This article derives the radiation impedance for two parallel pistons in two different configurations: baffled and unbaffled. For the former, the cylindrical system of coordinates is adapted to compute the self and mutual impedance of a piston between perfectly rigid infinite parallel planes. For the latter, spheroidal coordinates are used and declined for the case of two disks in free-space. The formulae for Green's function that respects Neumann boundary conditions are established in both configurations in order to compute the pressure field using the Huygens-Fresnel principle. Contrasting with the radiation impedance obtained for one single radiator, both systems exhibit resonant frequencies. The result of this study, validated in each case by a numerical method, can be used to predict the acoustic interaction of two vibrating systems for which their radiation surfaces are facing each other.
References
1.
V.
Doutaut
, D.
Matignon
, and A.
Chaigne
, “Numerical simulations of xylophones. ii. time-domain modeling of the resonator and of the radiated sound pressure
,” J. Acoust. Soc. Am.
104
(3
), 1633
–1647
(1998
).2.
M.
Junger
, “Sound radiation by resonances of free-free beams
,” J. Acoust. Soc. Am.
52
(1B
), 332
–334
(1972
).3.
I.
Bork
, “Practical tuning of xylophone bars and resonators
,” Appl. Acoust.
46
(1
), 103
–127
(1995
).4.
P.
Rucz
, M.Á.
Ulveczki
, J.
Angster
, and A.
Miklós
, “Simulation of mallet percussion instruments by a coupled modal vibroacoustic finite element model
,” J. Acoust. Soc. Am.
149
(5
), 3200
–3212
(2021
).5.
T. A.
Johansson
and M.
Kleiner
, “Theory and experiments on the coupling of two Helmholtz resonators
,” J. Acoust. Soc. Am.
110
(3
), 1315
–1328
(2001
).6.
E. M.
Arase
, “Mutual radiation impedance of square and rectangular piston in a rigid baffle
,” J. Acoust. Soc. Am.
36
, 1521
–1525
(1964
).7.
R.
Pritchard
, “Mutual acoustic impedance between radiators in an infinite rigid plane
,” J. Acoust. Soc. Am.
32
(6
), 730
–737
(1960
).8.
K.
Sha
, J.
Yang
, and W.-S.
Gan
, “A simple calculation method for the self-and mutual-radiation impedance of flexible rectangular patches in a rigid infinite baffle
,” J. Sound Vib.
282
(1–2
), 179
–195
(2005
).9.
P.
Stepanishen
, “Evaluation of mutual radiation impedances between circular pistons by impulse response and asymptotic methods
,” J. Sound Vib.
59
(2
), 221
–235
(1978
).10.
11.
T.
Hélie
and X.
Rodet
, “Radiation of a pulsating portion of a sphere: Application to horn radiation
,” Acta Acust. united Ac.
89
(4
), 565
–577
(2003
).12.
A.
Lefebvre
, G. P.
Scavone
, and J.
Kergomard
, “External tonehole interactions in woodwind instruments
,” Acta Acust. united Ac.
99
(6
), 975
–985
(2013
).13.
J.-P.
Dalmont
, C. J.
Nederveen
, and N.
Joly
, “Radiation impedance of tubes with different flanges: Numerical and experimental investigations
,” J. Sound Vib.
244
(3
), 505
–534
(2001
).14.
P.
Rucz
, “A finite element approach for the calculation of self and mutual radiation impedances of resonators
,” J. Acoust. Soc. Am.
143
(4
), 2449
–2459
(2018
).15.
E. A.
Skelton
, “Free-space Green's functions of the reduced wave equation
,” Technical Memo (Admiralty Marine Technology Establishment
, Teddington, UK
, 1982
).16.
S.
Kirkup
, “The boundary element method in acoustics: A survey
,” Appl. Sci.
9
(8
), 1642
–1690
. (2019
).17.
A.
Van Buren
and B.
King
, “Acoustic radiation from two spheroids
,” J. Acoust. Soc. Am.
52
(1B
), 364
–372
(1972
).18.
B.
King
and A.
