UIUC Physics 406 Acoustical Physics of Music
Time Domain vs. Frequency Domain Sound Field Quantities
For “everyday” sound pressure levels SPL r 134 dB in air (at NTP), the purely real,
instantaneous sound field S r , t at a listener point r in space is uniquely/fully specified iff
(if and only if) two physical measurements are simultaneously made at the listener’s point r :
the purely real, instantaneous acoustic over-pressure p r , t {n.b. a scalar quantity} and the
purely real, instantaneous 3-D particle velocity u r , t {n.b. a vector quantity}. Both of these
instantaneous quantities are also known as time-domain quantities.
Note that simultaneous measurement of these two physical quantities {alone} at the listener
point r is not sufficient to enable extrapolation of knowledge of the instantaneous sound field
S r , t to other space points r and/or other times, t . The gradient of scalar p r , t – and the
divergence and curl of 3-D vector u r , t – must also be known/measured…
Instantaneous vs. Complex Pressure and 3-D Particle Velocity:
For a monochromatic harmonic sound field – associated with a single frequency 2 f ,
the physical, purely real, instantaneous acoustic over-pressure at the space point r is:
p r , t po r , cos t p r , {SI units = Pascals}, the physical, purely real,
instantaneous 3-D particle velocity {SI units = m/sec} at the space point r is:
u r , t u x r , t xˆ u y r , t yˆ u z r , t zˆ
uox r , cos t ux r , xˆ
uoy r , cos t u y r , yˆ
uoz r , cos t uz r , xˆ
Both the physical, purely real instantaneous acoustic over-pressure and instantaneous
particle velocity are manifestly what we refer to as a time-domain quantities.
Using the trigonometric relation: cos A B cos A cos B sin A sin B we can rewrite these
two physical, purely real instantaneous time-domain quantities as:
p r , t po r , cos t p r , po r , cos t cos p r , sin t sin p r ,
uk r , t uok r , cos t uk r , uok r , cos t cos uk r , sin t sin uk r ,
where k x, y, z . Note that a “generic” physical, purely real instantaneous monochromatic
harmonically-varying/time-dependent signal V t can similarly be written as:
V t Vo cos t V Vo cos t cos V sin t sin V
-1-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
A dual-channel lock-in amplifier {such as the Stanford Research Systems SRS-830 that we
routinely use in the UIUC P406 POM lab} measures the in-phase and 90 out-of-phase (aka
quadrature) amplitude components Vx Vo cos V and: Vy Vo sin V ,
respectively, of whatever “generic” purely real, instantaneous monochromatic harmonic signal
V t Vo cos t V is input to the dual-channel lock-in amplifier.
However, order to obtain such measurements, the dual-channel lock-in amplifier must be
phase-referenced relative to a periodic/harmonic reference signal {of the same frequency }
– e.g. a sine-wave signal output from a driving function generator. Detailed information on how
a dual-channel lock-in amplifier works is discussed in P406 POM Lect. Notes XIII – Part 2.
We can thus form the complex “generic” amplitude (aka phasor): V Vx iVy where:
Vx Re V Vo cos V and: Vy Im V Vo sin V .
Using the Euler relation: ei cos i sin , we see that we can also equivalently write the
complex “generic” amplitude as:
i
i
V Vx iVy Vo cos V i sin V Vo e V V e V .
Note further that the complex “generic” amplitude V Vo eiV is what we refer to as
a frequency domain quantity – i.e. a dual-channel lock-in amplifier manifestly/intrinsically carries
out measurements of complex “generic” amplitudes in the frequency domain.
Thus, we can similarly construct complex time-domain acoustic over-pressure and complex
time-domain acoustic 3-D particle velocity as follows:
p r , t po r , cos t p r , i sin t p r ,
i t p r ,
i r ,
po r , e
po r , e p
eit p o r , eit
p o r ,
and:
u r , t u x r , t xˆ u y r , t yˆ u z r , t zˆ
uox r , cos t ux r , i sin t ux r , xˆ
uoy r , cos t u y r , i sin t u y r , yˆ
uoz r , cos t uz r , i sin t uz r , zˆ
i t u y r ,
i t u x r ,
i t u z r ,
uox r , e
xˆ uoy r , e
yˆ uoz r , e
zˆ
iu r ,
i r ,
i r ,
eit zˆ
uox r , e u x
eit xˆ uoy r , e y
eit yˆ uoz r , e uz
uox r ,
uo y r ,
-2-
uoz r ,
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Thus:
iu r ,
i r ,
i r ,
u r , t uox r , e ux
xˆ uoy r , e y
yˆ uoz r , e uz
zˆ eit
uox r , xˆ uoy r , yˆ uoz r , zˆ eit uo r , eit
The complex scalar quantity:
i r ,
p o r , po r , e p
po r , cos p r , i sin p r ,
Re p o r , i Im p o r ,
is known as the frequency domain complex over-pressure amplitude. We can experimentally
measure complex p o r , e.g. using a dual-channel lock-in amplifier.
Likewise, e.g. in Cartesian coordinates, the complex 3-D vector quantity:
iu r ,
i r ,
i r ,
uo r , uox r , e ux
xˆ uoy r , e y
yˆ uoz r , e uz
zˆ
uox r , cos ux r , i sin ux r , xˆ Re uox r , i Im uox r ,
Re uoy r , i Im uoy r ,
uoy r , cos u y r , i sin u y r , yˆ
uoz r , cos uz r , i sin uz r , zˆ
Re uoz r , i Im uoz r ,
xˆ
yˆ
zˆ
is the frequency domain complex 3-D particle velocity amplitude. We can experimentally
measure complex 3-D vector uo r , e.g. using three dual-channel lock-in amplifiers.
The relationship between time-domain vs. frequency-domain complex acoustic quantities
such as complex over-pressure and complex particle velocity, associated with a single-frequency/
monochromatic/harmonic sound field are:
p r , t p o r , eit and: u r , t uo r , eit
We thus see that the frequency-domain complex acoustic quantities p o r , and uo r ,
physically represent the complex amplitudes of these acoustic quantities:
i r ,
p o r , p or r , ip oi r , p o r , e p
and:
i r ,
i r ,
i r ,
uo r , uor r , iuoi r , uox r , e ux xˆ uoy r , e u y
yˆ uoz r , e uz
zˆ
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Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Instantaneous Physical vs. Complex 3-D Vector Acoustic Intensity:
The instantaneous physical 3-D acoustic intensity (aka sound intensity) {n.b. a purely real 3-D
vector quantity} is a time-domain quantity, defined as the product of the instantaneous physical
scalar over-pressure p r , t and the instantaneous 3-D vector particle velocity u r , t :
I a r , t p r , t u r , t p r , t u x r , t xˆ u y r , t yˆ u z r , t zˆ
p r , t u x r , t xˆ p r , t u y r , t yˆ p r , t u z r , t zˆ
I ax r , t xˆ I a y r , t yˆ I az r , t zˆ
Since p r , t has SI units of Pascals (= Newtons/m2) and u r , t has SI units of m/s,
we see that I a r , t has SI units of (Newton-m)/m2-s = (Joules)/m2-s = (Joules/s)/m2 = Watts/m2.
Since the purely real instantaneous time-domain over-pressure and 3-D particle velocity are:
p r ,t
ux r , t
uy r , t
uz r , t
po r , cos t p r ,
uox r , cos t ux r ,
uoy r , cos t u y r ,
uoz r , cos t uz r ,
po r , cos t cos p r , sin t sin p r ,
uox r , cos t cos ux r , sin t sin ux r ,
uoy r , cos t cos u y r , sin t sin u y r ,
uoz r , cos t cos uz r , sin t sin uz r ,
Then {temporarily suppressing the arguments r , t and r , for notational clarity}, the
k x, y, z component of the purely real instantaneous time-domain 3-D acoustic intensity is:
I ak po cos t p uok cos t u
po uk cos t p cos t u
po uok cos t cos p sin t sin p cos t cos uk sin t sin uk
po uok cos 2 t cos p cos uk sin 2 t sin p sin uk sin t cos t sin p cos uk cos p sin uk
Now: sin 2 t 1 cos 2 t and thus for the first two terms on the RHS:
cos 2 t cos p cos uk sin 2 t sin p sin uk cos 2 t cos p cos uk 1 cos 2 t sin p sin uk
cos 2 t cos p cos uk sin p sin uk sin p sin uk cos 2 t cos p uk sin p sin uk
But: sin p cos uk cos p sin uk sin p uk , and: sin t cos t 12 sin 2t and then, using:
sin A sin B 12 cos A B cos A B then: sin p sin uk 12 cos p uk cos p uk
2
1
1
1
Additionally: cos t 2 2 cos 2t 2 1 cos 2t , thus we have:
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Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
I ak 12 po uok 1 cos 2t cos p uk cos p uk cos p uk sin 2t sin p uk
12 po uok cos 2t cos p uk cos p uk sin 2t sin p uk
12 po uok cos 2t cos p uk sin 2t sin p uk cos p uk
Again using: cos A B cos A cos B sin A sin B , the above term in the curly brackets can be
rewritten as: cos 2t cos p uk sin 2t sin p uk cos 2t p uk
Thus:
I ak 12 po uok cos 2t p uk cos p uk
Next, we define: p uk p uk
Thus:
and then:
p
I ak 12 po uok cos 2t 2 p p uk cos p uk
12 po uok cos 2t 2 p p uk cos p uk
12 po uok cos 2 t p p uk cos p uk
Then using: cos A B cos A cos B sin A sin B :
cos 2 t p p uk cos 2 t p cos p uk sin 2 t p sin p uk
Hence:
I ak 12 po uok cos 2 t p cos p uk sin 2 t p sin p uk cos p uk
Now, note that:
Re e
i puk
1 e 2it p cos 2 t cos
p
p uk sin 2 t p sin p uk cos p uk
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uk 2 p uk p 2 p p uk .
