PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Physical realization of the radiation of complex multipoles
Rafael PISCOYA1; Martin OCHMANN2
University of Applied Sciences Berlin, Germany

ABSTRACT
Complex multipoles are sound sources located at positions having an imaginary component. They are also
solutions of the Helmholtz equation, but their radiation patterns are different from those of their realccounterparts. If rq is the complex source position, the source preferably emits the sound in a direction

ccindicated by Im( rq ) and the degree of focusing depends on the amplitude of Im( rq ). A loudspeaker in a box
behaves at low frequencies like a monopole, a loudspeaker without a box like a dipole and a tuning fork likea longitudinal quadrupole. In contrast, complex multipoles do not have simple physical equivalents, since
they are pure mathematical constructions. In the present work, the sound radiation of complex multipoles will
be modelled by an array of loudspeakers whose amplitudes are determined so, that their radiation pattern isreproduced. To obtain the amplitude of each loudspeaker, a combination of the Boundary Element Method(BEM) and the Equivalent Source Method (ESM) is applied. With the BEM the transfer function of each
loudspeaker is calculated and with the ESM their amplitude by minimizing the error at certain control points.
Keywords: Radiation pattern, Complex position, BEM, ESM

1. INTRODUCTION
Complex multipoles can be of interest for practical applications, since they possess a pronounced
directivity, which depends on two parameters, a scalar and a unit vector. But to be used, they need to bephysically constructable. Norris (1) showed that a monopole can be represented by a distribution ofmonopoles on the surface of a sphere of radius a, with a uniform surface density (4a 2 j0(ka)) -1. By
shifting the radius a into the complex plane, he demonstrated that the monopole could also berepresented as a distribution of complex monopoles with imaginary part Im(a) over the surface of asphere with radius Re(a) and unit vectors parallel to the radial unit vectors. Unfortunately, the opposite
situation, i.e. a representation of a complex monopole in terms of a uniform distribution of realmonopoles is not possible.
The next possible situation is the representation of a complex multipole with a non uniform surface
distribution. In practice, this can be achieved by an array of loudspeakers flush-mounted on the surface
of a radiating body. The radiation of each loudspeaker has to be controlled in such a way that the total
emitted sound pressure reproduces the radiation of the complex monopole. In the present work, wedevelop a procedure to obtain the normal velocity of the loudspeaker membranes of the radiator, basedon the “Dummy-source-method” conceived by Pavić and Lindberg (2). They replace the originalradiating body by a rigid dummy source with a simpler geometry and occupying (approximately) thesame volume, covered by several vibrating patches on the surface. The radiation of the patches isobtained using the equivalent source method (ESM). In our case, we are in principle not limited to any
particular object size, since our original source is a mathematical construction and we use acombination of BEM and ESM to compute the normal velocity of the loudspeaker membranes.

2. COMPLEX SOURCE POSITIONS
For the time convention e

12jt

c, a complex source position rq is defined as

piscoya@beuth-hochschule.deochmann@beuth-hochschule.de3235

rqc  rq  j uˆwith j ,

1(1)



where rq is the real part and  û ( û : unit vector) is the imaginary part.

c

The distance between a (real) field point r and the complex source position rq is defined by thedistance function R cr  r  r  r Rc cqcq(2)

In this case, R c vanishes not only in a point, but in a ring with radius  (singularity ring) thatlies in the planer  r  uˆ  0q(3)In order to have outgoing waves, Re( R c )>0. This condition is fulfilled by choosing the branch cut2  2

 arg R c (4)As a result of the choice, arg( R c ) changes from +/2 at points slightly above, to -/2 at points slightly
below a circular disk bounded by the singularity ring, i.e. a jump of Im( R c ) is obtained (Fig. 1).

