Extensions of the Born approximation for acoustic radiation force and torqueto inhomogeneous objects and progressive spherical waves
Thomas S. JEROME(1) and Mark F. HAMILTON(1)(1) Applied Research Laboratories and Department of Mechanical Engineering, University of Texas at Austin, Austin, Texas, USA

Abstract
The Born approximation was used in recent work to develop simple expressions for the acoustic radiation force
and torque acting on objects of arbitrary shape [Jerome et al., J. Acoust. Soc. Am. 145, 36-44 (2019)]. Results
for spheres and prolate spheroids in plane standing wave fields were compared with the full solution based on
spherical harmonic expansions of the incident and scattered fields. The approximation was shown to be accurate
for objects with material properties similar to those of the surrounding fluid, and with dimensions up to about
one wavelength. Here, closed-form expressions based on the Born approximation are presented for a finite
cylinder with material inhomogeneity that varies linearly along the axis of the cylinder. The expressions reveal
the dependence of the radiation force and torque on the position and orientation of the inhomogeneous cylinder
with respect to an incident plane standing wave. Next, the Born approximation is used to obtain a closed-form
expression for the radiation force on a homogeneous compressible sphere as a function of its distance from thecenter of a diverging or converging spherical wave. The validity of this solution is assessed via comparison withthe full solution.
Keywords: Acoustic radiation force, Born approximation

1

INTRODUCTION

Simple volume integrals based on the Born approximation were developed previously for the acoustic radiation
force and torque exerted by a standing plane wave on a homogeneous object of arbitrary shape. [1] The principal
restriction is that the material properties of the object are similar to those of the surrounding fluid, in whichcase the approximation is reasonably accurate for objects with dimensions up to about one wavelength. Results
were presented for spheres, finite cylinders, and prolate spheroids.
Here the Born approximation is generalized as follows to describe the radiation force F and torque τ acting on
an inhomogeneous object of volume V in a multidimensional wave field:
F=−

Z
V

∇hE p i − 32 f2 (r)∇∇hEk i] dV[ f1 (r)∇

τ =−

Z
V

∇hE p i − 23 f2 (r)∇∇hEk i] dV
r × [ f1 (r)∇(1)

where hE p i and hEk i are the time-averaged potential and kinetic energy densities, respectively, in the incident
sound field. Material properties are taken into account by the quantities
f1 (r) =Ks (r) − K0
Ks (r)

f2 (r) = 2

ρs (r) − ρ02ρs (r) + ρ0(2)

where f1 and f2 are Gor’kov’s parameters [2] for the bulk modulus K and density ρ of the object, also referred
to as the scatterer (subscript s), relative to the corresponding properties of the surrounding fluid (subscript 0).
Two applications of these generalizations are presented below.
First, the effect of material inhomogeneity is analyzed using closed-form expressions obtained from Eqs. (1) for
the radiation force and torque exerted by a standing plane wave on a finite cylinder having material properties
that vary linearly along its axis. The expressions reveal the dependence of the force and torque on the positionand orientation of the object with respect to the incident field as a function of the degree of inhomogeneity.

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Second, instead of a standing plane wave, a progressive spherical wave is considered for the incident field.
The Born approximation relies on energy density gradients providing the dominant contribution to the radiation
force and torque. [1] This is always the case in a standing wave, whereas in a progressive plane wave thedominant contribution is associated with the momentum in the scattered wave. Gor’kov [2] demonstrated thatenergy density gradients determine the radiation force on a small sphere located near the center of a diverging
or converging spherical wave, and that scattering determines the force far from the center. Gor’kov’s analysis isextended here to assessment of the Born approximation applied to a homogeneous compressible sphere of finite
size in a diverging or converging spherical wave field. The assessment is based on comparison of a closed-form
expression for the radiation force obtained from Eqs. (1) and the full solution based on spherical harmonic
expansions of the incident and scattered fields [3, 4].

