Expression for the estimation of time-averaged three-dimensional acoustic

energy density spectral density

Running title: Expression for 3D acoustic energy density

Ben S. Cazzolato�

and Justin Ghan†

School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia.

PACS numbers: 43.58.-e, 43.60.Qv

�

Address all correspondence to this author. Phone: +61 8 8303 5449. Fax: +61 8 8303 4367; Electronic address:

benjamin.cazzolato@adelaide.edu.au†Electronic address: justin.ghan@adelaide.edu.au

1

Abstract

This paper builds on earlier work by the same authors to derive an expression for the time-

averaged three-dimensional acoustic energy density in the frequency domain using the auto-

and cross-spectral densities of multiple microphone elements. Expressions for the most com-

mon geometric arrangements are derived and validated numerically. It is shown that in the

presence of uncorrelated noise the various geometric configurations produce significantly dif-

ferent energy density estimates, unlike the perfectly correlated case when all configurations

yield almost identical energy density estimates.

2

I. INTRODUCTION

In several articles [1–5] an expression for the acoustic energy density spectral density along

a single axis using the two-microphone technique has been derived. This frequency domain

expression enables the calculation of acoustic energy density spectral density using the auto-

spectra and cross-spectrum of the two microphone signals. The advantage of this approach, as

opposed to the traditional approach of first calculating the mean pressure and particle velocity,

is that the energy density may be measured with a two-channel spectrum analyser with no

additional circuitry apart from the two microphone signals.

In this paper, the acoustic energy density estimate using spectral methods will be extended

to three dimensions. Several geometric configurations and spectral density estimation formula-

tions are discussed. Expressions involving the minimum number of terms necessary to estimate

the energy density spectral density are also derived to facilitate the calculation of the three-

dimensional energy density using a two-channel spectrum analyser. The resulting analytical

expressions are verified using a numerical simulation in the time domain. It is shown that under

certain conditions the different spectral density estimation formulations may vary significantly

from each other.

II. PHYSICAL 3D ENERGY DENSITY CONFIGURATIONS

Since the measurement of the three-dimensional acoustic energy density requires the same

set of measurements as the acoustic intensity, namely acoustic pressure and the three orthogonal

particle velocities, then any sensor capable of measuring one quantity can also be used to mea-

sure the other. There exists several different geometrical arrangements for which 3D acoustic

intensity and energy density may be measured[6–8]. This paper will loosely focus on the two

most common arrangements found in commercially available 3D sound intensity probes. These

are the four-microphone tetrahedral arrangement like those found in the Ono Sokki Tetra-phone

MI-6420 (Figure 1 a) and the six-microphone arrangement like those of the Brüel & Kjær Type

5356 Intensity Probe (Figure 1 b) or the GRAS Vector Intensity Probe (Figure 1 c).

A number of variants of the four-microphone sensor exist, the specific details of which are

discussed in detail in the following sections. A slight variant of the six-microphone sensor is the

3

seven-microphone sensor[9]. Both the six and seven-microphone sensors use three pairs of op-

posed closely spaced microphones to calculate the three orthogonal particle velocities. However

unlike its six-microphone sibling, which must estimate the pressure at the centre of the sensor

using a finite sum, the seven-microphone sensor uses the seventh microphone at its geometric

centre to directly measure the pressure. This is the most accurate of all the geometries[8] and

consequently has been used to accurately measure three-dimensional impulse response func-

tions for surround sound systems[9].

The bias errors, both inherent and systematic, for most of the three-dimensional geometric

arrangements have been documented by Cazzolato and Hansen [8], and by Parkins et al. [10]. It

should be noted that Parkins et al. [10] also investigated the errors arising from a six-microphone

arrangement with the microphones mounted on the surface of a rigid sphere (rather than the

open configurations shown in Figures 1 b&c). This mounted arrangement was employed in an

attempt to use the effects of diffraction caused by the sphere [11] to produce more favorable

bias conditions, and in doing so reduce the inherent errors. The effect of acoustic diffraction

around the rigid sphere caused the effective microphone spacing to be 50% greater than the

actual microphone spacing. When calculating the particle velocity estimate when using the

finite difference approximation, the effective microphone spacing should be used, rather that

the physical spacing.

