example # 3: point monopole sound source – spherical waves … · 2017-04-11 · lecture notes...

UIUC Physics 406 Acoustical Physics of Music

Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois

2002 - 2017. All rights reserved.

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Examples of Complex Sound Fields (Continued):

Example # 3: Point Monopole Sound Source – Spherical Waves Propagating in “Free Air”:

Note: there exists no electromagnetic analog for this acoustic example, due to the manifest vectorial nature of the EM field – which is mediated at the microscopic level by the spin-1 photon. So-called electric monopole {E(0)} and/or magnetic monopole {M(0)} EM radiation associated e.g. with a spherically-symmetric, radially oscillating electric charge distribution

3, i te or t q r e

and/or magnetic charge distribution 3, i tm or t g r e

cannot occur.

Imagine a spherically-symmetric, point sound source located at the origin of coordinates 0r

that isotropically emits monochromatic spherical acoustic waves into “free air”. The 3-D wave equation describing the behavior of the instantaneous/physical {i.e. purely real time-domain} over-pressure

,p r t

at the space-time point ,r t

is an inhomogeneous, linear 2nd-order differential equation:

2

2 32 2

,1, 4 coso

p r tp r t B r t

c t

The gradient

and Laplacian 2 operators in 3-D spherical-polar , ,r coordinates are:

1 1ˆ ˆˆsin

rr r r

and: 2

2 22 2 2 2 2

1 1 1sin

sin sinr

r r r r r

The RHS of this 3-D wave equation is originates from 2 31 4r r

, thus 34 cosoB r t

describes the point sound source located at the origin 0r

, radiating sound isotropically into 4 steradians. The function 3 r

is known as {Dirac’s} 3-D delta function, which has many intriguing

mathematical properties, one of which is that the 3-D delta function 3 r has an (infinite!) spike at

the origin 0r

and is 0 elsewhere. Thus, we can equivalently write 3 3 0r r

. Note that

in spherical-polar coordinates , ,r , that 2

3 14 r

r r

where 0r r is the 1-D delta

function in the radial (r) direction only. If we integrate the 3-D delta function over a volume V containing the origin 0r

, e.g. integrate over all space:

2

23 3 2 1

40 0 0sin

r

rV rr dV r r dr d d

2r r

2

0 0 0

21 1

4 2 20 0 0

sin

sin

r

r

r

r

dr d d

r dr d d

2

0 0

11 1 12 2 20 0 1 0

sin

cos

r

r

r u r

r u r

r dr d

r dr d r dr du

2 0

1r

rr dr

If the volume V does not contain the origin 0r

, then: 3 0V

r dV

.




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Note also that the 3-D {1-D} delta functions 3 r { r } have SI units of 3m { m },

respectively; the integrals 3 1V

r dV

and 0

1r

rr dr

respectively, are dimensionless.

If we now integrate the above inhomogeneous 2nd order linear differential equation over a {finite} arbitrary volume V but e.g. centered on, and thus containing the origin 0r

, where the

isotropic point sound source is located:

2

2 32 2

,1, 4 cosoV V V

p r tp r t dV dV B r dV t

c t

Then using the Gauss divergence theorem: 2 ˆ, , ,V V S

p r t dV p r t dV p r t ndS

where n is the outward-pointing unit normal to the surface S {which encloses/bounds the

volume V} and: 3 1V

r dV

, the above integral relation then becomes:

2

2 2

,1ˆ, 4 cosoS V

p r tp r t ndS dV B t

c t

A spherically-symmetric time-dependent scalar ,p r t

over-pressure has rotational

invariance/rotational symmetry and therefore cannot have any explicit - and/or -dependence

– only r-dependence. Thus , , ,p r t p r t fcn

, and hence for a spherically-symmetric

point sound source located at the origin of coordinates, then:

,1 1ˆ ˆˆ ˆ, , ,sin

p r tp r t p r t r p r t r

r r r r

and the 3-D integral wave equation for the scalar over-pressure ,p r t

associated with this

isotropic point sound source (for 0r ) becomes:

2

2 2

, ,1ˆ ˆ 4 cosoS V

p r t p r tr ndS dV B t

r c t

The instantaneous/physical (i.e. purely real time-domain) solution to this integral wave equation is a purely real spherical-outgoing harmonic over-pressure wave:

, cos oBp r t t kr Pascals

r . Note that the constant oB has SI units of -Pascal m .

The instantaneous/physical (i.e. purely real time-domain) particle velocity ,u r t

associated

with this problem is determined via use of the {linearized} Euler’s equation for inviscid fluid flow:

, 1,

o

u r tp r t

t




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Since , , ,p r t p r t fcn

, then ˆ, , ,p r t p r t p r t r r

and thus the

instantaneous/physical (i.e. purely real time-domain) particle velocity ,u r t

associated with a

spherically-symmetric monochromatic point sound source can only be oriented in the radial direction, i.e. r ˆ, , ,u r t u r t r fcn

:

, 1

,o

u r tp r t

t

r , ,1ˆ ˆ

o

u r t p r tr r

t r

r , ,1

o

u r t p r t

t r

Now: 2

,cos cos sino o o

p r t B B kBt kr t kr t kr

r r r r r

r2

, ,1 1Then: cos sin

1 cos sin

o o

o o

o

o

u r t p r t B kBt kr t kr

t r r r

Bt kr k t kr

r r

Thus:

r

1 1, sin cos cos sin o o

o o

B B ku r t t kr k t kr t kr t kr m s

r r r kr

Using c k and o oz c this relation becomes:

r

1, cos sin o

o

Bu r t t kr t kr m s

z r kr

We can then “complexify” the radial-outgoing spherical over-pressure and particle velocity waves:

