Models of spherical and rectangular cuboid loudspeaker arrays
Franz Zotter, Hannes Pomberger
Institute of Electronic Music and Acoustics
University of Music and Performing Arts Graz, Austria
zotter@iem.at, pomberger@iem.at

Abstract
Compact spherical/cubical loudspeaker arrays are used as beamformers to radiate focused sound with adjustabledirection. A common enclosure housing all the transducers is easier to manufacture but acoustically couples
their motion, which appears to make their independent control more challenging. However, simple analytic IIR
filters to control cubical arrays were recently published and motivate the effort to investigate their existence
for more general common-enclosure arrays in the future. This contribution describes the analytic model ofthe interior crosstalk in (i) rectangular cuboid enclosures housing multiple piston-shape transducers and (ii) in
spherical enclosures housing multiple cap-shape transducers. For both models, this contribution shows the accurate integrals over the interior modes that evaluate both the contribution of the transducers’ velocity boundary
conditions and the forces by which the produced pressure loads on active/passive transducers. A fully causal
model is obtained by assuming a damping constants for the interior modes. Both models are demonstrated andfor the cuboid model, the full electro-acoustic model is shown to compare well to laser velocity measurements
on a physical prototype. This verifies the model up to frequencies at which transducers break up into partial
vibrations. Higher-order interior modes often become irrelevant, as at high frequencies the transducer mass
starts to dominate. Simple, well-calibrated IIR control appear feasible, in general.
Keywords: acoustic coupling, common-enclosure cube/sphere loudspeaker arrays, cross-talk cancellation

1

INTRODUCTION

Sphere- and cube-shaped or platonic loudspeaker arrays have been discussed in various papers [3, 12, 5, 2, 13,
10, 9, 7, 1, 14, 11] that established the electroacoustic background and technology required to describe compact
spherical loudspeaker arrays built with electrodynamic transducers, in order to drive them with superdirectional
beamforming.

(a) Helmholtz and force integral(b) Cuboid enclosure(c) Spherical enclosure

Figure 1. The coupling forces between sending and a receiving parts of a coboid or spherical enclosure are
evaluated as impedance matrix using the Helmholtz integral over the vibrating surface Si and the resulting force
from the integrated pressure load on the receiver surface S j .
This work is inspired by the simple analytic models that works for a 4-channel horizontally beamforming loud-1085

speaker cube [4], and it refines the models [13, 9, 7]. This is done with the purpose to simplify velocity
control, and might yield simple time-domain multichannel filters to digitally equalize the transducer velocities
and remove the cross talk of the interior acoustic coupling.
Based on the refined model and measurements, we can show which model parts are often negligible (e.g. higher
interior modes) or would be much effort for limited benefit (modal break up of the loudspeaker cone vibration).

2

ACOUSTIC COUPLING BETWEEN RIGID ENCLOSURE PARTS

The motion of transducers mounted on a common enclosure is coupled by the impedance Za,i j
Fj = ∑ Za,i j vi .

(1)ithat yields a force Fj due to any transducer velocity vi . To get the impedance,
Za,i j , we calculate the force onRa resting, piston-shaped interior surface part S j by the integral Fi j = S j p(ssj )dss j of the sound pressure p(ss j )loading on S j . The integrated sound pressure p(ss j ) is caused by a piston-shaped surface part Si vibrating withRR
the velocity vi . It results from the Helmholtz integral p(ss j ) = Si ∂∂ np G dssi = iωρ vi Si G dssi using the interiorproblem Neumann Green’s function G(ssi − s j ). This function is observed at x with a source at x 0 , and it isdefined by the non-homogeneous Helmholtz equation with the wave number k = ωc
(4 + k2 )G = −δ 3 (xx − x 0 ),(2)

