acoustic wave equation. acoustic variables pressure density – condensation velocity (particle)...
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Acoustic Wave Equation
Acoustic Variables
• Pressure
• Density – Condensation
• Velocity (particle)
• Temperature
op P -Po
o
s
ut
Sound Speed
Bulk modulus Bc
density
Air Water SteelBulk Modulus 1.4(1.01 x 105) Pa 2.2 x 109 Pa ~2.5 x 1011 Pa
Density 1.21 kg/m3 1000 kg/m3 ~104 kg/m3
Speed 343 m/s 1500 m/s 5000 m/s
Please Memorize!!!
Necessary Differential Equations to Obtain a Wave Equation
• Mass Continuity• Equation of State• Force Equation – N2L
Assumptions: homogeneous, isotropic, ideal fluid
Equations of State
Ideal Gasses: k= rTP
o
o
PP
o
o
PP
Real Fluids: o o
22
o o2
1...
2
o
P PP=P
Vp B B Bs
V
o
oB
P
Continuity Equation
u 0t
x xu x x dx
u
dxdy
dzyx z
dmdm dmdM
dt dt dt dt
y zx xx x dx
dm dmdMu dydz u dydz
dt dt dt
yx zx xx x
x
dmu dmdMu dydz u dx dydz
dt x dt dt
yx z
x zy
uu udMdxdydz dxdydz dxdydz
dt x y z
yx z
x zy
uu ud
dt x y z
Force Equation
xP x dx
P
dxdy
dz x x x dxdf dydz dydz
P P
x x xx x
df dydz dx dydz dxdydzx x
P PP P
x zy
ˆ ˆ ˆdf i j k dxdydz dmg dxdydz dmgx y z
P P PP
Fluid Accelerationu
dxdydz g dxdydz dxdydzt
P
u x, y, z, t u x dx, y dy, z dz, t dt
t 0
u x dx, y dy, z dz, t dt u x, y, z, ta lim
t
u u u uu x dx, y dy, z dz, t dt u x, y, z, t dx dy dz dt
x y z t
x y z
u u u uu x, y, z, t u dt u dt u dt dt
x y z t
x y z x y z
u u u u u u ua u u u u u u u u u
x y z t t x y z t t
Lagrangian and Eulerian Variables
• Eulerian – Fixed Moorings• Lagrangian – Drifting Buoys
Du ua u u
Dt t
Material, substantial or Lagrangian Derivative
Eulerian Derivative
ConvectiveTerm
Newton’s Second Law
udxdydz g dxdydz dxdydz u u
t
P
ug u u
t
P
u 0t
p Bso
oB
P
Linear Continuity Equation
o p P P
o os
o oo o
su s u 0
t t
oo
su 0
t
su 0
t
Linear Force Equation
ug u u
t
P
o p P P
o os
o o o o o
up g s p gs
t
P
o
up
t
o
up
t
su 0
t
2o
up
t
2
2
s u0
t t
2 22 o
o 2 2
s pp
t B t
p Bs
22
2 2
1 pp
c t
2
o
Bc
Linear Wave Equation
Velocity Potential
u
o o
up
t t
o op p 0t t
o p 0t
Variation of sound speed with temperature
o
2o oB c
o
PP 2
o
c
oP
k o ko= rT = rT oP P
2 o koo ko
o
rTc rT
2
kc rT
2C Ck k
2o ko
273 T Tc T T1
c T 273 273 273
Co
Tc c 1
273
Speed of sound in water-temperature, pressure, and salinity
2 2 4 3 2 2c t, z,S 1449.2 4.6t 5.5x10 t 2.9x10 t 1.34 10 t S 35 1.6x10 z
with the following limits:
