fundamentals of acoustics(13)

8/7/2019 Fundamentals of Acoustics(13)

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Fundamentals ofFundamentals ofAcousticsAcoustics

Some General WaveSome General Wave

PhenomenaPhenomena


2/18


3/18

This expression has two parts: first partThis expression has two parts: first part

is a standing wave field.is a standing wave field. 2 cosj t

iA p kxe[

It describes a waveform that does notIt describes a waveform that does notpropagate along the x direction, instead , thepropagate along the x direction, instead , thewaveform remains stationary.waveform remains stationary.

Such a wave is called a standing wave and isSuch a wave is called a standing wave and ismathematically characterized by an amplitudemathematically characterized by an amplitudethat depends on the position along the xthat depends on the position along the xdirection.direction.

,Tnkx ! ( 1, 2, )2x n nP! ! L

The positions of maximum pressure are

called antinodes of standing wave


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Where the pressure is zero at all time ,Where the pressure is zero at all time ,calledcalled nodes of standing wave.nodes of standing wave.

,

2

)12(T

! nkx ),2,1(

4

)12( .!! nnxP

when

The interference of sound waves

If sound waves of the same frequencyIf sound waves of the same frequencyand amplitude are superposed, theyand amplitude are superposed, theyeither neutralize or reinforce eacheither neutralize or reinforce eachotherothers effects. The phenomenon iss effects. The phenomenon is

described asdescribed as interferenceinterference


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This could be the case for two signalsThis could be the case for two signalsboth deriving from the same source,both deriving from the same source,

such as two speakers both being drivensuch as two speakers both being drivenby the same signal generator or lightby the same signal generator or lightfrom a single laser beam being split andfrom a single laser beam being split and

recombinedrecombined

1 1 1

2 2 2

cos( );

cos( ).

A

A

p p t

p p t

[ N

[ N

!

! 12 NN] !

Two signals that have a definite fixed relative

phase relation are called coherent.


6/18

1 2 1 1 2 2cos( ) cos( )

cos( )

A A

A

p p p p t p t

p t

[ N [ N

[ N

! !

!

!

!

.coscossinsintan

),cos(2

2211

22111

1211

2

2

2

1

2

NN NNN

NN

AA

AA

AAAAA

pppp

ppppp

,,4,2,0 .TT] ss!12 NN] !

1 2 A A A p p p!

When

Maximum cooperation

,,3, .TT] ss!When

1 2 A A A p p p! Maximum cancellation


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33--9 Spherical Acoustic9 Spherical Acoustic

WavesWavesA disturbance is produced by a pointA disturbance is produced by a point

source and propagated away from thesource and propagated away from the

sphere uniformly in all direction assphere uniformly in all direction asspherical wavesspherical waves, we have spherical, we have sphericalacoustic wavesacoustic waves

Expressed in spherical coordinates theExpressed in spherical coordinates thewave equation iswave equation is2 2

2 2

02 2 2 2 2 2

1 1 1[ ( ) (sin ) ]

sin sin

p p p pc r

t r r r r r U

U U U U Nx x x x x x

!

x x x x x x


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oy

x

z

U r

N

If the waves havespherical symmetry, theacoustic pressure p is afunction of radial

distance and time butnot of the angularcoordinates UN,


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Spherical acoustic waves do not changeSpherical acoustic waves do not changeshape as they spread out. Although theshape as they spread out. Although thewavefront of spherical acoustic waveswavefront of spherical acoustic waves

can be assumed plane at great distancescan be assumed plane at great distancesfrom the source, many acousticalfrom the source, many acousticalproblems are concerned with divergingproblems are concerned with diverging

spherical acoustic waves radiated from aspherical acoustic waves radiated from asimple source rather than plane acousticsimple source rather than plane acousticwaves.waves.


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In the case of spherical symmetry,In the case of spherical symmetry,

the wave equation simplifies tothe wave equation simplifies to

)](1

[2

2

2

02

2

r

pr

rrc

t

p

xx

xx

!

xx

2 22

02 2

2( )

p p pc

t r r r

x x x!

x x x

Xpr!

Rewriting the wave equation2 2

2

02 2

X Xc

t r

x x!

x x


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The equation is of the same form asThe equation is of the same form asthe plane wave equation with thethe plane wave equation with thegeneral solutiongeneral solution

The first term represents a spherical waveThe first term represents a spherical wavediverging from a point source at the origindiverging from a point source at the originwith speed cwith speed c00; the second term represents a; the second term represents awave converging on the originwave converging on the origin..

1 0 2 0( / ) ( / )X f t r c f t r c!

1 20 0( ) ( )

r rrp f t f tc c!

1 2

0 0

1 1( , ) ( ) ( )

r r p r t f t f t

r c r c!


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The most important diverging sphericalThe most important diverging sphericalwaves are harmonic. Such waves arewaves are harmonic. Such waves arerepresented in complex form byrepresented in complex form by

The wave diminish in amplitude as theThe wave diminish in amplitude as the

distance from the source increase.distance from the source increase.

)/(1

01 crtfr

p !

( ) j t kr Ap er

[

!

The converging wave has little application inacoustics while the diverging wave is

frequently produced by a small source andhas many uses.


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The acoustic impedance ofThe acoustic impedance ofspherical wavesspherical waves

Form the equation of motionForm the equation of motion

r

p

t

u

x

x!

x

x

0

1

V xx

! d

tr

p

u 0

1

V

( )

2

0 0

1(1 ) j t kr

A jkru jkr e p

j r j r

[

[V [V

! !

It is apparent that , in contrast with planewaves, the particle velocity is not in phase

with the pressure


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For acoustic impedanceFor acoustic impedancea

Z

pu !

2

0 0 0 0

2 2

( )

1 1 ( ) 1 ( )a

j r c kr r Z j

jkr kr kr

[ V V V[! !

Za will be found to be complex

a a a a aZ r jx r j m[! !

2

2

00

)(1

)(

kr

krcr

a !

V

2

00

)(1

)(

kr

krcx

a !

V2

0

)(1 kr

rm

a

!V

Where ra

is called the acoustic resistance and Xa

theacoustic reactance


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j

a a Z Z e N!

0 0

2

1, tan

1 ( )

a

a

a

c kr xZ

r krkr

VN! ! !

0 0

2cos , cos

1a

krZ ckr

V N N ! !

N1

kr

2 21 k r

N

1

kr

2 2k rA geometrical representation ofN

is given in Fig.


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0 1 2 3 4 5 6 7 0

0 .2

0 .4

0 .6

0 .8

1

kr

30o

60 o

90o

N0 0

ar

cV

0 0

ax

cV

0 0

ar

cV

0 0

ax

cV

N

When the distance from the source is only asmall fraction of a wavelength, the phasedifference between the complex pressureand particle speed is large

1, 0, 0, / 2a a

kr r x N Tp p p=


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When kr=1, both the acousticWhen kr=1, both the acoustic

resistance and reactance areresistance and reactance areequal toequal to

And the acoustic reactance has itsAnd the acoustic reactance has its

maximum value.maximum value. WhenWhen

0

0 0/ 2, 45c andV N !

0 01, , 0, 0

a akr r c xV Np p p?

At distances corresponding to a

considerable number of wavelengths, pand u are very nearly in phase, and thespherical wave then assumes the

characteristics of a plane wave.


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This behavior is to be expected , sinceThis behavior is to be expected , since

the wave fronts of all spherical wavesthe wave fronts of all spherical wavesbecome essentially plane at greatbecome essentially plane at greatdistances from their source.distances from their source.

fundamentals of acoustics(13)

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