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14-9-2010
Challenge the future
Delft University of Technology
Fundamentals of Acoustics
14-9-2010
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Delft University of Technology
Sound is a wave phenomenon
• Propagating disturbance
• Longitudinal waves
Transverse waves
Surface waves
14-9-2010
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Delft University of Technology
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”
14-9-2010
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Delft University of Technology
)(),( ctxftx
Propagating disturbance
14-9-2010
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Delft University of Technology
Plane harmonic waves
)cos(),( 0 tkxptxp
p0: amplitude
k: wavenumber
: radial frequency
kx- t+: phase
14-9-2010
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Delft University of Technology
Period and wavelength of the signal
Tkt
xc
14-9-2010
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Delft University of Technology
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
14-9-2010
Challenge the future
Delft University of Technology
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
14-9-2010
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Delft University of Technology
)(),( ctxftx
Propagating disturbance
14-9-2010
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Delft University of Technology
Wave equation for plane waves
• The particle displacement
• The density
• The pressure P
• The particle speed
• The particle acceleration
tv
2
2
ta
14-9-2010
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Delft University of Technology
Variation of displacement with x:
change in density
0x
(x+x+(x+x,t)
-x-(x,t))
x
01
14-9-2010
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Delft University of Technology
)(PP
0
100
'
10100 )()()(
d
dPPfffpP
2
1cp 0
2
d
dPcwith
Variation of density:
change in pressure
14-9-2010
Challenge the future
Delft University of Technology
P(x t, )
P x x t( , )
x
x x
x
txPxtxxPtxP
t
txx
),(),(),(
),(2
2
0
Inequalities in pressure:
movement of the medium
14-9-2010
Challenge the future
Delft University of Technology
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
14-9-2010
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Delft University of Technology
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
14-9-2010
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Delft University of Technology
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
14-9-2010
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Delft University of Technology
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
14-9-2010
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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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Delft University of Technology
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
14-9-2010
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Delft University of Technology
Spherical waves, continued
)()(),( ctrgctrftrpr
)(),( ctrfr
Atrp
)(),( tkrier
Atrp
Harmonic solution
14-9-2010
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Delft University of Technology
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
14-9-2010
Challenge the future
Delft University of Technology
2222
22
011),(
),(
rk
kri
rk
rkc
trv
trpZ
kr >> 1 (r >> ): Z = 0c
kr << 1 (r << ): Z = -i0ckr
Acoustic impedance, spherical harmonic waves
14-9-2010
Challenge the future
Delft University of Technology
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!
14-9-2010
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Delft University of Technology
Cylindrical waves
c
prI
0
2
0
2)(
c
prHP o
0
2
22
0
1~p
r
)(),( tkrier
Atrp
Harmonic cylindrical wave:
14-9-2010
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Delft University of Technology
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
14-9-2010
Challenge the future
Delft University of Technology
Sound absorption
Plane wave
xtkxietrp )(),(
xexe x 686.8)log20(log20 1010
x = 1 km = 1000 m: (in dB/km) = 8686 (m-1)
14-9-2010
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Delft University of Technology
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)
14-9-2010
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Delft University of Technology
Absorption coefficient
14-9-2010
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Delft University of Technology
Sound absorption
Frequency (kHz) (dB/km) (Thorpe) r10dB (km)
0.1 0.0012 8333
1 0.07 143
10 1.2 8.3
14-9-2010
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Delft University of Technology
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
14-9-2010
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Delft University of Technology
Propagation loss
)km()dB/km()km(log2060)( 10 rrdBPL
)km()dB/km()km(log1030)( 10 rrdBPL
spherical
cylindrical
Environment?
14-9-2010
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Delft University of Technology
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
14-9-2010
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Delft University of Technology
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(
14-9-2010
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Delft University of Technology
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:
14-9-2010
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Delft University of Technology
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!
14-9-2010
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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
14-9-2010
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Delft University of Technology
Critical angle: total reflection
If c2 > c1 :
1R for 0 c
R < 1 c . for
2
1arccosc
cc
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Delft University of Technology
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.
14-9-2010
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Delft University of Technology
General wave phenomena
• Diffraction
• Reflection
• Scattering
• Refraction
14-9-2010
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Delft University of Technology
Exercises
• Harmonic plane wave
• f = 1000 Hz
• Sound pressure level = 0 dB
• Undisturbed density = 1.21 kg/m3, c = 343 m/s
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
14-9-2010
Challenge the future
Delft University of Technology
Exercise 1
• Harmonic plane wave
• F = 1000 Hz
• Sound pressure level = 0 dB
• Undisturbed density = 1.21 kg/m3, c = 343 m/s
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!
14-9-2010
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Delft University of Technology
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
14-9-2010
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Delft University of Technology
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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