III. ANALYTICAL DERIVATION

The analytical derivation in this paper closely follows the derivation by Ghan et al. [5]. The

instantaneous acoustic energy density, ED�t � , at a point is defined as the sum of the acoustic

potential energy density and the acoustic kinetic energy density at that point, given by [7]

ED�t ��� p2 � t �

2ρc2 �ρ � v � t ��� 2

2 � (1)

where p�t � is the instantaneous pressure and � v � t ��� is the magnitude of the total instantaneous

particle velocity, at that point, c is the speed of sound, and ρ is the mean density of the fluid. In

practice the total particle velocity is estimated using the three orthogonal terms (v�t ��� ivx

�t � �

4

jvy�t � � kvz

�t � ), therefore Equation (1) can be rewritten as

ED�t � � p2 � t �

2ρc2 � ∑i � x � y � z

ρv2i

�t �

2� (2)

For a three-dimensional sensor the pressure is typically estimated using an average of all the

microphone elements, thus for an sensor containing n microphones the pressure estimate is

p�t � � 1

n ∑ni � 1 pi

�t � � (3)

The measurement of the particle velocity can be obtained through a finite difference approx-

imation using two microphones located on one of the three orthogonal axes, thus for a micro-

phone separation distance of 2h, the particle velocity components are approximated by [7]

vi�t � � 1

2ρh

� t

� ∞ � pi1

�τ �� pi2

�τ �� dτ � (4)

In the following sub-sections frequency domain expressions are derived for the single-sided

time-averaged acoustic energy density estimate for a number of three-dimensional sensor ge-

ometries. The expressions are based on single-sided spectral densities so they may be directly

used with real-time spectrum analysers which usually calculate single-sided spectra.

A. Multichannel Expressions

In the case where a multi-channel spectrum analyser (such as a Brüel & Kjær Pulse system) is

available, it is possible to obtain exact frequency domain expressions in real time for any number

of three-dimensional energy density sensors. Expressions for the most common geometries are

derived in the following sections. It should be noted that the following estimates for energy

density can be calculated using a two-channel spectrum analyser by recording all auto and

cross-spectra, and then post processing the spectra in a software package such as Matlab or

Excel.

1. Cubic Four-microphone sensor with pressure averaging

The following analysis is for the four-microphone configuration as shown in Figure (2 a).

Three microphones (marked 2, 3 and 4), each forming one of the ends of the 3 orthogonal axes

5

(marked x, y and z respectively). This particular geometric configuration is known as a “cubic”

arrangement since the microphones lie in the corner of a cube, where the origin microphone

is in one corner of the cube and the three other microphones are located on the corresponding

vertices of the cube.

The specifics of this particular four-microphone sensor when used for energy density sens-

ing is discussed in detail by Cazzolato and Hansen [8]. This energy density sensor arrangement

has been widely applied to the characterisation of reverberant sound fields [12–20]. A fre-

quency domain expression for the sound intensity spectral density from this arrangement has

been derived[21], although to the authors knowledge, no such equivalent expressions exists for

energy density.

The pressure estimate is the mean of the pressures measured by all four microphones. The

particle velocities are calculated using Equation (4). The distance from the “origin” microphone

(microphone 1) to the other microphones was 2h. It should be noted that this particular sensor

does not have a unique acoustic centre [8, 21, 22]. This means that the positions in which

the particle velocities are estimated are not coincident with each other or the location of the

pressure estimate. The alternative four-microphone arrangements shown in Figures (2 b & c)

and discussed in more detail in Sub-sections III A 2 and III A 3 have a unique acoustic centre

which is located at the centre of gravity of the microphones.

The total time-averaged acoustic energy density is given by[5]

ED � limT � ∞

1T

E

� � T

0

�1

2ρc2 � p2 � t � �ρ2 � � v � t ��� 2 � dt � (5)

where the operator E is the expectation and T is the record length.

Using Parseval’s theorem [23] it can be shown that the three-dimensional double-sided time-

averaged acoustic energy density spectral density is,

ED�ω � � lim

T � ∞

1T

E

�1

2ρc2 � �P � ω � T � � 2 � ρ2 � �Vx

�ω � T ���

2 �ρ2 � �Vy

�ω � T ���

2 �ρ2 � �Vz

�ω � T ���

2 � (6)

where P�ω � T � and Vi

�ω � T � are the Fourier transforms of the pressure and velocity estimates.

