In this work an expression for the solution of the Helmholtz equation for wedge spaces is derived. Such propagation spaces represent scenarios for many acoustical problems where a free field assumption is not eligible. The proposed sound field model is derived from the general solution of the wave equation in cylindrical coordinates, using sets of orthonormal basis functions. The latter are modified to satisfy several boundary conditions representing the reflective behaviour of wedge-shaped propagation spaces. This formulation is then used in the context of nearfield acoustical holography (NAH) and to obtain the expression of the Neumann Green function. The model and its suitability for NAH is demonstrated through both numerical simulations and measured data, where the latter was acquired for the specific case of a loudspeaker on a hemi-cylindrical rigid baffle.
References
1.
M. J.
Buckingham
, “Acoustic propagation in a wedge-shaped ocean with perfectly reflecting boundaries
,” in Hybrid Formulation of Wave Propagation and Scattering
(Springer Netherlands
, Dordrecht
, 1984
), pp. 77
–99
.2.
E. G.
Williams
, Fourier Acoustic: Sound Radiation and Nearfield Acoustical Holography
(Academic
, London
, 1999
), pp. 1
–295
.3.
J.
Meyer
, “Beamforming for a circular microphone array mounted on spherically shaped objects
,” J. Acoust. Soc. Am.
109
, 185
–193
(2001
).4.
T. D.
Abhayapala
and D. B.
Ward
, “Theory and design of high order sound field microphones using spherical microphone array
,” in 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing
(2002
), Vol. 2, pp. 1949
–1952
.5.
J.
Meyer
and G. W.
Elko
, “A highly scalable spherical microphone array based on an orthonormal decomposition of the soundfield
,” in 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing
(IEEE
, New York
, 2002
), pp. 1781
–1784
.6.
Z.
Li
, R.
Duraiswami
, E.
Grassi
, and L. S.
Davis
, “Flexible layout and optimal cancellation of the orthonormality error for spherical microphone arrays
,” in 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing
(2004
), pp. 1520
–6149
.7.
S.
Moreau
, J.
Daniel
, and S.
Bertet
, “3D sound field recording with higher order Ambisonics—objective measurements and validation of a 4th order spherical microphone
,” in Preprints of the 120th AES Convention
(2006
).8.
H.
Teutsch
and W.
Kellermann
, “Acoustic source detection and localization based on wavefield decomposition using circular microphone arrays
,” J. Acoust. Soc. Am.
120
, 2724
–2736
(2006
).9.
W.
Luo
and R.
Zhang
, “Exact solution of three-dimensional acoustic field in a wedge with perfectly reflecting boundaries
,” Sci. China Phys., Mech. Astron.
58
, 1
–10
(2015
).10.
J.
Daniel
, S.
Moreau
, and R.
Nicol
, “Further investigations of high-order ambisonics and wavefield synthesis for holophonic sound imaging
,” in 114th Convention of the Audio Engineering Society
(2003
).11.
J.
Ahrens
and S.
Spors
, “An analytical approach to sound field reproduction using circular and spherical loudspeaker distributions
,” Acta Acust. Acust.
94
, 988
–999
(2008
).12.
F. M.
Fazi
, M.
Noisternig
, and O.
Warusfel
, “Representation of sound fields for audio recording and reproduction
,” in Acoustics 2012: 11ème Congrès Français d'Acoustique. Annual Meeting of the Institute of Acoustics
, Nantes, France
(23–27 April 2012
).13.
J.
Ahrens
, “Analytic methods of sound field synthesis
,” in T-Labs Series in Telecommunication Services
(Springer
Verlag, Berlin
, 2012
), pp. 1
–272
.14.
P. N.
Samarasinghe
, T. D.
Abhayapala
, and M. A.
Poletti
, “Room reflections assisted spatial sound field reproduction
,” in Proceedings of the 22nd European Signal Processing Conference (EUSIPCO)
(2014
), pp. 1352
–1356
.15.
G.
Weinreich
and E. B.
Arnold
, “Method for measuring acoustic radiation fields
,” J. Acoust. Soc. Am.
68
, 404
–411
(1980
).16.