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Proof:
e
i puk
1 e 2it p Re ei puk
1 e 2it p i Im ei puk
1 e 2it p
cos p uk i sin p u 1 cos 2 t p i sin 2 t p
i sin
cos p uk cos 2 t p cos p uk sin 2 t p sin p uk
p uk
cos 2 t p sin p uk sin 2 t p cos p uk
i.e.
i p uk
1 e 2it p cos
p uk cos 2 t p cos p uk sin 2 t p sin p uk
i p uk
1 e 2it p sin
p uk cos 2 t p sin p uk sin 2 t p cos p uk
Re e
and:
Im e
Thus, we can equivalently write the purely real, instantaneous time-domain 3-D vector sound
intensity as:
2 i t p
i
I ak 12 po uok Re e puk 1 e
Explicitly reinstating the arguments r , t and r , , this is:
2 i t p r ,
i
r ,
I ak r , t 12 po r , uok r , Re e puk
1 e
i r , uk r ,
i r ,
i
i r ,
r ,
Since: p uk r , p r , uk r , , then: e puk
e p
e p
e uk
,
and thus we see that since:
i r ,
p r , t po r , e p
eit p r , eit
and:
hence:
i r ,
{where: p r , po r , e p
}
i r ,
i r ,
uk r , t uok r , e uk
eit uk r , eit {where: uk r , uok r , e uk
}
i r ,
i r ,
uk* r , t uok r , e uk
e it uk* r , e it {where: uk* r , uok r , e uk
}
the purely real, instantaneous time-domain 3-D vector sound intensity can also equivalently be
written as:
2 i t p r ,
I ak r , t Re 12 p r , uk* r , 1 e
-6-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Thus, we can now define the frequency-domain complex “amplitude” of the 3-D vector sound
intensity as:
Ia r , 12 p r , u * r ,
12 p r , u *x r , xˆ 12 p r , u *y r , yˆ 12 p r , u *z r , zˆ
Iax r , xˆ Ia y r , yˆ Iaz r , zˆ
Finally, we can thus explicitly show that the purely real, instantaneous time-domain 3-D
vector sound intensity I a r , t is related to the frequency-domain complex 3-D vector sound
intensity “amplitude” I a r , by:
2 i t p r ,
I a r , t Re Ia r , 1 e
Hence, the complex time-domain instantaneous 3-D vector sound intensity is thus:
2 i t p r ,
Ia r , t Ia r , 1 e
A phasor plot of the behavior of the complex time-domain 3-D vector sound intensity I a r , t is
shown below
in time at t = 0, for the simple 1-D case of a purely real frequency for a snapshot
domain I r , I r , zˆ I zˆ 1 p u zˆ :
a
a
o
2
o oz
2 i t p r ,
Ia r , t I o 1 e
I 1 cos 2 t p r , i sin 2 t p r ,
o
2 i
2 i
At t = 0: Ia r , 0 I o 1 e p I o I o e p I o I o cos 2 p iI o sin 2 p
n.b. the yellow sound
intensity phasor
component Ioe2ip
rotates CW in the
complex plane {about
the point (Io,0)} at
angular frequency 2
as time t increases!
Im
Io
I o cos 2 p
Re
2 p
I o sin 2 p
Ia r , 0
Ia r , t
rotates CW
-7-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
For completeness’ sake, note that each of the k x, y, z components of the purely real,
instantaneous time-domain 3-D vector acoustic intensity I a r , t , using the fact that
Re e i Re cos i sin cos can be written as:
I ak r , t p r , t uk r , t po r , cos t p uok r , cos t uk
Re p r , t Re uk r , t Re p r , t Re uk* r , t
Re p r , eit Re uk* r , e it
i
i
Re po r , e p eit Re uok r , e uk e it
Note also that each of the k x, y, z components of the frequency-domain complex 3-D vector
acoustic intensity “amplitude” I r , can be written as:
a
Iak r , 12 p r , uk* r ,
Re Iak r , i Im Iak r ,
I ark r , iI ai k r ,
i
Since complex p pr ipi p e p and complex {conjugated}
uk* urk iuik uk* e
iuk
Iak 12 p uk*
1
2
uk e
1
2
iuk
, then for each of the k x, y, z components of I a r , :
pr ipi ur
i
p e p uk* e
iuk
1
2
iuik 12 pr urk piuik i piurk pr uik
i
i
i
i
p uk* e p e uk 12 p uk* e p uk 12 p uk* e puk
k
i
Iak e Ik
Thus, we have learned several things:
a.) The phase of complex Iak r , is equal to the phase difference between complex p and uk ,
i.e. I k p uk p uk , and:
i
i
i
I ark Re Iak Re Iak e Ik 12 Re p uk* e puk 12 Re p uk* e p uk
Iak cos I a
k
and:
1
2
p uk* cos p uk
1
2
pu
r rk
piuik
i
i
I ai k Im Iak Im Iak e Ik 12 Im p uk* e puk 12 Im p uk* e p uk
Iak sin I a
k
1
2
p uk* sin p uk
-8-
1
2
pu
i rk
pr uik
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
b.) The time average of the time-domain, purely real, instantaneous 3-D vector sound intensity is:
2 i t p r ,
I a r , t Re Ia r , 1 e
t
t
1
one
cycle
2 i t p r ,
Re Ia r , 1 e
dt
2 i t p r ,
Re I a r , 1 e
dt
one
cycle
1
Re Ia r , 1 dt cos 2 t p r , dt i sin 2 t p r , dt
0
0
0
1
Re I a r , cos 2 t p r , dt i sin 2 t p r , dt
0
0
0 over one cycle
0
over
one
cycle
*
Re Ia r , Re 12 p r , u r ,
1
In words: The time average of the purely real, instantaneous time-domain 3-D vector sound
intensity I a r , t is equal to the real part of the frequency-domain complex 3-D vector sound
t
intensity “amplitude” Re Ia r , Re 12 p r , u * r , .
For a monochromatic/single-frequency harmonic sound field:
a.) The real part of the frequency-domain complex 3-D vector sound intensity “amplitude”
I r r , Re I r , is associated with propagating sound – i.e. sound radiation.
a
a
b.) The imaginary part of the frequency-domain complex 3-D vector sound intensity “amplitude”
I i r , Im I r , is associated with non-propagating sound energy – i.e. sound
a
a
energy that only “sloshes” back and forth {locally} during each cycle of oscillation.
Note that for a {lossless} standing acoustic wave consisting of two equal-amplitude counterpropagating traveling waves, the time average of the purely real, instantaneous time-domain
3-D vector sound intensity I a r , t 0 because the real part of the frequency-domain
t
complex 3-D vector sound intensity “amplitude” Re Ia r , Re 12 p r , u * r , 0 .
