Appendices Acoustics

8/13/2019 Appendices Acoustics

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Basic Acoustics

Robert Mannell

These notes are divided into two parts. The first part, or main body of the notes, providessome basic information on acoustics. This is supplemented by explanatory footnotes

which contain extra information which might aid in your understanding of the main text.

The second part of these notes are the appendices which contain detailed information thatyou might wish to examine if you are interested.

These notes do not attempt to cover basic issues such as the nature of sound itself. During

the lectures you will be referred to various references that should adeuately cover suchtopics. These notes particularly focus on sound !amplitude!, the velocity of sound, basic

units and measurements of sound and the calculation of resonant freuencies.

"#M$ %#T$" #% T$RM&%#'#()&f you choose to read about these topics further in other publications, be especially careful

about terminology as it tends to vary somewhat from one text to another. &n this course &

have chosen to use the word !AM*'&T+D$! when referring generically to the concept

that includes *ower, &ntensity and *ressure-.

&t is common for engineers to tal/ about !*#0$R! when discussing sound amplitude.*hysicists will tell you that *ower is the total wor/ done per second by a sound source.

0hat the engineers are often referring to when they use the word !power!, however, isacoustic !&%T$%"&T)! or power per unit area ie. per suare metre. The terms !power!

and !intensity! are used interchangeably by many authors. 1or this course we will use the

term !intensity! in preference to the term !power! unless we are explicitly discussing thetotal acoustic power output of a sound source.

0hat is measured directly at a microphone diaphragm is !*R$""+R$! or !"#+%D

*R$""+R$ '$2$'! "*' which is a measure of the slight fluctuations in the ambientpressure of the medium eg. air through which the sound is being conducted3. #nce

you have derived a pressure value you can then mathematically convert that into an

intensity value.&n these notes the following symbols have been used45*ower *wr

&ntensity &

*ressure *"ound *ressure 'evel "*'

B$0AR$4 &t is very common to find the symbol !*! referring to !*ower! in many

publications. Be -667 sure as to whether !*! refers to *ower or *ressure in anypublication you are reading. This is particularly important when engineers use !*ower!

to refer to !&ntensity! and so use the symbol !*! instead of !&! in their formulae. Thisusage has the potential to greatly confuse you, especially in dB formulae ie. what isreally an &ntensity5to5dB formula may seem to you to be a *ressure5to5dB formula.

A8#+"T&8 +%&T" #1 M$A"+R$M$%T9The main units of measurement of relevance to acoustics are as follows described in

detail elsewhere45
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%ameAbbreviation

Basic +nits:

non5Basic +nits

- 0avelength 4 metre m m

3 1reuency f4

;ertsecond m.s5-1rom the above it can be seen that the period and freuency of a wave are the inverse of

each other45

Before ma/ing suchcalculations be sure

that the values are in

;< and seconds. Milliseconds ms can be converted to seconds by dividing by -666.

Gilo;ert< /; 6.6- H -66 ;oxygen ;eliox mixtures used in deep sea diving. *ure oxygenslightly denser than air has a speed of sound which is slightly lower than for normal air,

whilst the speed of sound in ;eliox mixtures is considerably higher than for normal air,

owing to the very much lower density of helium when compared to nitrogen the gasnormally found with oxygen in air. The exact speed of sound in ;eliox mixtures depends

upon the ratio of helium to oxygen, but is around twice the speed of sound in normal air.

As we saw above, when the speed of sound is changed the apparent resonantcharacteristics of sounds emitted by an acoustic resonator such as the vocal tract are

changed. &n the case of speech in ;eliox, the fundamental freuency pitch and resonant

pea/ formant freuencies are shifted to much higher values approximately doubled to

give the speech of deep sea divers a !chipmun/! sound.&t should also be clear that as temperature increases then so will the speed of sound. &t is

most important to note that temperature is Absolute Temperature in G and so the ratio of45

3I8 > -I8 3Rather 3?G > 3:G H -.669?

0hat this means is that the effect of temperature change on the speed of sound within

temperature ranges at which speech is normally uttered say4 5?6I8 to J:?I8 onlyresults in moderate changes in the speed of sound.