6

From Equations (3) and (4),

P�ω � T �

� P1�ω � T � � P2

�ω � T � � P3

�ω � T � � P4

�ω � T �

4 � (7)

Vx�ω � T �

� 12ρh � P1

�ω � T � � P2

�ω � T �

jω � (8)

Vy�ω � T �

� 12ρh � P1

�ω � T � � P3

�ω � T �

jω � (9)

Vz�ω � T �

� 12ρh � P1

�ω � T � � P4

�ω � T �

jω� (10)

Equation (6) is valid for all three-dimensional sensor configurations, however the expressions

for the pressure and particle velocity differ for each sensor. Evaluating the expectation operator

in Equation (6) it can be shown that the single-sided time-averaged energy density spectral

density estimate for the four-microphone energy density sensor with pressure averaging is given

by

ED1-sided�ω � 4̄ � � 1

32ρc2 �3

8ρω2h2 � �G11 � �

� 132ρc2 �

18ρω2h2 � �

G22 � G33 � G44 �

� � 132ρc2 � 1

8ρω2h2 � �2Re � G12 � 2Re �G13 � 2Re � G14 � (11)

� � 132ρc2 � �

2Re � G23 � 2Re � G24 � 2Re � G34 �where Gxx is the single-sided auto-spectral density function for the pressure signal measured

by microphone x, and Re � Gxy is the real part of the single-sided cross-spectral density function

for microphone signals x and y. The subscript 4̄ represents the pressure estimate is obtained by

averaging all four microphone signals.

2. Regular Tetrahedral Four-microphone sensor with pressure averaging

Santos et al. [22] described a four-microphone arrangement in which the “acoustic centre”

is the same as the geometric centre of the four microphones (see Figure 2b). Their design

used microphones mounted on the corners of a regular tetrahedron. They combined pairs of

the microphone signals to estimate the pressure at the midpoint of each edge of the tetrahedron.

These six “virtual points” are then used to calculate the pressure gradient at the geometric centre

of the sensor.

7

This four-microphone sensor configuration uses the mean of all four microphones to provide

the pressure estimate. The double-sided time-averaged acoustic energy density spectral density

is given by Equation (6) where[22],

P�ω � T �

� P1�ω � T � � P2

�ω � T � � P3

�ω � T � � P4

�ω � T �

4 � (12)

Vx�ω � T �

� 1�

2ρh jω � � P1�ω � T � � P3

�ω � T �

2� P2

�ω � T � � P4

�ω � T �

2 � � (13)

Vy�ω � T �

� 1�

2ρh jω � � P1�ω � T � � P4

�ω � T �

2� P2

�ω � T � � P3

�ω � T �

2 � � (14)

Vz�ω � T �

� 1�

2ρh jω � � P1�ω � T � � P2

�ω � T �

2� P3

�ω � T � � P4

�ω � T �

2 � � (15)

It can be shown that the single-sided time-averaged energy density spectral density estimate

for the regular tetrahedron four-microphone energy density sensor with pressure averaging is

given by

ED1-sided�ω � Tetrahedron

� � 132ρc2 �

316ρω2h2 � �

G11 � G22 � G33 � G44 �

� � 132ρc2 � 1

16ρω2h2 � �2Re � G12 � 2Re � G13 � 2Re � G14 � (16)

�� 1

32ρc2 � 116ρω2h2 � �

2Re � G23 � 2Re � G24 � 2Re � G34 � �

3. Ono Sokki Four-microphone sensor with pressure averaging

The Ono Sokki four-microphone sensor [24–27] shown in Figure (1 a) also has the micro-

phones arranged in a regular tetrahedron. The weighting of the microphone signal differs from

that employed by Santos et al. [22] resulting in a different alignment of the three measurement

axes (see Figure 2c).

Using the approach employed by the Ono Sokki corporation[24–27] when processing the

data, the double-sided time-averaged acoustic energy density spectral density is given by Equa-

8

tion (6) where [26],

P�ω � T �

� P1�ω � T � � P2

�ω � T � � P3

�ω � T � � P4

�ω � T �

4 � (17)

Vx�ω � T �

� 12ρh jω � � P3

�ω � T � � P2

�ω � T � � � (18)

Vy�ω � T �

� 1

2�

3ρh jω � � 2P1�ω � T �� P2

�ω � T �� P3

�ω � T � � � (19)

Vz�ω � T �

� 1

2�

6ρh jω � � 3P4�ω � T �� P1

�ω � T �� P2

�ω � T �� P3

�ω � T � � � (20)

It can be shown that the single-sided time-averaged energy density spectral density estimate

for the regular tetrahedron four-microphone Ono Sokki energy density sensor with pressure

averaging is identical to that of the regular tetrahedral sensor given by Equation (16). This is to

be expected since the pressure and total velocity estimate is the same for both cases, only the

orientation with respect to the coordinate system has changed.