2

PLANE STANDING WAVE INCIDENT ON AN INHOMOGENEOUS CYLINDER

The standing plane wave acting on the inhomogeneous cylinder is described by the pressure fieldp = p0 cos[k(z + d)] cos ωt(3)where k = ω/c0 , c0 is the speed of sound in the fluid surrounding the object, and d is a parameter introduced
to shift the standing wave relative to the origin. Equations (1) for the force and torque become
Fz =

p20 k
4ρ0 c20

ZV

fG (r) sin[2k(z + d)] dV

τ=

p20 k
4ρ0 c20

Z
V(r × ez ) fG (r) sin[2k(z + d)] dV(4)

where F = Fz ez and fG (r) = f1 (r) + 23 f2 (r).
The cylinder has length L and diameter D, its axis of symmetry is designated as z0 , which is assumed to lie
in the x-z plane and form an angle θ with the z axis (with increasing θ corresponding to positive rotation
about the y axis in accordance with the right-hand rule), and the center of the cylinder is positioned at theorigin, (x, y, z) = (0, 0, 0). The material inhomogeneity is assumed to vary linearly along the axis of the cylinderaccording to fG (z0 ) = fm + (2z0 /L) fδ , which increases with respect to the mean value fm from fm − fδ at oneend (z0 = −L/2) to fm + fδ at the other (z0 = L/2). Because the geometry and material properties of the cylinder
are symmetric about the plane y = 0 the torque has only a y component, such that τ = τy ey . Evaluation of theintegrals in Eqs. (4) yields
Fz (d, θ ) =p20 kV 2J1 (kD sin θ )[ fm j0 (kL cos θ ) sin 2kd + fδ j1 (kL cos θ ) cos 2kd]
4ρ0 c20 kD sin θ(5)τy (d, θ ) =p20V[ fm σm (θ ) cos 2kd − fδ σδ (θ ) sin 2kd]
192ρ0 c20

(6)

where


8J2 (kD sin θ )2 2 2J1 (kD sin θ ) 3 j1 (kL cos θ )σm (θ ) = sin 2θ 3k2 D2
j
(kL
cos
θ
)
−
4k
L
(7)
0
(kD sin θ )2
kD sin θ
kL cos θ
8J2 (kD sin θ ) 3 j1 (kL cos θ )2J1 (kD sin θ )σδ (θ ) = k3 D2 L sin 2θ cos θ+ 8kL sin θ[ j0 (kL cos θ ) − 2 j2 (kL cos θ )] (8)(kD sin θ )2kL cos θ
kD sin θand V = 41 πD2 L is the volume of the cylinder, Jn the cylindrical Bessel function, and jn the spherical Bessel
function. For fδ = 0 the results obtained previously [1] for a homogeneous cylinder are recovered.

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The effects of inhomogeneity are illustrated most clearly in the limiting case of a small (kD, kL  1) thin
(D2  L2 ) rod, for which Eqs. (5) and (6) reduce at leading order toFz (d, θ ) =
p20 kV
3 fm sin 2kd + kL fδ cos 2kd cos θ2
12ρ0 c0τy (d, θ ) = −
p20 kV L
kL fm cos 2kd sin 2θ + 2 fδ sin 2kd sin θ2
48ρ0 c0(9)(10)

In this limit it is observed that the effect of the inhomogeneity fδ on the force Fz is smaller by a factor oforder kL than the force associated with the mean value fm of the material properties, whereas the reverse is true
for the effect of the inhomogeneity on the torque τy . The force is affected significantly by the inhomogeneity
only when the rod is located in the vicinity of a pressure node or antinode, corresponding to kd ' nπ/2. In
contrast, the torque is affected by inhomogeneity for all kd (except kd ' nπ/2) provided fδ / fm is of order kL.
The dependence of τy on sin θ resulting from the inhomogeneity, as opposed to sin 2θ for a homogeneous rod(corresponding to the term multiplied by fm ), is due to the inhomogeneity being an odd function of the axial
coordinate with respect to the midpoint of the rod.

