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(1)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 1

Time-Domain vs. Frequency-Domain Sound Field Measurements

Introduction:

In this lab handout, we discuss the physical meaning of, the relationship(s) between, and the experimental techniques associated with time-domain vs. frequency-domain measurements of over-pressure p and particle velocity u

associated with an arbitrary sound field.

For sound waves propagating in air, the instantaneous over-pressure p r t

 

, (n.b. a scalar quantity) and the instantaneous 3-D vector particle velocity u r t 

 

, obey their respective wave equations, which are {neglecting/ignoring (small) dissipative/energy loss effects) and for normal/everyday sound pressure levels

SPL134 dB

}:

 

2

 

2

2 2

1 ,

, p r t 0

p r t

v t

   

 

and: 2

 

2

 

2 2

1 ,

, u r t 0

u r t

v t

   

   

where v is the wave propagation speed in the medium (= air, in this case, here).

The instantaneous over-pressure p r t

 

, and particle velocity u r t 

 

, are not independent quantities. For e.g. traveling-type sound waves in air, neglecting/ignoring (small) dissipative/

energy loss effects and normal/everyday sound pressure levels

SPL134 dB

, the Euler equation for inviscid (i.e. dissipationless) fluid flow reasonably accurately describes the spatial- temporal relationship between these two instantaneous physical quantities:

 

, 1

 

,

o

u r t

p r t

t

   

   

where for sound propagation in air: oair 1.204kg m3@ NTP.

Sound Field Measurements in One Dimension:

For the sake of simplicity and clarity, we first discuss 1-D sound fields, such as that

associated with the propagation of a monochromatic (i.e. single-frequency) traveling plane wave in the ˆz direction, or e.g. the “far-field”

r

regime associated with the radial-outward propagation of {monochromatic/ single-frequency} spherical waves emanating from a point sound source, located at the origin. Please refer to the UIUC Physics 406 Lecture Notes XII and XII – Part 2 for discussion/details of the nature of these two 1-D type sound fields.

In order to completely describe an arbitrary, instantaneous monochromatic/single-frequency { 2 f }1-D sound field associated with a 1-D longitudinal sound wave (at least locally) propagating in the ˆz direction, we need to measure two physical quantities at the listener’s position, e.g. rzzˆ

:

a.) the instantaneous over-pressure, p r t

, ;

and

b.) the instantaneous 1-D particle velocity uz

r t, ;

.

(2)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 2 The most general mathematical description – in the time-domain – for these two instantaneous physical quantities, associated with a monochromatic/single-frequency traveling 1-D sound wave (at least locally) propagating in the ˆz direction, for a listener’s position at rzzˆ

are:

           

     

0 0

0

, ; , cos , ; , cos ,

, cos ,

p p

o

z p

p r t p r r t p r t r

p r t k r z

      

    

   

 

    

    

 

           

     

0 0

0

, ; , cos , ; , cos ,

, cos ,

z uz z z

z z

z u

o

z u

u r t u r r t u r t r

u r t k r z

      

    

   

 

    

    

 

Note that the amplitudes for over-pressure p0

r,

and longitudinal particle-velocity

 

0 ,

u z r are time-independent quantities {for a constant-amplitude sound source}. However, depending on the detailed nature of the specific sound source under consideration in a given physics situation, the amplitudes p0

r,

and 0

,

u z r in general can be/are position- .and.

frequency-dependent.

Note that the overall arguments of the cosine function(s), p

r t, ;

and uz

r t, ;

in the above ˆz 1-D traveling-wave expressions for instantaneous over-pressure p r t

, ;

and

1-D longitudinal particle velocity uz

r t, ;

are, in general not constants, due to {in general}

possible position- and frequency-dependence of the overall phases p

r,

and uz

r,

, the {longitudinal} wavenumber kz

r,

and the

r 0

phases   and op

 

  : uoz

 

, ;

 

,

 

,

o

 

p r t   t p r  t kz rz  p

        

, ;

 

,

 

,

  

uz z z

o

u z u

r t   t r  t k rz  

        

The overall phase(s) associated with the generalized over-pressure and {longitudinal}

1-D particle velocity waves are:

,

o

  

,

p r p kz r z

       

At the origin

r0

: p

0,

 po

 

,

   

,

z z

o

u r u kz r z

       

At the origin

r0

: uz

0,

 uoz

 

At

r0,t0

we also see that: p

0, 0;

 op

 

and: u

0, 0;

z

 

z

o

  u

  .