4. Six-microphone sensor with pressure averaging

Figure 2d shows an energy density sensor containing six microphones, with three pairs form-

ing three orthogonal axes. This sensor configuration uses the mean of all six microphones sig-

nals to provide the pressure estimate. The double-sided time-averaged acoustic energy density

spectral density is given by Equation (6) where,

P�ω � T �

� P1�ω � T � � P2

�ω � T � � P3

�ω � T � � P4

�ω � T � � P5

�ω � T � � P6

�ω � T �

6 � (21)

Vx�ω � T �

� 12ρh � P1

�ω � T � � P2

�ω � T �

jω � (22)

Vy�ω � T �

� 12ρh � P3

�ω � T � � P4

�ω � T �

jω � (23)

Vz�ω � T �

� 12ρh � P5

�ω � T � � P6

�ω � T �

jω� (24)

Therefore it can be shown that the single-sided time-averaged energy density spectral density

9

estimate for the six-microphone energy density sensor with pressure averaging is given by

ED1-sided�ω � 6̄ � � 1

72ρc2 �1

8ρω2h2 � �G11 � G22 � G33 � G44 � G55 � G66 �

� � 172ρc2 � 1

8ρω2h2 � �2Re � G12 � 2Re � G34 � 2Re � G56 � (25)

� � 172ρc2 �

�6

∑i � 3

�2Re � G1i � 2Re � G2i � �

6

∑i � 5

�2Re �G3i � 2Re �G4i ��� �

If the six microphones are mounted in a rigid sphere as described by Parkins et al. [10], then

the substitution h � 32r should be made, where r is radius of the sphere.

5. Seven-microphone sensor

With reference to the seven-microphone configuration in Figure 2(d), the additional “origin”

microphone (p0 � , located at the geometric centre of the six-microphone sensor, is used to mea-

sure the pressure for the sensor. By measuring the pressure directly rather than interpolating the

pressure measured by six microphones, the error in pressure associated with the finite-sum is

avoided.

The single-sided time-averaged energy density spectral density estimate for the seven-

microphone energy density sensor is given by

ED1-sided�ω � 7 � � 1

2ρc2 � �G00 �

� � 18ρω2h2 � �

G11 � G22 � G33 � G44 � G55 � G66 � (26)

� � 18ρω2h2 � �

2Re � G12 � 2Re � G34 � 2Re � G56 �

where G00 is the auto-spectral density function of the origin microphone signal. Note the sig-

nificant reduction in terms compared to the six-microphone sensor.

10

B. Reduced Order Expressions

Although it is desirable to calculate all the necessary auto- and cross-spectra simultaneously

on a multi-channel spectrum analyser, two-channel spectrum analysers (such as the HP 35665A)

may be used to measure the auto- and cross-spectral density estimates. The number of two-

channel measurements required is a function of the number of microphones and the particular

spectral density formulation. To calculate the energy density spectral density estimates using

Equations (11) and (16) requires 6 individual two-channel measurements and Equations (25)

and (26) would require 15 and 4 individual two-channel measurements respectively.

Clearly the large number of two-channel measurements for all but the seven-microphone

sensor would require a significant effort to obtain experimentally. It is therefore desirable to

have expressions requiring fewer measurements. The minimum number of separate measure-

ments is limited by the need for the cross-spectral density estimates used by the finite-difference

expression in the kinetic energy density estimate. In the following sections a number of ap-

proximations using less terms are derived to facilitate calculation using two-channel spectrum

analysers.

1. Cubic four-microphone sensor using origin microphone

An alternative formulation for the four-microphone sensor is to use the pressure at the “origin

microphone” of the cubic arrangement as the pressure estimate rather than the mean pressure

sensed by the four microphones as used previously. The particle velocities may be calculated

as before using Equations (8) to (10). This arrangement has been analysed by Cazzolato and

Hansen [8] and used experimentally for active noise control [28, 29].