, cosoBp r t t kr

r , cos sin i t kro oB B

p r t t kr i t kr er r

and: r

1 1, cos sino

o

Bu r t t kr t kr

z r kr

r

1 1 1, cos sin sin cos

1 1 cos sin cos sin

1 1 1 1

o

o

o

o

i t kr i t kro o

o o

Bu r t t kr t kr i t kr t kr

z r kr kr

Bt kr i t kr i t kr i t kr

z r kr

B B ie i e

z r kr z r kr

1 1 ,i t kr

o

ie p r t

z kr




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The relationships between the complex time-domain and complex frequency-domain over-pressure and radial particle velocity amplitudes for the point monopole sound source are:

, ,i t kr i toBp r t e p r e

r i.e. , ikroB

p r er

r r

1, 1 ,i t kr i to

o

B iu r t e u r e

z r kr

i.e. r

1, 1 ikro

o

B iu r e

z r kr

Note that the purely real “amplitude” oB is in general complex -BiB B e Pa m , in order to

accommodate an (arbitrary) absolute phase B , which we can always “rotate” away, simply by

re-defining the zero of time: Bt t }. So, to make life simpler, we can set 0B , then

oB B B is purely real. The magnitudes of the complex time-domain over-pressure and

radial particle velocity amplitudes then are:

, ,B

p r t p rr

and: 2

r r

1, 1 1 ,

o

Bu r t kr u r

z r

The phases of the complex pressure and radial particle velocity are:

0p B and: r

1 1 1tan 1 tan 1 tan 1u B Bkr kr kr

Thus, we also see that: r

1 1tan 1 tan 1 p u p u B B z Ikr kr

n.b. the phase difference r p u p u z I is independent of the absolute phase B .

Let us examine the behavior of time-domain ,p r t and r ,u r t as r increases from r = 0.

The complex over-pressure , i t krop r t B r e simply decreases with increasing r, modulated

by the {complex} oscillatory factor i t kre . The behavior of r

1, 1 i t kr

o

B iu r t e

z r kr

is

mathematically more interesting… When 0r {i.e. in proximity to the point sound source}, then 1kr , and the imaginary part of r ,u r t dominates, because then 1 1i kr kr , thus for 0r

1kr , r ,u r t will be ~ 90 out-of-phase/in quadrature with the complex over-pressure

,p r t , and hence in proximity to the point sound source, both the frequency domain complex

radial specific acoustic impedance r, , ,az r p r u r and the frequency domain complex

radial acoustic intensity *1r2, , ,aI r p r u r will be largely imaginary/reactive/non-

propagating – i.e. for 0r 1kr , sound energy is largely stored locally, oscillating/sloshing

back-and-forth during each cycle of oscillation…




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However, far from the sound source, when r 1kr , then 1 1i kr kr and

r ,u r t is predominantly real, and also in-phase with the complex over-pressure ,p r t , and

hence far from the point sound source, both the frequency-domain complex radial specific acoustic impedance

r r, , ,az r p r u r and frequency-domain radial acoustic intensity

r

*1r2, , ,aI r p r u r will be largely real quantities associated with active/propagating

sound radiation originating from the {distant} point sound source.

The “near-field” 1kr and the “far-field” 1kr behavior of the complex sound field

,S r t associated with an isotropic point sound source emitting monochromatic spherical

outgoing waves can be easily understood from a physically-intuitive perspective:

Far from the point sound source r , the spherical waves are to an increasing degree

nearly perfect plane waves – the curvature of the spherical wavefronts becomes increasingly neglectable as r increases far from the point sound source. As we have seen in the previous P406 Lecture Notes 12, for a monochromatic plane/traveling wave propagating in “free air”, the complex over-pressure ,p r t and {longitudinal} particle velocity ,u r t are perfectly

in-phase with each other; hence {here} both the frequency domain complex radial specific

acoustic impedance r r, , ,az r p r u r o oc z and the frequency domain radial

acoustic intensity r

*1r2, , ,aI r p r u r will be purely real quantities associated with

active/propagating sound radiation in the form of these monochromatic plane waves.

However, in proximity to the point sound source 0r , the curvature of the spherical

wavefronts becomes increasingly important as 0r , one consequence of which is that the radial particle velocity r ,u r t becomes increasingly more and more imaginary/out-of-phase

with the complex over-pressure ,p r t as the curvature of the spherical wavefronts becomes

more and more significant as 0r .

In the “intermediate field” region associated with an isotropic point sound source, this is where ~ 1kr , and the complex particle velocity r ,u r t has ~ equal real and imaginary

components, and hence is ~ 45 out of phase with the complex over-pressure ,p r t .

Thus, for a complex sound field ,S r t associated with an arbitrary sound source, if the

wavefronts are non-planar (such as would be expected in proximity to the sound source), the particle velocity ,u r t will acquire an increasingly large imaginary component as the wavefronts become

increasingly non-planar, ,u r t will become increasingly disparate in phase with respect to the

complex over-pressure ,p r t . Consequently, the complex specific acoustic impedance r

,az r and

complex acoustic intensity r

,aI r will acquire increasingly reactive/non-propagating/imaginary

components as the wavefronts become increasingly non-planar, in proximity to the sound source…




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The complex sound field ,S r t associated with a “point” monopole sound source radiating

monochromatic, radially-outgoing spherical waves thus has three basic regions, or zones:

(a) The “near” zone: 1kr , the radial particle velocity 2r , ~ ou r t i B z kr is largely

imaginary (i.e. reactive), decreasing as 2~ 1 r (while the pressure , ~p r t B r decreases

as ~ 1 r ), and where the particle velocity lags the pressure by

1 1 otan 1 ~ tan ~ 90p u kr in phase. This region of the complex sound field

,S r t is reactive, largely consisting of non-propagating acoustic energy, and is also

inertia-like (i.e. mass-like), because r rIm 0iu r u r for 1kr .