Thus the instantaneous over-pressure and {longitudinal} 1-D particle velocity at

r0,t0

are:

0, 0;

0,0

 

cos op

 

p   p   

0, 0;

0

0,

cos

 

z z

o

z u

u  u   

(3)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 3 Important Comment: The

r0,t0

phases   and op

 

  are defined relative uoz

 

e.g. to the sine-wave signal output from a sine-wave function generator that is used to produce the sound field in the first place. The sine-wave signal output from the function generator

 

o cos

FG FG

V  Vt thus provides the reference signal needed for defining, and determining/

experimentally measuring the

r0,t0

relative phases   and op

 

  . uoz

 

Note that for a constant/fixed value of the overall phase(s), the position z and the time t are related to each other via z t

 

v

r,

t

where v

r,

is the phase speed associated with the longitudinal propagation of the 1-D traveling wave: v

r,

 fz

r,

 kz

r,

{= the speed of propagation of surfaces of constant phase}, which again, depending on the detailed nature of the specific sound source under consideration in a given physics situation, the phase speed v

r,

, the {longitudinal} wavelength z

r,

and the {longitudinal}

wavenumber kz

r,

2 z

r,

can be/are in general both position- .and. frequency- dependent. Thus, we see that:

, ;

 

,

 

,

o

   

,

 

o

 

p r t   t p r   t z v r   pt z v r   p

            

, ;

 

,

 

,

    

,

   

uz z z z

o o

u u u

r t   t r   t z v r     t z v r   

            

The overall phase(s) associated with the generalized over-pressure and {longitudinal}

1-D particle velocity traveling waves are:

,

o

  

,

o

  

,

p r p kz r z p z v r

            

,

   

,

   

,

z z z

o o

u r u kz r z u z v r

            

A modern digital oscilloscope or e.g. a digital recorder (n.b. both are manifestly time-domain instruments!) can be used to measure the time-dependent voltages output from omni-directional pressure and/or 1-D particle velocity microphones, e.g. located at the “listener” point rzzˆ

in a sound field associated with a monochromatic/single-frequency traveling 1-D sound wave (at least locally) propagating in the ˆz direction. The instantaneous voltage signals output from the p- and u-mics will be of the form:

 

0

   

- , ; - , cos o

p mic p mic z p

V r t  V r  tk z  

 

0

   

- , ; - , cos

z

o

u mic u mic z u

V r t  V r  tk z  

The p- and u-mics must be absolutely calibrated to obtain their respective microphone sensitivity calibration constants Sp mic-

mV Pa

and Su mic-

mV Pa*

{or Su mic-

mV mm s

}

 

n b. . 1.0Pa* 2.42 mm s

, in order to {absolutely} convert their respective voltage signals

 

- , ;

p mic

V r t  and Vu mic-

r t, ;

into p r t

, ;

and uz

r t, ;

signals:

(4)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 4 p r t

, ;

Vp mic-

r t, ;

Sp mic-

 

Pa

, ;

-

, ;

-

* or

z u mic u mic

u r t  V r t  S Pa mm s

The absolute calibration of p- and u-mics that we routinely use in the UIUC Physics of

Music/Musical Instruments Lab is discussed in detail in the Lab Handout “Absolute Calibration of Pressure and Particle Velocity Microphones” – available on the UIUC Physics 193POM/

P406POM Lab Handout Webpages.

