Silence is the only homogeneous sound field in unbounded...

32
Noise Engineering / Acoustics -1- Chap.5 Sources of Sound Silence is the only homogeneous sound field in unbounded space Sound field with no boundaries and no incoming field 3 - d wave equation which satisfies the radiation condition is With the closer inspection at the point of r=0 , the wave equation is reduce to the Laplace equation because spatial derivatives are more singular 0 / 1 2 2 2 2 r c r t f t c r c r t f r c r t f t c / / 1 2 2 2 2 2 Since 2 term is more singular

Transcript of Silence is the only homogeneous sound field in unbounded...

Page 1: Silence is the only homogeneous sound field in unbounded spaceaancl.snu.ac.kr/aancl/lecture/up_file/_1554866619_chap.5... · 2019. 4. 10. · Noise Engineering / Acoustics - 1 - Chap.5

Noise Engineering / Acoustics - 1 -

Chap.5 Sources of Sound

Silence is the only homogeneous sound field in

unbounded space

Sound field with no boundaries and no incoming field

• 3- d wave equation which satisfies the radiation condition is

• With the closer inspection at the point of r=0 , the wave equation is

reduce to the Laplace equation because spatial derivatives are more

singular

0

/1 2

2

2

2

r

crtf

tc

r

crtf

r

crtf

tc

//1 22

2

2

2

Since ▽2 term is more singular

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Noise Engineering / Acoustics - 2 -

Sources of Sound

Silence is the only homogeneous sound field in

unbounded space• Consider the small volume surrounding the origin is finite

• 3-D delta function δ(x)

0 as 4

/1/

4

//

2

2

2

tf

ctr

fctf

dSr

crtf

nr

crtf

sV

0x

V

24 area of surface S

V

dVx 1)(

0

If V includes x = 0

otherwise

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Noise Engineering / Acoustics - 3 -

Sources of Sound

Silence is the only homogeneous sound field in

unbounded space• Both sides of equation is set to be zero on the homogeneous wave

equation

only homogeneous condition of - ① , f=0

• Generally, eq. ① can be written as

x tf

r

crtf

tc4

/1 2

2

2

2

- ①

),(''1 2

2

2

2txqp

t

p

c

┌ In sound field, q = 0

└ In source region, q ≠ 0

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Sources of Sound

The definition of a sound source

The sound with very weak disturbances

• Three dimensional wave equation is

• In the sound field, q must be zero. In the source region, q is non-

zero

• The wave field does not contain sufficient information for its

sources to be identified. But the sound field generated by a source

field q is unique.

qpt

p

c

2

2

2

2

1

qf

t

qf

cqqfp

t

qfp

c

2

2

2

2

2

2

2

2

11

Sound field :

Source field :

')(' pqfp

)(2)(12

2

2 qfqt

qf

c

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Noise Engineering / Acoustics - 5 -

Sources of Sound

The definition of a sound source

Retarded time

• The sound arriving at x at time t must have been launched from the

source at the previous time τ = t-|x-y|/c, which is usually called the

retarded time.

• If the source region(V) is compact, the source region is negligible

for a distant observer(ㅣxㅣ≫ㅣyㅣ).

tfx

tfpt

p

ci

i

,,div1 2

2

2

2xx

x

xx

4,

ctqtp

yyx

yxyx

3

4

,, d

ctqtp

V

V

dtqtq yyxyx3,,

yxy

y

tq

r

crtq

tc,4

/,1 2

2

2

2Ix-yI

x

y

V

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Sources of Sound

The definition of a sound source

Monopole and dipole source distribution

• The monopole source distribution q(x,t) is said to be degenerate

into a dipole distribution

• After differentiation with respect to the xi,

yyx

yxyx

3

4

,, d

ctf

xtp

V

i

i

i

i

V

i

i x

txfyd

ctf

xtc

),(

4

/,1 32

2

2

2

yx

yxy

Surface S encloses

volume V in which

lies all the region

where f≠0

Source

),(, txftq

x

),(

,

txf

tq

x : monopole distribution

: dipole distribution

yyx

yxyx 3

4

,, d

ctftp

V

So,

Equivalent monopole formula,

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Sources of Sound

The definition of a sound source

Point monopole and dipoles

• 1) point monopole

Source :

For compact source omni-directional

• 2) Point Dipole

r

crtQtp

4

/,

x

O

RObserver

yyx

yxyx

3

4

,, d

ctqtp

V

X

)()(),( xftQtxq

Source strength

))()(())()((),( xtFxtFtxq ixi

Dipole strength with direction

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Sources of Sound

The definition of a sound source

Combined source strength

Source : F(t)=Q(t)δl

r

tQ

axis dipole tQ

0 as div

0 as

(

3

3

2

2

1

1

lltQ

lx

lx

lx

ltQ

tQ

x

xxx

δlxx

r

crtF

xtp i

i

d4

/,x

y

yx

yxyx

3

4

,, d

ctf

xtp

V

i

i

t

F

crr

Ftrpd

1

4

cos,,

2

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Sources of Sound

The definition of a sound source

Note

• i) the sound field of dipole source has an angular dependence

• ii) no disturbance at 90˚ to the dipole axis(their sound exactly

cancel)

• iii) Pressure disturbance have a maximum magnitude on the dipole

axis

• iv) P`dipole = Near-field term (~1/r2)+ Far-field term (~1/r)

P`monopole = Far-field term only (~1/r)

• v) Dipole has some cancellation of the acoustic field,

so dipoles are less efficient radiations than monopoles. (by M2)

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Sources of Sound

The definition of a sound source

(Example)

• Two point sources ; (1) monopole, (2) dipole at r=0

• radiates sound of frequency ω.