There is no net sound propagation in this situation. {n.b. These are analogous to the real and
imaginary components of complex scalar electrical power Pe associated e.g. with an LCR
circuit – the real component is the electrical power dissipated in the resistance R; the imaginary
component is power transitorily stored/circulating in the inductance L and/or the capacitance C
of the LCR circuit!}
-9-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Note also that in this situation, the imaginary part of the frequency-domain complex 3-D vector
acoustic intensity “amplitude” Im I r , Im 1 p r , u * r , 0 , because the acoustic
a
2
energy associated with a standing acoustic wave {consisting of two equal-amplitude counterpropagating traveling waves} simply “sloshes” back and forth {locally} during each cycle of
oscillation. A {lossless} standing acoustic wave has no net sound propagation.
we see that the complex frequency-domain 3-D vector sound intensity “amplitude”
Thus,
Ia r , 12 p r , u * r , actually gives us more physical insight into what is going on than
just the time average of the purely real, instantaneous time-domain 3-D vector sound intensity
I a r , t , since I a r , t is equal to the real part of the frequency-domain complex 3-D
t
t
vector sound intensity “amplitude”: I a r , t Re Ia r , Re 12 p r , u * r , .
t
Complex 3-D Vector Specific Acoustic Immittances:
As we have previously discussed in considerable detail the manifestly frequency-domain
complex 3-D vector specific acoustic immittances in Physics 406 Lect. Notes XI, Part 2
(p. 16-27), for completeness’ sake, we summarize them again, here:
Complex 3-D Vector Specific Acoustic Admittance:
y a r , y ax r , xˆ y a y r , yˆ y az r , zˆ
u y r ,
u x r ,
u z r ,
u r ,
1
xˆ
yˆ
zˆ
p r ,
p r ,
p r ,
p r , za r ,
yar r , iyai r , Re y a r , Im y a r ,
yar x r , iyai x r , xˆ yar y r , iyai y r , yˆ yar z r , iyai z r , zˆ
i ya r ,
i y r ,
i r ,
y ax r , e yx
xˆ
y a y r , e y
yˆ
y az r , e az
zˆ
Complex 3-D Vector Specific Acoustic Impedance:
p r ,
1
za r , zax r , xˆ za y r , yˆ zaz r , zˆ
u r , y a r ,
p r , u *y r ,
p r , u x* r ,
p r , u z* r ,
p r , u * r ,
xˆ
yˆ
zˆ
2
2
2
2
u r ,
u r ,
u r ,
u r ,
zar r , izai r , Re za r , Im za r ,
zar x r , izai x r , xˆ zar y r , izai y r , yˆ zar z r , izai z r , zˆ
i za r ,
i z r ,
i z r ,
zaz r , e az
zˆ
zax r , e ax
xˆ
za y r , e y
yˆ
-10-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
where, for the k x, y, or z components {temporarily suppressing the r , arguments}:
uk urk iuik urk iik pr ipi pr urk piuik
2
p
pr ipi pr ipi pr ipi
p
y ak yar k iyai k
piur pr ui
k
k
i
2
p
and:
p u * pr ipi urk iuik
zak zar k izai k 2 k
2
u
u
p ip u
*
r
i
2
u
rk
iuik
p u
r rk
pi uik
2
u
pu p u
r ik
i i rk
2
u
i.e. for the k x, y, or z components of the immittances y a r , and za r , :
yar k Re y ak
pr urk piuik
p
2
and: yai k Im y ak
pr urk piuik
p
2
piurk pr uik
p
2
and:
pr urk piuik
piurk pr uik
and: zai k Im zak
2
2
u
u
However, in the frequency domain, for the k x, y, z components of I a r , we have:
zar k Re zak
p ip u iu p u
Iak 12 p uk* Re Iak i Im Iak I ark iI ai k
1
2
1
2
r
i
i
p e p uk* e
rk
iuk
ik
1
2
1
2
i
p uk* e p
piuik i piurk pr uik
i
i
i
e uk 12 p uk* e p uk 12 p uk* e puk
r rk
i I
Iak e ak
i.e. for the k x, y, or z components of the frequency domain complex 3-D vector acoustic
intensity, I a r , , we have:
pu
I ark Re Iak
1
2
r rk
piuik
pu
and: I ai k Im Iak
1
2
i rk
pr uik
Hence, we see that frequency domain relations exist between the complex 3-D vector specific
acoustic immittances za r , , y a r , and the complex 3-D vector acoustic intensity I a r , :
2
2
I ark Re Iak 12 pr urk piuik 12 u zar k and/or: I ark Re Iak 12 pr urk piuik 12 p yar k
2
2
I ai k Im Iak 12 piurk pr uik 12 u zai k and/or: I ai k Im Iak 12 piurk pr uik 12 p yai k
2
2
Iak 12 p uk* 12 u zak
and/or: Iak 12 p uk* 12 p y a*k
2
*
2 *
Ia 12 p u * 12 u za
and/or: Ia 12 p u 12 p y a
-11-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Note also that the real and imaginary frequency domain components of complex 3-D vector
acoustic intensity I a r , , the real and imaginary frequency domain components of complex
3-D vector specific acoustic immittances za r , , y a r , are proportional to linear
combinations of the real and imaginary frequency domain components of complex over-pressure
p r , and complex 3-D vector particle velocity u r , , i.e. that:
I ark , zar k , yar k pr urk piuik
and: I ai k , zai k , yai k piurk pr uik .
For a monochromatic/single-frequency harmonic sound field:
a.) The real part of the frequency-domain complex 3-D vector sound intensity “amplitude”
I r r , Re I r , , and the real part of the frequency-domain complex 3-D vector
a
a
specific acoustic immittances zar r , Re za r , and yar r , Re y a r , ,
are associated with propagating sound – i.e. sound radiation.
b.) The imaginary part of the frequency-domain complex 3-D vector sound intensity “amplitude”
i
I r , Im I r , , and the imaginary part of the frequency-domain complex 3-D vector
a
a
specific acoustic immittances zai r , Im za r , and yai r , Im y a r , ,
are associated with non-propagating sound energy – i.e. sound energy that only “sloshes” back
and forth {locally} 2× during each cycle of oscillation.
Acoustic Energy Densities:
Energy, W t (SI units: Joules) and energy density, w r , t (SI units Joules /m3) are always
purely real, scalar physical quantities. They also are additive in nature, in that different kinds of
energy/energy density – such as potential, kinetic, etc. energies can/must be linearly added
together to obtain e.g. the total energy of a system – since total energy Wtot t wtot r , t dV
v
(global) and total energy density wtot r , t (local) are conserved quantities.
The total instantaneous, physical acoustic energy density, watot r , t is the linear sum of the
instantaneous, physical acoustic potential and kinetic energy densities (time-domain quantities):
watot r , t wapotl r , t wakin r , t
z o oc =
characteristic
2
p r ,t 1 1
1
longitudinal
wapotl r , t o
p2 r , t
where:
2
2
specific acoustic
zo
2
2 o c
impedance of
1
1
kin
2
“free-air”
wa r , t ou r , t ou r , t u r , t
2
2
and:
1
o u x2 r , t u y2 r , t u z2 r , t
2
-12Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Note that the instantaneous, physical acoustic potential and kinetic energy densities,
wapotl r , t 12 1c2 p 2 r , t and: wakin r , t 12 ou 2 r , t
o
are respectively analogous e.g. to the instantaneous mechanical potential and kinetic energies for
a simple 1-D mass-spring system:
potl
kin
Wmech
t 12 kx 2 t and: Wmech
t 12 mv 2 t .
For a harmonic sound field, the instantaneous, physical purely real scalar over-pressure is:
p r , t po r , cos t p r , po r , cos t p r ,
and the k x, y, or z components of instantaneous, physical 3-D vector particle velocity are:
uk r , t uok r , cos t uk r ,
Hence:
1 1
1 1
wapotl r , t
p2 r , t
po2 r , cos 2 t p r ,
2
2
2 o c
2 o c
and:
1
1
wakin r , t o u 2 r , t o u r , t u r , t
2
2
1
o u x2 r , t u y2 r , t u z2 r , t
2
uo2 r, cos 2 t u r,
x
x
1 2
2
o uoy r , cos t u y r ,
2
u 2 r, cos 2 t r,
uz
oz
Now, we can also equivalently write these expressions in terms of complex instantaneous/timedomain over-pressure and 3-D particle velocity as:
1 1
1 1
2 i t p r ,
Re p 2 r , t
Re po2 r , e
wapotl r , t
2
2
2 o c
2 o c
and:
1
wakin r , t o Re u x2 r , t u y2 r , t u z2 r , t
2
u 2 r, e 2it ux r ,
ox
2
2 i t u y r ,
1
o Re uoy r , e
2
uo2 r , e 2it uz r ,
z
-13-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
The time averages of the instantaneous acoustic potential and kinetic energy densities are:
1 1
1 1
wapotl r , t
p2 r , t
po2 r , cos 2 t p r ,
2
2
t
t
2 o c
2 o c
t
1 2
1 1
1
1 1
po2 r , cos 2 t p r , dt
po2 r ,
2
2
0
t
4 o c
2 o c
t
2
wakin r , t
t
and:
1
o u x2 r , t u y2 r , t u z2 r , t
2
u 2 r, cos 2 t r,
ux
ox
1
o uo2y r , cos 2 t u y r ,
2
uo2z r , cos 2 t uz r ,
1
o uo2x r , uo2y r , uo2z r ,
4
t
t
t
Thus, the time average of the instantaneous total acoustic energy density is:
watot r , t wapotl r , t wakin r , t
t
t
t
1 1
1
po2 r , o uo2x r , uo2y r , uo2z r ,
2
4 o c
4
For a harmonic sound field: p r , t p o r , eit and u r , t uo r , eit , hence we can
“complexify” these relations to obtain the purely real, frequency-domain acoustic potential,
kinetic and total energy densities – note that they remain purely real quantities:
2
1 1
2 1 1
wapotl r ,
p
r
t
p
r
wapotl r , t
,
,
2
2
4 o c
4 o c
t
2 1
1
1
2
2
2
wakin r , o u r , t o u r , t u * r , t o u x r , t u y r , t u z r , t
4
4
4
2
2
2
2
1
1
1
o u r , o u r , u * r , o u x r , u y r , u z r ,
4
4
4
wakin r , t
t
watot r , wapotl r , wakin r ,
2
2
1 1
1
p
r
,
u
r
,
o
4 o c 2
4
wapotl r , t wakin r , t watot r , t
t
t
-14-
t
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Conservation of Energy:
Consider the following “gedanken” experiment: An idealized, infinitesimally-thin spherical
shell of radius R contains air @ NTP, but has a vacuum on the outside the spherical shell, as
shown in the figure below. The infinitesimally-thin spherical shell is made of piezoelectric
o
material, such that when an AC voltage VFG t VFG
cos t is applied to the piezoelectric
material e.g. using a sine-wave function generator, the spherical shell mechanically vibrates/
oscillates back and forth in the radial r̂ direction with angular frequency 2 f .
