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 The PhysicsHypertextbook

Opus in profectus

Intensity

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Discussion

welcome

 The amplitude of a sound wave can be quantied in at least three ways:

1 by measurin! the maximum chan!e in position of the particles thatmake up the medium "the maximum particle displacement#

$ by measurin! the maximum chan!e in density of the medium% by measurin! the maximum chan!e in pressure "the maximum !au!e

pressure#

&easurin! displacement mi!ht as well be impossible 'or typical sound

waves( the maximum displacement of the molecules in the air is only a

hundred or a thousand times lar!er than the molecules themselves ) andwhat technolo!ies are there for trackin! individual molecules anyway*

+ensity ,uctuations are equally minuscule and very short lived "The period

of sound waves is typically measured in milliseconds# There are some

optical techniques that make it possible to see the intense compressions are

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rarefactions associated with shock waves in air( but this will be dealt with

in another section of this book

Pressure ,uctuations caused by sound waves are much easier to measure

-nimals "includin! humans# have been doin! it for several hundred millionyears with devices called ears Humans have also been doin! itelectromechanically for about a hundred years with devices called

microphones

.n any case( the results of such measurements are rarely ever reported

.nstead( amplitude measurements are almost always used as the raw data in

some computation /hen done by an electronic circuit "like the circuits in a

level meter# the resultin! value is called the intensity /hen done by aneuronal circuit "like the circuits in your brain# the resultin! sensation iscalled the loudness

 The intensity of a sound wave is a combination of its rate and density of ener!y transfer .t is an ob0ective quantity associated with a wave oudness

is a perceptual response to the physical property of intensity .t is a

sub0ective quality associated with a wave and is a bit more complex -s a!eneral rule the lar!er the amplitude( the !reater the intensity( the louder

the sound 2ound waves with lar!e amplitudes are said to be 3loud3 2oundwaves with small amplitudes are said to be 3quiet3 or 3soft3 "The word 3low3is sometimes also used to mean quiet( but this should be avoided 4se 3low3

to describe sounds that are low in frequency# oudness will be discussed at

the end of this section

5y denition( the intensity "I# of any wave is the time6avera!ed power "P# it

transfers per area " A# throu!h some re!ion of space The traditional way to

indicate the time6avera!ed value of a varyin! quantity is to enclose it inan!le brackets 78 These look similar to the !reater and less than symbolsbut they are taller and less pointy That !ives us an equation that looks like

this9

I 7P8 A

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 The 2. unit of power is the watt( the 2. unit of area is the suqare meter( so

the 2. unit of intensity is the watt per square meter ) a unit that has no

special name;

<

/

 

/ =

>m$ m$

intensity and displacement

'or simple mechanical waves like sound( intensity is related to the density of 

the medium and the speed( frequency( and amplitude of the wave This canbe shown with a lon!( horrible( calculation ?ump to the

next hi!hli!hted equation if you don@t care to see the sausa!e bein! made

below

2tart with the denition of intensity Aeplace power with ener!y "both kinetic

and elastic# over time "one period( for convenience sake#

I 7P8 A

I 7E8BT 

 A

I  7K  C Us8BT 

 A

2ince kinetic and elastic ener!ies are always positive we can split the time6

avera!ed portion into two parts

7P8 = 7E8T 

7P8 =  7K  C Us8T 

7P8 = 7K 8

 C7Us8

T T 

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&echanical waves in a continuous media can be thou!ht of as an innite

collection of innitesimal coupled harmonic oscillators ittle masses

connected to other little masses with little sprin!s as far as the eye can seeOn avera!e( half the ener!y in a simple harmonic oscillator is kinetic and half 

is elastic The time6avera!ed total ener!y in then either twice the avera!ekinetic ener!y or twice the avera!e potential ener!y

7P8 = $7K 8

 $7Us8

T T 

et@s work on the kinetic ener!y and see where it takes us .t has two

important parts ) mass and velocity

K   Dmv $

 The particles in a lon!itudinal wave are displaced from their equilibrium

positions by a function that oscillates in time and space

E x " x (t # E x max  sin;< $F

G   ft  I

 x  J=K>L

 Take the time derivative to !et the velocity of the particles in the medium"not the velocity of the wave throu!h the medium#

v " x (t # M

 E x " x (t #Mt 

v " x (t # $FNE x max  cos;< $F

G   ft  I

 x  J=K>L

 Then square it

v $" x (t # F$ N $E x $max  cos$;< $F

G  ft  I

 x  J=K>L

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On to the mass +ensity times volume is mass The volume of material we@re

concerned with is a box whose area is the surface throu!h which the wave is

travelin! and whose len!th is the distance the wave travels

m  V    Ax 

.n one period a wave would move forward one wavelen!th .n the volume

spanned by a sin!le wavelen!th( all the bits of matter are movin! with

diQerent speeds Ralculus is needed to combine a multitude of varyin!values into one inte!rated value /e@re dealin! with a periodic system here(

one that repeats itself over and over a!ain /e can choose to do our

calculation at any time we wish as lon! as we nish at the end of one cycle

'or convenience sake let@s choose time to be Sero ) the be!innin! of asinusoidal wave

