Silence is the only homogeneous sound field in unbounded...
Transcript of Silence is the only homogeneous sound field in unbounded...
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
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
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
Noise Engineering / Acoustics - 4 -
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
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
Noise Engineering / Acoustics - 6 -
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,
Noise Engineering / Acoustics - 7 -
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
Noise Engineering / Acoustics - 8 -
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
Noise Engineering / Acoustics - 9 -
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)
Noise Engineering / Acoustics - 10 -
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
Noise Engineering / Acoustics - 11 -
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
Noise Engineering / Acoustics - 12 -
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)
Noise Engineering / Acoustics - 13 -
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
Noise Engineering / Acoustics - 14 -
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
Noise Engineering / Acoustics - 15 -
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
Noise Engineering / Acoustics - 16 -
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
Noise Engineering / Acoustics - 17 -
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
Noise Engineering / Acoustics - 18 -
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
Noise Engineering / Acoustics - 19 -
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
,,
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
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)
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
Noise Engineering / Acoustics - 23 -
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)
Noise Engineering / Acoustics - 24 -
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
Noise Engineering / Acoustics - 25 -
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
Noise Engineering / Acoustics - 26 -
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
,,
Noise Engineering / Acoustics - 27 -
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
Noise Engineering / Acoustics - 28 -
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
Noise Engineering / Acoustics - 29 -
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
Noise Engineering / Acoustics - 30 -
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
Noise Engineering / Acoustics - 31 -
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
Noise Engineering / Acoustics - 32 -
Sources of Sound