8hange in the speed of sound resulting from changes in temperature can be derived from

the following formula45


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where c- and - are the speed of sound in and temperature IG of condition -

and c3 and 3 are the speed of sound in and temperature IG of condition 3

1or example, c at 5?6I8 is K:7 of c at J:?I8, which would result in perceivabledifferences in the freuency of the same sound. "ee appendix for a further example

"#+%D !AM*'&T+D$!

The human ear and the microphone the main artificial transducer of sound both measurethe tiny changes in pressure that result from the passage of a longitudinal wave through a

medium.

The average air pressure at sea level is approximately euivalent to the pressure exertedby a column of mercury cm high in a barometer at 6I8 under standard gravity. This

approximation has been adopted as the definition of the standard measure of air pressure,

- atmosphere atm. This unit of measurement is is euivalent to45- atm -.6-9 x -6? *a

The sound pressure that is only Lust perceivable ie. the threshold of hearing for a -666

;< tone is ta/en to be45

3 x -65? *a ie. 36 C*a "tandard Reference "ound *ressure 'evel

nb. the actual threshold of hearing varies greatly from freuency to freuency as well asfrom person to person. This value is actually a better approximation of threshold at 9366

;?,666,666,666 atmospheric pressure. &t is common to uote sound pressure in C*a as

this measure is almost of the same order of magnitude as the minimum perceivable soundpressures.

The threshold of pain ie. the maximum sound pressure that can be perceived without

pain is about -66 *a or about ->-666 atm, which is ?,666,666 times the threshold soundpressure.

Thresholds have also been defined in terms of intensity, with the standard intensity

threshold of hearing beingK45 -65-3 0atts.m53 "tandard Reference "ound &ntensityThe intensity of a sound, with a sound pressure level of 36 C*a, is very close to -65-3

0atts.m53. These two reference values have been rounded off and so do not describe

precisely the same sound, but they are very close.The intensity of a sound is proportional to the suare of the sound pressure.

& *3

The following is approximately true for sound travelling through air at - atm and 36I845& *>363

The ratio of the intensity of sound - &- over the intensity of sound 3 & 3 euals the

suare of the ratio of the sound pressure of sound - *- over the sound pressure of

sound 3 * 3. nb. *- is the sound pressure of a sound when that soundNs intensity is & -,and * 3 is the sound pressure of a sound when that soundNs intensity is & 3.

&->&3 H *->*33

D$8&B$'" DB&t was noted many years ago 1echner, -K6 that the sensitivity of the ear to changes in

intensity was not related linearly to either intensity or pressure. &t was believed then that

the earNs sensitivity to sound intensity or sound pressure was an approximatelylogarithmic relationship. &nitially, it was proposed that a new measure of intensity be
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utilised which was derived from the log base -6 of the ratio of two intensities.

Bel &' H log-6 &->&3

The Bel scale named after Alexander (raham Bell was approximately linearly related tothe earNs sensitivity to sound intensity at intensities louder than a whisper so that eual

steps in Bels were close to eual perceptual steps. A step of - Bel was however about -6

times greater than the minimally perceivable step and so a new scale was devised, thedeciBel dB.

- dB H 6.- Bel or - Bel H -6 dB

"o that45dB H -6 x log-6 &->&3

%ote that it is common to refer to dB values as !&ntensity in dB! whether derived from

intensity or sound pressureO.

&f &3 is set to a standard reference intensity then dB measurements become readilycomparable from one publication to another. The usual reference level chosen is the

"tandard Reference "ound &ntensity of -65-3 0atts.m53 see previous page and is

indicated by the symbol !&6!. "uch dB values are referred to as dB &' to indicate that

the dB value has been determined from the ratio of the soundNs intensity to the standardreference intensity.

dB &'4 refH-65-3 0.m53 H -6 x log-6 &>&6DeciBel values can also be derived directly from sound pressure conveniently, as this is

the way sound amplitude is measured by a microphone-6. As with intensity, dB can be

calculated from the ratio of any two sound pressures *- and *345dB H 36 x log-6*->*3

