Analysis of Sound Waves

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Occupational Health & SafetyPractitioner

Reading

ANALYSIS OF SOUND WAVES

January 2009


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Contents

OVERVIEW.....................................................................................................................1

SECTION 1: INTRODUCTION.......................................................................................2

SECTION 2: DECIBELS.................................................................................................3

SECTION 3: TYPICAL SOUND PRESSURE LEVELS..................................................6

SECTION 4: RELATIONSHIP BETWEEN SOUND INTENSITY LEVEL, SOUNDPOWER LEVEL AND SOUND PRESSURE LEVEL.......................................................8

SECTION 5: ADDITION OF DECIBELS ......................................................................10

SECTION 6: SUBTRACTION OF DECIBELS..............................................................12

SECTION 7: FREQUENCY SPECTRUM OF A SOUND .............................................13

SECTION 8: FREQUENCY ANALYSIS .......................................................................15

SECTION 9: FREQUENCY WEIGHTING NETWORKS ..............................................18

APPENDICIES AND REVIEW EXERCISE ANSWERS................................................22

CONCLUSION ..............................................................................................................28

REFERENCES & FURTHER READING.......................................................................28


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READING ANALYSIS OF SOUND WAVES

JANUARY 2009 SAFETYLINE INSTITUTE PAGE 1

OVERVIEW

This reading covers the analysis of sound waves and includes thedecibel scale, how to add and subtract decibels, sound spectra -octave and one-third octave frequency analysis, and A- and C-frequency weighting networks.

ObjectivesAfter reading this information you should be able to explain:

the decibel scale;

how to add and subtract decibels;

the terms sound power level, sound intensity level and soundpressure level and the relationship between them;

the relationship between sound pressure, sound intensity andsound power;

sound spectra - octave and one-third octave frequencyanalysis; and

A- and C- frequency weighting networks.

Pre-requisitesTo successfully undertake this reading learners need to:

have completed the reading on Basic Acoustics;

have an understanding of algebra (integration and logarithmic

functions); andhave access to a scientific calculator.

AuthorPamela Gunn B.Sc.(Hons Physics) University of Western AustraliaM.Sc.(Occupational Hygiene) London University


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Section 1: INTRODUCTION

In the first reading on Basic Acoustics you learnt about some of theterms used to describe sound waves. In this reading you will beintroduced to further terms we use in the measurement of themagnitude of sound waves and their frequency content.

Before you progress to actual measurement and assessment ofnoise, you need to understand the decibel scale of measurementand how to do decibel addition and subtraction .

For those of you not used to calculations, this may look a little

heavy going - but tables have been given to help you work throughthe examples. Give the mathematical option a try as, with the aidof a scientific calculator, you will find this method much quickeronce you get the hang of it.

Frequency analysis is another concept covered in the reading.This is an important area to understand, as many noise controlmeasures work to a different extent depending on the frequencycontent of the noise.

An explanation of A- and C- frequency weighting networks isalso given, along with exercises so that you can practice bothweighting and decibel addition.

PAGE 2 SAFETYLINE INSTITUTE JANUARY 2009


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Section 2: DECIBELS

You have probably all heard of the term decibel (or dB for short)in connection with sound. We will now see how this relates to thequantities discussed in the first reading.

2.1 Large range of sound intensitiesThe human ear responds to sounds over a very large range ofsound intensities:

the sound intensity ofthe threshold ofhearing (the quietestsound we can hear) is0.00000000001watts/m 2 (often written10 -12 watts/m 2 );

whilst the sound intensityat the threshold of pain isabout 10 watts/m.

2.2 Decibel scale is used to cover therange of sound intensitiesTo handle this large range we make use of a logarithmic ratioscale called the decibel scale .

In general, a decibel scale for any quantity, q, is defined as:

ref

qqq

log10L dB (Equation 2.1)



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Note that the decibel is not an absolute measure but is referencedto a selected quantity, q ref .