L

K  U   dK " x (V#

V

L

K  

U

1

 " Adx # v $

" x (V#$V

L

K  U

1 " A#"F$ N $E x $max #cos$

;< I $F

 x  =>  dx 

$ LV

Rlean up the constants

1 " A#"F$ N $E x $max # $F$ A N $E x $max 

$

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 Then work on the inte!ral .t may look hard( but it isn@t ?ust visualiSe the

cosine squared curve traced out over one cycle 2ee how it divides the

rectan!le boundin! it into equal halves*

 The hei!ht of this rectan!le is one "as in the number 1 with no units# and itswidth is one wavelen!th That !ives an area of one wavelen!th and a half6area of half a wavelen!th

LU cos$

;< I $F

 x  =>   dx  

1 L

L $V

Put the constants to!ether with the inte!ral and divide by one period to !et

the time6avera!ed kinetic ener!y "Aemember that wavelen!th divided by

period is wave speed#

7K 8 

  ⎰⎱

"$F$ A N $E x $max #"1

L#  ⎱

⎰1

T  $   T 

7K 8  F$ A N $v E x $max 

T 

 That concludes the hard part +ouble the equation above and divide byarea9

I 7P8

 $7K 8BT 

 A A

I $"F$ A N $v E x $max #

 A

One last bit of al!ebra and we@re done /e now have an equation that relates

intensity to displacement amplitude

I  $F$N $v  W x $max 

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+oes this formula make sense* Rheck to see how each of the factors aQect

intensity

factor comments

I X

 The denser the medium( the more intense the wave That makes

sense - dense medium packs more mass into any volume than arareed medium and kinetic ener!y !oes with mass

I X N $

 The more frequently a wave vibrates the medium( the moreintense the wave is . can see that one with my mind@s eye -

lackluster wave that 0ust doesn@t !et the medium movin! isn@t!oin! to carry as much ener!y as one that shakes the medium like

craSy

I X v 

 The faster the wave travels( the more quickly it transmits ener!y

 This is where you have to remember that intensity doesn@t somuch measure the amount  of ener!y transferred as it measures

the rate at which this ener!y is transferred

I X E x $m

ax 

 The !reater the displacement amplitude( the more intense the

wave ?ust think of ocean waves for a moment - hurricane6driven(wall6of6water packs a lot more punch than ripples in the bathtub

 The metaphor isn@t visually correct( since sound waves are

lon!itudinal and ocean waves are complex( but it is intuitivelycorrect

'actors aQectin! the intensity of sound waves

YZA.'[ ZYZA[TH.\] 4P TO THZ \Z^T HZ-+.\]

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2ome people are interested in the way intensity relates to maximum velocity

"the velocity amplitude# and maximum acceleration "the acceleration

amplitude# as well as the maximum displacement "the displacementapmlitude# 'or sense of completeness( let@s also derive these relationships

2tart with diplacement

 W x   W x max  sin"$F"N t  I x 

 C _##L

'indin! the amplitude of this equation is trivial

 W x max   W x max 

/e 0ust derived the equation that relates intensity to displacement

amplitude

I  $F$N $v E x $max 

Yelocity is the time derivative of displacement

 Wv   M  W x max  sin"$F"N t  I  x   C _##Mt  L

 Wv   $FN W x max  cos"$F"N t  I x 

 C _##L

'ind the amplitude9

 Wv max   $FNW x max 

9solve for W x max  9 W x max  

 Wv max 

$FN 

9substitute9

I  $F$N $v   Wv max 

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G

J$

K$FN 

9and simplify

I  v   Wv $max 

$

\ow we have an equation that relates intensity to velocity amplitude

-ccleration is the time derivative of velocity

 Wa  M  $FNW x max  cos"$F"N t  I  x   C _##Mt  L

 Wa  F$ N $ W x max  sin"$F"N t  I x 

 C _##L

'ind the amplitude9

 Wamax   F$ N $ W x max 

9solve for W x max  9

 W x max    Wamax 

F$ N $

9substitute9

I  $F$N $v G

 Wamax  J$

KF$ N $

9and simplify

I  v   Wa$

max 

`F$ N $

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 This completes the set with an equation that relates intensity to acceleration

amplitude

intensity and pressure

 The amplitude of a sound wave can be measured much more easily withpressure "a bulk property of a material like air# than with displacement "the

displacement of the submicroscopic molecules that make up air# Here@s a

quick and dirty derivation of a more useful intensitypressure equation froman eQectively useless intensitydisplacement equation

2tart with the equation that relates intensity to displacement amplitude

I  $F

$

N 

$

v  W x 

$

max 

\ow let@s play a little !ame with the symbols ) a !ame called al!ebra \otethat many of the symbols in the equation above are squared &ake all of 

them squared by multiplyin! the numerator and denominator by $v 

I F$$ N $v $ W x $max 

$v 

/rite the numerator as a squared quantity

I  "$FN v  W x max #$

$v 

ook at the pile of numbers and letters in the parenthesis in the numerator

$FN v  W x max 

ook at the units of each physical quantity

;<

k! 

