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?