Two main methods have been proposed to derive the acoustical radiation force and torque applied by an arbitrary acoustic field on a particle: The first one relies on the plane wave angular spectrum decomposition of the incident field (see Sapozhnikov and Bailey [J. Acoust. Soc. Am. 133, 661–676 (2013)] for the force and Gong and Baudoin [J. Acoust. Soc. Am. 148, 3131–3140 (2020)] for the torque), while the second one relies on the decomposition of the incident field into a sum of spherical waves, the so-called multipole expansion (see Silva [J. Acoust. Soc. Am. 130, 3541–3544 (2011)] and Baresch, Thomas, and Marchiano [J. Acoust. Soc. Am. 133, 25–36 (2013)] for the force, and Silva, Lobo, and Mitri [Europhys. Lett. 97, 54003 (2012)] and Gong, Marston, and Li [Phys. Rev. Appl. 11, 064022 (2019)] for the torque). In this paper, we formally establish the equivalence between the expressions obtained with these two methods for both the force and torque.

1.
L.
Rayleigh
, “
On the pressure of vibration
,”
Philos. Mag.
3
,
338
346
(
1902
).
2.
L.
Rayleigh
, “
On the momentum and pressure of gaseous vibrations, and on the connection with the virial theorem
,”
Philos. Mag.
10
,
366
374
(
1905
).
3.
P.
Biquard
, “
Les ondes ultra-sonores
,”
Rev. D'Acous.
1
,
93
109
(
1932
).
4.
P.
Biquard
, “
Les ondes ultra-sonores II
,”
Rev. D'Acous.
1
,
315
355
(
1932
).
5.
L.
Brillouin
, “
Les tensions de radiation; leur interprétation en mécanique classique et en relativité
,”
J. Phys. Rad.
6
,
337
353
(
1925
).
6.
L.
Brillouin
, “
Sur les tensions de radiation
,”
Ann. Phys.
10
,
528
586
(
1925
).
7.
L. V.
King
, “
On the acoustic radiation pressure on spheres
,”
Proc. R. Soc. London
147
(
861
),
212
240
(
1934
).
8.
K.
Yosika
and
Y.
Kawasima
, “
Acoustic radiation pressure on a compressible sphere
,”
Acustica
5
,
167
173
(
1955
), available at https://www.ingentaconnect.com/content/dav/aaua/1955/00000005/00000003/art00004.
9.
T.
Hasegawa
and
K.
Yosika
, “
Acoustic radiation pressure on a solid elastic sphere
,”
J. Acoust. Soc. Am.
46
,
1119
1143
(
1969
).
10.
T.
Embleton
, “
Mean force on a sphere in a spherical sound field
,”
J. Acoust. Soc. Am.
26
,
40
45
(
1954
).
11.
X.
Chen
and
R.
Apfel
, “
Radiation force on a spherical object in the field of a focused cylindrical transducer
,”
J. Acoust. Soc. Am.
101
,
2443
2447
(
1997
).
12.
M.
Baudoin
and
J.-L.
Thomas
, “
Acoustic tweezers for particle and fluid micromanipulation
,”
Annu. Rev. Fluid Mech.
52
,
205
234
(
2020
).
13.
L.
Gor'ov
, “
On the forces acting on a small particle in an acoustic field in an ideal fluid
,”
Sov. Phys. Dokl.
6
,
773
775
(
1962
).
14.
F. H.
Busse
and
T. G.
Wang
, “
Torque generated by orthogonal acoustic waves–theory
,”
J. Acoust. Soc. Am.
69
(
6
),
1634
1638
(
1981
).
15.
L.
Zhang
and
P. L.
Marston
, “
Angular momentum flux of nonparaxial acoustic vortex beams and torques on axisymmetric objects
,”
Phys. Rev. E
84
(
6
),
065601
(
2011
).
16.
O. A.
Sapozhnikov
and
M. R.
Bailey
, “
Radiation force of an arbitrary acoustic beam on an elastic sphere in a fluid
,”
J. Acoust. Soc. Am.
133
(
2
),
661
676
(
2013
).
17.
Z.
Gong
and
M.