Van Buren
, “Fortran computer programs for calculating the axisymmetric acoustic radiation from two coaxial spheroids
,” Technical Report (Naval Research Lab
, Washington, DC
, 1974
).19.
N.
Baddour
, “Multidimensional wave field signal theory: Mathematical foundations
,” AIP Adv.
1
(2
), 022120
(2011
).20.
T.
Mellow
and L.
Kärkkäinen
, “On the sound field of an oscillating disk in a finite open and closed circular baffle
,” J. Acoust. Soc. Am.
118
(3
), 1311
–1325
(2005
).21.
E.
Skelton
, “Green's functions of the reduced wave equation between parallel planes
,” Technical Memo (Admiralty Marine Technology Establishment
, Teddington, UK
, 1985
).22.
W.
Rosenheinrich
, “Tables of some indefinite integrals of Bessel functions
,” (University of Applied Sciences
, Jena, Germany
, 2012
).23.
C.
Bouwkamp
, “Theoretical and numerical treatment of diffraction through a circular aperture
,” IEEE Trans. Antennas Propag.
18
(2
), 152
–176
(1970
).24.
T.
Mellow
, “On the sound field of a resilient disk in free space
,” J. Acoust. Soc. Am.
123
(4
), 1880
–1891
(2008
).25.
P. E.
Falloon
, P.
Abbott
, and J.
Wang
, “Theory and computation of spheroidal wavefunctions
,” J. Phys. A: Math. General
36
(20
), 5477
–5503
(2003
).26.
J.
Meixner
and F. W.
Schäfke
, “Mathieu Sphäroidfunktionen
” (“Mathieu spheroidal functions”), in Mathieusche Funktionen und Sphäroidfunktionen (Mathieu Functions and Spheroidal Functions)
(Springer
, New York
, 1954
), pp. 98
–221
.27.
C.
Flammer
, Spheroidal Wave Functions
(Courier Corporation
, North Chelmsford, MA
, 2014
).28.
J.
Boisvert
and A.
Van Buren
, “Accurate calculation of prolate spheroidal radial functions of the first kind and their derivatives
,” Q. Appl. Math.
60
, 589
–599
(2002
).29.
R.
Adelman
, N. A.
Gumerov
, and R.
Duraiswami
, “Semi-analytical computation of acoustic scattering by spheroids and disks
,” J. Acoust. Soc. Am.
136
(6
), EL405
–EL410
(2014
).30.
R.
Adelman
, N. A.
Gumerov
, and R.
Duraiswami
, “Software for computing the spheroidal wave functions using arbitrary precision arithmetic
,” arXiv:1408.0074 (2014
).31.
32.
A.
Van Buren
, “Mathieu and spheroidal wave functions fortran programs for their accurate calculation
,” http://mathieuandspheroidalwavefunctions.com/ (Last viewed May 9, 2021
).33.
P.
Morse
and H.
Feshbach
, Methods of Theoretical Physics Part I and II
(McGraw-Hill, Inc
., New York
, 1953
).34.
J.
Dalmas
and R.
Deleuil
, “Translational addition theorems for prolate spheroidal vector wave functions and
,” Quart. Appl. Math.
44
(2
), 213
–222
(1986
).35.
M.
Aussal
, “Openhmx, an open-source h-matrix toolbox in matlab
,” in Proceedings of Waves 2017
, Minneapolis, MN
(May 15–19
, 2017
).36.
I.
Terrasse
and T.
Abboud
, Modélisation des Phénomènes de propagation D'ondes (Modeling of Waves Propagation Phenomena)
(École Polytechnique
, Palaiseau, France
, 2007
).37.
M.
Aussal
, “The gypsilab open-source matlab toolbox
,” https://github.com/matthieuaussal/gypsilab/tree/master/nonRegressionTest/radiationImpedances (Last viewed July 23, 2021
).38.
O. R.
Cruzan
, “Translational addition theorems for spherical vector wave functions
,” Q. Appl. Math.
20
(1
), 33
–40
(1962
).© 2021 Acoustical Society of America.
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