Fundamentals of Acoustics - TU Delft OCW · Fundamentals of Acoustics . 14-9-2010 Challenge the future Delft University of Technology Sound is a wave phenomenon •Propagating …

14-9-2010

Challenge the future

Delft University of Technology

Fundamentals of Acoustics

14-9-2010



Sound is a wave phenomenon

• Propagating disturbance

• Longitudinal waves

Transverse waves

Surface waves

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Geo-acoustic modelling of the seafloor P- and S- waves

(sound) (only solids)

A geo-acoustic model of the seafloor comprises the following parameters

- compressional (P-) wave speed

- shear (S-) speed

- compressional wave attenuation

- shear wave attenuation

- density

as a function of depth.

Recall: Slide from previous lecture on “elements marine geology”

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)(),( ctxftx

Propagating disturbance

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Plane harmonic waves

)cos(),( 0 tkxptxp

p0: amplitude

k: wavenumber

: radial frequency

kx- t+: phase

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Period and wavelength of the signal

Tkt

xc

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Typical values

• Audible sound: 20 Hz-20000Hz

• Sound speed in water:

• Sound speed in air:

•

1500 m/s

343 m/s

Frequency-range Sonar

‘0’ - 1000 Hz passive

5 kHz - 10 kHz hull-mounted

40 kHz active torpedo

100 kHz echo-sounder, mine detection

400 kHz mine classification

10 MHz medical imaging

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Sound level

• 2x10-5 Pa (hearing threshold) – 100 Pa (pain threshold)

(1 atm = 101325 Pascals= 1.01325 bar)

• dB

with p the effective pressure prms

ref

10 log20p

pL

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)(),( ctxftx

Propagating disturbance

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Wave equation for plane waves

• The particle displacement

• The density

• The pressure P

• The particle speed

• The particle acceleration

tv

2

2

ta

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Variation of displacement with x:

change in density

0x

(x+x+(x+x,t)

-x-(x,t))

x

01

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)(PP

0

100

'

10100 )()()(

d

dPPfffpP

2

1cp 0

2

d

dPcwith

Variation of density:

change in pressure

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P(x t, )

P x x t( , )

x

x x

x

txPxtxxPtxP

t

txx

),(),(),(

),(2

2

0

Inequalities in pressure:

movement of the medium

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Wave equation

x

01

1

0

d

dPp

x

txP

t

tx

),(),(2

2

0 2

22

2

2

x

pc

t

p

0

2

d

dPc

2

22

2

2

xc

t

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Example

• Harmonic plane wave

• f = 1000 Hz

• Sound pressure level = 100 dB

• Undisturbed density = 1.21 kg/m3, c = 343 m/s

refrms pp 20

100

10 Pa210210 55

m34.01000

343

f

c

10-6 m

s

m105 3 rmsv

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Sound speed for ideal gas

0

2

d

dPc

constantP use

Pd

dPc

1

)0(

2 constant

RTPV

M

RT

V

RTc

s

m332

108.28

27331.84.13

c

air

For adiabatic processes

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Sound speed for liquids and solids

0

Kc

Pure water : K = 2x109 N/m2, 0 = 1000 kg/m3 : c = 1414 m/s

K is the bulk modulus=reciprocal of the compressibility

Empirical relation:

zSTTTTc 017.0)35)(01.034.1(00029.0055.06.42.1449 32

with

T the temperature in C

z the depth in m

S the salinity (salt content) in ppt

0

Ec

Steel E = 2x1011 N/m2 and 0 = 7800 kg/m3 : c = 5100 m/s.

E is Young’s elasticity modulus

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Acoustic intensity

Definition:

mean rate of flow of energy through a unit area normal

to the direction of the propagation

IF

A tpv

Harmonic wave: cv

p0

c

p

c

pI rms

0

2

0

2

ACOUSTIC IMPEDANCE

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Spherical waves

2

2

22

2

2

2

2

2 1

t

p

cz

p

y

p

x

p

2

2

22

2

2222

2 1

sin

1)(sin

sin

1)(

1

t

p

c

p

r

p

rrp

rr

2 2

2 2 2

1 1( )

prp

r r c t

or:

2

2

22

2 )(1)(

t

rp

crp

r

For a spherical wave, the pressure is independent of and :

The wave equation

The wave equation in spherical coordinates

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Spherical waves, continued

)()(),( ctrgctrftrpr

)(),( ctrfr

Atrp

)(),( tkrier

Atrp

Harmonic solution

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Acoustic impedance, spherical harmonic waves

x

p

t

2

2

0

r

p

t

v

0

vit

v

r

p

iv

0

1

k

r

itrptrv

0

),(),(

2222

22

011),(

),(

rk

kri

rk

rkc

trv

trpZ

with )(),( tkrie

r

Atrp

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2222

22

011),(

),(

rk

kri

rk

rkc

trv

trpZ

kr >> 1 (r >> ): Z = 0c

kr << 1 (r << ): Z = -i0ckr

Acoustic impedance, spherical harmonic waves

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Spherical wave intensity

IF

A tpv

kr >> 1 (r >> ): Z = 0c

Harmonic wave: c

v

p0 c

p

c

pI rms

0

2

0

2

2

22

02

22 r

Apprms

2

1~I

r

In accordance with the conservation of energy law!

