AcousticsAcoustics

Dr. Tamer Elnady – Dr. Wael Akl – Dr. Adel Elsabbaghaelsabbagh@gmail.com

Vibration of Continuous Systems and Wave Equation

#6: Wave Equation and Measurement Techniques

Continuous Systems

The Wave Equation in Fluids

Wave Equation in Fluids

Pressure p / p0

Direction of propagation

Wavelength lDirection of propagationRarified Compressed

Equation of Continuity

x

yz

x

t ux

t uxt ux x

y

z

x

Equation of Continuity

x

yz

x

t ux

t uxt ux x

y

z

x

xxxtxxtt zyuzyuzyxt )()()(

xzyux

zyu

zyuzyxt

xxtxxt

xxtt

)(

)()(

0

xtt u

xt

),(),( 0 trtrt

00

xu

tx

Equation of Motion

( ) ( )

x

yz

0p+p [ ]

x0p+p ( ) 0

p+p

y

z

x

y y z z x-

Equation of Motion

zyxxpzyx

xppzypF tt

ttx

)(

zyxxpFx

),(),( 0 trpptrpt

tu

xuu

tu

ttuxu

ttxua xx

xxxxx

x

))()((),( 21

Dm = (r0 + r)DV

00

xp

tux

State EquationMRTpp )()( 00

0

0

0

0

ppp

...21

00

2

22

00

tt t

t

t

tt

pppppp

0

tt

tpp

0p

0

0

tt

tp

The 1-D Wave Equation

02

02

2

txu

tx

00

xu

tx

02

22

0

xp

txux00

xp

tux

02

2

2

2

txp

02

20

2

2

tp

xp

012

2

22

2

tp

cxp

MRTc 0/ 2730 Tc

0p

The 3-D Wave Equation

012

2

22

tp

cp

2

2

2

2

2

22

zp

yp

xpp

Harmonic Solution of the Free 1-D Wave Equation

At a certain x1

)(cosˆ),( cxtptxp

ck /

))((cosˆ)(cosˆ 11 kxTtpkxtp

T/2

xktpkxtp 11 cosˆ)(cosˆ

2,2 kk

fc

At a certain t1

Harmonic Solution of the Free 1-D Wave Equation

p(x1,t)

p(x,t1) Period T

Wavelength l

t

x

p̂

p̂

Specific ImpedancexupZ

dtxpux 1

0

00

xp

tux

cZ 00

)()( ˆˆ),( kxtikxti epeptx

p

)()(

0

ˆˆ1, kxtikxtix ep

iikep

iiktx

u

cZ 00

ms.Pa 41500 cZ

.

For air at 20 C and 1 atmExample

Sound Intensity

),(),(),(x txutxptxI x

T

xx dttxutxpT

xI0

),(),(1)(

cppI x 022 2)ˆˆ(

cp

cpI x

0

2

0

2 ~

~

Sound and Vibration Measurement Techniques

Measurement Chain

Sound or vibration signal T ransducer E lectrical signal

Impedance m atching

A mplification

Signal conditioning

M icrophone

Accelerometer

Am plification

Freq uency

T ime

Signal

Time

D igital

filtering

Fast Fourier

Transform

An alog filter An alog-d igita l co nversion Frequency analysis

Am plitu de

Frequ ency

M easurem ent data processing

G ra ph

Low pass filtering

Example:

Averaging

RM S amplitude determination

A-, B -, C -w eighting

S ignal

1

Transducers

)(V)( tpCtU

•Sensitivity

•Frequency band

•Dynamic range:

Accelerometers

Förspänd fjäder

Rörlig massa

Piezoelektriska element

Bas/monteringsplatta

Vibrerande objektFästskruv

Elektriskutsignal

+_

Electrical Output signal

Pre-loaded spring

Seismic mass

Piezoelectric element

Baseplate / mounting plate

Vibrating objectThreaded stud

+-

F

F

q+-

+-+-+-

Accelerometers

Frequency (log) f f

Useful frequency range

Accelerometer with a large mass

with a small mass Accelerometer

Sensitivity

r r

~f /3 r

Mounting of Accelerometers

Axis of principal sensitivity (100%)

Transverse sensitivity (<4%)

A

B

C

D

Mounting of Accelerometers

Stålskruv Isolerad skruv Tunnt lager vax

Lim Limbricka med skruvtapp

Magnet

Mätspetsar

1 2 3

4 5 6

Threaded Stud Isolated threaded Stud Wax

Threaded-Cemented base Measuring Tips

Microphones

Backplate electrode

Diaphragm

Shield Cavity

Insulator Output terminal

Hål för statisk tryckutjämning

Isolator

Membran

Motelektrod Hylsa

Pressure relief vent

Backplate electrode

Membrane

Casing

Sound Level Meter