Baudoin
, “
Radiation torque on a particle in a fluid: An angular spectrum based compact expression
,”
J. Acoust. Soc. Am.
148
(
5
),
3131
3140
(
2020
).
18.
G. T.
Silva
, “
An expression for the radiation force exerted by an acoustic beam with arbitrary wavefront (L)
,”
J. Acoust. Soc. Am.
130
(
6
),
3541
3544
(
2011
).
19.
G.
Silva
,
T.
Lobo
, and
F.
Mitri
, “
Radiation torque produced by an arbitrary acoustic wave
,”
Europhys. Lett.
97
(
5
),
54003
(
2012
).
20.
D.
Baresch
,
J.-L.
Thomas
, and
R.
Marchiano
, “
Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere
,”
J. Acoust. Soc. Am
133
(
1
),
25
36
(
2013
).
21.
Z.
Gong
,
P. L.
Marston
, and
W.
Li
, “
T-matrix evaluation of three-dimensional acoustic radiation forces on nonspherical objects in Bessel beams with arbitrary order and location
,”
Phys. Rev. E
99
(
6
),
063004
(
2019
).
22.
Z.
Gong
,
P. L.
Marston
, and
W.
Li
, “
Reversals of acoustic radiation torque in Bessel beams using theoretical and numerical implementations in three dimensions
,”
Phys. Rev. Appl.
11
(
6
),
064022
(
2019
).
23.
D.
Baresch
,
J.-L.
Thomas
, and
R.
Marchiano
, “
Spherical vortex beams of high radial degree for enhanced single-beam tweezers
,”
J. Appl. Phys.
113
(
18
),
184901
(
2013
).
24.
G. T.
Silva
,
J. H.
Lopes
, and
F. G.
Mitri
, “
Off-axial acoustic radiation force of repulsor and tractor Bessel beams on a sphere
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
60
(
6
),
1207
1212
(
2013
).
25.
G. T.
Silva
,
A.
Baggio
,
J. H.
Lopes
, and
F. G.
Mitri
, “
Computing the acoustic radiation force exerted on a sphere using the translational addition theorem
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
62
(
3
),
576
583
(
2015
).
26.
Z.
Gong
,
P. L.
Marston
,
W.
Li
, and
Y.
Chai
, “
Multipole expansion of acoustical Bessel beams with arbitrary order and location
,”
J. Acoust. Soc. Am.
141
(
6
),
EL574
EL578
(
2017
).
27.
L.
Zhang
, “
A general theory of arbitrary Bessel beam scattering and interactions with a sphere
,”
J. Acoust. Soc. Am.
143
(
5
),
2796
2800
(
2018
).
28.
D.
Zhao
,
J.-L.
Thomas
, and
R.
Marchiano
, “
Computation of the radiation force exerted by the acoustic tweezers using pressure field measurements
,”
J. Acoust. Soc. Am.
146
(
3
),
1650
1660
(
2019
).
29.
P. J.
Westervelt
, “
The theory of steady forces caused by sound waves
,”
J. Acoust. Soc. Am.
23
(
3
),
312
315
(
1951
).
30.
P. J.
Westervelt
, “
Acoustic radiation pressure
,”
J. Acoust. Soc. Am.
29
(
1
),
26
29
(
1957
).
31.
G.
Maidanik
, “
Torques due to acoustical radiation pressure
,”
J. Acoust. Soc. Am.
30
(
7
),
620
623
(
1958
).
32.
Z.
Fan
,
D.
Mei
,
K.
Yang
, and
Z.
Chen
, “
Acoustic radiation torque on an irregularly shaped scattered in an arbitrary sound field
,”
J. Acoust. Soc. Am.
124
,
2727
2732
(
2008
).
33.
L.
Zhang
and
P.
Marston
, “
Acoustic radiation torque and the conservation of angular momentum (L)
,”
J. Acoust. Soc. Am.
129
(
4
),
1679
1680
(
2011
).
34.
T.
Hasegawa
,
T.
Kido
,
T.
Iusizka
, and
C.
Matsuoak
, “
A general theory of Rayleigh and Langevin radiation pressures
,”
J. Acoust. Soc. Am.
21
,
145
152
(
2000
).
35.
P.