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Cylindrical waves

c

prI

0

2

0

2)(

c

prHP o

0

2

22

0

1~p

r

)(),( tkrier

Atrp

Harmonic cylindrical wave:

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Geometrical loss

)(),( tkxieAtxp

)(),( tkrier

Atrp

)(),( tkrier

Atrp

Plane wave

Spherical wave

Cylindrical wave

c

AI

0

2

2

2

0

2

2 cr

AI

cr

AI

0

2

2

Up to now only geometrical loss

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Sound absorption

Plane wave

xtkxietrp )(),(

xexe x 686.8)log20(log20 1010

x = 1 km = 1000 m: (in dB/km) = 8686 (m-1)

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Absorption coefficient

2

2

2

2

2

0003.04100

44

1

11.0f

f

f

f

f

Thorpe:

f = frequency (kHz)

= absorption coefficient (dB/km)

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Absorption coefficient

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Sound absorption

Frequency (kHz) (dB/km) (Thorpe) r10dB (km)

0.1 0.0012 8333

1 0.07 143

10 1.2 8.3

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Propagation loss

)(

)1(log10

)(

)1(log10

)(

)1(log10

2

0

2

010

2

2

1010

rp

p

rp

p

rI

IPL

rms

rms

•Geometrical spreading loss

•Absorption 0 ~

rep

r

)log(20)log(20)log(20 101010 errerPL r

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Propagation loss

)km()dB/km()km(log2060)( 10 rrdBPL

)km()dB/km()km(log1030)( 10 rrdBPL

spherical

cylindrical

Environment?

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Example: Losses

f = 6 kHz

r (km) PL (dB)

0.1 60-20+0.05 = 40.05

1 60+ 0+0.5 = 60.5

10 60+20+5 = 85

100 60+40+50 = 150

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).(),(),,,( trkietrptzyxp

with zkykxkrk zyx

.

)sincos( 111 zxik

i ep

)sincos( 111 zxik

r eRp

)sincos( 222 zxik

t eTp

1

1

1 kc

k

2

2

2 kc

k

Reflection, refraction and

transmission

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Boundary conditions

(I) continuity of pressure tri ppp

Z = 0:

xkkieTR

)coscos( 1122)1(

0coscos 1122 kk

1

1

2

2 coscos

cc

Snell’s law:

TR )1(

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Boundary conditions

(II) continuity of the normal component of the particle velocity

z

p

iz

pp

i

tri

21

1)(1

Z = 0:

122

211

sin

sin1

c

cTR

12

12

ZZ

ZZR

1

111

sin

cZ

12

22

ZZ

ZT

2

222

sin

cZ

2

0 02

v p

t t x

vi v

t

Recall:

0

1 pv

i x

Giving:

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Typical values for geo-acoustic

parameters

Bottom

type porosity (%) (g/cm3) c (m/s)

clay 80 1.2 1470

Clayey silt 70 1.5 1515

Fine sand 45 1.95 1725

siltstone - 2.4 3800

basalt - 2.5 4800

Important note: for ocean bottom materials a good

approximation is that the absorption coefficient increases

linearly with frequency

)()(686.8)/( 1 mmdB : independent on frequency!

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Example: 2 = 1.9 g/cm3, = 0 dB/, c2 = 1650 m/s

1800 m/s 0.5 dB/

2.7 g/cm3

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Example

2 = 1.5 g/cm3, = 0 dB/, c2 = 1450 m/s

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Critical angle: total reflection

If c2 > c1 :

1R for 0 c

R < 1 c . for

2

1arccosc

cc

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Angle of intromission

R = 0

1

1

arctan2

11

22

2

1

2

0

c

c

c

c

0 exists if

121122 and cccc muddy ocean bottoms;

121122 and cccc never occurs, is not physical.

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General wave phenomena

• Diffraction

• Reflection

• Scattering

• Refraction

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Exercises


• f = 1000 Hz



Calculate the particle displacement

Plane harmonic waves, frequency of 1000 Hz, in air and in water.

Intensities are identical.

Sound pressure level of the wave in air = 120 dB.

Calculate the sound pressure level in water.

1

2

Calculate the intensity at the pain threshold in air 3

Consider an echosounder at 100 kHz, 4 km of water depth. What is the

loss expected due to absorption? 4

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Exercise 1


• F = 1000 Hz



Pa10210 520

0

refrefrms ppp

Estimate the particle displacement

m101000234321.1

102 115

0

c

prmsrms

Ten times smaller than the radius of an atom!

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Exercise 2

water

rms

air

rms

c

p

c

p

0

2

0

2

airrmsairrmswaterrms ppp ,,, 6034321.1

15001000

Pa2010log20120 ,

6

,

10

airrefairrms

airref

rms ppp

p

Plane harmonic wave, frequency of 1000 Hz, in air and in water.

Intensities are identical.

Sound pressure level of the wave in air = 120 dB.

Calculate the sound pressure level in water.

Pap waterrms 1200,

dB18210

1200log20

6

10

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Calculate the intensity at the pain threshold in air

2

2

0

2

m

Watt24

34321.1

100

c

p

Exercise 3

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At 100 kHz: absorption coefficient of 35 dB/km

Total loss due to absorption: 4*2*35 = 280 dB

Exercise 4

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Fundamentals of Acoustics - TU Delft OCW · Fundamentals of Acoustics . 14-9-2010 Challenge the future Delft University of Technology Sound is a wave phenomenon •Propagating …