Another reason for using this scale is that the ear itself 'hears'logarithmically and humans judge the relative loudness of twosounds by the ratio of their intensities, a logarithmic behaviour.

The Bel was named after Alexander Graham Bell (1847-1922), theScottish/American inventor of the telephone and worker for thedeaf.

As this unit is rather large it has been divided into ten smaller unitscalled decibels or dB . (deci = one tenth of).

A brief tutorial refresher on logarithms is available fromwww.comdis.wisc.edu/vcd202/Logs.html

When sound intensity is expressed as a decibel it is referred to assound intensity level and is given the symbol L I .

To convert sound intensity, I , to sound intensity level L I thefollowing relationship is used:

ref I

IIlog10L dB re I ref (Equation 2.2)

where I = sound intensity whose level is being specified, inwatts/m 2

and I ref = reference intensity = 10 -12 watts/m 2 (the threshold ofhearing)

Sound power, W , can also be expressed in decibels and is thenreferred to as the sound power level, L w .

ref W

WW

log10L dB re W ref (Equation 2.3)

where W = sound power of the source in watts

W ref = reference sound power = 10 -12 watts

http://www.comdis.wisc.edu/vcd202/Logs.htmlhttp://www.comdis.wisc.edu/vcd202/Logs.htmlhttp://www.comdis.wisc.edu/vcd202/Logs.html


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For sound pressure, p , we use the relationship:

Sound pressure level 2ref

2

p

p

plog10L

=ref p

plog20 dB re p ref (Equation 2.4)

where p = rms sound pressure in Pa

p ref = 2 x 10 -5 Pa

(sometimes written as 20 Pa = 20 x 10 -6 Pa, which is the soundpressure at the threshold of hearing at 1000 Hz)

The sound pressure level at the threshold of hearing is thus:

01log20102102

log20L 5--5

p dB

Review exercise 2.1Answers are at the end of this reading.

Using equation 2.4, calculate the sound pressure level of a soundwith a sound pressure of 2 Pa.



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Section 3: TYPICAL SOUND PRESSURE LEVELS

To get a feel for decibels, look at the table below which gives values for thesound pressure levels of common sounds in our environment. Also shown are thecorresponding sound pressures and sound intensities. From these you can seethat the decibel scale gives numbers in a much more manageable range.

Table of Sound Pressure Levels and Corresponding Sound Pressure and Sound Intensity

Examples Sound

Pressure

Level dB

SoundPressure

Pa

Sound Intensity watts/m

30m from jet aircraft 140 200 100

Threshold of pain 130 10

120 20 1

Chainsaw 110 0.1

Disco 100 2 0.01

90 0.001

Kerbside of busy road 80 0.2 0.0001

70 0.00001

Conversational speech 60 0.02 0.000001

50 0.0000001

40 0.002 0.00000001

Quiet bedroom at night 30 0.000000001

Background in TV studio 20 0.0002 0.0000000001

10 0.00000000001

Threshold of hearing 0 0.00002 0.000000000001



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From the above table you can see that a sound which has a soundpressure level 10 dB more than another has a sound intensity 10times as much which, as we saw in the first reading, means thatthe sound contains 10 times the energy.

A sound 20 dB more than another, has a sound intensity 100 timesas much and contains 100 times the energy.

A sound which is 3 dB more than another contains twice theenergy.

This is an important concept to remember when we come toassessing people's noise exposures, for if someone is receivingnoise 3 dB above the exposure standard, they are not merely 'justabove' the standard, but are receiving twice as much potentiallydamaging sound to their ears.

A different concept is that of loudness, a psychophysical sensationcaused by sound. Here an increase of 10 dB in sound pressurelevel causes the sound to be perceived as twice as loud. Adecrease of 10 dB is perceived as half as loud. So, if you are ableto introduce noise controls which make the sound half as loud, youhave actually reduced the energy getting to the ear by a factor of

10 - a very worthwhile effort.