&f *3 is set to a standard reference sound pressure then dB measurements become readily

comparable from one publication to another. The usual reference level chosen is the

"tandard Reference "ound *ressure of 36C*a see previous page and is indicated by thesymbol !*6!. "uch dB values are referred to as dB "*' to indicate that the dB value has

been determined from the ratio of the soundNs sound pressure level to the standard

reference sound pressure.dB "*'4 refH36C*a H 36 x log-6*>*6

&t is conventional to indicate whether dB was calculated from intensity or pressure mainly

because of the slight difference in the reference levels which could result in significantdifferences in the resultant dB value for very low test intensities and pressures. This is

usually indicated in one of the following ways45

dB &' H -6 x log-6&>&6

dB &'4 refH-65-30.m53 H -6 x log-6&>&6dB refH-65-30.m53 H -6 x log-6&>&6

dB "*' H 36 x log-6*>*6

dB "*'4 refH36C*a H 36 x log-6*>*6dB refH36C*a H 36 x log-6*>*6

The symbol dB without one of the ualifiers listed above could imply that any pressure or

intensity has been used as the reference.Another important thing to /now about deciBels is how to interpret values of different dB

relative to each other. 1or example, a dB rise implies a doubling of sound pressure

whilst a 9dB rise implies a doubling of intensity. 8onversely, a dB fall implies a halving

of sound pressure whilst a 9dB fall implies a halving of sound intensity.
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36 x log-63>- H dB "*'

36 x log-66.?>- H 5dB "*'

-6 x log-63>- H 9dB &'-6 x log-66.?>- H 59dB &'

where !3>-! implies that the new value pressure in the first case and intensity in the

second case is twice the old value and !6.?>-! implies that the new value is half the oldvalue. &n all cases the !old! value here is eual to -.

ADD&%( #R "+BTRA8T&%( T0# DB #R "#+%D *R$""+R$ 2A'+$"

)ou must be very careful when doing arithmetic on sound amplitudes. All calculationsshould be carried out on sound intensities, never on sound pressures or dB values except

for a small number of exceptions, and only when you are 2$R) clear about what you are

doing.1or example, when adding together two sounds45

dB4 dB- J dB- 3 x dB-

*ressure4 *- J *- 3 x *-

&ntensity4 &- J &- H 3 x &-

#nly the addition or multiplication of intensities results in a correct answer. 0hen twoidentical sounds are added together, the effect on dB and * values is as follows45

dB4 dB- J dB- H dB- J 9*ressure4 *- J *- H -.:-: x *- -.:-: H suare root of 3

"ince & *3 then the doubling of intensity only results in an increase in pressure by

-.:-:. Pie. 3>- H -.:-:>-3Q. 1urther, as outlined in the previous section, doubling asound results in an increase of 9 dB since the ratio of the new intensity to the old intensity

is 3>- and this results in a value of J9 dB.

To add together two sound pressures, convert to intensities by suaring, add the suarestogether and the ta/e the suare root of the result, as in the following example where two

sounds of * H - are added together--45

&f you are as/ed to addtogether two sounds with

/nown dB values and to determine the resultant amplitude in dB you must follow this

procedure45i 8onvert both dB values to intensity nb. it doesnNt matter whether the dBvalues were derived from &' or "*', the calculations M+"T be performed on intensities.

"ince4

dB H -6 x log-6&->&6 then &->&6 H -6dB>-6This formula can be simplified by arbitrarily assuming &6 H -. This gives45

&- H -6dB>-6

ii Add together the derived intensity values45

& H &- J &3iii 8onvert the intensity bac/ to dB remembering to utilise the same reference &6 H -

that was used in the first calculation, thus again removing &6 from the calculation45

dB H -6 x log-6&This procedure can be simply extended to deal with the addition of more than 3 sounds or

the subtraction of one sound from another.