For the case of the four-microphone energy density sensor with the pressure estimate given

by the microphone at the origin (p1), the single-sided time-averaged energy density spectral

11

density estimate is given by

ED1-sided�ω � 40

� � 12ρc2 �

38ρω2h2 � �

G11 �

� � 18ρω2h2 � �

G22 � G33 � G44 � (27)

� � 18ρω2h2 � � � 2Re � G12 � 2Re � G13 � 2Re � G14 � �

This formulation only requires three individual sets of measurements as compared to six for

Equations (11) and (16).

2. Six-microphone sensor - Reduced order

An approach commonly employed for three-dimensional sound intensity measurements [6,

30] is to use three one-dimensional probes. This approach was used by Parkins et al. [10] when

calculating the energy density for a six-microphone sensor. They used the mean of three two-

channel pressure estimates (obtained from the finite pressure sum of each of the three orthogonal

axes) to provide an overall pressure estimate. Thus, when using this approach the square of the

pressure magnitude in Equation (6) is given by

�P � ω � T ���2 � 1

3 ������P1�ω � T � � P2

�ω � T �

2 ����2

� ����P3�ω � T � � P4

�ω � T �

2 ����2

� ����P5�ω � T � � P6

�ω � T �

2 ����2 � �(28)

For this particular two-channel approximation, the single-sided time-averaged energy den-

sity spectral density estimate is given by

ED1-sided�ω � 63̄

� � 124ρc2 �

18ρω2h2 � �

G11 � G22 � G33 � G44 � G55 � G66 �

� � 124ρc2 � 1

8ρω2h2 � �2Re �G12 � 2Re � G34 � 2Re � G56 � � (29)

This formulation also only requires three individual sets of two-channel measurements com-

pared to 15 two-channel measurements needed when using Equation (25). This probably ex-

plains why this particular approach was used by Parkins et al. [10] when estimating the 3D

12

energy density using a two-channel spectrum analyser. It should also be noted that the expres-

sion derived by Parkins et al. [10] differs from Equation (29).

3. Cubic four-microphone sensor - Reduced order

The method used in Subsection III B 2 has been applied to a cubic four-microphone sensor

to derive an expression for the energy density using a reduced number of two-channel measure-

ments. Thus the square of the pressure magnitude in Equation (6) is given by

�P � ω � T � �2 � 1

3 � ����P1�ω � T � � P2

�ω � T �

2 ����2

� ����P1�ω � T � � P3

�ω � T �

2 ����2

� ����P1�ω � T � � P4

�ω � T �

2 ����2 �

(30)

For this four-microphone energy density sensor approximation, the single-sided time-

averaged energy density spectral density estimate is given by

ED1-sided�ω � 43̄

� � 18ρc2 �

38ρω2h2 � �

G11 �

� � 124ρc2 �

18ρω2h2 � �

G22 � G33 � G44 � (31)

� � 124ρc2 � 1

8ρω2h2 � �2Re �G12 � 2Re � G13 � 2Re �G14 � �

This formulation only requires three individual sets of measurements as compared to six

for Equations (11) and (16). It is interesting to note that Equation (31) contains the same

auto and cross-spectral density terms as Equation (27), however the weights of the individual

terms differ, therefore likely producing quite different results under certain conditions, which is

discussed further in Section III C.

The reason for the differing weights in the cubic four-microphone expressions given by

Equations (11), (27) and (31) is the location at where the pressure estimate is made. For Equa-

tion (11), the geometric centre of the four microphones is used, which is located a distance ofh � 3

2 from the origin microphone along a line normal to the plane intersecting the other three

points [28]. The pressure estimate using Equation (27) is obviously made at the origin micro-

phone. The estimate of the pressure using Equation (31) is made at a distance of h� 3

from the

13

origin microphone, which is half way between the origin microphone and the plane intersecting

the other three points.

Each component of the velocity is estimated halfway between the origin microphone and one

of the axis microphones. Given that the distance between the location of the pressure estimate

and velocity estimates using Equation (31) is slightly less that obtained by using Equation (11),

namely h � 2� 3

� 0 � 8165h compared to h � 32 � 0 � 8660h, it could be argued that the former is the

most accurate formulation. However, confirmation of this is beyond the scope of this paper.