The decibel scale some typical soundlevels



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Section 4: RELATIONSHIP BETWEEN SOUNDINTENSITY LEVEL, SOUND POWER LEVEL AND

SOUND PRESSURE LEVEL

The relationship between sound intensity level, sound power leveland sound pressure level for a point source in a free field (i.e. onewhere there are no reflecting surfaces) can now be considered.

We saw in the first reading that 2r 4W

I was the relationship of

intensity, I, to power, W

(where r = distance from the source).

Changing to levels

)r 4log(10WW

log10II

log10 2

ref ref

L I = L w - 10 log( r 2) - 10 log (4 )

= L w - 20 log r - 11 (Equation 2.5)

For air at 21 oC, we saw in the first reading that:

I = 0.0024 p2

So

12

2

ref I

10 p0024.0log10

IIlog10L

102

104 p

log10

= 2ref

2

p p

log10

= L p

The sign means approximately equal to. That is, the soundintensity level and the sound pressure level are approximatelynumerically equal.



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So using equation 2.5

L p L w - 20 log r - 11 (Equation 2.6)

From this you can see that if the distance from the source ( r ) isdoubled, the sound pressure level is decreased by (20 log 2),which is 6 dB - a relationship that was depicted in the first reading.

Review exercise 2.2Answers are at the end of this reading

If the sound power level of a machine is 102 dB what are thesound pressure levels:

at 2 m.at 4 m

assuming the source is in a free field?



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If more than 2 sounds are to be added you need to do it two at atime.

Example: Add 4 sounds of 90, 91, 86 and 88 dB

First place in numerically increasing order i.e. 86, 88, 90, 91 dB.Add 86 and 88 dB using the table. This gives 90.1 dB. Now add90 dB to 90.1 dB. This gives 93.1 dB. Next add 91 to 93.1 dB. Thisgives 95.2 dB. Always remember to round to the nearest wholedecibel. So the total is 95 dB.


1. Add together two sounds of 94 dB and 99 dB.2. Add together 3 sounds of 96, 89 and 92 dB.

If you want to learn how to do the calculations mathematically (withthe aid of your electronic calculator or computer) and how toaverage decibels, go to Appendix A of this reading. This isoptional, but you will find this method much quicker when you are

adding more than 2 levels, once you get the hang of it.



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Section 6: SUBTRACTION OF DECIBELS

L total - L p1 Amount to be Subtracted

from L total to get L p2

0 greater than 10

1 6.8

2 4.5

3 3.0

4 2.2

5 1.6

6 1.2

7 1.0

8 0.7

9 0.6

10 0.5

Example: If the total soundpressure level is 96 dB and

source 1produces 89 dB by itself, thesound pressure level of source2 is:L p2 = 96 - 1 = 95 dB.

Review exercise 2.4Answers are at the end of this reading

If the sound pressure level at a worker's location is 93 dB with onemachine and the ventilation system operating and it is 87 dB withthe machine switched off and the ventilation still on, what is thesound pressure level due to the machine by itself?



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Section 7: FREQUENCY SPECTRUM OF A SOUND

A sound that consists of a single frequency is called a pure tone.

Nearly all sound sources emit sounds of more than one frequency.

Musical sounds contain a fundamental frequency plus severalharmonics (i.e. sounds with frequencies that are integral multiplesof the fundamental frequency). This combination tends to soundpleasant to the ear.



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Noise produced by most mechanical sources, such as machines,is made up of many different unrelated frequencies (wide bandnoise), and tends to sound unpleasant.

The above graph of the magnitude of the sound pressure level ateach frequency is called the frequency spectrum of the sound.



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Section 8: FREQUENCY ANALYSIS

It is often necessary to obtain information about the frequencyspectrum of a sound to design effective noise control and to selectappropriate personal hearing protectors.