J. D.
Maynard
, E. G.
Williams
, and Y.
Lee
, “Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH
,” J. Acoust. Soc. Am.
78
, 1395
–1413
(1985
).17.
Y. T.
Cho
, J. S.
Bolton
, and J.
Hald
, “Source visualization by using statistically optimized near-field acoustical holography in cylindrical coordinates
,” J. Acoust. Soc. Am.
118
, 2355
–2364
(2005
).18.
E. G.
Williams
, N.
Valdivia
, P. C.
Herdic
, and J.
Klos
, “Volumetric acoustic vector intensity imager
,” J. Acoust. Soc. Am.
120
, 1887
–1897
(2006
).19.
E. G.
Williams
and K.
Takashima
, “Vector intensity reconstructions in a volume surrounding a rigid spherical microphone array
,” J. Acoust. Soc. Am.
127
, 773
–783
(2010
).20.
F.
Jacobsen
, G.
Moreno-Pescador
, E.
Fernandez-Grande
, and J.
Hald
, “Near field acoustic holography with microphones on a rigid sphere
,” J. Acoust. Soc. Am.
129
, 3461
–3464
(2011
).21.
D.
Colton
and R.
Kress
, Inverse Acoustic and Electromagnetic Scattering Theory
(Springer
, New York
, 1998
), pp. 1
–405
.22.
F. M.
Fazi
, “Sound field reproduction
,” Ph.D. thesis, University of Southampton
, 2010
.23.
M.
Kolundzija
, C.
Faller
, and M.
Vetterli
, “Baffled circular loudspeaker array with broadband high directivity
,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2010
, Sheraton Dallas Hotel, Dallas, Texas
(14–19 March 2010
), pp. 73
–76
.24.
M.
Kolundzija
, C.
Faller
, and M.
Vetterli
, “Design of a compact cylindrical loudspeaker array for spatial sound reproduction
,” in 130th Convention of the Audio Engineering Society
(Audio Engineering Society, 2011
).25.
A.-G.
Webster
, “Acoustical impedance, and the theory of horns and of the phonograph
,” Proc. Natl. Acad. Sci.
5
, 275
–282
(1919
).26.
27.
J. B.
Allen
and D. A.
Berkley
, “Image method for efficiently simulating small-room acoustics
,” J. Acoust. Soc. Am.
65
, 943
–950
(1979
).28.
L. E.
Kinsler
, A. R.
Frey
, A. B.
Coppens
, and J. V.
Sanders
, Fundamentals of Acoustics
, 4th ed., edited by B.
Russiello
(Wiley
, New York
, 2000
), pp. 1
–548
.29.
M. J.
Buckingham
, “Theory of acoustic radiation in corners with homogeneous and mixed perfectly reflecting boundaries
,” J. Acoust. Soc. Am.
86
, 2273
–2291
(1989
).30.
J. S.
Asvestas
, J. J.
Bowman
, P. L.
Christiansen
, O.
Einarsson
, R. E.
Kleinman
, D. L.
Sengupta
, T. B. A.
Senior
, F. B.
Sleator
, P. L. E.
Uslenghi
, and N. R.
Zitron
, Electromagnetic and Acoustic Scattering by Simple Shapes
, edited by J. J.
Bowman
, T. B. A.
Senior
, and P. L. E.
Uslenghi
(North-Holland
, Amsterdam
, 1970
), pp. 252
–282
.31.
E. G.
Williams
, “Regularization methods for near-field acoustical holography
,” J. Acoust. Soc. Am.
110
, 1976
–1988
(2001
).32.
E. G.
Williams
, B. H.
Houston
, and P. C.
Herdic
, “Fast Fourier transform and singular value decomposition formulations for patch nearfield acoustical holography
,” J. Acoust. Soc. Am.
114
, 1322
–1333
(2003
).33.
G. H.
Koopmann
, L.
Song
, and J. B.
Fahnline
, “A method for computing acoustic fields based on the principle of wave superposition
,” J. Acoust. Soc. Am.
86
, 2433
–2438
(1989
).34.
Z.
Wang
and S. F.