3

PROGRESSIVE SPHERICAL WAVE INCIDENT ON A HOMOGENEOUS SPHERE

The validity of the Born approximation for the radiation force on a homogeneous compressible sphere in adiverging or converging spherical wave field is assessed via comparison with the full solution based on sphericalharmonic expansions. For a sphere in a plane standing wave field, as considered previously [1], energy density
gradients provide the dominant contribution to the radiation force, as required by the Born approximation. In a
progressive spherical wave field, energy density gradients dominate only near the geometric center of the field,
whereas farther away the dominant contribution is associated with momentum in the scattered wave. Gor’kov [2]determined how close to the center of the spherical wave a small particle must be for energy density gradientsto dominate. The purpose of the following analysis is to determine this distance for a compressible sphere offinite size, and thus determine the region in which the Born approximation is valid for a progressive sphericalwave field.
3.1 Born approximation
Since the sphere is homogeneous (constant f1 and f2 ) there is no torque because of the symmetry of both theobject and the incident field, and the expression for the force in Eqs. (1) becomes
F=−

Z
V

∇ [ f1 hE p i − 23 f2 hEk i] dV(11)

The diverging or converging spherical wave incident on the sphere is centered along the z axis at r = r0 ≡ −r0 ez :p=

1
pin e−iωt + c.c.
2pin (s) = p0s0 ±ikses(12)

where s = |r − r0 | is distance from the center of the spherical wave, |pin | = p0 at s = s0 , and the plus sign ine±iks corresponds to a diverging spherical wave, the minus sign to a converging spherical wave. The energy
densities are the same in each case:
p20 s20
p20 s201hE p i =
(13)
hE
i
=
1
+
k
k2 s2
4ρ0 c20 s2
4ρ0 c20 s2
Therefore the radiation force in the Born approximation is the same, both magnitude and direction, for bothdiverging and converging spherical waves.

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The spherical wave is assumed to be incident on a spherical object of radius a that is centered along the z axis
at z = 0, such that the distance between the centers of the spherical wave and the spherical object is r0 . For
simplicity it is further assumed here that r0 > a so that the center of the spherical wave is not located within
the spherical object. In this configuration the net radiation force on the sphere has only a z component, andEq. (11) yields

π p20 s20
3
3 f2
f1 − f2 g(α) − 2 2 h(α)
(14)
Fz =
2
4ρ0 c20
k r0
where


1+α
g(α) = (1 + α ) ln
1−α
2


− 2α



1
1+α
α + α3− lnh(α) =(1 − α 2 )2 2
1−α

(15)

and α = a/r0 (< 1). When a/r0 is sufficiently small, α is approximately the angle arcsin(a/r0 ) subtended by
the z axis and the perimeter of the sphere as viewed from r = r0 .
For α  1 both g(α) and h(α) reduce to 38 α 3 + O(α 5 ) and Eq. (14) becomes
 3
2π p20 s20
3
a
2
Fz =
f1 − f2 1 + 2 2
r0  a(16)22
3ρ0 c0k r0
r03
Equation (16) is also obtained if Eq. (11) is replaced by Fz = −(d/ds)[ f1 hE p i − 23 f2 hEk i] 43 πa3 |s=r0 , which isconsistent with Eqs. (12) and (15) of Gor’kov [2] for a small sphere in a progressive spherical wave field. In
the opposite limit, r0 → a, the first term in h(α) determines the asymptotic behavior near the singularity at
r0 = a:Fz ∼ −

3π p20 s20
f2
8ρ0 c20 k2 (r0 − a)2r0 → a(17)

which depends only on the density contrast associated with f2 , and not on the compressibility contrast associatedwith f1 . This dependence is consistent with the nearly incompressible nature of a spherical wave field for small
ks.
3.2 Transition points
If f1 and f2 are either both positive or both negative, as is typically the case, and if f1 / f2 > 3/2, then Fz
passes through a zero as the distance r0 between the centers of the spherical wave and the spherical object isincreased. From Eq. (14), this distance r0 = r1 is determined by the implicit relation
1 f1 1 g(ka/kr1 ) −1/2
f1
3
kr1 =
−
>
(18)
3 f2 2 h(ka/kr1 )f2
2
Note that g/h → 1 as a/r1 → 0, and the explicit relation obtained from Eq. (16) for a small sphere, discussed
previously by Gor’kov [2], is recovered.
Of particular interest is the transition point r0 = r2 where the contribution to the radiation force due to momentum transfer associated with scattering becomes comparable to that due to the energy density gradients in
Eq. (11). The Born approximation for a sphere of finite size is valid only when energy density gradients provide the dominant contribution. An expression for the transition point r2 in a spherical wave field is available
only for small spheres, obtained by setting Eq. (16) equal to the result for a progressive plane wave incident
on a small sphere [2]:
kr2 =