(b) The “intermediate” zone: ~ 1kr , the radial particle velocity r , ~ 1 2 ou r t i B z r has ~

comparable real and imaginary components, decreasing somewhat/slightly faster than ~ 1 r

(while the pressure , ~p r t B r decreases as ~ 1 r ), and where the particle velocity lags

the pressure by 1 1 otan 1 ~ tan 1 ~ 45p u kr in phase. This region of the complex

sound field ,S r t is ~ an equal mix of propagating and non-propagating acoustic energy.

(c) The “far” (or radiation) zone: 1kr , the radial particle velocity r , ~ 1 ou r t B z r is

largely real (i.e. active), decreasing ~ 1 r (while the pressure , ~p r t B r also decreases

as ~ 1 r ), and where the radial particle velocity is in-phase with the pressure, i.e.

1 1 otan 1 ~ tan 0 ~ 0p u kr . This region of the complex sound field ,S r t is

active, dominated by propagating acoustic energy.

The radial complex specific acoustic impedance and its magnitude are:

r 2

r

1, 1,

, 1 1 1a o o

i krp rz r z z

u r i kr kr

and:

r

2

2 2

1 1 1,

1 1 1 1a o o

krz r z z

kr kr

Note that since r r

, ,a o az r c r these relations can be written in dimensionless form as:

r r

2

, ,1

1 1

a a

o

z r c ri kr

z ckr

and:

r r

2

, ,1

1 1

a a

o

z r c r

z ckr

The real and imaginary parts of the complex radial specific acoustic impedance associated with the point monopole sound source are thus:

r r

r

2

1, Re ,

1 1a a oz r z r z

kr

and: r r

i

2

1, Im ,

1 1a a o

krz r z r z

kr




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Likewise, the real and imaginary parts of the complex radial acoustic energy flow velocity are:

r r

r

2

1, Re ,

1 1a ac r c r c

kr

and: r r

i

2

1, Im ,

1 1a a

krc r c r c

kr

The phase of the complex radial specific acoustic impedance and complex radial acoustic energy flow velocity is:

1tan 1a az c p u p u kr

(a) In the “near” zone: 1kr , the complex radial specific acoustic impedance and the

complex radial energy flow velocity are both largely imaginary (i.e. reactive):

r r

21 1 1 ~a o az r z c r c i kr kr i kr , increasing ~ linearly with r, with

fractional magnitude r r

21 1 1 ~ 1a o az r z c r c kr kr and phase

1 1 otan 1 ~ tan ~ 90a az c p u kr . Again, the complex sound field ,S r t

in

this region is inertia-like (i.e. mass-like), because r r

i Im 0a az r z r for 1kr .

(b) In the “intermediate zone: ~ 1kr , the complex radial specific acoustic impedance and the complex radial energy flow are both ~ an equal mix of active (i.e. real) and reactive (i.e.

imaginary) components: r r

21 1 1 ~ 1 2a o az r z c r c i kr kr i , with

fractional magnitude r r

21 1 1 ~ 1 2a o az r z c r c kr and phase

1 1 otan 1 ~ tan 1 ~ 45a az c p u kr .

(c) In the “far” (or radiation) zone: 1kr , the complex radial specific acoustic impedance and the complex radial energy flow velocity are largely real (i.e. active):

r r

21 1 1 ~ 1a o az r z c r c kr with magnitude

r r

21 1 1 ~ 1a o az r z c r c kr and phase

1 1 otan 1 ~ tan 0 ~ 0a az c p u kr .

The frequency domain complex radial acoustic intensity associated with the point acoustic monopole also points outward in the radial ( r ) direction; it and its magnitude are:

r

2*r 2

1 1 1, , , 1

2 2o

ao

B iI r p r u r

z r kr

and: r

22

2

1 1, 1 1

2o

ao

BI r kr

z r

The real and imaginary parts of the frequency domain radial complex acoustic intensity are:

r r

2r

2

1 1, Re ,

2o

a ao

BI r I r

z r and:

r r

2i

2

1 1 1, Im ,

2o

a ao

BI r I r

z r kr




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The phase of the frequency domain radial complex acoustic intensity is:

1tan 1a a aI z c p u p u kr

(a) In the “near” zone: 1kr , the frequency domain complex radial acoustic intensity is largely

imaginary (i.e. reactive): r

2 3, ~ 2a o oI r i B z kr , with magnitude r

2 3, ~ 2a o oI r B z kr

(decreasing as 3~ 1 r ) and phase 1 1 otan 1 ~ tan ~ 90I z p u kr . Again, the

complex sound field ,S r t in this region is inertia-like (i.e. mass-like), because

r r

i Im 0a aI r I r for 1kr .

(b) In the “intermediate zone: ~ 1kr , the frequency domain complex radial acoustic intensity is ~ an equal mix of active (i.e. real) and reactive (i.e. imaginary) components:

r

2 2, ~ 1 4a o oI r i B z r , with magnitude r

2 2, ~ 4a o oI r B z r (decreasing slightly

faster than 2~ 1 r ) and phase 1 1 otan 1 ~ tan 1 ~ 45a a aI z c p u kr .

(c) In the “far” (or radiation) zone: 1kr , the frequency domain the complex radial acoustic intensity is largely real (i.e. active):

r

2 2, 2a o oI r B z r , with magnitude

r

2 2, 2a o oI r B z r (decreasing as 2~ 1 r ) and phase

1 1 otan 1 ~ tan 0 ~ 0a a aI z c p u p u kr

.