C. Discussion

It was shown by Cazzolato and Hansen [8] that in the absence of uncorrelated noise and for

small non-dimensional microphone separations (2kh � 1) that there is very little difference in

the accuracy of the energy density estimate given by the configurations in Sub-sections III A 1,

III A 4, III A 5 and III B 1.

It is beyond the scope of this article to determine which of the expressions in Sections III A

and III B are the most accurate since accuracy is a function of the sound field, orientation and

also more importantly the coherence between the individual microphone elements. Poor co-

herence between microphone elements may be caused by any number of factors, such as poor

signal to noise ratios when there are low sound pressure levels [31] or when the FFT length is

insufficient resulting in resolution bias errors [23]. Even so, some general comments may be

made. If one takes the worst case scenario when there is no correlation between the microphone

elements, and assuming that the spectral densities at the microphones are equal (to Gpp), then

the following expressions can be obtained:

ED1-sided�ω � 4̄ � � 1

8ρc2 �3

4ρω2h2 � Gpp (32)

ED1-sided�ω � Tetrahedron � � 1

8ρc2 �3

4ρω2h2 � Gpp (33)

ED1-sided�ω � 6̄ � � 1

12ρc2 �3

4ρω2h2 � Gpp (34)

ED1-sided�ω � 7 � � 1

2ρc2 �3

4ρω2h2 � Gpp (35)

14

ED1-sided�ω � 40 � � 1

2ρc2 �3

4ρω2h2 � Gpp (36)

ED1-sided�ω � 43̄

� � 14ρc2 �

34ρω2h2 � Gpp (37)

ED1-sided�ω � 63̄

� � 14ρc2 �

34ρω2h2 � Gpp � (38)

Although the likelihood of the microphone signals being uncorrelated is quite unrealistic, it

can be seen that the estimates of the energy density spectral densities derived above can be quite

different when the microphone elements are poorly correlated.

IV. SIMULINK SIMULATION

Using the same technique employed by Ghan et al. [5] several Simulink models were created

to validate the three-dimensional energy density expressions derived above. The models were

used to obtain the time-averaged acoustic energy density estimate both by directly calculating

the average pressure and particle velocity, and also by using the expressions derived in Sections

III A and III B. Figure 3 shows the Simulink model for the four-microphone sensor without

pressure averaging. This particular model was chosen to illustrate the models as this is the

simplest of all the formulations.

A simulated acoustic environment was created with a monopole white noise point source

in free space. The four microphone pressure signals were then calculated using the following

expression for the transfer function from a monopole point source to a sensor at distance r:

G�s � � 1

re� r

c s

� (39)

where c � 343m � s� 1 is the speed of sound. These transfer functions were implemented using

a second order Padé approximation in series with a gain. A number of different sensor orien-

tations were trialled to ensure that the spectral density estimates held for any orientation. For

the results presented below the origin microphone was placed at distance r1 � 5 � 00m from the

monopole noise source and the remaining three microphones (separation distance, 2h � 50mm)

15

are r2 � 3&4 � 5 � 0289m from the source. The source produces continuous white noise of 1 � 0Pa

at 1m (94dBre20µPa). The density of air was ρ � 1 � 21kg � m� 3.

Figure 4 shows the sub-model used to calculate the energy density using the traditional

time-domain approach. The instantaneous pressure and particle velocities were estimated using

Equations (3) and (4) respectively (as shown in Figure 5). The velocity values were multiplied

by ρc so that all the outputs were dimensionally consistent. They were sampled at 1000Hz, and

a frequency spectrum of each was obtained by performing a 512 point FFT (see Figure 4). The

potential and kinetic energy auto-spectral densities were calculated, then summed and scaled

by the factor 12ρc2 to produce the time-domain based acoustic energy density spectral density

estimate (see Figure 4).

Figure 6 shows the implementation in the frequency domain. To compute an estimate di-

rectly in the frequency domain, a spectrum of each of the microphone pressure readings was

obtained by performing a 512 point FFT (see Figure 6). The auto- and cross-spectral densities

were then calculated, and Equation (27) was applied to produce the frequency-domain based

acoustic energy density spectral density estimate.