G
and G fulfills the boundary condition ∂∂n(ss) = 0 everywhere perpendicular to the given enclosure s ∈ S. It is
composed of interior modes G =RRR∑lmn γ ψlmn (xx). These interior modes are countable, have 3 indices in 3D,
they are orthogonal (normalized)
ψl 0 m0 n0 (xx)ψlmn (xx)d3 x = δll 0 δmm0 δnn0 eigenfunctions to the Laplacian 4ψlmn =22−klmn ψlmn , with the real-valued wave number klmn
as eigenvalue. The required modes and their eigenvaluesi ∂ ψlmn (ss)support the homogeneous normal velocity boundary conditions vn = ωρ∂ n(ss) = 0 everywhere on the enclosure223s ∈ S. Inserted into the Helmholtz equation ∑lmn (k − klmn ) γ ψlmn (xx)d k = −δ 3 (xx − x 0 ) and using orthonormality(xx )0 n0 0by multiplication with ψl 0 m0 n0 and integration over x , the unknown coefficient becomes γ = − kl 02m−k
so that2

ψlmn

Green’s function becomesψlmn (xx) ψlmn (xx0 )ψlmn (xx) ψlmn (xx0 )= − lim c2 ∑.
2
2 − k2+0k(ω− iσlmn )2 − ωlmn
σ
→
lmn
lmn
lmn
lmn

⇒ G(xx − x 0 ) = − ∑(3)A mode-specific positive and real damping constant σlmn has been introduced for a causal solution.
Coupling forces: Apart from obtaining problem-specific modes ψlmn , obtaining the coupling impedance Za,i j
requires to integrate the source mode ψlmn (ssi ) over Si and the field mode ψlmn (ss j ) over S jiωρc2 Si ψlmn (ssi ) dssi S j ψlmn (ss j ) dss jR

F j = − ∑ vi ∑
i

lmn

R

2
(ω − iσlmn )2 − ωlmn

(lmn)

= − ∑ vi ∑ Za,i j = ∑ vi Za,i j .
i

R

lmn

(4)

i

R

Lower modes integrate to a sample-and-weight approximation limlmn→0 S ψlmn (ss)dss = −ψlmn (xx) S dss = ψlmn (xx) A
because they do not vary much over positioniωρc2 ψlmn (xxi ) Ai A j ψlmn (xx j )
.
(5)
2
lmn→0
(ω − iσlmn )2 − ωlmn
√ −1
Without approximation, the constant 0 Hz mode ψ000 = V , ωlmn = 0 responds with the impedance to the
2 2transducer velocity, with the known stiffness ρcVA , and for simplicity σ000 = 0,(lmn)

lim Za,i j = −(000)

Za,i j = −1086iωρc2 Ai A jρ c2 Ai A j
=
.
ω2 V
iω V

(6)

2.1 Acoustic coupling between vibrating pistons of a rectangular cuboid enclosure
The interior modes for a rigid cuboid rectangular box uses cosine modes with corresponding wave numbers klmncos( Lπx l x) cos( Lπy m y) cos( Lπz n z)
ψlmn (xx) = p,
Lx Ly Lz (2 − δl )−1 (2 − δm )−1 (2 − δn )−12klmn= ( Lπx l)2 + ( Lπy m)2 + ( Lπz n)2 ,∂ ψlmn∂ ψlmnand ensure zero-velocity boundaries ∂ ψ∂lmn
x x=0,Lx = ∂ y y=0,Ly = ∂ z z=0,Lz = 0, l, m, n ∈ N0 . Appendix A
solves the integrals of the modes ψ(xx) over both the emitting and receiving piston.

Solution: For evaluation at constant z and excitation at constant x, for instance, assuming a piston radius Rp ,(lmm)
we get the modal coupling impedance Za,i j with all scalars and constants (ωlmn = klmn c)
q
q


m
n
l
m
2 Rp J1 π Rp ( L )2 + ( L )2 2 Rp J1 π Rp ( L )2 + ( L )2
y
z
x
y
q
q
ψlmn (xx j )iωρc2 ψlmn (xxi )
m 2
n 2
l 2
m 2
(lmn)

Za,i j = −( Ly ) +( Lz )( Lx ) +( Ly ).

2(ω − iσlmn )2 − ωlmn

(7)

The sample-and-weight approximation for low orders / small arguments J1 (x) ≈ 2x verifies
 the solution by apq
2 Rp J1 Rp π ( Llx )2 +( Lmy )2q
proaching Eq. (5) for both modal multipliers; exemplarily: limlm→0( Llx )2 +( Lmy )2= π R2p = A.