The frequency domain complex acoustic power and its magnitude associated with the point monopole sound source are:

r

2

ˆ, , 2 1oa aS

o

B iP r I r r dS

z kr

and: 22 1

, 2 1oa

o

BP r

z kr

Notice that both the frequency domain point monopole complex acoustic power and its magnitude do indeed have an explicit r-dependence associated with them – we are used to thinking/told that they are not supposed to have an explicit r-dependence, because this is only true for the propagating portion of the complex acoustic power – i.e. the real acoustic power!!!

The real and imaginary parts of the frequency domain complex acoustic power are:

2

r , Re , 2 oa a

o

BP r P r constant fcn r

z

and:

2

i 1, Im , 2 !!!o

a ao

BP r P r explicit fcn r

z kr

The phase associated with the frequency domain complex acoustic power is:

1tan 1a a a aP I z c p u p u kr




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In our previous discussions of acoustic power Pa associated e.g. with “point” sound sources, the context was always with regard to propagating sound – i.e. sound radiation – purely real (and/or time-averaged) acoustic power.

We see from above that the real component of the complex acoustic power of a monopole sound source – which is associated with propagating sound energy – is indeed a constant, whereas the imaginary component of the complex acoustic power of a monopole sound source – which is associated with non-propagating acoustic energy – is explicitly r-dependent, and especially so in the “near” zone, when 1kr .

The purely real, scalar frequency domain potential, kinetic and total acoustic energy densities associated with a “point” monopole sound source radiating monochromatic, radially-outgoing spherical waves are:

2

2 23

2 2 2 2 2

,1 1 1,

4 4 4potl o o oa

o o o

p r B Bw r Joules m

c c r z r

22

2* 3r r r 2 2

1 1 1, , , , 1

4 4 4kin o oa o o

o

Bw r u r u r u r Joules m

z r kr

2 22 2 2

32 2 2 2 2 2

1 1, , , 1 2

4 4 4tot potl kin o o o o o oa a a

o o o

B B Bw r w r w r Joules m

z r z r kr z r kr

Note that the ratio of potential energy density to kinetic energy density is:

2 2

2 22

2

r

,14, 4

1, ,4

o opotl

oa okina

o

Bp rz rw r c

w r u r

2

2 24o o

o

B

z r

2 2

11

1 11 1

kr kr

In the “near” zone, 1kr where the complex radial particle velocity, specific acoustic impedance, energy flow velocity, acoustic intensity and power are largely reactive (i.e. imaginary), , ,potl kin

a aw r w r . Only in the “far” (i.e. radiation) zone, when 1kr , and

moreover, when kr (i.e. in free-field conditions) does , ,potl kina aw r w r .

Please see/look at plots of the above complex acoustic quantities associated with the point acoustic monopole, available on the Physics 406 Software web-page, at the following URL:

http://online.physics.uiuc.edu/courses/phys406/406pom_sw.html




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Example # 4: The Physical “Point” Monopole Sound Source:

At very low frequencies, a loudspeaker mounted in a fully-enclosed/sealed cabinet of characteristic dimension a, with 1ka (i.e. 2f c a {using 2k , 2 f and

c f k }) approximates a “point” monopole sound source – the directivity factor, Q of a

typical enclosed loudspeaker is very nearly 1 (i.e. isotropic) at frequencies 2f c a . For example, for a typical “bookshelf”-type loudspeaker with a ~ 1 ft ~ 0.3 m, then

2 343 0.6 ~ 180 f c a Hz , or equivalently 1.9 m .

If we use the “spherical cow” approximation, i.e. model a physical monopole sound source as a radially-pulsating sphere of radius a, subject to the restriction 1ka , then the acoustical properties of such a device (for r > a) will closely approximate that of an ideal, point monopole sound source.

How do we characterize the strength of a physical monopole sound source – i.e. a radially- pulsating sphere of radius a? Typically, this is done by considering the {complex} volumetric velocity (aka volume velocity) of the physical monopole source, evaluated at the radius a of the sphere – the {radial} outward volume rate (or flow ) of fluid (i.e. air) from this sphere:

3r ˆ ˆ, 4 1 i t ika i t

a So

B iQ e u r a t r ndS a e e m s

z ka

Thus, the {complex} source strength/volume velocity of a physical monopole is:

34 1 ikaa

o

B iQ a e m s

z ka

Since 1ka , then 1 1ka , we can therefore approximate this expression as:

3

1 0

4 cos sin 4 4 4 ao o o o

B B B BQ i ka i ka i i i m s

z k z k ck

Thus, we see that: 2 -4

oaB i Q Pascal m kg s

Expressed in terms of the complex source strength/volume velocity 3 aQ m s of the physical

monopole sphere of radius a, the time-domain and frequency-domain complex pressure and radial particle velocity associated with this sound source are (for r > a):

, ,4

i t kr i t kr i to aQBp r t e i e p r e

r r

i.e. ,

4ikro aQ

p r i er

r r

1, 1 1 ,

4i t kr i t kr i ta

o

QB i iu r t e i e u r e

z r kr c r kr

i.e. r , 1

4ikraQ i

u r i ec r kr




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The complex radial specific acoustic impedance and energy flow velocity are (unchanged):

r r

2

, ,11

1 1 1

a a

o

z r c ri kr

z i kr ckr

The frequency-domain complex radial acoustic intensity (for r > a) is:

r

222

*r 2 2 2

1 1 1, , , 1 1

2 2 32

aoa

o

QB i iI r p r u r

z r kr c r kr

Note that r

2,aI r f - i.e. an acoustic monopole sound source is not efficient at very low

frequencies in terms of generating sound….