A comparison of the results is shown in Figure 7. The acoustic energy density spectral

densities obtained via each method of calculation are identical, verifying the derived Equation

(27). This same method of comparison was repeated for the expressions in Section III A. For

the two remaining reduced order frequency domain expressions shown in Equations (29) and

(31) it was not possible to obtain a time domain equivalent. Instead the results were compared

against the multi-channel equivalents shown in Equations (25) and (11) respectively. Although

the frequency domain spectral estimates were not exactly identical to the time domain esti-

mates, they were extremely close, with the difference being attributed to the differing weighted

contributions of the individual spectra and cross-spectra.

V. CONCLUSIONS

Several expressions for the time-averaged three-dimensional acoustic energy density esti-

mate have been derived using the auto- and cross-spectral densities between several closely

spaced microphones. These were validated numerically using a time domain simulation in

Simulink. The results obtained using these expressions were identical to the results using the

16

traditional method of using the weighted sum of the squared pressure and particle velocities. It

is shown that in the presence of uncorrelated noise the expressions may result in significantly

different spectral density estimates. Expressions containing a reduced number of terms were

also derived to facilitate the calculation of the three-dimensional energy density spectral den-

sity using two-channel spectrum analysers.

Acknowledgments

The authors gratefully acknowledge the financial support for this work provided by the Aus-

tralian Research Council.

17

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20

List of Figures

1 Commercially available sound intensity sensors suitable for 3D energy density

measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 Multi-microphone three-axis energy density sensors. . . . . . . . . . . . . . . 23

3 Simulink model of a four-microphone energy density sensor. The “Time Do-

main Calculation” sub-model is shown in Figure 4, and the “Frequency Domain

Calculation” sub-model is shown in Figure 6. . . . . . . . . . . . . . . . . . 24

4 Simulink model for the calculation of the time-averaged acoustic energy density

estimate as the weighted sum of the auto-spectral densities of the pressure and

the three particle velocities. The block titled Energy Density Sensor (see Figure

5) calculates the time-domain quantities. . . . . . . . . . . . . . . . . . . . 25

5 Simulink model for the calculation of the pressure and the three particle veloc-

ities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Simulink model for the calculation of the time-averaged acoustic energy density

estimate using the derived expression in terms of the auto- and cross-spectral

densities of the four pressure readings. . . . . . . . . . . . . . . . . . . . . 27

7 Time-averaged acoustic energy density estimates using the time domain method

and the frequency domain expression. . . . . . . . . . . . . . . . . . . . . . 28

21

(a) Ono Sokki Tetra-phone MI-6420.

Photo courtesy of Ono Sokki

(b) B&K Type 5356. Photo

courtesy of Brüel and Kjær

(c) GRAS Vector

Intensity Probe. Photo

courtesy of GRAS

Figure 1: Commercially available sound intensity sensors suitable for 3D energy density measurement.

22

1 2

4

3

x

y

z

2h

2h

2h

(a) Cubic four-microphone sensor

h

h

h

h

hh

h

h

1

2

3

4y

z

x

(b) Four-microphone regular tetrahedon

sensor. Small black circles represent the

virtual microphone locations.

2h

2h

2h

2h

3

2

1

4y

z

x

(c) Four-microphone Ono Sokki sensor

y

hh

h

h

h

h

x

z

12

3

45

6

0

(d) Six-microphone and

seven-microphone sensor

Figure 2: Multi-microphone three-axis energy density sensors.

23

Figure 3: Simulink model of a four-microphone energy density sensor. The “Time Domain Calculation”

sub-model is shown in Figure 4, and the “Frequency Domain Calculation” sub-model is shown in Figure

6.

24

Figure 4: Simulink model for the calculation of the time-averaged acoustic energy density estimate as the

weighted sum of the auto-spectral densities of the pressure and the three particle velocities. The block

titled Energy Density Sensor (see Figure 5) calculates the time-domain quantities.

25

Figure 5: Simulink model for the calculation of the pressure and the three particle velocities.

26

Figure 6: Simulink model for the calculation of the time-averaged acoustic energy density estimate using

the derived expression in terms of the auto- and cross-spectral densities of the four pressure readings.

27

0 50 100 150 200 250 300 350 400 450 500−85

−80

−75

−70

−65

−60

−55

−50

Frequency (Hz)

Aco

ustic

Ene

rgy

Den

sity

(dB

re

1 J/

m3 )

Time Domain MethodFrequency Domain Method

Figure 7: Time-averaged acoustic energy density estimates using the time domain method and the fre-

quency domain expression.

28