NEAR-FIELD SOUND PROPAGATIONBASED ON A CIRCULAR AND LINEAR ARRAY COMBINATION
Takuma Okamoto
National Institute of Information and Communications Technology
3-5, Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-0289, Japanokamoto@nict.go.jp
ABSTRACTThis paper proposes a new method for realizing 3D near-fieldsound propagation. Its main concept is that the sound pressure radiated from a circular sound source outside the circle
is completely canceled out using another linear sound source.
The appropriate driving functions and the reproduced soundpressures of both sound sources are analytically derived basedon the two-dimensional spatial Fourier transform. The radiation property reproduced by a circular source outside thecircle is different from that inside the circle. As a result,three-dimensional near-field sound propagation can be realized within the radius of a circular source because the total
radiated sound pressure outside the circle can only be completely canceled out. Compared with previous methods, thepropagation distance can be controlled by changing the circle’s radius. The results of computer simulations suggest thatthe proposed method can realize effective three-dimensional
near-field sound propagation.
Index Terms— Near-field sound propagation, circular
loudspeaker array, linear loudspeaker array, spatial Fouriertransform, helical wave spectrum
1. INTRODUCTION
Personalizing a listening area is one important and attractive
acoustic communication technique. By the superposition ofvarious localized listening areas, multiple listening zones canalso be realized [1,2]. These techniques are useful not only for
personal sound systems [2–6] but also for multilingual guideservices and other virtual reality applications.
There are two active control approaches of a sound field
for personalizing a listening area. A number of methods havebeen proposed that control both the acoustic contrast or theenergy between two spaces [1,2,4–10] and reproduce multiple
sound fields [11–16] using multiple loudspeakers. In theseapproaches, the control areas are typically set away from theloudspeakers. In another approach [17–20], a localized soundThis study was partly supported by JSPS KAKENHI Grant Number
25871208.

area is generated near the loudspeakers so that the reproduced
sound is only propagated near them, and the sound energy isquite low apart from the loudspeakers. In this paper, the latterapproach is called near-field sound propagation.
Some methods of near-field sound propagation have been
proposed [17–20]. One is based on evanescent wave [21] reproduction using linear or circular arrays of loudspeakers [17,18]. In this method, however, the attenuation property of theradiated sound cannot be controlled because the propagation
property of the evanescent wave is dependent on the wave
length [21]. Another method [19, 20], which uses double circular arrays, is based on least squares approaches [2, 4–7, 9,
12–14]. However, such methods are quite unstable because
the acoustic inverse problem is very ill-conditioned [21] andonly consider two-dimensional reproduction.
To solve these problems, this paper proposes a new
approach for achieving three-dimensional near-field soundpropagation. The primary concept of the proposed method
is that the sound pressure radiated from a circular soundsource outside the circle is completely canceled out using
another linear sound source. To realize this situation, an adequate arrangement of both circular and linear sound sources
are firstly proposed. Then the appropriate driving functionsof the sound sources and the three-dimensional reproduced
sound pressures are analytically derived from the analytical approaches of the sound field reproduction based on thetwo-dimensional spatial Fourier transform [21, 22] and thespectral division method (SDM) [23]. The key point is thatthe radiation property reproduced by a circular source outside the circle is different from that inside the circle, and thetotal radiated sound pressure outside the circle can only be
perfectly canceled out.
Compared with previous methods [17–20], the propagation distance can be controlled by changing the radius of acircular source, and the driving functions of both the circularand linear sources are based on analytical solutions instead ofthe least squares approach.

