An alternative pressure-sensor based method for estimating the acoustic intensity, the phase and amplitude gradient estimation (PAGE) method, is presented. This method uses the same hardware as the standard finite-difference method, but does not suffer from the frequency-dependent bias inherent to the finite-difference method. A detailed derivation of the PAGE method and the finite-difference method is presented. Both methods are then compared using simple acoustic fields. The ability to unwrap the phase component of the PAGE method is discussed, which leads to accurate intensity estimates above previous frequency limits. The uncertainties associated with both methods of estimation are presented. It is shown that the PAGE method provides more accurate intensity estimates over a larger frequency bandwidth.

1.
IEC 1043:1993, “
Electroacoustics—Instruments for the measurement of sound intensity—Measurement with pairs of pressure sensing microphones
” (
International Electrotechnical Commission
,
Geneva, Switzerland
,
1993
).
2.
ANSI/ASA S1.9-1996
,
Instruments for the Measurement of Sound Intensity
(
Acoustical Society of America
, Melville,
NY
,
1996
).
3.
ISO 11205:2003,
Acoustics—Noise emitted by machinery and equipment—Engineering method for the determination of emission sound pressure levels in situ at the work station and at other specified positions using sound intensity
(
International Organization for Standardization
,
Geneva, Switzerland
,
2003
).
4.
ISO 15186-1:2000,
Acoustics—Measurement of sound insulation in buildings and of building elements using sound intensity—Part 1: Laboratory measurements
(
International Organization for Standardization
,
Geneva, Switzerland
,
2000
).
5.
ISO 15186-1:2003,
Acoustics—Measurement of sound insulation in buildings and of building elements using sound intensity—Part 1: Field measurements
(
International Organization for Standardization
,
Geneva, Switzerland
,
2003
).
6.
ISO 9614-1:1993,
Acoustics—Determination of sound power levels of noise sources using sound intensity—Part 1: Measurement at discrete points
(
International Organization for Standardization
,
Geneva, Switzerland
,
1993
).
7.
ISO 9614-2:1996,
Acoustics—Determination of sound power levels of noise sources using sound intensity—Part 2: Measurement by scanning
(
International Organization for Standardization
,
Geneva, Switzerland
,
1996
).
8.
ISO 9614-3:2002,
Acoustics—Determination of sound power levels of noise sources using sound intensity—Part 3: Precision method for measurement by scanning
(
International Organization for Standardization
,
Geneva, Switzerland
,
2002
).
9.
F.
Fahy
,
Sound Intensity
(
Spon
,
London
,
2002
), pp.
1
295
.
10.
F.
Jacobsen
, “
Sound intensity
,” in
Springer Handbook of Acoustics
, edited by
T. D.
Rossing
(
Springer
,
New York
,
2007
), pp.
961
1017
.
11.
F.
Jacobsen
, “
Sound intensity measurements
,” in
Handbook of Noise and Vibration Control
, edited by
M. J.
Crocker
(
Wiley
,
Hoboken, NJ
,
2007
), pp.
534
548
.
12.
M. J.
Crocker
, “
Measurement of sound intensity
,” in
Handbook of Acoustical Measurements and Noise Control
, edited by
C. M.
Harris
(
McGraw-Hill
,
New York
,
1991
), pp.
14.1
14.17
.
13.
U. S.
Shirahatti
and
M. J.
Crocker
, “
Two-microphone finite difference approximation errors in the interference fields of point dipole sources
,”
J. Acoust. Soc. Am.
92
,
258
267
(
1992
).
14.
R.
Raangs
,
W. F.
Druyvesteyn
, and
H. E.
De Bree
, “
A low-cost intensity probe
,”
J. Audio Eng. Soc.
51
,
344
357
(
2003
).
15.
J. H.
Giraud
,
K. L.
Gee
, and
J. E.
Ellsworth
, “
Acoustic temperature measurement in a rocket noise field
,”
J. Acoust. Soc. Am.
127
,
EL179
EL184
(
2010
).
16.
K. L.
Gee
,
J. H.
Giraud
,
J. D.
Blotter
, and
S. D.
Sommerfeldt
, “
Energy-based acoustical measurements of rocket noise
,” AIAA Paper 3165-2009.
17.
K. L.
Gee
,
J. H.
Giraud
,
J. D.
Blotter
, and
S. D.
Sommerfeldt
, “
Near-field acoustic intensity measurements of a small solid rocket motor
,”
J. Acoust. Soc. Am.
128
,
EL69
EL74
(
2010
).
18.
T. A.
Stout
,
K. L.
Gee
,
T. B.
Neilsen
,
A. T.
Wall
,
D. W.
Krueger
, and
M. M.
James
, “
Preliminary analysis of acoustic intensity in a military jet noise field
,”
Proc. Mtgs. Acoust.