"ome tric/s that may simplify dB calculationsi 0hen adding together two sounds of the same dB value, simply add 9 dB
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ii 0hen doubling intensity, simply add 9 dB ie. same as i

iii 0hen halving intensity, simply subtract 9 dB

iv Multiplying intensity by four is the same as doubling twice, so add 9 dB twice, etc.v Dividing intensity by four is the same as halving twice, so subtract 9 dB twice, etc.

vi 0hen adding together two sounds the resultant dB value is somewhere between the

higher original dB value and 9 dB above that value. That is for two sounds of dB- and dB3 where the higher value is dB- the resultant value is between dB - and dB -J9. The

new value is only eual to dB-J9 when dB3 exactly euals dB-. &f your result is outside

this range then your calculations are wrong.eg. ?6 dB J ?6 dB H ?9 dB

?6 dB J : dB H ?-.K dB

?6 dB J :6 dB H ?6.: dB

?6 dB J 36 dB H ?6.66: dB ?6 dB J 6 dB H ?6.6666: dB nb. 6 dB is not & H 6

R##T M$A% "+AR$D RM" A2$RA($ #1 "#+%D *R$""+R$

The R.M.". method is the only valid way to determine the !average! sound pressure of alength of speech signal. This is because pressures cannot be added together in a

straightforward way but must be first converted to intensities. &t must be rememberedthat45

& *3

This implies that suaring * will effectively convert it into an intensity value of arbitraryunits dividing by 363 would convert it to standard units but this is unnecessary as the 36

3 factor would be cancelled out in the later reconversion to pressure. Therefore, the

following formula can be used for the calculation of average pressure45

Average intensity can, on the

other hand, be derived as asimple mean or average45

The

tablebelow demonstrates how to calculate the R.M.". average of a set

of five %H? pressure values. The first column indicates the value

of !n! ie. this line refers to the -st, 3nd, ... ?th pressure value.The second column is the actual pressure value eg. *3H3. The

third column is the suare of the pressure value. The th line of the 9rd column is the sum

of the suared values. The th line is the R.M.". value which is calculated by dividing the

!"+M #1 T;$ "+AR$"! by the number of values %H? and then ta/ing the suareroot of the result. &n other words, the R.M.". value is the45

"+AR$ R##T #1 T;$ A2$RA($ #1 T;$ "+M #1 T;$ "+AR$" #1 T;$

#R&(&%A' *R$""+R$ 2A'+$".

n *n *n3

- 9 O

3 3 :


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9 6 6

: 53 :

? 59 O

*n3 3

*RM" 3>?

A8#+"T&8 &%T$%"&T) A%D T;$ &%2$R"$ "+AR$ 'A0-3The acoustic intensity, or average rate at which wor/ is being transferred through a unit

area on the surface of the spherical wave front radiating out from the source in all

directions diminishes with distance in accordance with the inverse suare law.

where45 & the intensity of a sound

r the distance from the source of the sound

1or the purposes of this course, we are mostly interested in comparing intensities andpressures at varying distances from the sound source. +se the following formula45

where45 &- the intensity of a sound at distance r- from the source &3 the intensity of the same sound at distance r3 from the source.

&f you are as/ed the effect of

increasing the distance by

multiplying by a factor of , the ratio on the right hand side of the last euation becomes->3 and so the value of &3 is simply &-multiplied by ->3. 1or example, if the new

distance r3 is twice the old distance r- from the sound source, then the new intensity &3 is

->33 or ->: times the old intensity &-. )ou donNt need to /now what the actual distancesare if the distances are referred to in any uestion as a ratio because the above euation is

expressed as a ratio of 3 distances and of 3 intensities.

. &f the intensity at a certain distance from a soundNs source is 3? 0.m53, what is theintensity if the distance is increased five5fold Pr3>r-Q H ?, so Pr->r3Q H ->?.

A. &3 H &- x r->r33 H 3? x ->?3 H 3? x ->3? H - 0.m53

. &f the intensity at a certain distance from a soundNs source is 36 0.m53, what is theintensity if the distance is decreased by one fifth Pr3>r-Q H ->?, so Pr->r3Q H ?.