2.2 Acoustic coupling between between vibrating caps of a spherical enclosure
The acoustic coupling in a spherical enclosure works similar as above, but uses interior modes of a sphere, adirection vector θ , the variable radius r and R is the radius of the enclosure,ψlmn (xx) = Nln Ynm (θθ ) jn kln r ,with klmn = kln such that jn0 (kln R) = 0.
(8)
0

0

mm , and the orthogonal spherical Bessel fucThe spherical harmonics Ynm (θθ ) are orthonormal S2 YnmYnm0 dθθ = δnn
0ntions jn (kln r) are normalized by Nln as derived in Appendix B. The wave numbers kln need to be found to as
roots of jn0 (kln R) = 0 to fulfill the boundary conditions; l, n, m ∈ N0 with 0 ≤ n, |m| ≤ n, and l counts the zerosof jn0 (kln R) = 0 upwards. Non-trivial modes jn (kln r) 6= 0 exist for n = 0, 1 using l ≥ 0, and n > 1 with l ≥ 0.

R

Spherical cap function: The integral of Ynm (θθ ) over a spherical cap at the radius R is known to be R2 Ynm (θ0 ) an
( α
cos 2 Pn (cos α2 )−Pn+1 (cos α2 ), for n > 0n(9)an = 2πα
1 − cos 2 ,for n = 0.
Solution: The response is of the same structure as for the cuboid, only the middle part with the cap shapes is2
2 j (k R)2 R3 Eq. (23) (shorthand N 2 j (k R)2 R3 = (2+δl )(kln R) )now exactly square and contains the term Nlnn ln
ln n ln(kln R)2 −n(n+1)
(an R2 )22m
223iωρc Yn (θθ i )(Nln jn (kln R) R ) Ynm (θθ j )R3(lmn)
Za,i j = −;(10)2(ω − iσln )2 − ωlnθTsummation over m could be further simplified by ∑m Ynm (θθ i )Ynm (θθ j ) = 2n+1
4π Pn (θ i θ j ). App. B Eq. (25) verifies a
α 2
2
2
2
2
sample-and-weight approximation [an R ] = [2πR (1 − cos 2 )] = A for α → 0, which yields with the spherical
(lmn)

4π
volume R13 = 3V
and the shorthand normalizer Za,i j

2

2

iωρc
Y m (θθ i ) 4πA= − (ω−iσ
3V)2 −ω 2 nln

ln

(2+δl )(kln R)2(kln R)2 −n(n+1)

Ynm (θθ j ). For1
the zeroth-order mode (Y00 )2 = 4π
this approximation is exact. For the modal frequency ω00 = 0 of l = 0 with
(000)

2 2

2 2

A
1 ρc A
σ00 = 0, the square brackets yields 3, and we get the known stiffness Za,i j = − iωρc
= iω
V .
ω 2V

1087

parameter

value

unit

parameter

value

unit

coil inductance Lc

0.6·10−3Hcoil resistance Rc

3.2·1

Ω

transduction constant Bl

4.4·1

Tm

effective piston area A
equivalent volume Vm
mechanical Q factor Qm

129·10−4
2.16·1

m2
m3-

dynamically moved mass Mm
2 A2 /V→ stiffness Sm = ρcm√→ damping Rm = Mm Sm/Q

12·10−3
618·1
1.26·1

kg
N/m
Ns/m

edge length Lxyzp
3 3V
→ sphere radius R = p/4π→ piston radius Rp = A/π

22·10−2
13.7·10−2
6.4·10−2mmm

3→ volume V = Lxyz

10.6·10−3m3

38·10−3→ cap size α = 2 arccos 1 − A/2πR2

54◦

Table 1. Electro-acoustic characteristics of the loudspeaker cube with Visaton WS170S 4Ohm transducers.

3

CROSS-TALK AND VELOCITY CONTROL: ELECTROACOUSTIC MODEL

Compact spherical loudspeaker array literature [13, 9, 8, 4] describes radiation as caused and controlled by
the boundary condition that the housing and transducer velocities v j impose. Transducers are fed by voltages,
therefore directivity/radiation control seeks the driving voltages Ui able to synthesize transducer velocities v j
in a desired constellation. We complement the acoustic coupling from above with a complete electro-acoustic
model. As outlined in [4, 8], the transducer voltage corresponds to its current times impedance (static-coil
impedance Zc,i consists of inductance and resistance) plus the voltage induced by the transducer’s velocity (weuse s = iω)Ui = Zc,i Ii +Uind,i ,with Zc,i = Rc,i + Lc,i s.