• Determine the ratio of maximum pressure disturbance at r=ε(≪c/ω)

to that at r=l≫c/ω) for the two sources.

• i) pressure perturbation of monopole

)(

4),(' c

rtie

r

Atrp

4

'

max

Ap

r

l

Ap

lr4

'

max

lRatio

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Sources of Sound

The definition of a sound source• ii) pressure perturbation of dipole

• εω/c≪1, so ≫

• Near-field disturbance decays very fast, so does not does not

radiate far away as sound.

)(

2

1

4

cos),(' c

rtie

rcr

iBtrp

2

'

max4

Bp

r

lc

Bp

lr4

'

max

clRatio

cl

l

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Sources of Sound

Acoustic source process

What mechanisms do produce these source distribution in

nature?

• Source process ≡ one in which there is a forcing term in the wave

equation

Violate the homogeneous wave equation!!

• Common sources

Vibration of the surface

Breakdown of the state equation p=f(ρ)

(unsteady heat addition to fluid)

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Sources of Sound

Acoustic source process

Combustion noise

• When heat is added to a fluid the density of the fluid a function of

two variables of pressure and heat ρ=ρ(p,h)

• Hence,

• For a perfect gas,

• Finally,

• The linearized equation describing motion of a perfect gas with

unsteady heat is

dhh

dpc

dhh

dpp

dpph

2

1

2

0

22

1

ccRT

p

h

T

RT

p

hRTp

ppp

dh

cdp

cd

2

0

2

11

2

2

2

02

2

2

2

11

t

h

cp

t

p

c

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Sources of Sound

Acoustic source process• The solution of wave equation is

• The sound field is forced by the rate of change in the heat addition

rate. Steady heating is silent

• Actually, the sound of the monopole source caused by unsteady

heating is purely due to the unsteady expansion of the fluid at

essentially constant pressure.

• ∴Monopole strength =

yyx

yxyx

3

2

2

2

0/,

4

1, d

cth

tctp

ptt

h

c 2

2

2

2

2

0 1

)(0 Vt

Rate of change in the fractional

rate of volumetric growth

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Sources of Sound

Acoustic source process

Sound generation by the linear creation of matter and

externally applied forces

• The acoustic sources by the mass creation

• The linearized momentum equation with applied forces

• The density would be function of a pressure and the newly created

mass. Suppose that total density volume is consisted of created

mass and rest of the fluid ,

mt

v div0

fm 1

fv

p grad0

)( mtm

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Sources of Sound

Acoustic source process• The density fluctuation with respect to time is

• So,

• The wave equation above these relation is

• It is only the volume occupied by that newly created mass that

drives sound field

02

11

tt

p

cm

ttt

fm

t

m

ttt

p

c

02

2

2

2

2

2

2

1

f div

12

2

0

2

2

2

2

tp

t

p

c

Acceleration term External Forcing term

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Sources of Sound

Acoustic source process

Note

• 1) steady creation of mass is silent

(only accelerating growth makes sound)

• 2) Monopole strength

• 3) dipole strength

The solution of wave equation is

• If the flow field has non-linear effects, sound waves are also

generated.(quadrupole)

y

yx

yxy

yyx

yxyx

3

3

2

2

0

4

,

4

,,

dctf

x

dct

ttp

V

i

i

V

volumeunitt

p2

2

0

f

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Sources of Sound

Lighthill’s acoustic analogy

Sound generation by flow

• Sound – “ Weak motion about a homogenous state of rest ”

• Source of sound by Lighthill (1951)

“ The difference between the exact statement of natural laws and

their acoustical approximations ”

• The exact statement mass conservation

• The exact statement of momentum conservation

• The manipulation of above two equations

0

i

i

vxt

0

jiij

j

i vvpx

vt

ijijijij pp

jiij

ji

vvpxxtt

2

2

2

2

2

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Sources of Sound

Lighthill’s acoustic analogy• Subtract density 2nd order derivatives and Lighthill’s equation is

derived

• Tij is the source and is called “ Lighthill stress tensor ”

• The sound field by the exact equations is

• Tij is of course zero in the sound field if we have linear motion only

• However, Tij cannot be vanished in the turbulent flows

ji

ij

xx

Tc

t

2

22

2

2

ijijjiij cpvvT 2

dVc

ctT

xxt

V

ij

ji

yx

yxyx

2

2

4

,,

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Noise Engineering / Acoustics - 20 -