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x

ω)

z

r0

2. SOUND FIELD REPRODUCED BY CONTINUOUS
CYLINDRICAL SECONDARY SOURCEFor describing three-dimensional sound propagation by acontinuous circular sound source in the next section, the reproduced sound field by a continuous cylindrical secondary
source is briefly introduced.
Cylindrical coordinates [r, ϕ, x]T relative to Cartesian
coordinates [x, y, z]T are defined in Fig. 1. Sound pressure
P (r, ω) synthesized at position r = [r, ϕ, x]T by a continuous cylindrical secondary source with radius r0 is given as
Z 2πZ ∞
P (r, ω) =D(r 0 , ω)G3D (r−r 0 , ω)r0 dx0 dϕ0 , (1)0−∞where ω = 2πf denotes the radial frequency, f is the temporal frequency, D(r 0 , ω) is the sound source driving function
at position r 0 = [r0 , ϕ0 , x0 ]T , and G3D (r − r 0 , ω) denotes
the transfer function of the secondary source placed at r 0 topoint r. Under the free-field assumption, this is the threedimensional free-field Green’s function, defined as
G3D (r − r 0 , ω) =−jk|r −r 0 |e,
4π|r − r 0 |(2)√
where j = −1, k = ω/c denotes the wavenumber and c isthe speed of sound.
When applying the two-dimensional spatial Fourier transform [21] to (1) with respect to ϕ and x, the convolution along
them is performed by the convolution theorem:P̃n (r, kx , ω) = 2πr0 D̃n (kx , ω)G̃n (r − r0 , kx , ω),(3)where kx denotes the spatial frequency in the direction of
x. From [21, 22] and [24], G̃n (r − r0 , kx , ω) is the twodimensional spatial Fourier transform of G3D (x−x0 , ω) withrespect to ϕ and x and is analytically derived asj= − Hn(2) (kr r0 ) Jn (kr r) , (4)4jG̃n (r − r0 , kx , ω)
= − Hn(2) (kr r) Jn (kr r0 ) , (5)4r>r0G̃n (r − r0 , kx , ω)

r<r0

L|
r>
+
C

P˜

x

P˜

Fig. 1. Definition of cylindrical coordinates [r, ϕ, x]T relative to Cartesian coordinates [x, y, z]T .

C

+
L|
r<

y

r0

=

0

y

0

z
y

!

=

φ = tan

−1

r0

C(


y2 + z2

r0

φ

r

r=

D

z

,ω

)

D

L(

x,

r = [r, φ, x]T

Fig. 2. Proposed near-field sound propagation using circularsound source with radius r0 centered origin on y-z plane andlinear sound source along x-axis. DC (r0 , ω) and DL (x, ω)are driving functions of circular and linear sound sources analytically derived in (6) and (18). Reproduced sound pressures
P̃C+L (r, kx , ω) for r > r0 and r < r0 are analytically calculated by (16) and (17).
p(2)
k 2 − kx2 , Hn 1 and Jn denote the n-th orwhere kr =der Hankel function of the second kind and the n-th order
Bessel function of the first kind, respectively [21]. Therefore, G̃n (r − r0 , kx , ω) is the transfer function in the hericalwave spectrum domain from secondary source cylinder r0 to
reproduced cylinder r. (4) and (5) indicate that the propagation property of a cylindrical secondary source with r < r0 isdifferent from that with r > r0 .
3. NEAR-FIELD SOUND PROPAGATION USINGCIRCULAR AND LINEAR SOUND SOURCES
The foundational concept of the proposed methods is that thesound pressure radiated from a circular sound source outside
the circle is canceled out by another linear sound source. To
realize this situation, a continuous circular sound source withradius r0 is centered on the origin on the y-z plane and acontinuous linear sound source is located along the x-axis(Fig. 2). This situation can also be realized with cylindricaland linear sources. However, for realistic implementations,the sound source amount should be reduced, and the proposed
method introduces a circular source instead of a cylindricalsource.
The sound pressure reproduced by linear sound source
PL (r, ω) is radiated axisymmetrically to the x-axis. Then thesound pressure reproduced by circular sound source PC (r, ω)
must also be radiated axisymmetrically to the x-axis. It is apparent that P̃C,n (r, kx , ω) only has a 0-th order component(n = 0), and the driving function of a circular source is then
1 For fitting with the previous research [1], the three-dimensional free-field
Green’s function is defined in (2) instead of ejk|r −r 0 | /4π|r − r 0 | defined(2)(1)
in [21, 22]. Then Hn is derived instead of Hn used in [21, 22].

independent on ϕ and given as 1
DC (r0 , ω) =0

It is important that D̃L (kx , ω) is independent on r and(x = 0)(6)(x ̸= 0).