We can gain some additional physical insight into the nature of these two instantaneous acoustic signals by using the trigonometric identity cos

AB

cosAcosBsinAsinB.

At the listener’s location rzzˆ

, for arbitrary time t:

             

             

       

 

0 0

0

0

Amplitude component @ in-phas

, ; , cos , , cos ,

, cos cos , sin sin ,

, cos ,

o

p z p

o o

z p z p

o

z p

r

p r t p r t r p r t k r z

p r t k r z t k r z

p r k r z

         

        

   

 

 

       

   

 

    

  

 

   

       

 

   

0

Amplitude component @

e with 90 out-of-phase with

; cos ; cos

cos , sin , sin

o

o o

FG FG FG FG

o

z p

r

V t V t V t V t

t p r k r z t

     

 

 

 

         

             

       

 

0 0

0

0

Amplitude component @ i

, ; , cos , , cos ,

, cos cos , sin sin ,

, cos ,

z z z z

z z z

z z

z u u

o o

z u z u

o

z u

r

u r t u r t r u r t r

u r t k r z t k r z

u r k r z

        

        

   

   

      

   

 

    

  

 

   

       

 

   

0

Amplitude component @

n-phase with 90 out-of-phase with

; cos ; cos

cos , sin , sin

z z

o

o o

FG FG FG FG

o

z u

r

V t V t V t V t

t u r k r z t

     

 

 

 

If we choose the listener’s position to be at the origin

r0

, at arbitrary time t, using the fact that cos x

sin x

is an even (odd) function of x, i.e. cos

 

 x cosx

sin

 

  x sinx

respectively, the above expressions become:

       

   

   

 

   

0 0

Amplitude component @ 0 Amplitude component @ 0

in-phase with 90 out-of-phase with

; cos ; cos

0, ; 0, cos cos 0, sin sin

o

o o

FG FG FG FG

o o

p p

r r

V t V t V t V t

p t p t p t

        

 

 

       

   

   

 

   

0 0

Amplitude component @ 0 Amplitude component @ 0

in-phase with 90 out-of-phase with

; cos ; cos

0, ; 0, cos cos 0, sin sin

z z z z

o

o o

FG FG FG FG

o o

z u u

r r

V t V t V t V t

u t u t u t

        

 

 

(5)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 5 As discussed in detail in Physics 406 Lecture Notes XIII – Part 2, experimentally, we can e.g.

use lock-in amplifier and/or spectral analysis cross-correlation techniques to obtain/measure/

determine the above in-phase and 90o out-of-phase amplitude components of over-pressure and 1-D particle velocity, phase-referenced relative to the sine-wave signal VFG

 

 VFGo cost output from the sine-wave function generator that is used to produce the monochromatic/single- frequency sound field in the first place.

Note that the above purely real mathematical expressions that describe the instantaneous over-pressure p r t

, ;

and 1-D particle velocity uz

r t, ;

are manifestly time-domain quantities. These expressions can be related to their frequency-domain counterparts as follows:

First, we “complexify” the above instantaneous time-domain over-pressure p r t

, ;

and

1-D particle velocity uz

r t, ;

expressions by adding an “imaginary”, 90o phase-shifted component to their purely real expressions. Defining i 1, with complex conjugation

* 1

i     i {hence i*     i i i* 1}, the complex instantaneous over-pressure p r t

, ;

and 1-D particle velocity uz

r t, ;

at the listener’s location rzzˆ

, for arbitrary time t are:

         

           

       

0 0

0 0

0

, ; , cos , , sin ,

, cos , , sin ,

, cos , sin ,

p p

o o

z p z p

o

z p z p

p r t p r t r i p r t r

p r t k r z i p r t k r z

p r t k r z i t k r z

        

         

       

   

        

   

         

 

       

    

   

  o

 

         

           

     

0 0

0 0

0

, ; , cos , , sin ,

, cos , , sin ,

, cos , sin

z z z z

z z z z

z z

z u u

o o

z u z u

o

z u

u r t u r t r i u r t r

u r t k r z i u r t k r z

u r t k r z i t

        