nˆ rˆ
Vacuum
Air @
NTP
R
S
V
The instantaneous mechanical power Pmech t associated with the mechanically oscillating
spherical shell is equal to the instantaneous time rate of change of the {total} mechanical work
done on the spherical shell: Pmech t Wmech t t . Neglecting any/all dissipative/frictional/loss
effects, by conservation of energy, the instantaneous mechanical power Pmech t being produced
at/on the surface of the spherical shell must equal the instantaneous acoustical power Pa t
associated with instantaneous acoustic energy flowing into the interior volume V of the spherical
shell {through the surface S of the spherical shell of radius R} via the relation:
ˆ , where n̂ is the outwardPa t I a r , t da , where the vector area element da nda
S
pointing unit normal to the surface S, which means that nˆ rˆ , Since the instantaneous acoustical
power flowing into the sphere must be a positive quantity, i.e. Pa t 0 , hence the need for the
sign in the above formula, since we are considering acoustical energy flowing {radially} into
the volume V of the spherical shell through the enclosing surface S of radius R.
However, using the divergence theorem: Pa t I a r , t da I a r , t d . The
S
V
instantaneous, physical time-domain vector acoustic intensity I a r , t p r , t u r , t , thus the
divergence of the instantaneous vector acoustic intensity, I a r , t p r , t u r , t .
Note that since the SI units of vector acoustic intensity I a are Watts m 2 , the SI units of I a
are Watts m3 , i.e physically, I a is an acoustic power density. Hence, the scalar
instantaneous, physical time-domain acoustic power density aP r , t is:
aP r , t I a r , t p r , t u r , t Watts m3
with:
Pa t aP r , t d I a r , t d 0 Watts
V
V
-15-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Next, we use the following product rule from vector calculus: fA f A f A .
Thus: pu pu p u . For “everyday” sound fields, with SPL ' s 134 dB (i.e.
p 100 Pascals ), using the {linearized} Euler’s equation: p o u t , and using the
{linearized} mass continuity equation: u 1o t . However, from the {linearized}
adiabatic relationship between pressure, p and mass density, we also have: t
1
c2
p t .
Thus:
u
1
p
I a pu pu p u o u
p
2
o c
t
t
2
u u u u
u
u
u
u u 2 u 2u
Now, note that:
t
t
t
t
t
t
P
a
p 2 p p p
p
p
p
p p
2
p 2p
t
t
t
t
t
t
and that:
Hence, we see that:
1
2
aP I a pu p u p u o
u 2
p2
1 p 2 1
2
u
o
2
t 2 o c 2 t
t 2
t 2 o c
But the instantaneous acoustic kinetic, potential and total energy densities, respectively are:
2
1 p r ,t
1
kin
2
potl
wa r , t ou r , t , wa r , t
, and: watot r , t wapotl r , t wakin r , t
2
2 o c
2
Thus, we see that the {negative} divergence of the instantaneous time-domain vector acoustic
intensity, i.e. the instantaneous time-domain acoustic power density is:
wakin r , t wapotl r , t watot r , t
r , t I a r , t
Watts m3
t
t
t
P
a
The explicit forms of the instantaneous time-domain potential and kinetic energy densities are:
1 1
1 1
wapotl r , t
p2 r , t
po2 r , cos 2 t p r ,
2
2
2 o c
2 o c
and:
uo2 r , cos 2 t u r,
x
x
1
1
1
wakin r , t o u 2 r , t ou r , t u r , t o uo2y r , cos 2 t u y r ,
2
2
2
u 2 r, cos 2 t r ,
uz
oz
-16-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Hence:
wapotl r , t
t
1 1
po2 r , cos 2 t p r ,
2
t 2 o c
and:
2
po r , sin t p r , cos t p r ,
2
o c
uo2 r, cos 2 t u r,
x
x
wakin r , t 1 2
2
o uo r , cos t u y r ,
t
t 2 y
2
2
uoz r , cos t uz r ,
uo2 r , sin t u r , cos t u r,
x
x
x
o uo2y r , sin t u y r , cos t u y r ,
u 2 r, sin t r, cos t r ,
uz
uz
oz
Using the trigonometric relation: sin cos 12 sin 2 with: t , we can rewrite the
above relations as:
wapotl r , t
po2 r , sin 2 t p r ,
2
t
2 o c
uo2 r, sin 2 t u r ,
x
x
wakin r , t
1
and:
o uo2y r , sin 2 t u y r ,
2
t
u 2 r , sin 2 t r,
uz
oz
Thus, the instantaneous, physical time-domain acoustic power density can be seen to be an
oscillatory function (i.e. oscillating above/below aP r , t 0 ) at angular frequency
2 2 2 f 4 f , i.e. 2 per cycle of oscillation:
wakin r , t wapotl r , t watot r , t
r , t I a r , t
t
t
t
uo2 r , sin 2 t u r ,
x
x
1
2
o uoy r , sin 2 t u y r ,
po2 r , sin 2 t p r ,
2
2 o c
2
u 2 r , sin 2 t r,
uz
oz
P
a
Note that this is not the kind of behavior we associate with propagating sound radiation –
rather it is that which we associate with the non-propagating portion of the acoustic energy!
-17-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
It can thus be seen that since the time averages of the time rates of change of the individual
instantaneous, physical time-domain acoustic potential and kinetic energy densities, and thus
the time average of the instantaneous, physical time-domain total acoustic energy density are
all zero:
wapotl r , t
po2 r , sin 2 t p r ,
2
t
t
2 o c
t
1 t
po2 r , sin 2 t p r , dt 0
2
t 0
2 o c
0
and:
w r , t
t
kin
a
and:
wakin r , t
t
t
u 2 r, sin 2 t r,
ux
ox
1
o uo2y r , sin 2 t u y r ,
2
2
uoz r , sin 2 t uz r ,
wapotl r , t
t
t
t
t
t
watot r , t
t
0
t
0
t
that we also see/learn that the time average of the instantaneous, physical time-domain acoustic
power density, which is equal to the time average of the {negative} divergence of the
instantaneous, physical time-domain vector acoustic intensity, is also zero:
wakin r , t
r , t t I a r , t
t
t
P
a
t
wapotl r , t
t
t
watot r , t
t
0
t
It is instructive to also work this out for the complex frequency-domain case – e.g. for a
harmonic/monochromatic sound field, because these relations must also hold for complex
* 12 p u * 12 p u * ,
I a r , 12 p r , u * r , 12 p r , t u * r , t : I a 12 pu
where the complex time-domain p r , t p r , eit and: u r , t u r , eit , thus:
u * r , t u * r , e it . Using the complex time-domain relations: p o u t ,
u * 1o * t and: * t c12 p * t we have:
*
*
1 u * 1 1
p *
*
1
1
1
I a 2 pu
2 p u 2 p u o u
p
2
t
2 o c 2 t
P
a
-18-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
For a harmonic sound field we have: p r , t p r , eit , p * r , t p * r , e it and
u r , t u r , eit , u * r , t u * r , e it . Thus, for a harmonic sound field we also have:
u r , t t i u r , t and: p * r , t t i p * r , t , and thus we see for the complex
frequency-domain, for a harmonic sound field, that:
1 p 2
1 p p *
2
1 *
1
P
a I a i ou u i
i o u i
2
2 o c 2
2
2
2 o c
However, the frequency-domain versions of the acoustic kinetic and potential energy densities
are, respectively:
2
1
1
1
2
2
2
wakin r , o u r , t o u r , t u * r , t o u x r , t u y r , t u z r , t
4
4
4
2
2
2
2
1
1
1
o u r , ou r , u * r , o u x r , u y r , u z r ,
4
4
4
wakin r , t
t
2
1 1
2 1 1
wapotl r ,
p r , t
p r , wapotl r , t
2
2
4 o c
4 o c
t
Thus, we see that for a harmonic sound field, in the complex frequency-domain representation
we have the relation:
1 p r, 2
2
1
P
2i wakin r, wapotl r,
a r , I a r , 2i o u r , 2i
2
4 o c
4
aP r , Ia r , 2i wakin r , wapotl r ,
wakin r , wapotl r , watot r ,
Watts m3
t
t
t
i.e.