The frequency-domain complex acoustic power associated with the physical monopole sound source (for r > a) is:

r

22

2ˆ, , 2 1 18

a

a a oSo

QB i iP r I r r dS

z kr c kr

The frequency-domain potential, kinetic and total energy densities associated with the physical monopole sound source (for r > a) are:

22

23

2 2 2 2

,1,

4 64

apotl oa

o

Qp rw r Joules m

c c r

2

222* 3

r r r 2 2 2

1 1 1, , , , 1

4 4 64

akin oa o o

Qw r u r u r u r Joules m

c r kr

2

223

2 2 2

1, , , 2

64

atot potl kin oa a a

Qw r w r w r Joules m

c r kr




-12-

Example # 5: The Compact Physical Dipole Sound Source:

By the principle of linear superposition {for 134 SPL s dB }, we can create a so-called compact/physical dipole sound source using two out-of-phase physical monopole sources, of

source strength/volume velocity aQ , and separated from each other by a distance 2d, and subject

to the requirement that 1kd (i.e. 2f c d or 2d c f ), as shown in the figure below:

The time-domain and the frequency-domain total/resultant complex over-pressure amplitudes at the observer/listener’s point ( )P r

in the above figure is the linear sum of the

individual complex over-pressures associated with each monopole source:

1 2 1 21 2

1 2 1 2

1 1 1 1, , , ,

4ikr ikr ikr ikri t i t i to a

tot tot

Qp r t p r t p r t B e e e i e e e p r e

r r r r

The time-domain and frequency-domain total/resultant complex particle velocity at the observation/listener’s point ( )P r

in the above figure is the vector sum of the individual complex

particle velocities associated with each monopole source:

1 2 1 1 2 2ˆ ˆ, , , , ,totu r t u r t u r t u r t r u r t r

z

y

x

2

1

r

2r

1r

aQ

aQ

P r

d

d




-13-

1 2

1 2

1 21 1 2 2

1 21 1 2 2

1 1ˆ ˆ, 1 1 using: and:

4

1 1ˆ ˆ 1 1

4

ikr ikr i t otot a o o

o

ikr ikra

B i iu r t e r e r e B i Q z c

z r kr r kr

Q i ii e r e r

c r kr r kr

,i t i t

tote u r e

The acoustic monopole sources, of source strength/volume velocity aQ are located at 1 ˆd dz

and 2 ˆd dz

with 1 2d d d

. Vectorially, we see that 1 1r d r

and also that 2 2r d r

, with

ˆr rr

and 2 2 2r r x y z

. In Cartesian coordinates ˆ ˆ ˆ ˆsin cos sin sin cosr x y z .

We also see that: 1 1r r d

and 2 2r r d

. Using the law of cosines 2 2 2 2 cosc a b ab :

2 21 1 2 cosr r r d rd

and 2 2 2 22 2 2 cos 2 cosr r r d rd r d rd

,

with 1 1 1 1 1 1ˆ ˆ ˆ ˆr r r r r r d rr dz

and 2 2 2 2 2 2ˆ ˆ ˆ ˆr r r r r r d rr dz

with 1 1ˆ ˆ ˆr rr dz r

and 2 2ˆ ˆ ˆr rr dz r .

Since 1 2ˆ ˆ ˆr r r (especially in the “near-field” region), the above expression for the total /

resultant complex vector particle velocity ,totu r t is not easy to evaluate, analytically.

However, note that it does simplify considerably in the “far”-field region, where 1 2ˆ ˆ ˆr r r and

1 2r r r d .

It is quite clear from the above formulae for ,totp r t and ,totu r t

{as well as from previous

P406 lectures on sound interference effects with 2 (or more) sources} that interference effects will indeed manifest themselves here in this situation, albeit in a much more complicated manner….

However, the nature of this problem is such that all of the above quantities that we have calculated analytically e.g. for the various simpler sound sources can be also easily coded up on a computer, e.g. using MATLAB, Mathematica or e.g. a C/C++ based-program coupled to a graphics software package for plots, not just for them, but for this problem as well…

Computational calculations can also be done for {arbitrarily} higher-order acoustic multipoles – e.g. linear {and/or crossed (aka lateral)} quadrupoles (the tuning fork is an example of a linear quadrupole), sextupoles, octupoles, hexadecapoles, arbitrary linear 1-D acoustic arrays, 2-D/3-D acoustic arrays, all using the principle of linear superposition for N monopole sound sources… an arbitrary sound source can always be decomposed into a linear combination of acoustic multipoles, with suitably chosen complex coefficients – the multipole strengths…

Some very nice animation demos of pressure fields for monopole, dipole, quadrupole… sound sources exist e.g. at Prof. Dan Russell’s website:

http://www.acs.psu.edu/drussell/Demos/rad2/mdq.html




-14-

Example # 6: The Plane Circular Piston (A Simple Acoustics Model of a Loudspeaker):

Consider a longitudinally oscillating piston of radius a mounted on an infinite, perfectly rigid {i.e. immovable} planar baffle {oriented in the x-y plane at z = 0} as shown in the figure below.

The surface of the plane circular piston oscillates harmonically back and forth along the

z -axis with complex velocity ˆ, i tp ou z t u e z . What is the resulting complex pressure

, , , 0,p r t p r t e.g. at an observation/listener point , , 0r r

lying in the x-z plane?

Note that the harmonically oscillating baffled plane circular piston is a spatially-extended sound source, i.e. it is not a point sound source. We can consider the harmonically oscillating baffled plane circular piston as a coherent linear superposition of infinitesimal, point sound sources, each of which isotropically radiate sound waves into the forward hemisphere { ˆ 0z } {since the piston is mounted on an infinite, rigid baffle lying in the x-y plane at z = 0}. Thus, each infinitesimal area element dS on the plane circular piston acts like a point sound

source, with infinitesimal volume velocity strength 3 odQ u dS m s .

