Directional beams have extensive applications in communication and sound reproduction. This paper investigates the theoretical maximum directivity of infinitely flanged open-ended waveguides and the radiation pattern synthesis. We derive a rigorous solution for the maximum directivity factor of a flanged aperture with arbitrary shape by projecting its surface velocity on the waveguide modes, enabling the creation of a directional beam in any desired direction. We present case studies for a three-dimensional circular waveguide and a bidimensional waveguide. The theoretical beam that is obtained in a subspace spanned by all the propagating modes can then be synthesized by a group of incident modes or a point-source array within the waveguide. The optimality of the beam is demonstrated by comparing it with Gaussian shaded modes radiated from the waveguide. If the evanescent modes are taken into account, the maximum directivity factor increases with considerable loss to the radiation efficiency. Nevertheless, the optimum aperture velocity dominated by its evanescent components is capable of precise beam steering in extreme directions and could be useful for designing material-filled horns. Our work provides benchmark directivity factors and patterns for the practical design of horn antennas. In addition, we present a generalized form of Bouwkamp's impedance theorem.
REFERENCES
1.
D. B.
Keele
, Jr.
“
Effective performance of Bessel arrays
,” J. Audio Eng. Soc.
38
(10
), 723
–748
(1990
).2.
B.
Rafaely
, “
Spherical loudspeaker array for local active control of sound
,” J. Acoust. Soc. Am.
125
(5
), 3006
–3017
(2009
).3.
P. J.
Westervelt
, “
Parametric acoustic array
,” J. Acoust. Soc. Am.
35
(4
), 535
–537
(1963
).4.
L.
Beranek
and
T.
Mellow
, Acoustics: Sound Fields, Transducers and Vibration
, 2nd ed. (
Academic
,
London
, 2019
), pp. 190
, 463–510, 690–691.5.
Y.
Xie
,
W.
Wang
,
H.
Chen
,
A.
Konneker
,
B.-I.
Popa
, and
S. A.
Cummer
, “
Wavefront modulation and subwavelength diffractive acoustics with an acoustic metasurface
,” Nat. Commun.
5
(1
), 5553
(2014
).6.
L.
Zhao
,
E.
Laredo
,
O.
Ryan
,
A.
Yazdkhasti
,
H.-T.
Kim
,
R.
Ganye
,
T.
Horiuchi
, and
M.
Yu
, “
Ultrasound beam steering with flattened acoustic metamaterial Luneburg lens
,” Appl. Phys. Lett.
116
(7
), 071902
(2020
).7.
E. E.
Altshuler
and
D. S.
Linden
, “
Wire-antenna designs using genetic algorithms
,” IEEE Antennas Propag. Mag.
39
(2
), 33
–43
(1997
).8.
R.
Pritchard
, “
Maximum directivity index of a linear point array
,” J. Acoust. Soc. Am.
26
(6
), 1034
–1039
(1954
).9.
J. L.
Butler
and
S. L.
Ehrlich
, “
Superdirective spherical radiator
,” J. Acoust. Soc. Am.
61
(6
), 1427
–1431
(1977
).10.
H.
Levine
and
J.
Schwinger
, “
On the radiation of sound from an unflanged circular pipe
,” Phys. Rev.
73
(4
), 383
–406
(1948
).11.
G.
Homicz
and
J.
Lordi
, “
A note on the radiative directivity patterns of duct acoustic modes
,” J. Sound Vib.
41
(3
), 283
–290
(1975
).12.
A.
Snakowska
,
H.
Idczak
, and
B.
Bogusz
, “
Modal analysis of the acoustic field radiated from an unflanged cylindrical duct—Theory and measurement
,” Acta Acust. united Acust.
82
(2
), 201
–206
(1996
).13.
G.
Gabard
and
R.
Astley
, “
Theoretical model for sound radiation from annular jet pipes: Far- and near-field solutions
,” J. Fluid Mech.
549
, 315
–341
(2006
).14.
B.
Baddour
,
P.
Joseph
,
A.
McAlpine
, and
R.
Leung
, “
Acoustic radiation characteristics of cutoff modes from ducts
,” J. Sound Vib.
541
, 117306
(2022
).15.
Y.
Nomura
,
I.
Yamamura
, and
S.
Inawashiro
, “
On the acoustic radiation from a flanged circular pipe
,” J. Phys. Soc. Jpn.
15
(3
), 510
–517
(1960
).16.
D.
Homentcovschi
and
R.
Bercia
, “
Re-expansion method for generalized radiation impedance of a circular aperture in an infinite flange
,” J. Acoust. Soc. Am.
144
(1
), 32
–40
(2018
).17.
S.
Félix
,
J.-B.
Doc
, and
M. A.
Boucher
, “
Modeling of the multimodal radiation from an open-ended waveguide
,” J. Acoust. Soc. Am.
143
(6
), 3520
–3528
(2018
).18.
J. W. S.
Rayleigh
, The Theory of Sound
(
Dover
,
New York
, 1945
), Vol. 2, pp. 487
–491
.19.
C.
Morfey
, “
A note on the radiation efficiency of acoustic duct modes
,” J. Sound Vib.