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3| f1 − 23 f2 |1

f12 + f1 f2 + 34 f22 (ka)3

ka  1(19)

The corresponding value of r2 for a sphere of finite size must be determined numerically by comparison withthe full solution for the radiation force due to a spherical wave incident on the sphere, but Eq. (19) neverthelessprovides a useful point of reference.
3.3 Comparison with the full solution for the force on a sphere
The full solution for the radiation force due to a spherical wave incident on the compressible sphere is [3, 4]
Fz =

∞
iπ
(n + 1)a∗n an+1
(A∗ + An+1 + 2A∗n An+1 ) + c.c.
∑22
ρ0 c0 k n=0 (2n + 1)(2n + 3) n

(20)

(1)

where an = i(−1)n (2n + 1)hn (kr0 )ks0 p0 is the coefficient in the spherical harmonic expansion of the incident(1)
diverging spherical wave for r0 > a (with its complex conjugate used for the converging spherical wave), hn isthe spherical Hankel function, and An is the coefficient in the expansion of the scattered wave.
Results for the radiation force are normalized by 2
πa2 p20s
1
F0 =1 + 2 2 02(21)
2
2ρ0 c0
2k r0 r0
This reference force follows from the definition F0 = S0 hEin i, where S0 = πa2 is the cross-sectional area of thesphere, and hEin i is the total energy density hE p i + hEk i in the incident field evaluated at s = r0 (the center ofthe sphere). To maintain constant pressure amplitude |pin | = p0 at s = r0 we set s0 = r0 , such that for kr0  1 thequantity Fz /F0 is equivalent to the acoustic radiation force function Yp traditionally employed for a progressive
plane wave with pressure amplitude p0 incident on a sphere of radius a.
Shown in Fig. 1 are results for f1 = 0.15 and f2 = 0.05 ( f1 / f2 = 3), which are nominal values for material
properties corresponding to soft biological media in water, and for ka = 0.4 (first column), ka = 0.7 (second
column), and ka = 1 (third column). The dashed curves are the full theory given by Eq. (20), with black dashes(upper curves) corresponding to diverging spherical waves, green dashes (lower curves) to converging sphericalwaves. The solid red curves are the Born approximation, Eq. (14), and the solid blue curves are Gor’kov’s
result for a small sphere, Eq. (16). The force is plotted on a linear scale in the first row of Fig. 1 and on alogarithmic scale in the second row, with the blue curves in the second row terminated at kr0 = ka, showing
only the range r0 > a encompassed by the present analysis.
For the smallest sphere considered, ka = 0.4, all four curves appear close together on the linear scale inFig. 1(a), and for ka <∼ 0.2 (not shown) they are virtually indistinguishable on the linear scale. However, differences become apparent on the corresponding logarithmic scale in Fig. 1(d). Below the zero crossing at kr1 = 1.5
predicted by Eq. (18), i.e., in the region kr0 < kr1 where the radiation force pushes the sphere toward the centerof the spherical wave, the Born approximation is in very close agreement with the full solution. In contrast,
Gor’kov’s result for a small sphere deviates from the full solution in this region. Above the zero crossing(kr0 > kr1 ), where the force initially pushes the sphere away from the center of the spherical wave, the Born
approximation and Gor’kov’s result become indistinguishable from one another, and both analytical results start
to depart from the full solution at kr0 ∼ 10. The increasing discrepancy beyond kr0 ∼ 10 is consistent withthe approach to the transition point kr2 = 110 predicted by Eq. (19) where the energy density gradients nolonger provide the dominant contribution to the radiation force, because the wave field increasingly resembles a
progressive plane wave.
For the larger spheres, the aforementioned discrepancies with the full solution are increased. Here we discuss
only the logarithmic plot for the largest sphere, ka = 1 in Fig. 1(f). Note that the Born approximation remains
in good agreement with the full solution in the region kr0 < kr1 , including at the location kr1 = 1.8 of thezero crossing predicted by Eq. (18). Above the zero crossing (kr0 > kr1 ) the Born approximation and Gor’kov’s
result again become indistinguishable, but they immediately diverge from the full solution due to the proximityof the transition point kr2 = 7 predicted by Eq. (19). For the intermediate sphere, ka = 0.7, the transition points
are kr1 = 1.6 and kr2 = 21.