(11)

Induction by velocity and the relation between current and force are described by the transduction constant Bli
Fi = Bli Ii ,and Uind,i = Bli vi ,(12)and the force is proportional to the mechanical transducer impedance (Zm,i , mass, damping, and stiffness)with Zm,i = Mm,i s + Rm,i + Sm,i s−1

Fi = Zm,i vi + Fa,i , ,(13)plus the acoustic force Fa.i loading on the transducer, which couples the transducer velocities, see Eq. (4),L

Fa,i = ∑ Za,i j v j .

(14)

j=1

The exterior acoustic impedance is typically negligible and therefore disregarded. Altogether, we get a system
mapping the desired output velocities to the required input voltages
Fi+ Bli vi = Zc,i Bli−1 Zm,i vi + ∑ Za,i j v j + Bli vi(15)
Ui = Zc,i Ii +Uind , i = Zc,i
Bli
j



(lmn)
= ∑ Bli δi j + Zc,i Bli−1 Zm,i δi j + Zc,i Bli−1 ∑ Za,i j
v j.
j

lmn

Stacking velocities and voltages into vectors v = [v j ], u = [Ui ], and transduction factors into a matrix C =(lmn)[Bli δi j + Zc,i Bli−1 Zm,i δi j + Zc,i Bli−1 ∑lmn Za,i j )]i j , we write the multiple-input-multiple-output system (MIMO)
compactly as u = C v. Of the acoustical Eq. (4), electrical Eq. (11), and mechanical impedances Eq. (13),the cross-talk-cancelling, velocity-controlling MIMO system C only contains parallel combinations, i.e. causal,
parallel filters. Consequently, if all electro-acoustic components Rc,i , Lc,i s, (=Zc,i ), Mm,i , Rm,i , Sm,i , (=Zm,i ), Bli ,Za,i , α, R a, V , σlmn are identified, we get a well-defined control system delivering the voltages u required toobtain the desired velocities v. It can directly be implemented, as is, by parallel, digital IIR filters.

1088

4

EXPERIMENTAL STUDY: CUBOID AND SPHERE ARRAY

To inspect the coupling and electro-acoustic model, we consider a concrete example geometry based on an
IEM loudspeaker cube [6, 4]. It houses transducers at 0◦ , 90◦ , 180◦ , −90◦ in azimuth at zero elevation. First
we inspect the coupling model assuming either a cube housing or an equivalent-volume sphere housing (withequivalent-piston-area cap sizes). Tab 1 specifies the parameters.
Fig. 2 shows the frequency response of the force F1 . The geometry and parameters are Lx = Ly = Lz = 23 cm/(2π300)(cube), radius 14.3 cm (sphere), piston R p = 6.4 cm, ρ = 1.2, c = 343, σlmn = 1+ωlmn0.14
3 ln 10. Monopole[vi ] = [1, 1, 1, 1], dipole [vi ] = [1, 0, −1, 0] and quadrupole [vi ] = [1, −1, 1, −1] velocity patterns were inserted,
modes used up to 15 kHz and maximum indices of the simulated modes were N = 18. Regardless of the
cube or sphere enclosure: Between simplified sample-and-weight Eq. (5) (left) and proper piston/cap-integrals
Eq. (7)/(10) (right), differences are drastic: Bessel-function/sphere-cap factors attenuate higher-order modes andreduce their long low-frequency skirts, which is more accurate if assuming piston/cap loudspeakers. At 0 Hz,(000)
only the velocity monopole pattern is loaded by volume stiffness Za,i j , therefore equivalent cube/sphere volumes and equivalent piston/cap areas were chosen for easy comparison. The high-frequency amplitude drop forthe sphere enclosure stems from the limit kR < N at 6.3 kHz, i.e. N = 18 was chosen slightly too low.
50

50

40

40
cub:mon@ls1
cub:dipx@ls1
cub:quad@ls1
sph:mon@ls1
sph:dipx@ls1
sph:quad@ls1
mech.imp.