8.1 Octave bandsIn most cases it is sufficient to measure the sound pressure levelin bands of frequencies, rather than at individual frequencies. Thewidth of the band usually chosen is the octave band - this is a

band where the upper frequency is twice that of the lower. Eachband is denoted by its centre frequency. Those usually measuredare the Internationally Preferred Frequencies of:

31.5Hz

63Hz

125Hz

250Hz

500Hz

1kHz

2kHz

4kHz

8kHz

16kHz

(k is a short way of writing kilo, which is 1000).and results are plotted on a graph like this:

This is a typical spectrum of a circular saw cutting aluminium.(Note that the line is simply a guide to move your eye from oneband to another and does not imply the magnitude at thefrequencies between the centre frequencies.)



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The limits of the octave bands are shown in the Table below:

Centre frequency, Hz Limits of band, Hz

31.5 22 - 4563 45 - 89

125 89 - 177

250 177 - 353

500 353 - 707

1000 707 - 1414

2000 1414 - 2828

4000 2828 - 5657

8000 5657 - 1131416000 11314 - 22627

Note that the centre frequency is the geometric mean and not theaverage of the band limit frequencies.

The lower and upper frequencies in the band can be found fromthe centre frequency by:

414.1f

2f f centrecentrelower (Equation 2.9)

and centrecentreupper f 414.12f f (Equation 2.10)

8.2 One-third octave bandsSometimes we want more detailed information than the octave

band analysis will give. This can be provided by selecting narrowerbands, such as one-third octave bands .

See Appendix B of this reading if you want more information onone-third octave bands. This is optional.



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8.3 Narrow band analysisFor even more detailed examination, narrow band analysis ispossible. This is used to pick up tonal noises with very narrowfrequency bandwidths that would be missed by octave or one-thirdoctave band analysis.

Note that in this example of noise from a fan the bandwidths areonly one-eighth of a hertz wide and we are only looking at therange 70 Hz to 80 Hz.



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Section 9: FREQUENCY WEIGHTING NETWORKS

9.1 A & C weighting networksSince the ear does not have an equal response to sounds ofdifferent frequencies we need to take this into account when wemeasure sound in relation to its effect on people. This is done bybuilding into sound measuring equipment an electrical networkwhich gives less weight to the frequencies to which the ear is lesssensitive. Several different weighting networks have beendeveloped over the years. The one which has been found to bestdescribe the damaging effect of noise is the A-weighting

network . This reduces the low frequency response and some ofthe high frequencies as shown in the diagram below.

Another weighting, which is useful when measuring peak noise orwhen estimating the attenuated noise when personal hearingprotectors are used, is the C-weighting network .

When the A- and C-weighting networks have been used in ameasurement, we denote this by writing the unit dB(A) or dB(C )respectively. When no weighting network is used we can denotethis by dB or dB(lin).



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9.2 Adjustments at octave bandcentre frequencies

From the weighting curves above, you can see that the followingare the adjustments at octave band centre frequencies:

Octave BandCentreFrequencies, Hz

31.5 63 125 250 500 1000 2000 4000 8000 16000

A-weightingadjustment, dB -39 -26 -16 -9 -3 0 +1 +1 -1 -7

C-weightingadjustment, dB -3 -1 0 0 0 0 0 -1 -3 -8

To obtain the overall A- or C-weighted sound pressure levels fromthe octave band sound pressure levels, the above adjustments aremade before adding, using one of the decibel addition methodsdescribed earlier.

EXAMPLE

What is the A-weighted sound pressure level in the cab of a diesellocomotive where the following octave band sound pressure levelswere measured?

Octave Band CentreFrequencies, Hz 31.5 63 125 250 500 1000 2000 4000 8000 16000

Diesel locomotiveoctave band levels, dB 104 102 92 90 87 82 80 68 60 57

A-weighting adjustment -39 -26 -16 -9 -3 0 +1 +1 -1 -7

A-weighted octave bandlevels, dB(A) 65 76 76 81 84 82 81 69 59 50



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Using the mathematical addition method or adding two at a timeusing the Table gives an A-weighted sound pressure level of88.7 dB(A), which we round to the nearest whole decibel giving89 dB(A).