From Example # 4 above, we learned that the complex time-domain over-pressure amplitude associated with an unbaffled physical point sound source {located at the origin} of volume

velocity strength aQ harmonically radiating sound waves isotropically into 4 steradians (with

directivity 1 ) a distance r away from this sound source is:

- ,

4i t kr i t kro a

un bafl

QBp r t e i e

r r

However, for a baffled physical point sound source {located at the origin} of volume velocity

strength aQ harmonically radiating sound waves isotropically only into the 2 steradians of the

{ ˆ 0z } forward hemisphere (with directivity 2 ) the complex time-domain over-pressure a distance r away from this sound source is 2 this, i.e.:

z

y

x

a ˆ, i tp ou z t u e z

r

r

, , , 0,p r t p r t

dS




-15-

-, 2 , 2 2 ,

4 2i t kr i t kr i t kr i t kro a o a

bafl un bafl o

Q QBp r t p r t e i e i e ip r e

r r r

The amplitude of the complex over-pressure a distance r away from the baffled point sound

source {located at the origin} is: , 2o o ap r Q r .

Hence, the amplitude of the complex over-pressure a distance r away from the baffled infinitesimal point sound source associated with the infinitesimal area element dS located at the

point on the surface of the plane circular piston {see figure above} is: , 2o o ap r Q r .

Hence the infinitesimal contribution to the complex time-domain over-pressure amplitude a distance r away from the baffled infinitesimal point sound source associated with the infinitesimal area element dS located at the point on the surface of the plane circular piston

is: , 2 2o o odp r dQ r u dS r where in the last step, we used the infinitesimal

relation odQ u dS .

The corresponding infinitesimal contribution to the complex time-domain over-pressure is:

, ,2 2

i t kr i t kr i t kro o oudQdp r t dp r t e i e i e dS

r r

Then, we simply need to sum up all of the individual infinitesimal contribution(s) ,dp r t

i.e. we need to integrate ,dp r t over the area of the plane circular piston, in order to obtain the

total complex time-domain over-pressure amplitude a distance r away from the center of the baffled harmonically oscillating plane circular piston, {which is taken to be the local origin of coordinates – see above figure}:

1,

2 2i t kr i t kro o o ou u

p r t dp r t i e dS i e dSr r

Referring to the above figure, we need to express r in terms of r , and angles and . Vectorially r r

, hence r r

, then:

2 2 22 2 22 2 2r r r r r r r r r r r r r

The vector lies in the x-y plane, and has components: ˆ ˆ ˆ ˆ ˆcos sinx yx y x y

.

It has magnitude: 2 2 2 2 2 2 2 2cos sin cos sinx y

.

The vector r

{n.b. fixed in space} has components {for the observation/listener position at r

lying off-axis, somewhere in the x-z plane}: ˆ ˆ ˆˆ ˆ ˆ0 sin cosx zr r x r z xx zz r x r z

and

has magnitude: 2 2 2 2 2 2 2 2 2 2sin cos sin cosx yr r r r x y r r r r




-16-

Thus, the dot product ˆ ˆ ˆ ˆcos sin sin cos sin cosr x y r x r z r .

Hence: 2 2 2 2 22 2 sin cosr r r r r ,

Or: 2 2 2 2 22 2 sin cosr r r r r r

Next, if we carry out the integration of the above integral over the surface area of the plane circular piston of radius a using e.g. the method of annular strips, the area element can be written as dS d d . The surface integral becomes:

2 2

2

0 0

2 2 sin cos

2 20 0

1 1,

2 21

2 2 sin cos

ai t kr i t kro o o o

a ik r ri to o

u up r t i e dS i e d d

r ru

i e e d dr r

This “off-axis” integral is not easy to carry out – analytically – it is a so-called elliptic integral (of the “würst” kind). It can be done e.g. numerically, on a computer. If the observer/listener position is located somewhere on the z -axis, i.e. ˆr zz

{when 0 }, the “on-axis” integral is

much easier to carry out – analytically, because the dot product r vanishes, the “on-axis” problem has axial symmetry {i.e. has rotational invariance about the z -axis}:

2 22

2 20 0

1,

2

2

a ik zi to oup r z t i e e d d

z

i

2o ou

2 2

2 20

a ik zi te e dz

Note that the integrand of the -integral is a perfect differential:

2 2

2 2

2 2

ik zik z d e

ed ikz

Noting also that c k , o oz c and o o o o op z u cu , the on-axis time-domain complex

over-pressure is thus:

2 2 2 2

0 0,

ik z ik za ai t i t

o o o o

d e ep r z t i u e d i u e d

d ik ik

i

o ou

i

2 22 2

22

0

1 1 1

1

1 ,

o

a ik z a z i t kzik zi to o

z

ikz a z ikz a zi t kz ikz i t i to o

e e cu e ek

p e e p e e e p r z e




-17-

The real and imaginary components of the on-axis frequency-domain complex over-pressure are:

2

r , Re , cos cos 1op r z p r z p kz kz a z

2

i , Im , sin sin 1op r z p r z p kz kz a z

The magnitude of the on-axis frequency-domain complex over-pressure amplitude is:

1 22* 2 1 cos 1 1op r z p r z p r z p kz a z

Using the trigonometric identity 2cos 2 1 2sinA A , this can be equivalently written as:

212sin 1 1op r z p kz a z

For z a , the Taylor series expansion for 2 2121 1a z a z .and. if additionally

2 2 21 12 2 2 z ka a a Rayleigh length , then the magnitude of the “far-field”

on-axis frequency-domain complex over-pressure amplitude is:

212far o op r z p a z ka p a z

Note that the magnitude of the “far-field” on-axis frequency-domain complex over-pressure is bounded by:

0 2far op r z p for 0 r z

Extrema of p r z occur when: 21 12 21 1 , 0,1, 2,3, 4,5,...kz a z m m

Solving the

quadratic equation for z, p r z extrema occur when: 4mz a a m m a , 1, 2,3, 4,5,...m