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Figure 1. Comparisons of the acoustic radiation force Fz as a function of kr0 for f1 = 0.15 and f2 = 0.05
obtained using Eq. (14) (Born approximation, solid red curves), Eq. (16) (Gor’kov’s result for a small sphere,solid blue curves), and Eq. (20) (full solution, dashed black curves for diverging spherical wave, dashed green
curves for converging spherical wave) for ka = 0.4 (first column), ka = 0.7 (second column), and ka = 1 (thirdcolumn). The force is plotted on a linear scale in the first row, and on a logarithmic scale in the second row.
The corresponding curves for ( f1 , f2 ) = (−0.15, −0.05), for which the values of kr1 and kr2 are the same as
those for ( f1 , f2 ) = (0.15, 0.05), are very close to the curves in the second row of Fig. 1, and they are virtually
mirror images with respect to Fz = 0 of the curves in the first row. The main difference is that the curves
for the diverging and converging waves based on the full solution are reversed because for sufficiently large
kr0 , where the field resembles a progressive plane wave, the diverging wave must eventually push the sphere
outward, and the converging wave must eventually push the sphere inward, regardless of the material properties.
The analysis in the preceding two paragraphs is unaffected by this distinction.
3.4 Discussion
In summary, for f1 and f2 either both positive or both negative, the Born approximation of the radiation force
on a soft compressible sphere in both diverging and converging spherical wave fields is in good agreement withthe full solution in the region a < r0 < r1 for ka as large as O(1). In the vicinity of its singularity at r0 = a, theBorn approximation agrees very well with the full solution. For ka <∼ 0.4, the Born approximation is accurate
beyond r0 = r1 out to kr0 ∼ 10.
While the case r0 < a was not considered here mainly to simplify the analysis, it corresponds to the situation
where the center of the spherical wave is located inside the spherical object. In this case there is little physicalsense to the assumption of purely progressive spherical waves, because a diverging wave would require a point
source to be located within the object, and a purely converging wave cannot exist without disappearing upon
passage through the focus inside the object. The case r0 < a is therefore most relevant to standing sphericalwaves, for which energy density gradients provide the dominant contribution to the radiation force everywherein the field, and this restriction on the Born approximation is not in question.

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The advantages of the Born approximation are that it applies to objects of any shape—subject to certain restrictions on material properties and size of the object—and that torque as well as force can be calculated rather
easily. [1] Although the accuracy of the Born approximation for an object in a diverging or converging spherical wave field is assessed here only for a spherical object, the nominal domain of validity should apply tononspherical objects of comparable size.

ACKNOWLEDGMENTS
TSJ was supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics, and by the Cockrell School of Engineering Charles F. VanBerg Graduate Fellowship in Engineering.

REFERENCES[1] Jerome, T.S.; Ilinskii, Yu.A.; Zabolotskaya, E.A.; Hamilton, M.F. Born approximation of acoustic radiation
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2019, pp. 36–44.
[2] Gor’kov, L.P. On the forces acting on a small particle in an acoustical field in an ideal fluid. Soviet Physics–
Doklady, Vol. 6, 1962, pp. 773–775.
[3] Ilinskii, Yu.A.; Zabolotskaya, E.A.; Treweek, B.C.; Hamilton, M.F. Acoustic radiation force on an elastic
sphere in a soft elastic medium. Journal of the Acoustical Society of America, Vol. 144, 2018, pp. 568–576.
[4] Sapozhnikov, O.A.; Bailey, M.R. Radiation force of an arbitrary acoustic beam on an elastic sphere in a
fluid. Journal of the Acoustical Society of America, Vol. 133, 2013, pp. 661–676.

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