A. &3 H &- x r->r33 H 36 x ?3 H 36 x 3? H ?66 0.m53

&t can also be readily shown that45

"o, if we double the distance from the

source we only halve the sound

pressure although we divide the intensity by four.

. &f the sound pressure at a certain distance from a soundNs source is 3? *a, what is thesound pressure if the distance is increased five5fold Pr3>r-Q H ?, so Pr->r3Q H ->?.

A. *3 H *- x r->r3 H 3? x ->? H ? *a.
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1igure -4 A two dimensional simulation of the inverse suare law.

1igure - simulates the euivalent of the inverse suare law in a 3 dimensional universe.&n this diagram eual amounts of sound intensity are represented by each of the radiallines. At distance -m - lines pass through an arc of length '. At distance 3m only K lines

pass through an arc of length ' and the - lines now pass through an arc of length 3'.

This means that in a 3 dimensional universe sound intensity halves every time distancefrom the source is doubled ie. an inverse law rather than an inverse suare law.


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1igure 34 A three dimensional simulation of the inverse suare law at distance R.

1igure 3 simulates the inverse suare law in a 9 dimensional universe. &n this diagram the

suares represent areas on the surface of a sphere at a distance R from a sound source inthe centre of the sphere. The dots are the euivalent of the lines in figure - and indicate

the points at which -666 lines of sound intensity pass through the -66 unit sided suare

on the surface of the sphere.


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1igure 94 A three dimensional simulation of the inverse suare law at distance 3R.1igure 9 simulates the inverse suare law in a 9 dimensional universe. &n this diagram the

suares represent areas on the surface of a sphere at a distance 3R from a sound source in

the centre of the sphere. The dots are the euivalent of the lines in figure - and indicatethe points at which -666 lines of sound intensity pass through the 366 unit sided suare

on the surface of the sphere. %ote, however that only 3?6 intensity lines pass through the

inner -66 unit sided suare. This is one uarter the number of intensity lines that passthrough the -66 unit sided suare at half this distance ie. at distance R from the central

sound source. At twice the distance the length of each side of the suare containing a

fixed measure of sound intensity has doubled as happened for the arcs in the the 3dimensional world in figure -. "ince each of the sides of this suare doubles with a

doubling of the distance from the centre, the si


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transducer surface of a microphone.

T;$ D#**'$R $11$8T0hen both a sound source " and an observer # are stationary, the freuency observed

by # can be readily determined from the wavelength of the sound emitted by " and the

speed of sound according to the following formula.

1igure -4 &llustration of the doppler effect with stationary sound source " and movingobserver #. The concentric circles represent the cycle pea/s of the radiating sound

waves.

0hen an observer # is moving towards a sound source ", #Ns ear intersects with eachcycle pea/ more rapidly than would be predicted from the wavelength and the speed of

sound. This has the same effect as would an increase in the speed of sound. That is, there

is an increase in the observed freuency of the sound. The reverse occurs when theobserver is moving away from the sound ie. it ta/es longer for each cycle pea/ to reach


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#Ns ear and so the observed freuency is lower. &n these cases the effective speed of

sound can be determined by adding 2o to c when # is moving towards the sound and by

subtracting 2o from c when # is moving away from the sound.

1igure 34 &llustration of the doppler effect with moving sound source " and stationary

observer #.

0hen a sound source " is moving towards an observer # each wave cycle is initiateda bit closer to # than was the preceding cycle. &n this diagram the dots between "- and

"3 represent the position of the source at the time of the propagation of the wave cycles

represented by the circles. This has the effect of reducing the wavelength in thedirection that " is moving. Reducing the wavelength increases the observed freuency of

the sound. The reverse is true for an observer that the sound source " is moving directlyaway from. %ote that whilst this actually effects the soundNs effective wavelength, theobserved freuency of the sound can be more easily determined by adding 2s to c when

" is moving towards # and by subtracting 2s from c when " is moving away from #.

1##T%#T$"

-. "ome authors seem to use the term !amplitude! as a synonym of !pressure! ie. !soundpressure level! 5 "*'. #ther seem to use the term both generically and as a synonym for

!sound pressure!.