Coming in from z , the first maxima 1 maxp r z occurs at: 1 4z a a a , the

next maxima 3 maxp r z occurs at: 3 3 3 4z a a a , and so on. Zeroes of mp r z

occur in between the local maxima, when 2,4,6,8,...m

Thus, having obtained the on-axis complex time-domain over-pressure amplitude ,p r z t ,

we next use the {linearized} Euler’s equation for inviscid fluid flow to obtain the on-axis

complex time-domain particle velocity , ,u r z t :

, ,1 1ˆ,

o o

u r z t p r z tp r z t z

t z




-18-

Since the observer/listener position is on-axis at ˆr z zz

, and the on-axis complex time-domain over-pressure ,p r z t has axial symmetry (i.e. no -dependence), the gradient

operator ˆ ˆr r z z

, and (after some derivative-taking and some algebra), the on-axis complex time-domain particle velocity is:

2 2 2 2

2

2

2 2

1 1

2

1

2

, 1 1

1 1

1

1 ,

1

ik z a z ik z a z i t kzz o

ikz a zi t kz

o

ikz a zikz i to z

zu r z t u e e e

z a

u e ea z

u e e e u r za z

i te

The real and imaginary components of the on-axis frequency-domain complex particle velocity are:

r

2

2

1, Re , cos cos 1

1z z ou r z u r z u kz kz a z

a z

and:

i

2

2

1, Im , sin sin 1

1z z ou r z u r z u kz kz a z

a z

The on-axis complex specific longitudinal acoustic impedance, using o o o oz c p u is:

, ,,

,z

i t

az

p r z t p r z ez r z

u r z t

, i tzu r z e

2 2

2 2

1 1 1 1

1 1 1 1

2 2

,

,

1 1

1 1

1 11 1

z

ikz a z ikz a z

o

o

ikz a z ikz a z

o

p r z

u r z

p e e

z

u e ea z a z




-19-

Using the relations z za o az c and o oz c the above relation can be rewritten as a

dimensionless quantity:

2

2

1 1

1 1

2

1, ,

11

1

z z

ikz a z

a a

o ikz a z

ez r z c r z

z ce

a z

The frequency-domain on-axis complex longitudinal acoustic intensity is:

2 2

2 2

*12

1 1 1 112 2

1 1 1 112 2

, , ,

1 1 1

1

1 1 1

1

za z

ikz a z ikz a z

o o

ikz a z ikz a z

o

I r z p r z u r z

p u e ea z

I e ea z

2 21 1 1 1

2

1 1 1

1

ikz a z ikz a zrmsoI e e

a z

We leave it as an exercise for the interested/motivated reader to explicitly obtain the real and imaginary parts of on-axis ,az r z and ,aI r z , calculate the {purely real} on-axis

potential/kinetic/total acoustic energy densities, etc.

Please see/look at plots of the above complex acoustic quantities associated with the baffled plane circular piston, available on the Physics 406 Software web-page, at the following URL:

http://courses.physics.illinois.edu/phys406/406pom_sw.html




-20-

Example # 7: The Uniform Planar Rigid-Walled 2-D Duct:

The final example we wish to discuss is that of sound propagation of monochromatic waves in an infinitely-long uniform planar duct consisting of two infinite, parallel, perfectly reflecting and rigid walls separated by a perpendicular distance a, as shown in the figure below:

The wave equation for the complex scalar over-pressure field is:

22

2 2

,1, 0

p r tp r t

c t

The sound propagation direction is in the x -direction; note that the sound waves are not constrained in the z-direction ( z points out of the page), whereas sound waves are constrained in the y-direction, being allowed only in the region between the two infinite, parallel walls: 0 y a . Thus, this problem is only a 2-D problem in ,x y rectangular coordinates.

The gradient

and Laplacian 2 operators in 2-D Cartesian/rectangular coordinates are:

ˆ ˆx yx y

and:

2 22

2 2x y

The 2-D wave equation for the complex time-domain scalar pressure field is thus:

2 2 2

2 2 2 2

, , , , , ,10

p x y t p x y t p x y t

x y c t

We seek x -propagating wave product-type solutions of the general form:

, , yxik yik x i tp x y t X x Y y T t Ae e e

The homogeneous wave equation is separable in these variables; the resulting characteristic equation for the wavenumber k is: 2 2 2

x yk k k , with accompanying dispersion relation 2 2 2k c .

{The details of this separation-of-variables technique for the 2-D wave equation are given in the P406 Lecture Notes on “Mathematical Musical Physics of the Wave Equation” p. 12-13}.




-21-

The boundary condition on the pressure at the two infinite, rigid parallel walls in the y - direction is that there are pressure anti-nodes at y = 0 and y = a. Mathematically, this requires Neumann-type boundary conditions on the walls, i.e. , 0, , , 0p x y t y p x y a t y ,

requiring cosine-type solutions for yik yY y e , i.e. ~ ~ cosy yik y ik y

yY y e e k y such that:

0,cos 1y y a

k y

with 0, 0,

cos sin 0y yy a y ak y y k y

, 0,1, 2,3...yk a n n

Thus we see that: yk a n or: , 0,1, 2,3....yk n a n Thus, for any given frequency

2f , there are an infinite number of possible solutions (aka eigenmodes) for this wave equation, each one of the general form:

, , cos ni t k xn np x y t A n y a e where: 22 2 2 x y nk k k k k n a

The transverse pressure distribution ~ cos n y a for 0 y a is a characteristic of the wall

geometry associated with this problem – i.e. a duct; and one which is caused by multiple, perfect (i.e lossless) reflections of the pressure waves off of the duct walls as they propagate in the x -direction. The integer n denotes the {duct-} mode of propagation.