Now, since energy densities such as wapotl , wakin are always/must be purely real, positive
quantities, we see that since aP r , I a r , is a purely imaginary quantity for a
harmonic sound field, this means that:
Re aP r , Re I a r , 0 Watts m3
and that:
Im aP r , Im I a r , 2 wakin r , wapotl r , Watts m3
-19-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Since the real (imaginary) parts of the complex frequency-domain vector acoustic sound
intensity I a r , are physically associated with propagating sound/sound radiation (nonpropagating acoustic energy, locally sloshing back and forth each cycle) respectively, we
see/learn that for propagating sound waves, the complex frequency-domain acoustic power
density, equal to the {negative} divergence of the complex frequency-domain vector acoustic
intensity is zero:
P
Active Acoustic Power Density: Re a r , Re I a r , 0
whereas for the non-propagating portion of the acoustic energy, the complex frequency-domain
acoustic power density, equal to the {negative} divergence of the complex frequency-domain
vector acoustic intensity is non-zero, it is:
P
kin
potl
Reactive Acoustic Power Density: Im a r , Im I a r , 2 wa r , wa r ,
For a harmonic sound field, we can thus also consider the physical meaning of the complex
frequency-domain representation of the mechanical and acoustic power:
Pa aP r , d I a r , d I a r , da
V
V
S
kin
potl
2i wa r , wa r , d Watts
V
Again, since energy densities are always purely real, positive quantities, we see/learn that the
complex frequency-domain representation of the mechanical and acoustic powers are such that:
Active Acoustic Power: Re Pa 0
and:
kin
potl
Reactive Acoustic Power: Im Pa 2 V wa r , wa r , d
The above results for active vs. reactive power have important physical ramifications as to how
the mechanical energy is able to be injected into and/or extracted from the acoustical system!
Since we have discussed the nature of the {negative} divergence of the vector acoustic
intensity, I a , for completeness’ sake, we also discuss the nature of the curl of the vector
acoustic intensity, I a , the vorticity associated with 3-D acoustic intensity.
In the instantaneous, physical time-domain representation: I a r , t p r , t u r , t .
Using another product rule from vector calculus: fA f A A f , we have:
I a p u p u u p u p
0
-20-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
where u 0 for an inviscid (i.e. dissipationless) fluid – no vorticity exists, such as is the case
for {still/calm} air @ NTP. The {linearized} Euler equation is: p o u t , thus:
u
I a p u u p ou
t
The instantaneous, physical time-domain representation of the vector particle velocity is:
u r , t u x r , t xˆ u y r , t yˆ u z r , t zˆ
uox r , cos t ux r , xˆ uoy r , cos t u y r , yˆ uoz r , cos t uz r , zˆ
and thus the time rate of change of the instantaneous, physical time-domain representation of
the vector particle velocity is:
u y r , t
u r , t u x r , t
u z r , t
xˆ
yˆ
zˆ
t
t
t
t
uox r , sin t ux r , xˆ uoy r , sin t u y r , yˆ uoz r , sin t uz r , zˆ
Thus, we need to work out the vector cross product: u u t . Recall that the cross product of
two arbitrary vectors A and B is defined as:
xˆ
yˆ
zˆ
A B Ax xˆ Ay yˆ Az zˆ Bx xˆ By yˆ Bz zˆ Ax Ay Az
Bx
By
Bz
Ay Bz Az By xˆ Az Bx Ax Bz yˆ Ax By Ay Bx zˆ
Thus, temporarily suppressing r , t and r , arguments for the sake of notational clarity:
u
uoy uoz cos t u y sin t uz uoz uoy cos t uz sin t u y
u
t
uoz uox cos t uz sin t ux uox uoz cos t ux sin t uz
uox uo y
cos t sin t u
ux
uy
oy
uox cos t u y
xˆ
yˆ
sin t zˆ
ux
cos t sin t yˆ
cos t sin t ẑ
uoy uoz cos t u y sin t uz cos t uz sin t u y xˆ
cos t sin t
uoz uox cos t uz sin t ux
uo x uo y
ux
-21-
uy
ux
uz
uy
ux
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Defining: uk t uk , k x, y, z we can rewrite this relation as:
u
uoy uoz cos u y sin uz cos uz sin u y xˆ
u
t
uoz uox cos uz sin ux cos ux sin uz yˆ
uox uoy cos ux sin u y cos u y sin ux zˆ
uoy uoz sin u y cos uz cos u y sin uz xˆ
uoz uox sin uz cos ux cos uz sin ux yˆ
uox uoy sin ux cos u y cos ux sin u y zˆ
Using the trigonometric relation: sin A B cos A sin B sin A cos B , and noting that:
uk u j t uk t u j uk u j uk u j , for k j x, y, z this relation can be
further rewritten as:
u
uoy uoz sin u y uz xˆ uoz uox sin uz ux yˆ uox uoy sin ux u y zˆ
u
t
or, more explicitly:
uoy r , uoz r , sin u y uz r , xˆ
u r , t
I a r , t ou r , t
o uoz r , uox r , sin uz ux r , yˆ
t
uox r , uoy r , sin ux u y r , zˆ
Thus, we see that the curl of the instantaneous, physical time-domain vector acoustic
intensity, I a r , t has no explicit time dependence, and note also that I a r , t has no
dependence on the over-pressure, p r , t . It does depend on the {sine of} the k j x, y, z
component phase differences in particle velocity, uk u j r , and bi-linear products of
k j x, y, z frequency-domain particle velocity amplitudes uok r , uo j r , . If all phase
difference components uk u j r , 0o , 180o , then I a r , t 0 . Maximum vorticity
associated with the 3-D vector acoustic intensity occurs when uk u j r , 90o .
Note that the SI units of I a r , t , as for I a r , t are Watts m3 . From Stoke’s theorem:
I
r
,
t
da
I
a
S
C a r , t d where S {here} is an open surface is enclosed {i.e. bounded}
by the contour C. The SI units of I a r , t da I a r , t d are thus Watts m .
S
-22-
C
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Again, it is instructive to work this out for the complex frequency-domain case – e.g. for a
harmonic/monochromatic sound field, because these relations must also hold for complex
I a r , 12 p r , u * r , 12 p r , t u * r , t :
* 12 p u * 12 u * p 12 u * p
I a 12 pu
0
where the complex time-domain p r , t p r , eit and: u r , t u r , eit , thus:
u * r , t u * r , e it . Using the complex time-domain relations p o u t ,
u t i u , hence: u * t i u * , we have:
*
* u 1
*
1
1
1
I a 2 pu
2 u p 2 o u
2 i o u * u
t
Now:
iu r ,
i r ,
i r ,
u r , t u r , eit uox r , e ux
xˆ uoy r , e y
yˆ uoz r , e uz
zˆ eit
And:
iu r ,
i r ,
i r ,
u * r , t u * r , e it uox r , e ux
xˆ uoy r , e y
yˆ uoz r , e uz
zˆ e it
*
*
Thus, we need to work out the complex cross product u u u u (n.b. which is not zero –
because u * u !):
iu
iu
i
i
i
i
u u * eit uox e ux xˆ uoy e y yˆ uoz e uz zˆ uox e ux xˆ uoy e y yˆ uoz e uz zˆ e it
u eiux xˆ u eiu y yˆ u eiuz zˆ u e iux xˆ u e iu y yˆ u e iuz zˆ
oy
oz
oy
oz
ox
ox
u u eiu y uz e iu y uz xˆ
o y oz
iu z u x
iu z u x
e
use: sin 21i ei e i
yˆ
uo z uo x e
i
i
uox uoy e ux u y e ux u y zˆ
2i uoy uoz sin u y uz xˆ uoz uox sin uz ux yˆ uox uoy sin ux u y zˆ
Hence:
1 * u 1
I a 2 ou
2 i o u * u 12 io u u *
t
12 i 2io uoy uoz sin u y uz xˆ uoz uox sin uz ux yˆ uox uoy sin ux u y zˆ
o uo uo sin u
y
z
y u z
xˆ uoz uox sin uz ux yˆ uox uoy sin ux u y zˆ
Note that this result is identical to that obtained for the curl of the physical purely real
instantaneous time-domain 3-D vector acoustic intensity, I a r , t .
-23-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Hence we see/learn that:
* u 1
1
2 i o u * u 12 i o u u *
Re I a 2 o u
t
o uoy uoz sin u y uz xˆ uoz uox sin uz ux yˆ uox uoy sin ux u y zˆ
and that:
Im I a 0
The above results make it clear that the curl of the 3-D vector acoustic intensity – the vorticity
associated with 3-D vector acoustic intensity is associated only with propagating sound radiation.