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3. The vibrating air particles cause the microphone diaphragm to move in and out and

sound pressure can be determined if the acceleration of the diaphragm is measured and

the mass and area of the diaphragm are /nown. Microphones can be calibrated so that"*' can be measured directly.

9. "ee appendix 9andappendix :for detailed descriptions of the Basic *hysical +nits of

Measurement and specifically of the units of measurement of particular relevance toacoustics. &n these notes the MG" system of measurement is used, and this is described in

these appendices. &n many older publications you may find reference to the now

discarded 8(" system of units and even to the British &mperial system of units. The 8("system is also described in the appendices.

:. The basic units of length metre m, time second s, and mass /ilogram /g form the

basis of most other units of measurement and of all common acoustic units of

measurements.?. "ee appendix for a detailed explanation of the speed of sound in different media and

conditions.

. - atmosphere H -.6-9 x -6 dynes.cm53 in the 8(" system

. Threshold sound pressure level is 3 x -65: dynes.cm53 in the 8(" systemK. Threshold acoustic intensity level is -65- 0atts.cm53 in the 8(" system

O. This is not strictly correct as the dB is a ratio, and so is not made up of basic units ofmeasurement ie. &ntensity /g.s59 whilst the dB is not

-6. "ee appendix ?for the derivation of the formula for calculating dB from sound

pressure level.--. "ee the next section on RM" for more details.

-3. "ee appendix for further information on the inverse suare law, including

derivations of the formulae.

Appendices

&n the following appendices you will find detailed information that supplements the

material supplied in the main body. The first two appendices explain basic concepts ie.scientific notation and logarithms that are assumed for this course. )#+ 0&'' %#T B$

$AM&%$D #% A%) MAT$R&A' 0;&8; &" #%') 1#+%D &% T;$ A**$%D&8$".

This material is supplied for thosestudents who wish to have a more detailed /nowledgeof general acoustics over and above the reuirements of this course.

A**$%D& -

"cientific %otation"cientific notation is used to ma/e very large or very small numbers more manageable.

The principle is described below by example.

-,666 H -69

-6,666 H -6:-,666,666 H -6

3,666,666 H 3 x -6

?,:96,666 H ?.:9 x -6-6 H -6-

- H -66

6 H 66.- H -65- nb. 5)->)
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6.666- H -65:

6.6669 H 9 x -65:

6.6669: H 9.: x -65:

A**$%D& 3

'ogarithmsThe two most common types of logarithm are !natural! logarithms base e and base -6

logarithms. #nly base -6 logarithms are of interest in this course as they form the basis of

the deciBel.&f -6A H B

then log-6B H A

&n other words, determining a logarithm as/s to what exponent the value -6 must beraised in order for the result to eual the value B.

log-6-66 H 3

log-6-,666 H 9

log-63 H 6.9

log-63,666 H 9.9%ote that45

log-6A x B log-6A J log-6B eg. log-6-,666 x 3 log-6-,666 J log-63log-6A>B log-6A 5 log-6B

log-6AB B x log-6A

A**$%D& 9Basic *hysical +nits of Measurement

There are two main metric systems of measurement based on the choice of the basic units

which ma/e them up45MG" Metres, Gilograms, "econds

8(" 8entimetre, (rams, "econds

There is also the British &mperial system, which has been gradually phased out over thepast ?6 years and which was based on the basic units45

1*" 1eet, *ounds, "econds

The MG" system has now been recognised as the international system for science and

has also been increasingly adopted for day5to5day use. )ou will, however, still comeacross 8(" units in various boo/s and articles on acoustics and so the various 8("

measurements will also be listed below.

&% T;&" 8#+R"$, #%') T;$ MG" ")"T$M &" +"$D MG" 8("

Mass /ilogram /g gram g

'ength metre m centimetre cm

Time second sec,s second sec,sMost, but not all, other physical measurements are made up of some combination of these

three basic units of mass M, length ', and time T.