The driving function of a circular source in the helical wave
spectrum domain with n = 0 is calculated from the spatial
Fourier transform to (6) with respect to x. Then DC (r0 , ω)just corresponds to delta function δ(x) in the Fourier transform [21] and is obtained as
Z ∞D̃C,0 (kx , ω) =DC (r0 , ω)e−jkx x dx = 1.
(7)−∞

From (3), (4), (5), and (7), the sound pressure reproduced bya circular source in the helical wave spectrum domain withn = 0 is analytically derived asjπr0 (2)H0 (kr r0 ) J0 (kr r) ,=−P̃C,0 (r, kx , ω)2r<r0
jπr0 (2)=−P̃C,0 (r, kx , ω)
H0 (kr r) J0 (kr r0 ) .
2
r>r0

(8)
(9)

(8) and (9) also suggest that the propagation property of acircular source for r < r0 is different from that for r > r0 .
P̃C,0 (r, kx , ω) is obviously equivalent to spatial Fourier transform coefficient P̃C (r, kx , ω) with respect to x and can describe the three-dimensional sound propagation reproduced
by a circular source.
In this paragraph, the driving function of linear soundsource DL (x, ω), which is appropriate for canceling out
P̃C,0 (r, kx , ω)|r>r0 in (9), is analytically derived. The soundpressure reproduced by a linear source along the x-axis
PL (r, ω) is also given as
Z ∞PL (r, ω) =DL (x0 , ω)G3D (r − x0 , ω)dx0 . (10)−∞Then, as in (3), the spatial Fourier transform of (10) with respect to x is derived asP̃L (r, kx , ω) = D̃L (kx , ω)G̃L (r, kx , ω),(11)where G̃L (r, kx , ω) is the transfer function in the wavenumber domain from linear sound source r = 0 to reproduced
cylinder r [23, 24] and is given as
j (2)G̃L (r, kx , ω) = − H0 (kr r) .
4
For canceling P̃C,0 (r, kx , ω)

r>r0

(12)

in (9) by D̃L (kx , ω),P̃L (r, kx , ω) = − P̃C,0 (r, kx , ω)

r>r0

,

(13)

and D̃L (kx , ω) can be analytically derived by the SDM [23]asD̃L (kx , ω) =−P̃C,0 (r, kx , ω)
= −2πr0 J0 (kr r0 ).
G̃L (r, kx , ω)

(14)P̃L (r, kx , ω) =jπr0 (2)H0 (kr r) J0 (kr r0 ) ,2(15)

can be completely reproduced.
Finally, from (9) and (15), the total sound pressure reproduced by both the circular and linear sources for r > r0 is
completely canceled out with all r0 and kx
P̃C+L (r, kx , ω)

r>r0

= P̃C,0 (r, kx , ω)

r>r0

= 0.

+ P̃L (r, kx , ω)
(16)

On the other hand, the reproduced sound pressure for r < r0
is calculated from (8) and (15) as+ P̃L (r, kx , ω)
= P̃C,0 (r, kx , ω)P̃C+L (r, kx , ω)
r<r0
r<r0