Sources of Sound

Lighthill’s acoustic analogy

Note

• i) No approximation made so far

• ii) Real material motion ↔ acoustic motion

Quadrupole ↔

• iii) The strength of the quadrupole is Tij

• iv) Double divergence - double tendency for source elements to

cancel

Dipole field - the divergence of a monopole field

Quadrupole field - the divergence of a dipole field

- the double divergence of a monopole field

Monopole

Dopole

Non- linear Linear

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Noise Engineering / Acoustics - 21 -

Sources of Sound

Lighthill’s acoustic analogy

Jet Noise

• Consider a jet flow of velocity U and diameter D

• Characteristic frequency and wavelength is defined by dimensional

coefficient of jet of flow and diameter

1 DMU

Dc

Turbulent eddies of

characteristic speed U

and scale D generate

sound of scale λ=D/M

Sound waves propagate to

(x,t) at speed c

Jet speed U

density ρ0

Tij at (y,t=lx-yl/c)

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Noise Engineering / Acoustics - 22 -

Sources of Sound

Lighthill’s acoustic analogy• For low Mach number (λ≫D) , the source region is compact.

→ the retarded time can be negligible

• The sound field is dimensionalized by belows :

• Together these give the estimate for low Mach number

• Low Mach number jet are acoustically very inefficient, as are all

compact sources

D

M

xi

or 1

3DdV 2

0UTij

2

282

0

24

0 xx

DM

DM

“Eighth power low”

when M<<1

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Sources of Sound

Lighthill’s acoustic analogy• For high Mach number, supersonic jets (λ≪D),

the sound field is dominated by retarded time

Far-field approximation

2

1

xx

xyxyx O

ryx

yryx r

11

as r→∞

dSr = surface

yr=constant

yr

r0

observer at (x,t)

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Sources of Sound

Lighthill’s acoustic analogy• The sound field with far-field approximation is

• In order to bring out the retarded time variations,

• The characteristic magnitude of the density perturbation in the

sound induced by supersonic jets is estimated by setting

• Density and the mean square noise level is

dVT

rcrr

,

4

12

2

2y ijjirr TxxxT

2

ddST

rcrrrr ,

4

12

2

y rtcyyy srs ,,y

1

Dr

2DdS

2

222

0

2

0 r

DM

r

DM

2

0UTij

U

Dd

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Sources of Sound

Lighthill’s acoustic analogy• Actually, there is one other effect to do with the fact that the

number of eddies that can be heard at any time increases with Mach

number which changes this law to the velocity cubed dependence

that is in agreement with the experimental observation

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Sources of Sound

The sound from the stationary foreign bodies

The foreign bodies in the flow

• Evaluate

Through some mathematical procedures, (pp. 163-164, text)

dVc

ctT

xxt

V

ij

ji

yx

yxyx

2

2

4

,,

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Sources of Sound

The sound from the stationary foreign bodies• When there are foreign bodies in the flow it is sometimes

convenient to lump together the quadrupoles distributed over the

interior of foreign bodies.

• First, the derivatives of x with respect to the observer position are

rewritten in terms of the derivatives of y with respect to the source

position

• Tij can be re-arranged as,([] implies that function of retarded time)

• Differentiate with respect to the xj gives

yx

yxy

yxyx

yxy ctT

y

yTctT

x

ij

i

iijij

i

,,

j

j

ji

i

ij

i

vtrr

p

xc

r

vv

yr

T

x

12

j

jjj

ji

ji

ij

ji

vtrxr

p

xxc

r

vv

xyr

T

xx

122

22

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Sources of Sound

The sound from the stationary foreign bodies• The procedure described above shows that

• After using the equation of mass conservation. Hence it follows

that

• Then, integrate over a stationary volume,

j

j

j

j

j

j

j

j

vtrytr

vtry

vytr

vtrx

11

111

2

2

2

j

j

ijji

ji

ij

ji

vtryr

pvv

xyr

T

xx

122

dSrt

dSr

pvv

xdV

r

T

xx SS

ijji

jV

ij

ji

nv2

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Sources of Sound

The sound from the stationary foreign bodies

The result allows Lighthill’s acoustic analogy

Note

• 1) The quadrupole field interior to the body = monopole + dipole

dSrtc

dSr

pvvn

xc

dVr

T

xxctx

S

S

ijji

i

j

V

ij

ji

nv

2

2

2

2

4

1

4

1

4

1, …..quadrupole

…..dipole

…..monopole

dSrt

dSr

pvvn

xdV

r

T

xx

S

S

ijji

i

jV

ij

ji

nv

0

2

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Sources of Sound

The sound from the stationary foreign bodies• 2) If the foreign bodies are motionless and rigid (n∙v=0)

in compact body,

Fi is “Forces applied” to the body => Dipole sources

Body forces are more efficient to generate sound by a factor of

M-2 than low Mach number free turbulence

r

F

xc

i

i

24

1

dSpnF

Sii

1

22

0

D

M

x

DUF

i 2

262

0

2

3

0

r

DM

r

DM

dsr

pn

xc si

i

24

1

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Sources of Sound

The sound from the stationary foreign bodies• 3) monopole source

More effective than “dipole”

r

Q

tc24

1

dSvQ

Sn 0

2

242

0

22

0 and r

DM

r

DM

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Sources of Sound