Martin
, “
On the far-field computation of acoustic radiation forces
,”
J. Acoust. Soc. Am.
142
,
2094
2100
(
2017
).
36.
P.
Martin
, “
Quadratic quantities in acoustics: Scattering cross-section and radiation force
,”
Wave Motion
86
,
63
78
(
2019
).
37.
J.-L.
Thomas
,
R.
Marchiano
, and
D.
Baresch
, “
Acoustical and optical radiation pressure and the development of single beam acoustical tweezers
,”
J. Quant. Spectrosc. Radiat. Transf.
195
,
55
65
(
2017
).
38.
K.
Melde
,
A.
Mark
,
T.
Qiu
, and
P.
Fischer
, “
Hologram for acoustics
,”
Nature
537
,
518
522
(
2016
).
39.
M.
Baudoin
,
J.-L.
Thomas
,
R. A.
Sahely
,
J.-C.
Gerbedoen
,
Z.
Gong
,
A.
Sivery
,
O.
Matar
,
N.
Smagin
,
P.
Favreau
, and
A.
Vlandas
, “
Spatially selective manipulation of cells with single-beam acoustical tweezers
,”
Nat. Commun.
11
,
4244
(
2020
).
40.
N.
Jimenez
,
R.
Pico
,
V.
Sanchez-Morcillo
,
V.
Romero-Garcia
,
L.
Garcia-Raffi
, and
K.
Staliunas
, “
Formation of high-order acoustic Bessel beams by spiral diffraction gratings
,”
Phys. Rev. E
94
(
5
),
053004
(
2016
).
41.
A.
Riaud
,
M.
Baudoin
,
O.
Bou Matar
,
L.
Becerra
, and
J.-L.
Thomas
, “
Selective manipulation of microscopic particles with precursors swirling Rayleigh waves
,”
Phys. Rev. Appl.
7
,
024007
(
2017
).
42.
N.
Jimenez
,
V.
Romero-Garcia
,
L.
Garcia-Raffi
,
F.
Camarena
, and
K.
Staliunas
, “
Sharp acoustic vortex focusing by fresnel-spiral-zone plates
,”
Appl. Phys. Lett.
112
(
20
),
204101
(
2018
).
43.
M.
Baudoin
,
J.-C.
Gerbedoen
,
A.
Riaud
,
O.
Bou Matar
,
N.
Smagin
, and
J.-L.
Thomas
, “
Folding a focalized acoustical vortex on a flat holographic transducer: Miniaturized selective acoustical tweezers
,”
Sci. Adv.
5
,
eaav1967
(
2019
).
44.
Z.
Gong
, “
Study on acoustic scattering characteristics of objects in Bessel beams and the related radiation force and torque
,” Ph.D. dissertation,
Huazhong University of Science and Technology
,
Wuhan, China
,
2018
.
45.
Z.
Gong
,
W.
Li
,
F. G.
Mitri
,
Y.
Chai
, and
Y.
Zhao
, “
Arbitrary scattering of an acoustical Bessel beam by a rigid spheroid with large aspect-ratio
,”
J. Sound Vib.
383
,
233
247
(
2016
).
46.
Z.
Gong
,
W.
Li
,
Y.
Chai
,
Y.
Zhao
, and
F. G.
Mitri
, “
T-matrix method for acoustical Bessel beam scattering from a rigid finite cylinder with spheroidal endcaps
,”
Ocean Eng.
129
,
507
519
(
2017
).
47.
W.
Li
,
Y.
Chai
,
Z.
Gong
, and
P. L.
Marston
, “
Analysis of forward scattering of an acoustical zeroth-order Bessel beam from rigid complicated (aspherical) structures
,”
J. Quant. Spectrosc. Radiat. Transf.
200
,
146
162
(
2017
).
48.
P. L.
Marston
, “
Axial radiation force of a Bessel beam on a sphere and direction reversal of the force
,”
J. Acoust. Soc. Am.
120
(
6
),
3518
3524
(
2006
).
49.
G. B.
Arfken
,
H. J.
Weber
, and, and
F. E.
Harris
,
Mathematical Methods for Physicists
, 7th ed. (
Academic Press
,
New York
,
2013
), pp.
756
765
.
50.