The Complex Acoustic 3-D Velocity of Energy Flow:
In Physics 406 Lecture Notes 12 (p. 3-6) we will see/learn that for a 1-D monochromatic
traveling wave propagating in “free air”, the complex longitudinal acoustic specific impedance, –
a physical property associated with the medium (“free air”) in which the 1-D monochromatic
traveling wave is propagating in – is a purely real constant: za r , o c zo . The so-called
characteristic longitudinal specific impedance of “free air”, zo o c 415 a @ NTP. Note
that the constant zo is thus independent of position r and of {angular} frequency 2 f .
For an arbitrary complex harmonic sound field S r , t , but one with “everyday” sound
pressure levels SPL r 134 dB ( p r 100 Pascals ) in air @ NTP, in general the 3-D
vector specific acoustic impedance za r , will be complex, and depend on both position in
space, r and {angular} frequency, 2 f . Thus, the above specific acoustic impedance
relation for the “free air” case za r , o c zo can be generalized to za r , o ca r , ,
where ca r , m s is the complex acoustic 3-D velocity of energy flow. Note that, like
za r , , ca r , is manifestly a frequency-domain quantity, since it is simply-related to
za r , by the equilibrium mass density of the medium o kg m3 , a scalar quantity.
The complex acoustic 3-D vector velocity of energy flow ca r , must not be confused
with the purely real, scalar adiabatic/thermodynamic speed of sound of the medium c, nor with
the purely real, scalar phase speed of propagation of the wave v r , k r , , nor with
1
the purely real, 3-D vector group velocity vg r , dk r , d . In various wave-type
physics situations, there can in fact be several/many different definitions /kinds of propagation
speeds…. {See e.g. S.C. Bloch, “Eighth Velocity of Light”, p. 538-549, Am. J. Phys., Vol. 45,
No. 6, June 1977}.
-24-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Since complex 3-D vector specific acoustic admittance is the reciprocal of the complex 3-D
vector specific acoustic impedance, we have the paired reciprocal relations:
p r ,
u r ,
1
1
1
za r ,
o ca r , a and: ya r ,
a1 .
p r , za r , o ca r ,
u r , ya r ,
However, we have the complex frequency-domain relation:
*
1
1 p r , u r , u r ,
*
I a r , p r , u r ,
2
2
u r ,
2 p
2
2
r , 1
1
1
u r ,
u r , za r , o u r , ca r ,
2
2
u r , 2
But, the {purely real} frequency-domain acoustic kinetic energy density
2
wkin r , 1 u r , , hence: I r , 2c r , wkin r , .
a
4
o
a
a
a
However, we also have the complex frequency-domain relation:
*
*
1
1 p r , p r , u r ,
*
I a r , p r , u r ,
p * r ,
2
2
2
*
p r ,
2 u r ,
2 *
1
1
1
p r , *
p r , y a r ,
p r , 2
2
2 o ca* r ,
2
2
c2
1 p r ,
1 p r ,
c r ,
c
r
,
a
a
2
2
2
2 c r ,
2 o c
r ,
c
o a
a
But, the {purely real} frequency-domain acoustic potential energy density
2
2
wapotl r , 14 p r , o c 2 , hence: I a r , 2ca r , c 2 ca r , wapotl r , .
Thus, we see that:
c2
kin
c r, w potl r,
I a r , 2ca r , wa r , 2
a
2
c r , a
a
Note also that: I a r , za r , ca r , , and additionally note that the ratio of acoustic
potential to kinetic energy density is:
2
2
za r ,
wapotl r , ca r ,
wakin r ,
c2
zo2
-25-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
As mentioned above, for the “free-field” propagation of monochromatic plane traveling
waves za r , o c zo , hence for the “free-field” propagation of monochromatic plane
2
2
traveling waves, we see/learn that: wapotl r , wakin r , ca r , c 2 za r , zo2 1 .
Note that in general these 3 ratios are not equal to unity for an arbitrary complex sound field.
We can also define a physically-related quantity, known as the complex 3-D vector acoustic
“index of refraction” of energy flow:
ca r , za r ,
na r ,
c
zo
2
2
ca r ,
za r ,
2
wapotl r ,
na r ,
kin
c2
zo2
wa r ,
thus:
Hence for the “free-field” propagation of monochromatic plane traveling waves, we see/learn
2
2
2
that: n r , w potl r , wkin r , c r , c 2 z r , z 2 1 .
a
a
a
a
a
o
Since the {purely real} frequency-domain acoustic total energy density is the sum of
acoustic kinetic and potential energy densities, we see that:
Ia r ,
Ia r ,
kin
potl
w r , wa r , wa r ,
2ca r , 2 c 2 c r, 2 c r,
a
a
tot
a
2
2
I
r
,
c
r
,
c2
a
a
Ia r ,
1 ca r , I a r ,
1
ca r,
c2
2
2ca r ,
2ca r ,
Thus, we also see that:
ca r , watot r ,
Ia r ,
2
1
c2
2 1 ca r ,
where:
za r ,
ca r ,
o
or equivalently:
na r , c watot r ,
Ia r ,
2
1
2 1 na r ,
where:
ca r , za r ,
na r ,
c
zo
From energy {density} conservation: watot r , wakin r , wapotl r , , we can also define
{purely real} frequency-domain fractional acoustic kinetic and potential energy densities:
f akin r , wakin r , watot r , and: f apotl r , wapotl r , watot r ,
with:
f akin r , f apotl r , wakin r , watot r , wapotl r , watot r , 1
-26-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
2
Using the relation: na r , wapotl r , wakin r , , we see that:
2
wapotl r , wapotl r , watot r , f apotl r ,
na r , kin
kin
wa r , wakin r , watot r ,
fa r ,
Or equivalently:
2
2
f apotl r , na r , f akin r , and/or: wapotl r , na r , wakin r ,
As we have discussed before, the physical meaning of the real parts of I a r , , za r ,
and hence the real parts of ca r , and na r , are associated with propagating sound
radiation, whereas the imaginary parts of I a r , , za r , and hence the imaginary parts of
c r , and n r , are associated with non-propagating acoustic energy, locally sloshing
a
a
back and forth each cycle of oscillation.
Since energy and energy densities are additive scalar, purely real, positive quantities, we have
from energy conservation:
watot r , wakin r , wapotl r ,
But we also have the additive scalar, purely real relation: watot r , warad r , wavirt r ,
where warad r , is the acoustic energy density associated with propagating sound radiation and
wavirt r , is the acoustic energy density associated with non-propagating acoustic energy,
locally sloshing back and forth each cycle of oscillation.
We can thus define: f arad r , warad r , watot r , and f avirt r , wavirt r , watot r ,
as the {purely real} frequency-domain fractional energy densities associated with propagating
sound radiation and non-propagating acoustic energy, respectively.
Note also that we must have: f arad r , f avirt r , warad r , wavirt r , watot r , 1 .