jπr0
(2)
(2)
=−
H0 (kr r0 ) J0 (kr r) − H0 (kr r) J0 (kr r0 ) .
2(17)
As a result, (16) and (17) suggest that the proposed method
can realize three-dimensional near-field propagation withinr < r0 and the propagation distance can be controlled bychanging the radius of circular source r0 .
To calculate stable driving signals, D̃L (kx , ω) = 0 for
kr2 = k 2 − kx2 < 0, corresponding to the evanescent wave
components [22]. Then, the inverse spatial Fourier transformof (14) is analytically calculated from the relation of (3.876-1)
in [24] and given as
 p
sin k x2 + r02
p
DL (x, ω) = −2r0(|k| > |kx |).
x2 + r02(18)
In a practical implementation, a circular and a linear arrayof loudspeakers is used instead of continuous circular and linear sound sources. Then (6) and (18) must be discretized and(18) must be truncated.
4. EXPERIMENTS
Computer simulations were conducted to evaluate the proposed method. In all the calculations, speed of sound c was343.26 m/s and a three-dimensional free-field was assumed.
For a realistic implementation, the channel numbers of
the circular and linear arrays were M =32 and N =64, respectively. The distance between the adjacent loudspeakers of thelinear array was ∆x=0.05 m.
Figs 3(a) to (c) show the results of the sound pressures
reproduced by a circular array, a linear array and both arrays,
respectively. The radius of the circular array is r0 =0.25 m,
and f = 1 kHz. These results suggest that the propagatedsound pressure by circular array PC (r, ω) for r > r0 is reproduced by a linear array with antiphase as PL (r, ω), and thez = 0 [m]z = 0 [m]
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Fig. 3. Results of (a)-(c) reproduced sound pressure and (d) sound pressure level with radius of circular array r0 =0.25 m for
f =1 kHz. R {·} denotes complex number’s real part. Blue circles are loudspeakers. Channel numbers of circular and linear
arrays are M =32 and N =64. Distance between adjacent loudspeakers of linear array is ∆x=0.05 m. Sound pressure level at
r = [0.1, 0, 0]T is set to 0 dB in (d).
z = 0 [m]z = 0 [m]z = 0 [m]

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Y axis [m](a) Monopole

2

[dB]

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Y axis [m][dB](b) f =3 kHz, r0 =0.25 m

−1

0

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2

Y axis [m](c) f =5 kHz, r0 =0.25 m[dB]

X axis [m]2

X axis [m]0X axis [m]X axis [m]z = 0 [m]
2

−1

0

1

2

Y axis [m]

−40

[dB]

(d) f =1 kHz, r0 =0.4 m

Fig. 4. Results of sound pressure level 20 log10 |P (r, ω)| reproduced by (a) monopole point source and (b)-(d) proposed methodwith M =32, N =64, and ∆x=0.05 m. Blue circles are loudspeakers. Sound pressure level at r = [0.1, 0, 0]T is set to 0 dB.
total reproduced sound pressure can be efficiently canceled atr > r0 .

5. CONCLUSION

The results of the sound pressure level of the total reproduced sound by the proposed method are depicted in
Figs. 3(d) and 4(b) to (d) and compared with that of amonopole point source shown in Fig. 4(a). Especially for
f =3 kHz in Fig. 4(b), these results indicate that the proposed
method can realize near-field propagation PL+C (r, ω) withinr < r0 relative to a monopole. Moreover, the results in
Fig. 3(d) for r0 =0.25 m and in Fig. 4(d) for r0 =0.4 m suggestthat the propagation distance can be controlled by changing
the radius of a circular array.

This paper proposed a novel three-dimensional near-fieldsound propagation method whose elementary concept is thatthe sound pressure radiated from a circular sound source outside the circle is perfectly canceled out using a linear soundsource. For describing three-dimensional sound propagation
by a circular sound source, a reproduced sound field by acylindrical secondary source was firstly introduced. Then theadequate arrangement of circular and linear sound sources
was proposed and the appropriate driving functions of both
sources and the three-dimensional reproduced sound pressures were analytically derived. Not only the analyticallyderived total sound pressures reproduced by both sources
but also the computer simulation results indicated that the
proposed method realizes effective three-dimensional nearfield sound propagation and the propagation distance can becontrolled by changing the radius of a circular sound source.

The reproduced sound pressures for f =1 kHz in Fig. 3(d)and in Fig. 4(d), however, are slightly radiated at r > r0 .
These artifacts are caused by the effects of the truncation of alinear array and discarding the evanescent component because
these effects are not negligible at low frequency. In addition,
the reproduced sound pressure for f =5 kHz in Fig. 4(c) is
obviously radiated at r > r0 by spatial aliasing because thespatial Nyquist frequency of both arrays is about 3.4 kHz.

Future work will apply the proposed method in actual environments and conduct implementations and evaluations inreverberant environments.

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