.edu/~wmr/log_4.htm http://www.phys.unsw.edu.au/jw/dB.html#soundfiles http://www.theage.com.au/news/reviews/deaf- sentence/2006/03/01/1141095746578.html http://www2.glenbrook.k12.il.us/gbssci/phys/Class/sound/u11l2b.html http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_ uids=9097241&dopt=Abstract http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf References / Sources: http://www.mansw.nsw.edu.au/members/reflections/vol21no1coupland.htm http://physics.mtsu.edu/~wmr/log_4.htm http://www.phys.unsw.edu.au/jw/dB.html#soundfiles http://www.theage.com.au/news/reviews/deaf- sentence/2006/03/01/1141095746578.html http://www2.glenbrook.k12.il.us/gbssci/phys/Class/sound/u11l2b.html http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_ uids=9097241&dopt=Abstract http://www.liquor.qld.gov.au/_Documents/Fact+sheets/Noise+restrictions.pdf

39. The Decibel Scale39 Sources of video clips: http://www.youtube.com/watch?v=RavdHOj-BA8 http://www.youtube.com/watch?v=6653_PHmf3o http://www.youtube.com/watch?v=tiBZC_nXnww&feature=PlayList &p=A214ED9F6E5E925E&playnext=1&playnext_from=PL&index =82 http://www.youtube.com/watch?v=mGgRdmEutmI http://www.youtube.com/watch?v=Z6GzD92P3X8&feature=related http://www.youtube.com/watch?v=AX0ODTqWp1Y http://www.youtube.com/watch?v=GpYd2DLe7XM&NR=1 Sources of video clips: http://www.youtube.com/watch?v=RavdHOj-BA8 http://www.youtube.com/watch?v=6653_PHmf3o http://www.youtube.com/watch?v=tiBZC_nXnww&feature=PlayList &p=A214ED9F6E5E925E&playnext=1&playnext_from=PL&index =82 http://www.youtube.com/watch?v=mGgRdmEutmI http://www.youtube.com/watch?v=Z6GzD92P3X8&feature=related http://www.youtube.com/watch?v=AX0ODTqWp1Y http://www.youtube.com/watch?v=GpYd2DLe7XM&NR=1