The n = 0 mode is known as the axial plane-wave eigenmode of propagation. The 1n modes are collectively known as transverse duct eigenmodes. At a given frequency f, if a specific duct eigenmode n is excited, it may only propagate along the duct with a unique axial wavenumber

given by 22nk k n a .

Note that for each/every duct eigenmode n of propagation, there is an (angular) frequency

for which the axial wavenumber 2 2 22 0nk k n a c n a . The so-called

cutoff frequency for the nth mode is: cutoffn n c a or: 2cutoff

nf nc a . Below this cutoff

frequency, the duct eigenmode n cannot propagate – it becomes an evanescent mode because the axial eigen-wavenumber nk becomes purely imaginary for 2cutoff

nf f nc a – i.e. the duct

eigenmode n is exponentially attenuated by a factor of nk xe when 2cutoffnf f nc a .

A plot of the dispersion curves - axial wavenumber x nk k vs. angular frequency showing

the effect of the cutoff frequency vs. mode number n = 0, 1, 2, 3, … is shown in the figure below.




-22-

For a given (angular) frequency , in general, the total/net complex pressure amplitude is a sum over all modes – the allowed/propagating individual complex pressure eigenmodes

cutoffn n , where cutoffn is the highest eigenmode number n such that

220x n cutoffk k c n a , i.e. int floorcutoffn a c a c , .and. the

individual non-propagating modes cutoffn n :

0 0

, , , , cos ni t k xtot n n

n n

p x y t p x y t A n y a e

{n.b. Far from the sound source, this reduces to the sum over propagating modes cutoffn n .}

The complex time-domain 2-D particle velocity ,u r t associated with this problem is

obtained via use of Euler’s equation for inviscid fluid flow. Since , , ,p r t p x y t fcn z ,

then ˆ ˆ, , , , , , ,p r t p x y t p x y t x x p x y t y y , thus the particle velocity can

only be in the ,x y direction(s), i.e. , , ,u r t u x y t fcn z

since:

, 1,

o

u r tp r t

t

, , , , , ,1ˆ ˆ

o

u x y t p x y t p x y tx y

t x y

Euler’s equation holds for each/every duct eigenmode n. With , , cos ni t k xn np x y t A n y a e ,

the general form of the complex time-domain 2-D particle velocity for the nth duct eigenmode is thus:

1ˆ ˆ, , cos sin ni t k x

n n no

u x y t A k n y a x i n a n y a y e

where: 2 2 22 x nk k k n a c n a




-23-

For a given (angular) frequency , for each of the propagating duct eigenmodes cutoffn n ,

the axial component of the particle velocity 1ˆ ˆ, , cos n

x

i t k xn n n

o

u x y t x A k n y a e x

is

in-phase with the complex pressure , , cos ni t k xn np x y t A n y a e , whereas the transverse

component of the particle velocity ˆ ˆ, , sin n

y

i t k xn n

o

iu x y t y A n a n y a e y

is in

quadrature (i.e. 90o out of phase) with the complex over-pressure amplitude.

The total/net complex time-domain 2-D particle velocity is likewise given by:

0 0

1ˆ ˆ, , , , cos sin ni t k x

n n nn no

u x y t u x y t A k n y a x i n a n y a y e

where: 2 2 22 x nk k k n a c n a and: int floorcutoffn a c a c .

The total/net complex pressure wave 0

, , , ,tot nn

p x y t p x y t

and 2-D particle velocity

wave 0

, , , ,nn

u x y t u x y t

that propagate in a duct depend on the details of the coupling of

the sound source to that duct. For example, a 2-D “line” monopole source of volumetric velocity

per unit length 2 a aQ Q L m s located e.g. at , 0, ox y y in the duct will produce modal

pressure amplitudes of:

coso an o

n

QA n y a

k a

This relation predicts that the nth modal pressure amplitude nA becomes infinite at the cutoff

frequency for that mode, cutoffn n c a when 2 2

0x n cutoffk k c n a !!! However,

in the real world, nothing becomes infinite – e.g. the finite impedance of a real/physical acoustic source precludes infinite acoustic energy transfer to the duct. Nevertheless, the experimentally-

measured modal pressure amplitudes nA do indeed become large at/near the cutoff frequency!

The method of (an infinite set of) acoustic images can be used to model sound sources inside of (perfectly reflecting – even for partially reflecting) ducts – the planar walls of the duct act like mirrors, thus virtual “images” of the sound source in the duct are produced outside of the duct, as shown in the figure below:




-24-

The pressure/particle velocity fields in proximity to the actual sound source inside the duct are determined largely by the image source(s) nearest to the actual sound source; the solution converges rapidly as the number of image sources is increased. However, accuracy in calculating the pressure / particle velocity fields far from the actual sound source (i.e. further down the duct,

x a ) requires increasingly larger numbers of image sources to be included. The image source technique is widely used e.g. in computational modeling of room acoustics.

The complex frequency-domain 2-D vector specific impedance/admittance, acoustic energy flow velocity, acoustic intensity and purely real, scalar energy densities, etc. inside the duct can all be computed (most easily accomplished e.g. via numerical computation…) from their definitions, for a given sound source & frequency:

*

2

, , , , , ,, ,

, , , ,a a

p x y p x y u x yz x y

u x y u x y

and:

1, ,, ,

, , a

u x yy x y

p x y

1, , , ,a a

o

c x y z x y m s

and: * 21

, , , , , , 2aI x y p x y u x y Watts m




-25-

2

32

, ,1, ,

4potla

o

p x yw x y Joules m

c

231

, , , , 4

kina ow x y u x y Joules m

3, , , , , , tot potl kina a aw x y w x y w x y Joules m




-26-

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example # 3: point monopole sound source – spherical waves … · 2017-04-11 · lecture notes...

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