Area '3 m3 cm3


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2olume '9

m9 cm9

2elocity '>T or 'T Distance travelled per unit time.

m.s5- cm.s5-

Acceleration '>T>T or '>T3 or 'T53 Rate at which velocity changes.

m.s53 cm.s53

1orce M'T53 M x acceleration A force causes a mass to accelerate or decelerate

/g.m.s53 g.cm.s53

/nown as

%ewton%

dyne- dyne -65? %ewtons

0or/ M'3T53 force 1 x ' PM'T53.'Q A measure of energy transfer from one body

to another. 0or/ only occurs when there is a transfer of energy. 0or/ can be related tothe distance an obLect is moved ' by a force 1.

/g.m3.s53

g.cm3.s53

or%ewton.

mdyne.cm

/now

n as@oule erg

- erg -65 Loules

*ower M'3T59 wor/>T PM'3T53 x T5-Q force x velocity PM'T53 x 'T5-QThe rate at which wor/ is done ie. wor/ per second.

That power which gives rise to the production of energy at the rate of - Loule per

second or -6ergs per second.

/g.m3.s5

9

g.cm3.s5

9

or @oule.s5- erg.s5-/now

n as0att 0 ...

-6 erg.s5- - 0att

&ntensity MT59 power>area PM'3T59.'53Q wor/>area>sec PM'3T53.'53.T5-QRate at which wor/ is done across a unit area per second.

*ower per unit area.

/g.s59 g.s59

[email protected]

3.s5-

erg.cm5

3.s5-

or 0att.m530att.cm53

- 0att.cm53 -6:0att.m53

*ressure M'5-T53 force>area PM'T53.'53Q

The amount of force per unit area.


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/g.m5-.s5

3

g.cm5

-.s53

or%ewton.m53

dyne.cm53

/now

n as

*ascal

*a

...

A**$%D& :

Acoustic +nits of MeasurementAcoustic units of measurement are a sub5set of the full list of units of measurement.

0avelength '

m cm

1reuency T5-

2elocity speed of sound divided by distance wavelength 'T5-.'5-

;ert< ;sec sec5-

/ilo;ert< /;area PM'3T59.'53Q wor/>area>sec PM'3T53.'53.T5-Q

Amount of power per unit area. /g.s59 g.s59

[email protected]

3.s5-

erg.cm5

3.s5-

or0att.m53

0att.cm53

- 0att.cm53 -6:0att.m53

Acoustic or "ound *ressure M'5-T53 force>area PM'T53.'53QThe amount of force per unit area.

/g.m5

-.s53

g.cm5

-.s53

or%ewton.m53

dyne.cm53

/nown

as

*ascal

*a... - dyne.cm53 6.- *a

Acoustic *ressure is commonly referred to as "ound *ressure 'evel or "*'.

A**$%D& ?

"ome $xtra %otes on deciBelsThe formula for deriving the Bel from a ratio of intensities is45


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Bel H log-6&->&3

"ince there are -6 dB to every Bel, the formula for deriving the deciBel dB is45

dB H -6 x log-6&->&3The formula for determining dB from pressure ratios is derived as follows45

"ince45

log-63 H 3 x log-6and45

&->&3 H *->*33

Then45log-6&>&6 H log-6*>*63 H 3 x log-6*>*6

and so45

-6 x log-6&>&6 H 36 x log-6*>*6

iff if and only if45* is the pressure of a sound whose intensity is &

*6 is the pressure of a sound whose intensity is &6

The second condition is only approximately true when using the standard reference

values as 36C*a is only approximately the pressure of a sound whose intensity is -65-30.m53

A**$%D& T;$ 2$'#8&T) #1 "#+%D

The velocity or speed of sound c is the number of metres that a wavefront travels in one

second.The speed of sound in a gas is dependent on a number of factors45

the composition of the gas eg. air K67 oxygen J 367 nitrogen

pressuretemperature

c velocity of sound *6 static pressure 6 static density

ratio of specific heats

The ratio of specific heats varies with gas and is proportional to temperature, so45

absolute temperature G4 degrees Gelvin

nb. 6I8 H 39.-?IG 39IG, and so IG H I8 J 39.These relationships are uite complex as an increase in temperature affects both 6 and