19
,
040074
(
2013
).
19.
J. H.
Giraud
,
K. L.
Gee
,
S. D.
Sommerfeldt
,
T.
Taylor
, and
J. D.
Blotter
, “
Low-frequency calibration of a multidimensional acoustic intensity probe for application to rocket noise
,”
Proc. Mtgs. Acoust.
14
,
040006
(
2013
).
20.
F.
Jacobsen
and
H.-E.
de Bree
, “
A comparison of two different sound intensity measurement principles
,”
J. Acoust. Soc. Am.
118
,
1510
1517
(
2005
).
21.
F.
Jacobsen
,
V.
Cutanda
, and
P. M.
Juhl
, “
A numerical and experimental investigation of the performance of sound intensity probes at high frequencies
,”
J. Acoust. Soc. Am.
103
,
953
961
(
1998
).
22.
G. W.
Elko
, “
Frequency domain estimation of the complex acoustic intensity and acoustic energy density
,” Ph.D. thesis,
The Pennsylvania State University
,
1984
.
23.
J. W.
Parkins
,
S. D.
Sommerfeldt
, and
J.
Tichy
, “
Error analysis of a practical energy density sensor
,”
J. Acoust. Soc. Am.
108
,
211
222
(
2000
).
24.
R.
Hickling
and
A. W.
Brown
, “
Determining the direction to a sound source in air using vector sound-intensity probes
,”
J. Acoust. Soc. Am.
129
,
219
224
(
2011
).
25.
G.
Rasmussen
, “
Measurement of vector fields
,” in
Proceedings of the 2nd International Congress on Acoustic Intensity
(
1985
), pp.
53
58
.
26.
B. S.
Cazzolato
and
C. H.
Hansen
, “
Errors arising from three-dimensional energy density sensing in one-dimensional sound fields
,”
J. Sound Vib.
236
,
375
400
(
2000
).
27.
B. S.
Cazzolato
and
C. H.
Hansen
, “
Errors in the measurement of acoustic energy density in one-dimensional sound fields
,”
J. Sound Vib.
236
,
801
831
(
2000
).
28.
C. P.
Wiederhold
, “
Analytical comparison of multimicrophone probes in measuring acoustic intensity
,” Master's thesis,
Brigham Young University
, Provo, UT,
2011
.
29.
C. P.
Wiederhold
,
K. L.
Gee
,
J. D.
Blotter
,
S. D.
Sommerfeldt
, and
J. H.
Giraud
, “
Comparison of multimicrophone probe design and processing methods in measuring acoustic intensity
,”
J. Acoust. Soc. Am.
135
,
2797
2807
(
2014
).
30.
J. A.
Mann
 III
,
J.
Tichy
, and
A. J.
Romano
, “
Instantaneous and time-averaged energy transfer in acoustic fields
,”
J. Acoust. Soc. Am.
82
,
17
30
(
1987
).
31.
J. A.
Mann
 III
and
J.
Tichy
, “
Acoustic intensity analysis: Distinguishing energy propagation and wave-front propagation
,”
J. Acoust. Soc. Am.
90
,
20
25
(
1991
).
32.
J. A.
Mann
 III
and
J.
Tichy
, “
Near-field identification of vibration sources, resonant cavities, and diffraction using acoustic intensity measurements
,”
J. Acoust. Soc. Am.
90
,
720
729
(
1991
).
33.
J.-C.
Pascal
and
J.-F.
Li
, “
A systematic method to obtain 3D finite-difference formulations for acoustic intensity and other energy quantities
,”
J. Sound Vib.
310
,
1093
1111
(
2008
).
34.
C. P.
Wiederhold
,
K. L.
Gee
,
J. D.
Blotter
, and
S. D.
Sommerfeldt
, “
Comparison of methods for processing acoustic intensity from orthogonal multimicrophone probes
,”
J. Acoust. Soc. Am.
131
,
2841
2852
(
2012
).
35.
K. H.
Miah
and
E. L.
Hixon
, “
Design and performance evaluation of a broadband three dimensional acoustic intensity measuring system
,”
J. Acoust. Soc. Am.
127
,
2338
2346
(
2010
).
36.
H.
Suzuki
,
S.
Oguro
,
M.
Anzai
, and
T.
Ono
, “
Performance evaluation of a three dimensional intensity probe
,”
J. Acoust. Soc. Jpn. (E)
16
,
233
238
(
1995
).
37.
C. A.
Szuberla
and
J. V.
Olson
, “
Uncertainties associated with parameter estimation in atmospheric infrasound arrays
,”
J. Acoust. Soc. Am.
115
,
253
258
(
2003
).
38.
S. T.
Smith
, “
Covariance, subspace, and intrinsic Cramér-Rao bounds
,”
IEEE Trans. Signal Process.
53
,
1610
1630
(
2005
).
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