Wu
, “Helmholtz equation–least-squares method for reconstructing the acoustic pressure field
,” J. Acoust. Soc. Am.
102
, 2020
–2032
(1997
).35.
A.
Schuhmacher
, J.
Hald
, K. B.
Rasmussen
, and P. C.
Hansen
, “Sound source reconstruction using inverse boundary element calculations
,” J. Acoust. Soc. Am.
113
, 114
–127
(2003
).36.
F.
Jacobsen
and V.
Jaud
, “Statistically optimized near field acoustic holography using an array of pressure-velocity probes
,” J. Acoust. Soc. Am.
121
, 1550
–1558
(2007
).37.
Y.-B.
Zhang
, F.
Jacobsen
, C.-X.
Bi
, and X.-Z.
Chen
, “Near field acoustic holography based on the equivalent source method and pressure-velocity transducers
,” J. Acoust. Soc. Am.
126
, 1257
–1263
(2009
).38.
M. R.
Bai
, C.
Wang
, and S.-W.
Juan
, “Optimal two-layer directive microphone array with application in near-field acoustical holography
,” J. Acoust. Soc. Am.
132
, 862
–871
(2012
).39.
J.
Hald
, “Fast wideband acoustical holography
,” J. Acoust. Soc. Am.
139
, 1508
–1517
(2016
).40.
W.
Klippel
and C.
Bellmann
, “Holographic nearfield measurement of loudspeaker directivity
,” in 141st Convention of the Audio Engineering Society
(2016
).41.
P. C.
Hansen
, Rank-Deficient and Discrete Ill-Posed Problems
(Society for Industrial and Applied Mathematics
, Philadelphia, PA
, 1998
), pp. 1
–247
, http://epubs.siam.org/doi/pdf/10.1137/1.9780898719697.42.
F. M.
Fazi
and P. A.
Nelson
, “The ill-conditioning problem in sound field reconstruction
,” in 123rd Convention of the Audio Engineering Society
(2007
).43.
G.
Chardon
, L.
Daudet
, A.
Peillot
, F.
Ollivier
, N.
Bertin
, and R.
Gribonval
, “Near-field acoustic holography using sparse regularization and compressive sampling principles
,” J. Acoust. Soc. Am.
132
, 1521
–1534
(2012
).44.
J.
Antoni
, “Synthetic aperture acoustical holography
,” in PROCEEDINGS OF ISMA2012-USD2012
(2012
), pp. 3525
–3538
.45.
E.
Kreyszig
, Introductory Functional Analysis with Application
(Wiley
, New York
, 1978
), pp. 1
–688
.46.
A.
Cruz
and J.
Sesma
, “Zeros of the Hankel function of real order and of its derivative
,” Math. Comput.
39
, 639
–645
(1982
).47.
P. C.
Hansen
, “The L-curve and its use in the numerical treatment of inverse problems
,” in Computational Inverse Problems in Electrocardiology
, edited by P.
Johnston
, Advances in Computational Bioengineering (WIT Press
, Southampton, 2000
), pp. 119
–142
.48.
I. N.
Bronstein
, K. A.
Semendjajew
, G.
Musiol
, and H.
Mühlig
, Taschenbuch der Mathematik (Handbook of Mathematics)
(Wissenschaftlicher Verlag Harri Deutsch GmbH
, Frankfurt am Main, Germany
, 2006
), pp. 926
–927
.49.
F.-M.
Hoffmann
, F. M.
Fazi
, and S.
Fontana
, “Sound field control with hemi-cylindrical loudspeaker arrays
,” in Audio Engineering Society Conference: 2016 AES International Conference on Sound Field Control
(2016
).50.
G.-B.
Stan
, J. J.
Embrechts
, and D.
Archambeau
, “Comparison of different impulse response measurement techniques
,” J. Audio Eng. Soc.
50
, 249
–262
(2002
).51.
F.
Jacobsen
and Y.
Liu
, “Near field acoustic holography with particle velocity transducers
,” J. Acoust. Soc. Am.
118
, 3139
–3144
(2005
).© 2017 Acoustical Society of America.
2017
Acoustical Society of America
You do not currently have access to this content.