G. T.
Silva
and
B. W.
Drinkwater
, “
Acoustic radiation force exerted on a small spheroidal rigid particle by a beam of arbitrary wavefront: Examples of traveling and standing plane waves
,”
J. Acoust. Soc. Am.
144
(
5
),
EL453
EL459
(
2018
).
51.
E. B.
Lima
,
J. P.
Leao-Neto
,
A. S.
Marques
,
G. C.
Silva
,
J. H.
Lopes
, and
G. T.
Silva
, “
Nonlinear interaction of acoustic waves with a spheroidal particle: Radiation force and torque effects
,”
Phys. Rev. Appl.
13
,
064048
(
2020
).
52.
D.
Baresch
,
J.-L.
Thomas
, and
R.
Marchiano
, “
Orbital angular momentum transfer to stably trapped elastic particles in acoustical vortex beams
,”
Phys. Rev. Lett.
121
(
7
),
074301
(
2018
).
53.
A.
Boström
, “
Multiple scattering of elastic waves by bounded obstacles
,”
J. Acoust. Soc. Am.
67
(
2
),
399
413
(
1980
).
54.
G. T.
Silva
and
H.
Bruus
, “
Acoustic interaction forces between small particles in an ideal fluid
,”
Phys. Rev. E
90
(
6
),
063007
(
2014
).
55.
J. H.
Lopes
,
M.
Azarpeyvand
, and
G. T.
Silva
, “
Acoustic interaction forces and torques acting on suspended spheres in an ideal fluid
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
63
(
1
),
186
197
(
2016
).
56.
Z.
Gong
and
M.
Baudoin
, “
Particle assembly with synchronized acoustic tweezers
,”
Phys. Rev. Appl.
12
(
2
),
024045
(
2019
).
57.
Z.
Gong
and
M.
Baudoin
, “
Three-dimensional trapping and assembly of small particles with synchronized spherical acoustical vortices
,”
Phys. Rev. Appl.
14
,
064002
(
2020
).
58.
P. L.
Marston
, “
Acoustic beam scattering and excitation of sphere resonance: Bessel beam example
,”
J. Acoust. Soc. Am.
122
(
1
),
247
252
(
2007
).
59.
P. L.
Marston
, “
Erratum: Acoustic beam scattering and excitation of sphere resonance: Bessel beam example
,”
J. Acoust. Soc. Am.
125
(
6
),
4092
(
2009
).
60.
W.
Li
,
Q.
Gui
, and
Z.
Gong
, “
Resonance scattering of an arbitrary Bessel beam by a spherical object
,”
IEEE Trans. Ultrason. Ferroelectr. Freq. Control
66
(
8
),
1364
1372
(
2019
).
61.
L.
Zhang
and
P. L.
Marston
, “
Geometrical interpretation of negative radiation forces of acoustical Bessel beams on spheres
,”
Phys. Rev. E
84
(
3
),
035601
(
2011
).
62.
X.-D.
Fan
and
L.
Zhang
, “
Trapping force of acoustical Bessel beams on a sphere and stable tractor beams
,”
Phys. Rev. Appl.
11
(
1
),
014055
(
2019
).
63.
D.
Baresch
,
J.-L.
Thomas
, and
R.
Marchiano
, “
Observation of a single-beam gradient force acoustical trap for elastic particles: Acoustical tweezers
,”
Phys. Rev. Lett.
116
(
2
),
024301
(
2016
).
64.
Z.
Gong
,
Y.
Chai
, and
W.
Li
, “
Reversals of acoustic radiation force and torque in a single Bessel beam: Acoustic tweezers numerical toolbox
,” Acoustofluidics2018, Lille, France (
2018
).
65.
T. A.
Nieminen
,
V. L.
Loke
,
A. B.
Stilgoe
,
G.
Knöner
,
A. M.
Brańczyk
,
N. R.
Heckenberg
, and
H.
Rubinsztein-Dunlop
, “
Optical tweezers computational toolbox
,”
J. Opt. A Pure Appl. Opt.
9
(
8
),
S196
S203
(
2007
).
66.
J. D.
Jackson
,
Classical Electrodynamics
, 3rd ed. (
Wiley
,
New York
,
1999
), p.
428
.
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