Now:
tot
I a r , na r , c wa r ,
1
2
1
Ia r ,
1
n
r
,
a
2
Or:
na r , c watot r ,
2
1
2 1 na r ,
w r ,
1
w r ,
tot
a
tot
a
1
1
-27-
1
1
2
na r ,
2
na r ,
watot r ,
watot r ,
2
na r ,
2
na r ,
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
We use watot r , warad r , wavirt r , for the numerator of the LHS of this relation, and
2
2
2
write na r , Re na r , Im na r , nar 2 r , nai 2 r , for the numerator of
the RHS of this relation:
1 nar 2 r , nai 2 r ,
warad r , wavirt r ,
rad
virt
fa r , fa r ,
watot r ,
1 n r, 2
a
1 n r, 2 f rad r, f virt r , 1 n r 2 r , ni 2 r,
Then:
a
a
a
a
a
2
Or: f arad r , f avirt r , na r , f arad r , f avirt r , 1 nar 2 r , nai 2 r ,
1
2
1 na r , f arad r , f avirt r , 1 nar 2 r , nai 2 r ,
Thus:
2
na r , f arad r , f avirt r , nar 2 r , nai 2 r ,
Or:
Or:
f
rad
a
nar 2 r , nai 2 r , nar 2 r , nai 2 r ,
virt
1
r , fa r , 2 2
2
na r ,
na r ,
na r ,
Thus we see that:
f
rad
a
f
virt
a
warad r ,
r , tot
wa r ,
nar 2 r ,
car 2 r ,
zar 2 r ,
I ar 2 r ,
2
2
2
2
za r ,
na r ,
ca r ,
Ia r ,
wavirt r , nai 2 r ,
cai 2 r ,
zai 2 r ,
I ai 2 r ,
r , tot 2 2 2 2
wa r , n r ,
za r ,
ca r ,
Ia r ,
a
Physical understanding of these relations can also be gained from the phasor diagram of the
frequency-domain n r , {or equivalently, that for c r , , z r , or I r , }:
a
a
Im na nai
a
a
na
nar 2 nai2
Re na nar
-28-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
While the real (imaginary) parts of the frequency-domain 3-D vector na r , amplitude (or
equivalently, the real (imaginary) parts of the frequency-domain 3-D vector ca r , , za r ,
or I a r , “amplitudes”) are physically associated with propagating sound radiation (non-
propagating acoustic energy), respectively, the fractional amounts of propagating versus non
propagating purely real, scalar acoustic energy density ( f arad r , vs. f avirt r , ) must be
based on the additive component^2 nature associated with the right triangle relation of the above
phasor diagram for na r , - i.e. Pythagoras’ theorem:
nr 2
ni 2
2
na nar 2 nai2 f arad f avirt a 2 a 2
na
na
Next, we explore relations between f akin r , , f apotl r , and f arad r , , f avirt r , , or
equivalently between wakin r , , wapotl r , and warad r , , wavirt r , . From conservation
of energy, we have:
watot r , wakin r , wapotl r , warad r , wavirt r ,
Or equivalently:
1 f akin r , f apotl r , f arad r , f avirt r ,
2
Using the relation: f apotl r , na r , f akin r , we can rewrite the above relation as:
2
1 f akin r , na r , f akin r , f arad r , f avirt r ,
1 n r, 2 f kin r , f rad r, f virt r,
a
a
a
a
Thus:
Or:
f akin r ,
1
f arad r , f avirt r ,
2
1 n r,
a
1
1
1 n r , 2
a
Or equivalently:
wakin r ,
1
warad r , wavirt r ,
2
1 n r,
a
watot r ,
1
watot r ,
2
1 n r,
a
2
Using the (same) relation for: f akin r , f apotl r , na r , , we also have:
2
1 f apotl r , na r , f apotl r , f arad r , f avirt r ,
-29-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
1 n r, 2
1
a
potl
f potl r, f rad r, f virt r,
Thus:
1 f r ,
a
a
n r , 2 a
n r , 2 a
a
a
n r, 2
n r, 2
a
f rad r , f virt r , a
f apotl r ,
a
2
2
a
1 n r,
1 n r ,
a
a
1
Or:
Or equivalently:
n r, 2
n r, 2
a
wrad r , wvirt r , a
wtot r ,
wapotl r ,
a
a
2
2
1 n r , 1 n r , a
a
a
watot r ,
Then as a check: f
kin
a
r , f apotl r ,
2
na r ,
1
1
2
2
1 na r ,
1 na r ,
1
2
na r ,
1
2
na r ,
Thus, we have obtained the relations:
f akin r ,
1
1
f rad r , f avirt r ,
2 a
1 n r , 1 n r , 2
a
a
1
and:
n r, 2
n r, 2
a
f rad r, f virt r, a
f apotl r ,
a
2
a
1 n r, 2
1 n r ,
a
a
1
Or equivalently:
wakin r ,
potl
a
w
1
1
wrad r , wavirt r ,
watot r ,
2 a
2
1 n r ,
1 n r ,
a
a
watot r ,
n r, 2
n r, 2
a
a
rad
virt
r , 2 wa r , wa r , 2 watot r ,
1 n r ,
1 na r , tot
a
wa r ,
Physically, from these relations we learn that the energy densities associated with propagating
sound radiation and non-propagating acoustical energy {locally sloshing back and forth each
cycle of oscillation} contribute linear-proportionally to the acoustic kinetic and potential energy
densities.
-30-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
We can equivalently write these relations in matrix form as:
1
f akin r ,
1
2
potl
2
f a r , 1 na r , na r ,
1
f rad r,
a
2
virt
na r , f a r ,
Note that the determinant of this 22 matrix is zero, i.e. this 22 matrix is singular, thus it has
no inverse…
warad r , nar 2 r ,
wavirt r , nai 2 r ,
rad
virt
However, since: f a r , tot
and: f a r , tot
wa r , n r , 2
wa r , n r, 2
a
a
Or, equivalently: w
rad
a
nar 2 r , tot
r , 2 wa r ,
na r ,
and: w
virt
a
nai 2 r , tot
r , 2 wa r ,
na r ,
Then, using: watot r , wakin r , wapotl r , on the RHS of both relations, we have:
rad
a
w
and:
virt
a
w
nar 2 r , tot
r , 2 wa r ,
na r ,
nar 2 r ,
wakin r , wapotl r ,
2
na r ,
nai 2 r , tot
r , 2 wa r ,
na r ,
nai 2 r ,
wakin r , wapotl r ,
2
na r ,
Dividing both sides of these relations by watot r , , we equivalently have:
and:
f
rad
a
nar 2 r ,
r , 2 f akin r , f apotl r ,
na r ,
1
nar 2 r ,
2
na r ,
f
virt
a
nai 2 r ,
r , 2 f akin r , f apotl r ,
na r ,
1
nai 2 r ,
2
na r ,
Then as a check, we can easily see that:
f
rad
a
2
nar 2 r , nai 2 r , nar 2 r , nai 2 r , na r ,
virt
1
r , fa r , 2 2
2
2
na r ,
na r ,
na r ,
na r ,
-31-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
We can equivalently write these relations in matrix form as:
f arad r ,
nar 2 r , nar 2 r , f akin r ,
1
virt
potl
2
i2
i2
f a r , na r , na r , na r , f a r ,
Note that the determinant of this 22 matrix is zero, i.e. this 22 matrix is singular, thus it has
no inverse…
Physically, we learn from these relations that the acoustic kinetic and potential energy densities
contribute linear-proportionally to the energy densities associated with propagating sound
radiation and non-propagating acoustical energy {locally sloshing back and forth 2× each cycle
of oscillation}.
We can now see/understand better why the 22 matrices in both cases are singular – because the
physics associated with how (kinetic vs. potential) energy density is parceled out into (propagating
vs. non-propagating) energy density {and/or vice-versa} is independent of the form/types of the
energy densities!
Conservation of Acoustic Linear Momentum:
In the linearized “every-day” sound field regime (sound pressure levels SPL r 134 dB ,
corresponding to p r 100 Pascals , and neglecting dissipative effects), if a complex
i r ,
harmonic sound field S r , t ; exists, with: p r , t ; p r , eit p r , e p
eit
iu y r ,
iu x r ,
i r ,
it
ˆ
and: u r , t ; u r , e u x r , e
x uy r , e
yˆ u z r , e uz
zˆ eit
in a volume V (with associated enclosing surface S) of interest, if no acousto-mechanical
excitation sources (or sinks) exist within the volume V/enclosing surface S, the net/total complex
acoustic force acting on the air within V , by Newton’s 2nd law is:
Fa t ; dGa t ; dt Newtons where Ga t ; Newton-sec kg -m sec is the total
complex acoustic linear momentum contained within the volume V/enclosing surface S.
We can write: Fa t ; f a r , t ; d , where f a r , t ; N m3 is the complex acoustic
V
force density at r , t ; . Likewise: Ga t ; g a r , t ; d , where
V
3
2
g a r , t ; N -sec m kg m -sec is the complex acoustic linear momentum density at
r , t; . Thus: fa r , t; g a r , t; t .
It can be shown that the complex acoustic force density
f a r , t ; g a r , t ; t Ta r , t ; , where Ta r , t ; is the so-called complex
acoustic stress tensor, a Hermitian rank-2 tensor (a 3×3 matrix), i.e. Ta r , t ; Ta† r , t ;
-32-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
where Ta† r , t ; is the conjugate transpose of Ta r , t ; . The acoustic stress tensor Ta r , t ;
has SI units of pressure – i.e. N m 2 Pascals {n.b. same units as energy density ( Joules m3 ) }.
Temporarily suppressing the space-time-frequency argument r , t ; , the acoustic stress tensor
Ta is constructed as follows (n.b. stress tensors are {always} expressed in Cartesian (x, y, z)
components}: Ta 12 ou u * La 1̂ where the symbol represents the tensor product
(aka outer, or dyadic product), the Lagrangian acoustic density La wakin wapotl
Joules m
3
{n.b. a purely real quantity) and 1̂ is the 3×3 unit matrix.