Question 11. What is the C-weighted sound pressure level in the above

case?2. Is it higher or lower than the A-weighted level?

Question 2Now try the following addition of octave band levels of noise from acircular saw:

Octave Band CentreFrequencies, Hz 31.5 63 125 250 500 1000 2000 4000 8000 16000

Circular saw octave bandlevels, dB 73 75 77 80 87 85 88 96 92 87

1. What is the A-weighted level?2. What is the C-weighted level?3. Which is the higher?



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Question 3Look at the answers to questions 1 and 2 above and the octaveband frequency spectra of the diesel locomotive and the saw anddecide if the following statements are True or False.

1. The C-weighted level of a sound is always greater than the A-weighted level.

2. If the C-weighted level of a sound is significantly greater thanthe A-weighted level, the sound contains high levels of lowfrequency components.

3. If the A-weighted level of a sound is greater than the C-weighted level, the sound contains high levels of componentsin the 1000 - 4000 Hz range.

Once you have understood the answers to this exercise you willhave learnt a quick way (i.e. without having to do a full octaveband analysis) of checking where the main frequency componentsof a sound lie - which is very important in considering the types ofnoise reduction strategies to apply, as we shall see in laterreadings.



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APPENDICIES AND REVIEW EXERCISE ANSWERS

Appendix A mathematical addition and averagingof decibels

Addition of decibelsThe sound pressure levels must first be converted back to p 2 andthen added.

2ref

21

1 p p p

log10L and 2ref

22

2 p p p

log10L

For L p1 = 100 dB and L p2 = 98 dB

101 p2ref

21 10110loganti

10100

loganti10

Lloganti

p p

102 p2ref

2

2 1063.08.9loganti1098

loganti10

Lloganti p

p

The total mean-square sound pressure ratio is 1.63 x 10 10.

The total sound pressure level is:

)total(L p = 10 log (1.63 x 10 10) = 10 log 1.63 + 10 log 10 10

= 2.1 + 100= 102.1 dB= 102 dB to the nearest whole dB.

Note: We did not actually need to find the pressures, but only theirratios to the reference pressure.

In general, if you want to add any number n of sound pressurelevels ( L p1 , L p2 to L pn ) the formula is

L p(total) = 10 log (10

Lloganti......

10

Lloganti

10

Lloganti pn2 p1 p )

(Equation 2.7)



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Review Exercise A1

Add together two sounds of 94 dB and 99 dB

Review Exercise A2

Add together 3 sounds of 96, 89 and 92 dB.

Averaging Decibels

The average sound pressure level is found using -

L p(av) = 10 log 10

Lloganti......10

Lloganti10

Lloganti(n

1 pn2 p1 p ) (Equation 2.8)

Review Exercise A3

(i) What is the average sound pressure level of the 4 sounds withthe following sound pressure levels: 82, 90, 88 and 80 dB?

(ii) What (erroneous) answer would you get if you just averagedthe decibel values arithmetically?



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Appendix B One-third octave bandsOne-third octave band centre frequencies in the audible range are:25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500,630, 800, 1k, 1.25k, 1.6k, 2k, 2.5k, 3.15k, 4k, 5k, 6.3k, 8k, 10k,12.5k, 16k, 20k Hz.

The lower and upper frequencies of the one-third octave bands aregiven by:

122.1f

2f

f centre61centre

lower (Equation 2.11)

f upper = f centre x 2 = 1.122 x f centre (Equation 2.12)

Review Exercise B1

What are the lower and upper frequencies of the one-third octaveband with a centre frequency of 500 Hz?



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Review exercise answers2.1

Using equation 2.4, calculate the sound pressure level of a soundwith a sound pressure of 2 Pa.