1 

m 

m =>m% s s 1

+o some more ma!ic ) not al!ebra this time( but dimensional analysis

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;<

k! 

k! m 

\  Pa

=>m s$ m$ s$ m$

 The units of that mess are pascals( so the quantity in the numerator is

pressure squared ) maximum !au!e pressure squared to be more precise/e now have an equation that relates intensity to pressure amplitude

I  WP$

max 

$v 

/here9

I  intensity /Bm$

 WPmax   pressure amplitude Pa

density k!Bm%

v   wave speed mBs

 The intensity of a sound wave depends not only on the pressure of the wave(but also on the density of the medium and speed of sound in the medium

Hi!her density and hi!her sound speed both !ive a lower intensity /ater is

about `VV times more dense than air and has a speed of sound timesfaster Thus( sounds with the same pressure amplitude are about %VV times

more intense in air than in water This is one of the reasons humans hear so

poorly underwater "The other reason is that our ears are really desi!ned to

work with air as the drivin! ,uid( not water#Here@s a slow and clean derivation of a the intensitypressure equation 2tart

from the version of Hooke@s law that uses the bulk modulus "K #F 

  K  EV 

 A V V

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 The fraction on the left is the compressive stress( also known as the pressure

"P# The fraction on the ri!ht is the compressive strain( also known as the

fractional chan!e in volume "# The latter of these two is the one we@reinterested in ri!ht now .ma!ine a sound wave that only stretches and

compresses the medium in one direction .f that@s the case( then thefractional chan!e in volume is eQectively a fractional chan!e in len!th

EV 

 MW x 

V V M x 

/e have to use calculus here to !et that fractional chan!e( since the

ininitessimal bits and piece of the medium are squeeSin! and stretchin! at

diQerent rates at diQerent points in space en!th chan!es are described by

a one6dimensional wave equation .ts spatial derivative is the same as thefractional chan!e in volume

 W x    W x max  sin"$F"N t  I x 

 C _##L

MW x   I

$F W x max  cos"$F"N t  I

 x  C _##

L LM x 

.t@s interestin! to note that the volume chan!es are out of phase from the

displacements Yolume chan!es are gV behind displacement The extreme

volume chan!es occur at places where the particles are sittin! at theirori!inal positions .nterestin!( but not so useful ri!ht now /e care more

about what  these extreme values are than where they occur 'or that( we

replace the ne!ative cosine expression with its extreme absolute value C1 This leaves us with this

$F W x max 

L

Plu!!in! this back into the bulk modulus equation !ives us9

Pmax   K  $F

 W x max L

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-nd now for the dirty work Aecall these two equations for the speed of 

sound

v   NL

1

 

 N 

L   v 

 

v   j  K 

  K   v $

2ubstitute into the previous equation9

Pmax   v $$FN 

 W x max 

v 

9and simplify to !et the pressure6displacement amplitude relation

 WPmax   $FN v  W x max 

'amiliar*

intensity and density

YZA.'[ ZYZA[TH.\] 4P TO THZ \Z^T HZ-+.\]

 The density chan!es in a medium associated with a sound wave are directlyproportional to the pressure chan!es The relationship is as follows9

v   j WP

 W

 This looks similar to the \ewton6aplace equation for the speed of sound inan ideal !as but it@s missin! the heat capacity ratio "!amma# /hy*

v   jP

-ssumin! the rst equation is the ri!ht one( solve it for W W WP

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v $

 Take the pressure6displacement amplitude relation9

 WPmax   $FN v  W x max 

9subsitute9

 Wmax  $FN v  W x max 

v $

9and simplify to !et the density6displacement amplitude relation

 Wmax  $FNW x max 

v 

&ildly amusin! et@s try somethin! else

-!ain( assumin! the rst equation is the ri!ht one( solve it for WP

 WP  Wv $

 Take the equation that relates intensity to pressure amplitude9

I  WP$

max 

$v 

9make a similar substitution9

I "Wmax v $#$

$v 

9and simplify to !et the equation that relates intensity to density amplitude

I  W$

max v %

$

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\ot very interestin!( but now our list is complete

-mplitude6.ntensity Aelationships

amplitude   intensity connection

displacement 

I  $F$N $v E x $max  

velocity 

I  v   Wv $max 

  $ Wv max   $FNW x max 

acceleration 

I  v   Wa$

max 

  `F$ N $ Wamax   $FNWv max 

pressure 

I  WP$

max 

  $v  

density 

I  W$

max v %

  $ W

 WP

v $

v   wave speed( Wv   particle velocity

levels