*6, but under conditions where pressure and density remain constant eg. same gas

under same pressure conditions, such as air at sea level the following relation holds45

To use this euation all

temperature values in I8 must

be converted to IG see aboveThe exact velocity of sound in

air at - atmosphere and 68 is 99- ms5- although this is often rounded off to 996 ms5- to

simplify calculations. This is the same as saying that the velocity of sound at 39IG is99- ms5-, and for air at the same pressure but different temperatures, the velocity of


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sound can be calculated by the above formula nb. if the gas changes, or if the pressure

changes, then the calculations will be inaccurate. +sing this formula it can be shown that

for air at - atmosphere pressure45if H 3O9IG 36I8 then

"imilarly45

if H 3OKIG 3?I8 then c H 9: ms5-

if H 969IG 96I8 then c H 9:O ms5-

The following table compares the velocity of sound and the density of three gasses at 6I8and - atmosphere45

(as 6 c

;ydrogen 6.6O -3K#xygen -.: 9-

Air -.9 99-&t can be clearly seen from this table that the velocity of sound in very low density gassessuch as hydrogen tends to be higher than it is in higher density gasses such as oxygen

and air. The main cause of the difference in the velocity of sound in oxygen and air

would be the effect that the density of nitrogen has on the density of air a mix of 367oxygen and K67 nitrogen.

The velocity of sound in liuids and solids depends upon the elasticity and density.

density / elastic or bul/ modulus

&t must be noted that the elastic or

bul/ modulus increases as the compressibility or deformability of the material decreases,so the elastic or bul/ modulus is a measure of the resistance of a material to deformation.

That is, the elastic or bul/ modulus is the inverse of the common notion of elasticity and

so the lower / the higher the elasticity. &n other words, increases in either elasticitydecrease in / or in density lead to a decrease in the speed of sound.

The following table indicates some relationships between density and velocity of sound

for various solids and liuids45

Material Temp I8Densityx -69 /gm59

c ms5-

0ater -? -.6 -:?6

vulcanised

rubber 6 -.- ?:granite 6 3. 666

aluminium 36 3. ?-66iron 36 .O ?-96

copper 36 K.O 9?6

lead 36 --.9 -396The table can be interpreted if it is realised that the compressibility of the metals copper,

iron, and lead is fairly similar so that the main trend in these metals is a decrease in the


20/21

velocity of sound as the density increases. Aluminium and iron on the other hand have

very different densities but sound travels through them at about the same velocity. &f they

were eually compressible then sound would travel at a much higher velocity throughlower density aluminium. The fact that it does not indicates that the velocity of sound in

aluminium is reduced because it is more compressible lower bul/ modulus than iron.

(ranite, on the other hand has the same density as aluminium but sound travels at agreater velocity because its compressibility is less higher bul/ modulus. 0ater has a

very low density, but sound has a similar velocity to what it has in lead. This suggests that

water is much more compressible than lead has a lower bul/ modulus. 2ulcanisedrubber is extremely compressible very low bul/ modulus and so, regardless of its low

average density, sound travels very slowly.

A**$%D& A8#+"T&8 &%T$%"&T) A%D T;$ &%2$R"$ "+AR$ 'A0

The acoustic intensity, or average rate at which wor/ is being transferred through a unit

area on the surface of the spherical wave front radiating out from the source in all

directions diminishes with distance in accordance with the inverse suare law.

That is, if you multiply the distance

from the source by two, you divide theintensity by four. This law is derived

from45

where45 & intensity of the sound per unit area at distance r from the source

*wr total power of the sound

r radius of sphere distance from sound source :r3 surface area of a sphere of radius r

1or the purposes of this course, we are more interested in comparing intensities and

pressures at varying distances from the sound source. 1rom the first euation above wecan derive45

or

where45 &- the intensity of a sound at distance r- from the source

&3 the intensity of the same sound at distance r3 from the source.

&f we remember that45

then we can also say45

or

&n other words45

That is sound pressure is inversely

proportional to the distance of the

point of measurement from thesource, so that if you double the distance you halve the sound pressure.


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Appendices Acoustics

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