For reference purposes, the tensor/outer/dyadic product of any two arbitrary 3-D space
vectors a ax xˆ a y yˆ az zˆ and b bx xˆ by yˆ bz zˆ , representing each of the two 3-D space
ax xˆ
bx xˆ
ˆ
vectors as 3×1 column matrices: a a y y and b by yˆ is:
a zˆ
b zˆ
z
z
ˆ ˆ ax by xy
ˆˆ ax bz xz
ˆˆ
ax bx xx
ax xˆ
ˆ ˆ
T
ˆ ˆ a y by yy
ˆ ˆ a y bz yz
ˆˆ
a b a b a y y bx x by yˆ bz zˆ a y bx yx
a zˆ
az bx zx
ˆ ˆ az by zy
ˆ ˆ az bz zz
ˆˆ
z
The i-jth component of the complex acoustic stress tensor (i, j = 1, 2, 3 {= x, y, z}) is thus:
Taij 12 oui u *j iˆ ˆj 14 o
p
2
zo2
u
2
iˆ ˆj
ij
1
2
o ui u *j iˆ ˆj 14 o
p
2
o2 c 2
u
2
iˆ ˆj
ij
where: ij 10 ifif ii jj is the Kroenecker -function. Explicitly writing out Ta in matrix form:
u 2 u 2 u 2 p 2 xx
ˆˆ
ˆˆ
2u x u*y xy
y
z
o2 c 2
x
2
u 2 u 2 u 2 p yy
ˆˆ
ˆˆ
2u y u *x yx
Ta 14 o
2
z
x
o c 2
y
u 2
ˆˆ
ˆˆ
2u z u *x zx
2u z u *y zy
z
ˆˆ
2u y u *z yz
2
2
2
p
ˆˆ
u x u y 2c2 zz
o
ˆˆ
2u x u *z xz
The 3 diagonal elements of the acoustic pressure tensor Taii iˆˆi {n.b. purely real quantities} are
physically interpreted as pressures acting in the ith direction on a surface with outward-pointing
unit normal in the ith direction, whereas the 6 off-diagonal elements Taij iˆ ˆj Ta*ji ˆj iˆ {n.b. in
general complex} are physically interpreted as shears – an areal force density (i.e. force per unit
area) acting in the ith direction on a surface with outward-pointing unit normal in the jth direction.
-33-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
Then:
Ta 14 o xˆ
yˆ
zˆ
y
z
x
u 2 u 2 u 2 p 2 xx
ˆˆ
ˆˆ
ˆˆ
2u x u*y xy
2u x u *z xz
y
z
o2 c 2
x
2
*
u 2 u 2 u 2 p yy
ˆˆ
ˆ
ˆ
ˆ
ˆ
u
u
yz
2u y u *x yx
2
z
x
y
z
o2 c 2
y
2
2
2
2
p
u u u
zz
ˆˆ
ˆˆ
ˆˆ
2u z u*x zx
2u z u *y zy
2 2
z
x
y
c
o
2
2
2
2
p
u
u
u
xˆ 2u y u *x xˆ 2u z u *x xˆ
2
x
y
z
o c 2
y
z
x
2
2
2
2
p
14 o 2u x u *y yˆ u y u z u x 2c2 yˆ 2u z u *y yˆ
o
y
z
x
2
2
2
2
2u x u *z zˆ 2u y u *z zˆ u z u x u y p2c2 zˆ
o
y
z
x
After some more work on carrying out the –ve divergence of Ta , one {indeed} obtains:
1
Ta 2
c t
1
2
1 I a
*
p u 2
c t
But the complex acoustic force density f a g a t Ta
1
c2
I a t N m3 , thus we see
that the complex acoustic linear momentum density is related to the complex acoustic intensity
2
via: g a I a c , the acoustic analog of that e.g. for propagation of EM waves in
electrodynamics, where the EM field linear momentum density g
N -sec m3 kg m2 -sec is
EM
2
related to Poynting’s vector S EM = 12 E H * Watts m 2 via: g EM S EM c .
So: I a c 2 g a , but we also have the relation: I a
ca watot
za
where: ca
, thus:
2 2
1
o
c
2 1 ca
ca watot
or:
2 2
1
1
c
c
a
2
2 g a 1 2 2 g a ca* 1 2 2 g a ca* 1
2 *
tot
2
1 2
wa 2 c ca
c
c
2 c ca
1
* 2 c ca
a g a ca
2
c
c c
2
a
a
a
ca
c g a
2
-34-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
2
2
2
14 u 12 1 c 2 g a ca* 12 g a ca* 12 c 2 g a ca* . If we {tentatively}
ca
ca
2
2
12 g a ca* and: wakin 14 u 12 c 2 g a ca* , then we can check these relations
But: watot wapotl wakin
assign: wapotl
2
1 p
4 c2
o
2
1 p
4 c2
o
ca
za
wapotl ca
with the ratio: kin 2 2
wa
c
zo
2
2
2
ca
g a ca*
* c 2 .
c2
g
2
a ca
1
2
1
2
ca
Thus, we also have the relations:
watot wapotl wakin
2
1 p
4 c2
o
2
14 u 12 1
c2
2
ca
* 1 * 1
g a ca 2 g a ca 2
c2
2
ca
g a ca*
where:
wapotl
p
2
1
4 c2
o
2
12 g a ca* and: wakin 14 u
2
1 c
2
ca
2
2
2
za
wapotl ca
*
2 2 .
g a ca , with:
wakin
c
zo
We also have the relations:
z
nr 2
cr 2
ni 2
ci 2
c
warad a 2 watot a 2 watot and: wavirt a 2 watot a 2 watot where: na a a .
c zo
na
ca
na
ca
Thus, we also see that:
rad
a
w
car 2 tot
2 wa
c
a
1
2
car 2
2
c
a
1
c2
2
c
a
cai 2 tot
*
virt
g a ca and: wa 2 wa
ca
1
2
cai 2
2
c
a
1
c2
2
ca
*
g a ca
Acoustic Angular Momentum Density and Conservation of Acoustic Angular Momentum:
We can now also define the complex acoustic angular momentum density:
a r , t ; r g a r , t ; N -sec m 2 kg m-sec
Note that this quantity is defined with respect to the local origin associated with r .
The complex acoustic force density f a r , t ; g a r , t ; t Ta r , t ; .
The complex acoustic torque density a r , t ; r f a r , t ; N m 2 , but the complex
acoustic torque density is also equal to the time rate of change of the complex acoustic angular
momentum density, i.e. a r , t ; a r , t ; t . Since a r , t ; r g a r , t ; , if r is a
constant vector {i.e. r fcn t }, then:
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Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
a r , t ; r g a r , t ;
g a r , t ;
r
a r , t;
N m2
t
t
t
Hence {for r fcn t }, we see that the complex acoustic torque density is:
a r , t ; g a r , t ;
a r , t; r f a r , t;
r
r Ta r , t ;
t
t
N m
2
This relation is in fact an expression for conservation of angular momentum associated with
the complex acoustic field. We can {perhaps} see this more clearly in an equivalent manner:
We can write the kth component {k = 1, 2, 3 (= x, y, z)} of the complex acoustic angular
momentum density ak r , t ; and its associated cross product r g a r , t ; , temporarily
k
i j
j i
suppressing the r , t ; argument(s), as: ak 12 kij x g a x g a where:
r xxˆ yyˆ zzˆ x1eˆ1 x 2 eˆ 2 x 3eˆ3 = constant vector, and kij is the so-called totally
antisymmetric rank-3 Ricci tensor, whose properties are such that: kij 1 for kij = even
permutation of (1, 2, 3), kij 1 for kij = odd permutation of (1, 2, 3) and kij 0 for kij = not a
permutation of (1, 2, 3). The notation {here} adopts the so-called “Einstein summation
i j
j i
convention”, i.e. that repeated indices are summed over. Hence, in ak 12 kij x g a x g a ,
we must sum over {both} indices i, j = 1, 2, 3 (= x, y, z) in this expression.
It is then also possible to relate ak to a new, rank-two anti-symmetric tensor Lija via:
ak 12 kij Lija . Conservation of complex acoustic angular momentum density (i.e. conservation
of complex angular momentum in differential form) can then be expressed as the –ve divergence
of a rank-3 complex acoustic “moment-of-a-force-density” tensor M akij :
M a1ij M a2ij M a3ij
Lija
M aij where: k 1 2 2
1
2
2
x x x
t
x
x
x
The rank-3 complex acoustic “moment-of-a-force-density” tensor M aij is related to the rank-2
ij
i j
j i
complex acoustic stress tensor T {for r fcn t } via: M T x T x .
a
a
a
a
Then in integral form, conservation of complex acoustic angular momentum can be written as:
d
d
a r , t ; d 12 kij eˆ k Lija r , t ; d
dt V
dt V
k
12 kij eˆ M aij r , t ; d 12 kij eˆ k n M aij r , t ; da
V
S
where we have used the divergence theorem in the last term, and nˆ n eˆ is the th outward
pointing unit normal { = 1, 2, 3 (= x, y, z)} associated with surface S, enclosing volume V.
-36-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
An explicit experimental demonstration of the acoustic angular momentum density/acoustic
torque was published in the paper “Circularly Polarized Acoustic Field: The Number Theory
Connection”, M.R. Schroeder, Acustica 75, p. 94-98 (1991). It’s a nice acoustic analog of EM wave
angular momentum density/EM wave torque: see e.g. “Mechanical Detection and Measurement of
the Angular Momentum of Light”, R.A. Beth, Physical Review 50, p. 115−125 (1936). See also “A
Radiation Torque Experiment”, P.J. Allen, Am. J. Phys. 34, p. 1185−1192 (1966).
-37-
Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
UIUC Physics 406 Acoustical Physics of Music
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Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois
2002-2016. All rights reserved.