         

     

   

        

   

         

 

     

    

   

  kz

r,

z uoz

 

We then use the Euler relation ei cos isin to equivalently write these expressions in complex exponential notation:

, ;

0

,

,   0

,

  , 0

,

,

o o

z p p z p

i t k r z i k r z i t i r i t

p r t   p r  e  p r  e  ep r  e e

, ;

0

,

,   0

,

   , 0

,

,

o o

z uz uz z uz

z z z

i t k r z i k r z i t i r i t

uz r t  u r  e  u r  e  eu r  e e Now for any complex quantity z x iy, the magnitude of the complex quantity z is

   

* 2 2

z  z z   x iy  x iyxy {n.b. a purely real quantity}, the phase ztan1

y x

, cos z

x z  andy zsinz. Thus, we can equivalently write the complex quantity z as:

 

cos z sin z cos z sin z i z

z  x iy z  i z   z  i   z e

(6)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 6 A phasor diagram of the “generic” complex quantity z  x iy z coszi zsinzz eiz in the complex plane is shown in the figure below:

The real part (or component) of z , xRe

 

z  zcosz lies on the horizontal axis.

The imaginary part (or component) of z , yIm

 

z  zsinz lies on the vertical axis.

Thus, complex z lies somewhere in the complex plane, oriented at an angle ztan1

y x

, referenced to the horizontal axis. In an acoustical physics situation, the physical meaning of the real (imaginary) part of the complex quantity z {e.g. complex p r t

, ;

or uz

r t, ;

} is that component of z which is in-phase (90o out-of-phase) with the {purely real} reference signal {VFG

 

 VFGo cost}, respectively.

Since z  x iy z eiz, we can write the (purely real amplitude).(complex overall phase) amplitude products in the above time-domain expressions for complex instantaneous over- pressure p r t

, ;

and 1-D particle velocity uz

r t, ;

as complex amplitudes:

   

,

 

  ,

0 , 0 , 0 ,

o

p z

p i k r z

i r

pr   p r  e p r  e 

   

,

 

  ,

0 , 0 , 0 ,

o

u z

uz z

z z z

i k r z

i r

u r  u r  e u r  e 

Note that the above complex amplitudes for over-pressure and 1-D particle velocity are time- independent {for a constant-amplitude sound source}, and in fact are none other than the frequency-domain representations of the time-domain expressions for complex instantaneous over-pressure p r t

, ;

and 1-D particle velocity uz

r t, ;

!

As discussed in Physics 406 Lecture Notes XIII – Part 2, the complex frequency-domain vs.

complex time-domain representations of acoustical quantities such as complex over-pressurep

and/or complex particle velocity u are related to each other by Fourier transforms of each other.

sin z y z

cos z x z

z

sin z z

y z

Real Axis Imaginary Axis

2 2

z  xy

(7)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 7 For any continuous complex time-domain function f t

 

:

The Fourier transform of f t

 

to the frequency domain is: f

 

 f t e

 

i tdt

The inverse Fourier transform of f

 

to the time domain is: f t

 

21  f

 

ei td



 

Thus, it should now be clear to the reader that for a harmonic/single-frequency sound field, we can write the time-domain complex instantaneous over-pressure p r t

, ;

and 1-D particle velocity uz

r t, ;

at the listener’s location rzzˆ

for arbitrary time t in an elegant and compact manner - as the product of the (complex frequency-domain amplitude).(complex ei t factor):

   

 

 

 

 

 

0

, ,

,

0 0 0

, ; ,

, , ,

o o

p z z p

p

i t

i k r z i t k r z

i r i t i t

p r t p r e

p r e e p r e e p r e

   

 

 

 

       

 

 

  

   

 

 

 

 

 

0

, ,

,

0 0 0

, ; ,

, , ,

z

o o

u z z u

uz z z

z z z

i t z

i k r z i t k r z

i r i t i t

u r t u r e

u r e e u r e e u r e

   