0 t 35 C
0 S 45 p.s.u.
0 z 1000 meters
Sound Speed Variations with Temperature and Salinity (z = 0 m)
13801400142014401460148015001520154015601580
0 5 10 15 20 25 30 35 40
Temperature (C)
So
un
d S
pe
ed
(m
/s)
0
30
35
40
ppt salinity
Class Sound Speed Data
Class Sound Speed in Water Data
y = 0.0004x3 - 0.0807x2 + 6.2061x + 1393.4
1400
1420
1440
1460
1480
1500
1520
0 5 10 15 20 25
Temp (C)
So
un
d S
pee
d (
m/s
)
Series1
Poly. (Series1)
Harmonic 1-D Plane Waves2
22 2
1 pp
c t
j t kx j t kxp Ae Be
o
up
t
j t kx j t kx j t kx j t kxo
uAe Be Ake Bke
t
j t kx j t kx
o
1 uAke Bke
t
j t kx j t kx j t kx j t kx
o o
1 k k 1u A e B e Ae Be
c
Condensation and Velocity Potential
2
p ps
B c
j t kx j t kx
o
1u Ae Be
c
j t kx j t kx
o o o
1 1 1 p jpA e B e
c jk jk j ck
Specific Acoustic Impedancef
Zu
Mechanical Impedance
Z f / A pz
A u u
Pa s
raylm
For a plane wave:
j t kx
oj t kx
o
j t kx
oj t kx
o
Aec
1Ae
cpz
u Bec
1Be
c
In general:r specific acoustic resistance
z r jxx specific acoustic reactance
Sound Speed
Bulk modulus Bc
density
Air Water SteelBulk Modulus 1.4(1.01 x 105) Pa 2.2 x 109 Pa ~2.5 x 1011 Pa
Density 1.21 kg/m3 1000 kg/m3 ~104 kg/m3
Speed 343 m/s 1500 m/s 5000 m/s
Spec. Ac. Imp. 415 Pa-s/m 1.5 x 106 Pa-s/m 5 x 107 Pa-s/m
z cAnalogous to E-M wave impedance
Z
Plane wave in an arbitrary direction
x y zj t k x k y k zp Ae
2
22 2
1 pp
c t
x y z
x y z
j t k x k y k z2
j t k x k y k z22 2
Ae1Ae
c t
x y z x y z
2j t k x k y k z j t k x k y k z2 2 2
x y z 2k k k Ae Ae
c
22 2 2x y z 2
k k kc
Shorthandx y zˆ ˆ ˆk k i k j k k
ˆ ˆ ˆr xi yj zk
x y z j t k rj t k x k y k zp Ae Ae
x
y
z
xkcos
k
ykcos
k
zkcos
k
k
Direction Cosines
k r const
k r k Surfaces (planes) of constant phase
Propagation Vector
2 2 2 2 2 2x y zk k k k k cos cos cos
-1
-0.5
0
0.5
1 -1
-0.5
0
0.5
1
-2
-1
0
1
2
-1
-0.5
0
0.5
1
-1
-0.50
0.51
-1-0.500.51
0
5
10
-1
-0.50
0.51
-1-0.500.51
k in x-y plane
x
z
ky x n2
k
k
ˆ ˆk k cos i k sin j
j t kx cos kysinp Ae
Energy
2 2k o o
1 1E mu V u
2 2
0 0
x V
p
x V
E Fdx pdV
o
2o
pB c
P
V const. dV Vd 0
2
d dpdV V V
c
0
pV 2
p o2 2oV 0
dp 1 pE pdV pV V
c 2 c
Energy Density2 2
2 2k p o o o o o2 2 2
o o
1 1 p 1 pE E E u V V u V
2 2 c 2 c
2 22 2
i o o2 2 2o o o
E 1 p pu u
V 2 c c
E
p P cos t kx o
Pu cos t kx U cos t kx
c
2 2T T 2
i i 2 2To o0 0
P cos t kx1 1 Pdt dt
T T c 2 c
E E E
2o o
o
U U P PU
2 2 c 2c
E
Average Power and Intensity
A
cdt
iT TdE Acdt E
TT
dEAc
dt E
2 2T
oTo
U 1 P 1I c c PU
A 2 2 c 2
E
For plane waves
Effective Average - RMS
T
2e rms
0
1F F f t dt
T
e
UU
2 e
PP
2
22 e
o e e eTo
PI cU P U
c
Intensity of sound• Loudness – intensity of the wave. Energy
transported by a wave per unit time across a unit area perpendicular to the energy flow.
Source Intensity (W/m2) Sound Level
Jet Plane 100 140
Pain Threshold 1 120
Siren 1x10-2 100
Busy Traffic 1x10-5 70
Conversation 3x10-6 65
Whisper 1x10-10 20
Rustle of leaves 1x10-11 10
Hearing Threshold 1x10-12 1
2 2
o o
p PI
c 2 c
Sound Level - Decibel
ref
IIL 10log
I
12
ref 2
WI 1x10
m
• Ears judge loudness on a logarithmic vice
linear scale
• Alexander Graham Bell
• deci =
• 1 bel = 10 decibel
Why the decibel?
1
10
ref
I"bel" log
I
ref
IIL(in dB) 10log
I
Reference Level Conventions
LocationReference Intensity
Reference Pressure
Air 1 x 10-12 W/m2 20 Pa
Water 6.67 x 10-19 W/m2 1 uPa
2ref
refo
pI =
c
Historical Reference
• 1 microbar• 1 bar = 1 x 105 Pa• 1 bar = 1 x 105 Pa
• So to convert from intensity levels referenced to 1 bar to intensity levels referenced to 1 Pa, simply add 100 dB
510 Pa20log 100 dB
1 Pa
Sound Pressure Level
ref
IIL 10log
I
2
rms
ref ref
p pSPL 20log 20log
p p
Mean Squared Quantities:Power, Energy, Intensity
Root Mean Squared Quantities:Voltage, Current, Pressure
“Intensity Level”
“Sound Pressure Level”
Example
• Tube with a piston driver– a=2.5 cm– f = 1 kHz– 154 dB in air
• What are the– rms piston displacement– intensity– power
Spherical Waves2
22 2
1 pp
c t
2 22 2
2 2 2 2 2 2 2
1 p 1 p 1 p 1 pp r sin
r r r r sin r sin c t
n m jm jwt
nn
j krp P cos e e
y kr
Standing waven=0,1,2,3,…m=-n,…,+n
(1)n m jm jwt
n(2)n
h krp Y cos e e
h kr
(1),(2)n n nh j jy
Traveling wave
Spherical Waves For Usjkr
(1),(2)n
eh
r
j t krAp e
r
o
1 1 jkru p
c jkr
2 2j
o 2 2
k r jkrz c ze
1 k r
o 2 2
krz c
1 k r
1
tankr
z
acr
acx
z
acr
acx