30
20
10
0
-10
25

50

100

200

400

800

1.6k 3.15k

6.3k 12.5k

cub:mon@ls1
cub:dipx@ls1
cub:quad@ls1
sph:mon@ls1
sph:dipx@ls1
sph:quad@ls1
mech.imp.

30
20
10
0
-10
25

50

100

200

400

800

1.6k 3.15k

6.3k 12.5k

Figure 2. Frequency response in dB of the acoustic coupling force F1 at the first transducer of four (horizontally
every 90◦ ) of a loudspeaker cube (solid) or equivalent-volume sphere (dashed) for specific velocity patterns:monopole (mon), dipole in x direction (dipx), or quadrupole (quad); the dotted response is the added force ofthe mechanical load Zm Eq. (13) for transducers of Tab. 1. The left response considers a simplified sample-andweight solution, the right one uses proper Bessel-function/sphere-cap factors.
For the cube housing, we compare the electro-acoustic model with open Laser-Doppler vibrometry measurements1 of the transducer velocities when driving every single input voltage of a multi-channel amplifier.
It is important to note that the measurements of the loudspeaker cube contains a 101 Hz, 3rd order Butterworthhigh pass of the Laser vibrometer that was removed from the responses shown below. When transformed to the
DFT domain, the measurements describe the inverse control system at every frequency bin u = T v . This matrix
T −1 is inverted to obtain the control system C meas at every frequency bin, which is comparable to the model
C . Figure 3 compares measurement and model in terms of the voltage frequency response of the first of theloudspeaker cube’s input channels, when desiring different velocity patterns as in the simulation above.
The comparison is done either when using only the first mode at 0 Hz (left) or all the interior modes up to
15 kHz (right). It becomes clear that above 800 Hz, the required voltages drops considerably, when comparedto the model. Obviously the loudspeaker does not exhibit the mechanical-impedance related rise by Lc Mm s2 /Bl,
cp. Fig. 2. This happens as the transducer cone does not move as a solid mass anymore, and 800 Hz appears tobe the modal break-up frequency of the cone. Above this frequency, precise modeling would be difficult, but isalso unimportant for beamforming that only requires precision below the spatial aliasing limit, i.e. f <746 kHz.
1 http://phaidra.kug.ac.at/o:67626

1089

60

60

mon@ls1
dipx@ls1
quad@ls1
model:mon@ls1
model:dipx@ls1
model:quad@ls1
Bl+ZcZm/Bl

50
40
30

mon@ls1
dipx@ls1
quad@ls1
model:mon@ls1
model:dipx@ls1
model:quad@ls1
Bl+ZcZm/Bl

50
40
30

20

20

10
25

50

100

200

400

800

1.6k 3.15k

6.3k 12.5k

10
25

50

100

200

400

800

1.6k 3.15k

6.3k 12.5k

Figure 3. Model voltages of the loudspeaker cube array in dB over frequency, when the array is driven
in the velocity patterns [1, 1, 1, 1] (monopole, mon), [1, 0, −1, 0] (dipole in x direction, dipx), or [1, −1, 1, −1](quadrupole, quad), compared to the frequency-domain matrix-inverted velocity measurements from voltages.
left: acoustic crosstalk only considers 0 Hz compression mode, right: all the modes up to 15 kHz. Above
800 Hz the loudspeaker cone oscillates in partial vibrations.
Between 300 and 800 Hz, the required voltage is modeled sufficiently well by the acoustic system without
coupling. The mode at 746 Hz, see Fig. 2 (right), obviously does not have much influence (2dB on the dipole
response in measurement and model, between 600 Hz and 800 Hz). In this regard, the simple impedancecoupling of only the 0 Hz mode Eq. (6) without any further term is sufficient, as in [8, 4] and Fig 2 (left).
The remaining deviations between model and measurement below 200 Hz that amount to a couple of dB are
not explained well, however the model parameters from the technical specifications were not individually tuned.

