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Mechanical Systems Laboratory ME 4053
Acoustics Lecture 1
Fall 2010
©2010 Peter Rogers
Georgia Tech
ACOUSTICS
• Generation
• Propagation
• Detection
• Effects
of compressional waves
in gases, liquids and solids
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
SOUND is a WAVE
A disturbance which propagates
Sound is a Disturbance in:
• Presure
• Density
• Velocity (longitudinal)
• Temperature
Pressure Range
10-5 Pa 108 Pa threshhold explosion of hearing
[105 Pa = 1 Atm]
Frequency Range
10-2 Hz 109 Hz gravity wavelength waves comparable
to light
Hearing range ~ 20 Hz 20,000Hz
infrasonic < audible < ultrasonic
1-D Wave Equation DerivationFluid in a Pipe
x x + dx
0
t
ux x
x
puu
xu
t xxx
Conservation of mass
Conservation of momentum
Sound is a disturbance from ambient state
1-D Wave Equation DerivationLinearization
),(),(
),(),(
),(),(
0
0
0
txuutxu
txpptxp
txtx
Axx
A
A
Keep only terms linear in acousticquantities
t
ux x
x
puu
xu
t xxx
x
p
t
u AAx
0
tx
u AAx
0
x
p
t
u AAx
0
tx
u AAx
0
AcousticContinuity Equation
Acoustic Euler Equation
2
22
0 txt
u AAx
2
22
0 x
p
tx
u AAx
t
Subtract 2
2
2
2
x
p
tAA
x
2
2
2
2
x
p
tAA
Need a relationship between pA and ρA
c is the speed of sound
€
pA = c 2ρA
€
1
c 2
∂ 2 pA
∂t 2 =∂ 2 pA
∂x 2
Wave Equation
€
c =p0
ρ0
€
p
p0
=ρ
ρ0
⎛
⎝ ⎜ ⎞
⎠ ⎟
γ
€
p0 + pA
p0
=ρ0 + ρA
ρ0
⎛
⎝ ⎜ ⎞
⎠ ⎟
γ
€
1 +pA
p0
= 1 + γρA
ρ0
+K
Sound propagation is adiabatic
Air is an ideal gas
€
p = ρRT
linearize
€
pA =γ p0
ρ0
ρA
Acoustic equation of state
xLaplace 1816
William Derham (1709)
c= 345.6 m/s
c= 289 m/s
€
1
c 2
∂ 2 pA
∂t 2 =∂ 2 pA
∂x 2
€
pA = c 2ρA
Boyles Law1662
€
p
ρ=
p0
ρ0
€
pA =p0
ρ0
ρA
Newton 1686
€
c =γ p0
ρ0
with0
02
p
c
but
€
p0 = ρ0RT
€
c 2 = γ RT
Wave Equation
2
2
2
2
2
1
x
p
t
p
cAA
AA
pp
0
0
€
c = γ RTsound speed
Proof:
€
pA (x, t) = f (x − ct)
2
2
2
2
2
1
x
p
t
p
cAA
so
Wave Equation
)(),(
)(),(
2
2
ctxfx
txp
ctxfx
txp
)(),(
)(),(
22
2
ctxfct
txp
ctxfct
txp
2
2
2
2
2)(
1
x
pctxf
t
p
cAA
solution
€
1
c 2
∂ 2 pA
∂t 2 =∂ 2 pA
∂x 2
Wave Equation
solution
• is also a solution
€
p(x, t) = g(x + ct)
• f can be any function
• uAx, ρA and TA also satisfy the wave equation
€
pA (x, t) = f (x − ct)
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = 0
x
pf(x)
x
pf(x-ct1)
At t = t1 ct1
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = 0
x
p
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t1
x
p
ct1
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t2
x
p
ct2
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t3
x
p
ct3
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t4
x
p
ct4
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t5
x
p
ct5
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t6
x
p
ct6
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t7
x
p
ct7
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t8
x
p
ct8
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t9
x
p
ct9
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t10
x
p
ct10
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t11
x
p
ct11
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t12
x
p
ct12
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t13
x
p
ct13
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t14
x
p
ct14
€
p(x, t) = f (x − ct)
Why is this a propagating wave?
At t = t15
x
p
ct15
€
p(x, t) = g(x + ct) propagates in the = -x direction
€
p(x, t) = f (x − ct) + g(x + ct)Most General Solution
x
p€
p(x, t) = g(x + ct) propagates in the = -x direction
c c
€
f
€
g
€
p(x, t) = f (x − ct)Propagating Wave:
At t = 0
x
f(x)p
pf(x-ct1)
At t = t1 ct1
x
At x = 0
t
f(-t)p
pf(x1/c-t)At x = x1 x1/c
t
Sinusoidal Waves
TAfge
2cos)(..