 

 

 

       

 

 

  

Sound Field Measurements in Three Dimensions:

The mathematical description and experimental measurement of 3-D sound fields is a straight-forward generalization of the above-discussed 1-D situation. For a monochromatic/

single-frequency 3-D sound field, the physical {purely real}, instantaneous time-domain scalar over-pressure p r t

, ;

and 3-D instantaneous vector particle velocity u r t 

, ;

are, e.g. in Cartesian coordinates:

           

     

0 0

0

, ; , cos , ; , cos ,

, cos ,

p p

o p

p r t p r r t p r t r

p r t k r r

      

    

   

 

    

    

   

       

           

     

0 0 0

0 0

ˆ ˆ ˆ

, ; , ; , ; , ;

ˆ ˆ ˆ

, cos , ; , cos , ; , cos , ;

ˆ ˆ

, cos , , cos ,

x ux y uy z uz

x z y y

x y z

u u

u r t u r t x u r t y u r t z

u r r t x u r r t y u r r t z

u r t r x u r t r y

   

     

       

  

     

   

    

     

   

     

           

0

0 0

, cos , ˆ

ˆ ˆ

, cos , , cos ,

z z

x x y y

u

o o

u u

u r t r z

u r t k r r x u r t k r r y

   

         

 

 

 

         

 

 

     

 

     

0 ˆ

, cos ,

z z

o

u rt k rr  u z

    

   

where the 3-D vector wavenumber k r 

,

kx

r,

xˆky

r,

yˆkz

r,

zˆ, withkk  kx2ky2kz2

.

(8)

 Professor Steven Errede, Dept. of Physics, The University of Illinois at Urbana-Champaign, 2000-2016. 8 The listener’s position vector {referenced to the sound source located at the origin}

is rxxˆyyˆzzˆ

, rr  x2y2z2

. Thus, the 3-D dot-product factor:

,

x

,

y

,

z

,

k r   rk rxk ryk rz. Note that cosx, cosy, cos are the 3-D z direction cosines. Then, since k rˆ ˆ cosx  x, ˆ ˆk ry cosy and k rˆ ˆ cosz  z the above 3-D dot-product factor can also equivalently be written as:

,

x

,

cos x y

,

cos y z

,

cos z

k r   rk rr  k rr  k rr  .

Thus, we see that for a monochromatic/single-frequency 3-D sound field, the physical, instantaneous time-domain scalar over-pressure p r t

, ;

is similar in form to that for the monochromatic/single-frequency 1-D sound field case, whereas for the physical, instantaneous time-domain 3-D vector particle velocity u r t 

, ;

ux

r t, ;

x uˆy

r t, ;

y uˆz

r t, ;

zˆ

, we have three {orthogonal/x-y-z} components to deal with/measure for the 3-D sound field case vs. only one component for the 1-D case. Three independent, orthogonally-oriented particle velocity microphones, located at the listener’s position r

are thus needed to measure/completely specify the 3-D vector particle velocity u r t 

, ;

at that location.

For the monochromatic/single-frequency 3-D sound field we again “complexify” the physical, instantaneous time-domain scalar over-pressure p r t

, ;

and 3-D vector particle velocity

, ;

u r t   following the above-described prescription for the 1-D sound field case. Then, the relations between complex time-domain scalar over-pressure p r t

, ;

, complex time-domain 3-D vector particle velocity u r t 

, ;

and their complex frequency-domain counterparts,

,

p r  and u r 

,

are also elegantly/compactly given by:

, ;

 

,

i t

p r t   p r  e

 

         

     

ˆ ˆ ˆ

, ; , ; , ; , ; ,

ˆ ˆ ˆ

, , ,

i t

x y z

i t i t i t

x y z

u r t u r t x u r t y u r t z u r e

u r e x u r e y u r z e

    

  

    

     

      

  

  

References

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