5

CONCLUSION

This work demonstrated how to analytically model physical acoustic forces coupling multiple loudspeakers
whose back sides are connected via the common cuboidal Eq. (7) or spherical enclosure Eq. (10).
For most cases, a drastic simplification will be sufficient as in [8, eq.31]: the most relevant mode to consider
is the 0 Hz compression mode Eq. (6) of the air enclosed. It yields the known volume stiffness as impedanceiρ c2 A A

Zi j = iω Vi j of the acoustic coupling, to which the net of all surface-weighted loudspeaker velocities contribute constructively/destructively. This is because at frequencies above the loudspeaker resonance frequency,
the transducer mass dominates the impedances, what makes coupling by higher modes negligible.
The full electro-acoustic model for frequencies below which the loudspeaker cone breaks up into partial vibrations promises a full, causal, feed-forward IIR transducer velocity control, without the need for MIMO system
inversion. However it is assumed that transducer and enclosure parameters need to be identified accurately andindividually than given in the specificantions, when employed in superdirectional beamformers.
In addition to the electro-acoustic components requiring a third-order control system for every transducer, theacoustic compression mode increases the system order by one and couples the transducers at low frequencies.

REFERENCES[1] R. M. Aarts and A. J. Jannsen. Comparing sound radiation from a loudspeaker with that from a flexible
spherical capon a rigid sphere. J. Audio Eng. Soc., 59(4):201–212, 2011.
[2] R. Avizienis, A. Freed, P. Kassakian, and D. Wessel. A compact 120 independent element spherical loudspeaker array with programmable radiation patterns. In prepr. 6783, 120th AES Conv., Paris, May 2006.
[3] J. L. Butler and S. L. Ehrlich. Superdirective spherical radiator. J Acoust. Soc. Am., 61(6):1427–1431, 1977.

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[4] T. Deppisch, N. Meyer-Kahlen, F. Zotter, and M. Frank. Surround with depth on first-order beamcontrolling loudspeakers. In prepr. 9977, 144th AES Conv., May 2018.
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San Francisco, CA, 2004.
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[10] H. Pomberger. Angular and radial directivity control for spherical loudspeaker arrays. M. thesis, Institut
für Elektronische Musik und Akustik, Kunstuni Graz, Technical University Graz, Graz, A, 2008.
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E-Brief 444, 144th AES Conv., May 2018.
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The icosahedral loudspeaker. Computer Music Journal, 41(3), 2017.

A

CUBOID PISTON INTEGRALS

Integrating ψlmn (xx) over a disk, e.g., at z = 0, or z = Lz of the radius R and centered at x = a, y = b, we get
p
p
p
p
−1
−1
with scalars (2 − δl )(2 − δm )(2 − δn ) Lx Ly Lz or (−1)n (2 − δl )(2 − δm )(2 − δn ) Lx Ly Lz excluded
Irect,ϕ,r =ZZ
D(R,a,b)cos[ Lπx l x] cos[ Lπy m y] dx dy =Z R Z 2π00cos[kx (r cos ϕ − a)] cos[ky (r sin ϕ − b)] dϕ rdr(16))
By cos α cos β = cos(α+β )+cos(α−βthe term cos[kx (r cos ϕ − a)] cos[ky (r sin ϕ − b)] becomes2

Z R Z 2π0012 [cos(kx r cos ϕ + ky r sin ϕ − kx a − ky b) + cos(kx r cos ϕ − ky r sin ϕ − kx a + ky b)] dϕ rdr√
where obviously we can gather the two sinusoids to one by A cos x + B sin x = A2 + B2 cos(x + arctan AB ) above,and essentially reformulate both expressions to cos[C cos(x + D) + E] whose integral over x becomes
Z 2π
0

1
1
2 cos[C cos(x ± D) + E ± F] dx = cos(E ± F) π J0 (A),(17)

and therefore in sum [cos(E + F) + cos(E − F)] π1 J0 (A) = cos E cos F π2 J0 (A). The inner integral in Eq. (16) isqIrect,ϕ = cos( Lπx l a) cos( Lπy m b) π2 J0 r π ( Llx )2 + ( Lmy )2 ,
(18)

1091

RR

and is integrated

0 r dr; DLMF says

Rx

0 x J0 (x) dx = x J1 (x), soRR

1 R kRR
0 r J0 (kr) dr = k2 0 kr Jn (kr) dkr = k J1 (kR),q
2 R J1 Rπ ( Llx )2 + ( Lmy )2q.
Irect,ϕ,r =Irect,ϕ rdr = cos( Lπx l a) cos( Lπy m b)0
( Llx )2 + ( Lmy )2
Z R