€
pA (x, t) = Acos2π
Tt −
x
c ⎛ ⎝
⎞ ⎠
⎛ ⎝ ⎜
⎞ ⎠ ⎟ = Acos 2π f t −
x
c ⎛ ⎝
⎞ ⎠
⎛ ⎝ ⎜
⎞ ⎠ ⎟
x
A
λ
t=0
€
pA (x,0) = Acos2πx
cT ⎛ ⎝
⎞ ⎠
λ is the wavelength (spatial period) λ =cT λ distance propagated in one period
temporal angular frequency
€
pA (x, t) = Acos ωt − kx( )
€
ω =2π f
€
k = ω /c wavenumber (spatial angular freq)
T period (s)
f = 1/T frequency (Hz)(# cycles/sec)
T
A
t
p
x=0
€
pA (0, t) = Acos2πt
T ⎛ ⎝
⎞ ⎠
x=x0
t
x0
c
A
Areas and Applicationsof
Acoustics
1) Noise
· usually something you want to get rid of
· Noise pollution- highways, aircraft, sonic
boom- machinery-OSHA
· submarines- quiet <----> undetectable
2) Audio
· loudspeakers· earphones· microphones· telephones· recording (tape, CD etc)· sirens and alarms
3) Environmental Science• Oceanography Ocean is opaque to EM radiation but transparent to sound
- remote sensing of temperature & current - current meters- ocean acoustic tomography (T & C over
wide areas)- global warming (Heard Island
experiment) - monitoring surface and internal gravity
waves - telemetry (for oceanographic
instrumentation)- tracking of marine animals- acoustic releases
• Geophysics- seismology (compressional waves)- ocean bottom mapping
• Meteorology- atmospheric sounding
4) Physical Science
• chemical and structural relaxation (sound absorption)• chemistry (enhanced reactions)• liquid helium (“second sound” etc)• Brillouin scattering (scattering of sound by light GHz region)• solid state physics (phonons)• photoacoustic spectroscopy•· cavitation •· sonoluminescence (fusion)
5) Naval Applications
(exploit “transparency” of ocean) • undersea surveillance• detection and localization
-active or passive• mine warefare
-detection-activation
• depth sounding - bottom mapping• obstacle avoidance• ice thickness • communication• submarine stealth (TS and noise reduction)• homing devices - torpedo
6) Other Military Applications• artillery localization• intruder detection• land mine localization• treaty verification
7) Architectural• room acoustics• theater, auditoriums etc• sound isolation
8) Bioacoustics (animals & sound ) • animal sonar (dolphins & bats)
-countermeasures• acoustic behavior (bird songs, mating calls,
animal sounds, avoidance reflex etc)• weapon? (whales & dolphins)• communication (whales, elephants)• physiological (how do fish, insects, frogs,
whales hear?)• Effects of noise on animals
9) Hearing and Speech
· physiology
· audiology (quantitative measurement of hearing capacity)
· speech synthesis· speech recognition· speech therapy· hearing aids· effects of noise on hearing and speech· psychoacoustics (psychology of hearing)
10)Music
· instruments · synthesizers· musical theory· reproduction of music
11) Medicine ultrasound therapy
- provide heat - destroy tumors etc
acoustic imaging- scanning- echocardiograms- fetal images- tomography- holography
blood flow (Doppler devices) stethoscopes
-fetal heartbeat•lithotripsy (shock waves break up kidney and bladder stones)•drug delivery• sonic therapy for cystic fibrosis• intraocular pressure•effects of sound and vibration on organs and tissue
12) Resource management• fish finding fish counting fish and manatee “scaring” seismic prospecting oil well logging
13) Microscopy• Better than optical - approach electron microscopes, Biological
-“different” contrasts Integrated circuit inspection
- ability to examine individual layers
14) Commercial
ultrasonic cleaningsonic drills
burglar alarmsinsect repellantsultrasonic atomizers -inkjet printersrange finders for camerasmeasurement of distance and heightAcoustic refrigerators
15) Electrical Engineering (Signal processing )
· SAW devices· acoustic delay lines
16) Quality control ultrasonic flaw detection acoustic emissions
-reactors-bone
“Weevil” detector
17) Miscellaneous
· enhancement of combustion· undersea search (Nessie, Titanic etc.)· acoustic levitation (processing in zero g)· sensors for control systems· tornado detector · acoustic aids for the blind· espionage
Others?
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