9
(3
), 367
–372
(1969
).20.
W. E.
Zorumski
, “
Generalized radiation impedances and reflection coefficients of circular and annular ducts
,” J. Acoust. Soc. Am.
54
(6
), 1667
–1673
(1973
).21.
J.
Kemp
,
D.
Campbell
, and
N.
Amir
, “
Multimodal radiation impedance of a rectangular duct terminated in an infinite baffle
,” Acta Acust. united Acust.
87
(1
), 11
–15
(2001
).22.
E. R.
Geddes
, “
On the use of the Hankel transform for sound radiation
,” in Proceedings of Audio Engineering Society Convention 93
, San Francisco (October 1–4, 1992
).23.
X.
Chen
,
H.
Feng Ma
,
X.
Ying Zou
,
W.
Xiang Jiang
, and
T.
Jun Cui
, “
Three-dimensional broadband and high-directivity lens antenna made of metamaterials
,” J. Appl. Phys.
110
(4
), 044904
(2011
).24.
B. N.
Nagarkar
and
R.
Finch
, “
Sinusoidal horns
,” J. Acoust. Soc. Am.
50
(1A
), 23
–31
(1971
).25.
L.
Polo-López
,
J.
Córcoles
,
J. A.
Ruiz-Cruz
,
J. R.
Montejo-Garai
, and
J. M.
Rebollar
, “
On the theoretical maximum directivity of a radiating aperture from modal field expansions
,” IEEE Trans. Antennas Propag.
67
(4
), 2781
–2786
(2019
).26.
W. L.
Stutzman
and
G. A.
Thiele
, Antenna Theory and Design
, 3rd ed. (
Wiley
,
New York
, 2012
), pp. 349
–351
, 361–362.27.
L.
Polo-López
,
J.
Córcoles
, and
J. A.
Ruiz-Cruz
, “
Contribution of the evanescent modes to the power radiated by an aperture
,” in Proceedings of the 2021 IEEE MTT-S International Microwave Symposium (IMS)
, Atlanta, GA (June 7–25, 2021
), pp. 474
–477
.28.
E. G.
Williams
, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography
(
Academic
,
London
, 1999
), pp. 15
–55
.29.
P. M.
Morse
and
K. U.
Ingard
, Theoretical Acoustics
(
McGraw-Hill
,
New York
, 1968
), pp. 492
–522
.30.
A. D.
Pierce
, Acoustics: An Introduction to Its Physical Principles and Applications
, 3rd ed. (
Springer Nature
,
Cham, Switzerland
, 2019
).31.
K. A.
Cunefare
and
M. N.
Currey
, “
On the exterior acoustic radiation modes of structures
,” J. Acoust. Soc. Am.
96
(4
), 2302
–2312
(1994
).32.
M. N.
Currey
and
K. A.
Cunefare
, “
The radiation modes of baffled finite plates
,” J. Acoust. Soc. Am.
98
(3
), 1570
–1580
(1995
).33.
C.
Bouwkamp
, “
A contribution to the theory of acoustic radiation
,” Philips Res. Rep.
1
(4
), 251
–277
(1946
).34.
R. A.
Horn
and
C. R.
Johnson
, Matrix Analysis
, 2nd ed. (
Cambridge University
,
New York
, 2012
).35.
See supplementary material at https://doi.org/10.1121/10.0020052 for further details on formulations of the circular waveguide regarding the transverse modes, generalized radiation impedances, and parameters used for implementing the PML-multimodal method; formulations of the bidimensional waveguide regarding the transverse modes, directivity factor, generalized radiation impedances, and perfectly matched layer (PML)-multimodal method; the results of the directivity maximization and radiation pattern synthesis; and condition number analysis of the transfer matrix.
36.
R. W.
Wood
, “
XLII. On a remarkable case of uneven distribution of light in a diffraction grating spectrum
,” London, Edinburgh Dublin Philosoph. Mag. J. Sci.
4
(21
), 396
–402
(1902
).37.
V.
Pagneux
,
N.
Amir
, and
J.
Kergomard
, “
A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation
,” J. Acoust. Soc. Am.
100
(4
), 2034
–2048
(1996
).38.
A.
Kamel
and
L. B.
Felsen
, “
On the ray equivalent of a group of modes
,” J. Acoust. Soc. Am.
71
(6
), 1445
–1452
(1982
).39.
S.
Félix
and
V.
Pagneux
, “
Ray-wave correspondence in bent waveguides
,” Wave Motion
41
(4
), 339
–355
(2005
).40.
K. A.
Cunefare
, “
The minimum multimodal radiation efficiency of baffled finite beams
,” J. Acoust. Soc. Am.
90
(5
), 2521
–2529
(1991
).41.
H.
Dong
,
H.
Gao
,
X.
Feng
, and
Y.
Shen
, “
Shape optimization of acoustic horns for improved directivity control and radiation efficiency based on the multimodal method
,” J. Acoust. Soc. Am.
149
(3
), 1411
–1424
(2021
).© 2023 Acoustical Society of America.
2023
Acoustical Society of America
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