B

(19)

SPHERICAL MODE NORMALIZATION AND SMALL CAP

Orthogonality of the spherical Bessel functions fulfilling the boundary conditions jn0 (α) = jn0 (β ) = 0 is typically
proven by inserting two different eigenvalues α, β , into the differential equation, and by integration over the
spherical Bessel function with the respective other eigenvalue, yielding a difference of both equations
α R jn0 (α) jn (β ) − β R jn (α) jn0 (β ) =−α 2 + β 2
R2

Z R
0

jn ( αR r) jn ( βR r) r2 dr.

(20)

Inserting α = kln R and β = kl 0 n R zeros the left side by the rigid condition jn0 (kln R) = jn0 (kl 0 n R) = 0. WithR
l 6= l 0 , the factor −α 2 + β 2 is non-zero so that the integral 0R jn ( αR r) jn ( βR r) r2 dr = 0 must be zero, verifying theorthogonality of jn (kln r) and jn (kl 0 n r). The integral needn’t vanish for l = l 0 : its factor −α 2 + β 2 = 0 vanishes.
2

We also need to know the missing normalization term Nln , which we find by multiplying Eq. (20) with β 2R−α 2
and setting β = kln R, so that jn0 (β ) = 0
1
= lim
2
α→β
Nln

Z R
0

β jn (α) jn0 (β ) − α jn0 (α) jn (β ) 3−α jn0 (α) jn (β ) 3
R
R .
=
lim
β =kln R
α2 − β 2α2 − β 2α→βα→βjn ( αR r) jn ( βR r) r2 dr = lim

Letting α → β , we get a 00 expression as limα→β jn0 (α) = 0. We derive numerator and denominator (de l’Hopital)
with regard to α and get[−α jn0 (α) − α 2 jn00 (α)] jn (β ) 3
1[− jn0 (α) − α jn00 (α)] jn (β ) 3
R
=
lim
R .
=
lim
2
2α
2α 2
α→β
α→β
Nln

(21)

The spherical Bessel differential equation helps simplifying −α 2 jn00 (α) = 2α jn0 (α) + [α 2 − n(n + 1)] jn (α)

 2
 0
1 − n(n+1)
jn (β )R3
α jn (α) + [α 2 − n(n + 1)] jn (α) jn (β ) 3 [β 2 − n(n + 1)] jn2 (β ) 3
1
β2
=
lim
R
=
R
=
,
2
2α 2
2β 2
2
α→β
Nln
which is verified by the indefinite integral

R

(22)

3

jn (kr) r2 dr = r2 [ jn2 (kr) − jn−1 (kr) jn+1 (kr)] using jn−1 = jn0 + n+1
x jn

and jn+1 = − jn0 − nx jn , where for k = kln only − n(n+1)
jn2 remains as jn0 vanishes. The exceptional constant modex2
j2 (0)R3

β → 0 for n = 0 is not allowed above, but j0 (0) = 1 makes it easily solved N12 = j02 (0) 0R r2 dr = 0 3
R, so that00

2 2
jn (kln R) R3 =
Nln

(
R)2
2 (k R)(k2ln−n(n+1), unless kln = 0
ln

3,

(23)

for kln = 0, i.e. n = l = 0.

Small-cap approximation Using Pn (1) = 1 and Pn0 (1) = n(n+1)
yields lim∆x→0 Pn (1 − ∆x) = 1 − n(n+1)
2
2 ∆x withα∆x = 1 − cos 2 , here α denotes the aperture, to approximate the cap weights
Z 1

an = 2πP

Pn (x)dx = −2π n+1
α

cos 2

by lim an = 2π
α→0

1092

(cos α2 )−cos α2 Pn (cos α2 )P (cos α2 )−Pn+1 (cos α2 )= 2π n−1,n2n+1−(n − 1)n∆x + (n + 1)(n + 2)∆x2(2n + 1)= 2π∆x = 2π (1 − cos α2 ).
2(2n + 1)2(2n + 1)(24)(25)