Answer100dB

2.2

If the sound power level of a machine is 102 dB what are the

sound pressure levels:

at 2m.at 4m

assuming the source is in a free field.

Answer:85dB79dB

2.3

1. Add together two sounds of 94 dB and 99 dB.Answer: 100 dB

2. Add together 3 sounds of 96, 89 and 92 dB.Answer: 98 dB

2.4

If the sound pressure level at a worker's location is 93 dB with onemachine and the ventilation system operating and it is 87 dB withthe machine switched off and the ventilation still on, what is thesound pressure level due to the machine by itself?

Answer: 92 dB



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2.5

Question 1105 dB(C)

Higher

Question 2 99 dB(A)98 dB(C)The A-weighted level is higher

Question 3FalseTrueTrue

A1

Add together two sounds of 94 dB and 99 dB.

Answer

100dB

A2

Add together 3 sounds of 96, 89 and 92 dB.

Answer

98dB



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A3

(i) What is the average sound pressure level of the 4 sounds withthe following sound pressure levels? 82, 90, 88 and 80 dB?Answer87dB

(ii) What (erroneous) answer would you get if you just averagedthe decibel values arithmetically?Answer

85dB

B1

What are the lower and upper frequencies of the one-third octaveband with a centre frequency of 500 Hz?Answer

f lower = 446 Hz

f upper = 561 Hz



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CONCLUSION

Now you have completed this reading, you should be familiar withthe terms used to analyse the magnitude of sound waves and theirfrequency components.

The decibel scale a logarithmic ratio scale was introduced toyou, along with the terms for the magnitude of a sound (soundintensity level, L I, sound power level, L W and sound pressurelevel, L p,) and their relationships.

Examples of sound pressure levels of everyday sounds were given

for you to become accustomed to the decibel scale.

You have also learnt how to add and subtract decibels, which willbe useful in assessing people's noise exposures to multiplesources and working out the most cost effective way of reducingnoise exposure in these situations.

The frequency spectrum of a sound has been discussed, followedby what we mean by frequency analysis into octave bands, one-third octave bands and narrow bands.

Finally the concept of frequency weighting has been presented,along with the two weightings most used in occupational noisemeasurements the A-weighting and the C-weighting.

Your feedbackWorkSafe is committed to continuous improvement. If you takethe time to complete the online Feedback Form at the SafetyLineInstitute website you will assist us to maintain and improve ourhigh standards.


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REFERENCES & FURTHER READING

Beranek, L. L, Noise and Vibration Control, Revised Edition,Institute of Noise Control Engineering, Washington, 1988.

Bies, D.A. and Hansen, C.H., Engineering Noise Control: Theory and Practice, 3rd Edition, E & FN Spon, London, 2003.

Norton, M. P., Fundamentals of noise and vibration analysis for engineers, Cambridge University Press, Cambridge, 1989.

01dB, Mediacoustic - Teaching Acoustics by Computer, 01dB,

France 1996. (An interactive CD-ROM)

A brief tutorial refresher on logarithms at the University ofWisconsin:www.comdis.wisc.edu/vcd202/Logs.html

Information on Alexander Graham Bell (1847-1922), theScottish/American inventor of the telephone and worker for thedeaf:www.sjsu.edu/depts/Museum/bel.html

On-line calculator for addition of decibels:www.sengpielaudio.com/calculator-spl.htm
http://www.comdis.wisc.edu/vcd202/Logs.htmlhttp://www.comdis.wisc.edu/vcd202/Logs.htmlhttp://www.sjsu.edu/depts/Museum/bel.htmlhttp://www.sjsu.edu/depts/Museum/bel.htmlhttp://www.sengpielaudio.com/calculator-spl.htmhttp://www.sengpielaudio.com/calculator-spl.htmhttp://www.sengpielaudio.com/calculator-spl.htmhttp://www.sjsu.edu/depts/Museum/bel.htmlhttp://www.comdis.wisc.edu/vcd202/Logs.html

Analysis of Sound Waves

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