Figure 1 – Real and imaginary part of R c with uˆ  1,1,0  / 2

3. COMPLEX MULTIPOLES
Complex multipoles can be generated by moving the source position in the complex plane. Hence,one defines the complex radiator of zero-th order, the complex monopole, ascp000 e  jkR
4R c(5)With the choice of the branch cut (4), one recognizes that the complex monopole has a singularity
ring and radiates with a higher amplitude on the positive side of the jump ( Im( R c )  0 ) than on thenegative side ( Im( R c )  0 ). So, the complex monopole shows a pronounced sound radiation in thedirection of the unit vector û . The focusing of the radiated sound rises with increasing k as well as
the size of the singularity ring. Fig. 2 shows the dependence of the focusing on  and on thefrequency.
Complex multipoles can be generated from the complex monopole. After Shin and Felsen (3), thecomplex multipole of order (m, n, l) can be obtained through repeated differentiation of the monopole3236

( p000 ) with respect to x, y and z.

pnml  n  m lp000
x n x m x l(6)Figure 2 – Radiation pattern in the far field of a complex monopole - left: fixed  ; right: fixed frequency
As example, Fig. 3 shows the radiation patterns of the dipole p100 and the quadrupole p110
whose formulas are given in Eqs. (7) and (8)1  x  xq
p100   jk  c p000
R  Rc

cc
 2 3 jk
3  x  xq
p110   k  c 2c
RR c  R (7) y  y  pcqRc000(8)

Figure 3 – Radiation pattern of complex multipoles - left: dipole p100 ; right: quadrupole p110

4. REALIZATION OF THE RADIATION PATTERN
Complex multipoles are interesting, because they can focus the acoustic energy in a specified
direction, even at low frequencies. Since they are mathematical constructions, there are no physicalmechanisms that can produce such radiation patterns. However it is possible to reconstruct their soundfields with an array of loudspeakers. Existing arrays are mostly constructed to radiate the sound in ahalf space. In the present work, an array is developed that can focus the sound at any direction in space.
That is why we consider spherical polyhedra. Three models are investigated: a) dodecahedron; b)
icosahedron and c) truncated icosahedron (“football”). The 3 models have a diameter of about 50 cm
and are equipped with loudspeakers. The number of loudspeakers (L) and their diameter (DL ) are: L=123237

and D L =18.3 cm for the dodecahedron, L=20 and D L =13.7 cm for the icosahedron and L=32 and D L=11 cm for the truncated icosahedron (Fig. 4).

Figure 4 Different models of the loudspeaker array
It is supposed that each loudspeaker has following radial normal velocity distribution  r 2 1   vn  DL / 22   DL / 2  23(9)For the reconstruction of the radiation pattern of the complex multipoles, the amplitude and phaseof each loudspeaker will be determined. The method is a variation of the„Dummy-source-method“ and consists of the following steps:
Simulation of the sound pressure at C control points around the model (Fig. 5), produced by
one loudspeaker when all others are turned off, using the BEM. A CL matrix (T) is built withthe values of sound pressure from all loudspeakers.

Figure 5 Control points for the three models
Computation of the exact values of the sound pressure at the control points using theanalytical formulas. These values build a vector p of length L
Calculation of the amplitude and phase of each loudspeaker by solving an overdetermined
system of equations (10). These values build the vector of unknowns q of length L.
Tq  p(10)To evaluate the quality of the simulation, we compute the averaged relative error percentage  C atthe control points
DC pi 1(s)iDpi 13238

 pi( e )(e) 2i2

 100(11)

(s )

(e )

where pi and pi
point respectively.

are the simulated and exact values of the sound pressure at the i-th control

5. RESULTS
A simulation with the three models for a complex monopole with imaginary amplitude   0.2
was performed. This value of  ensures that the singularity ring is completely contained inside themodels. The control points are located at surfaces that have the same form of the models but double thesize. Hence, a good agreement between simulation and exact values is expected in the region outside
the control points.
Table 1 shows the error percentage obtained with each of the described models.
Table 1 – Relative error percentage at the control points CNr. loudspeakers

125 Hz250 Hz500 Hz1000 Hz2000 Hz

122.6

3.3

9.3

54.2

66.2

20

1.5

1.9

5.8

42.7

78.5

32

0.3

0.3

0.5

4.9

72.7

The errors below 10% are presented in blue and the errors above 10% in red. The errors drop withincreasing number of loudspeakers when the models have at least two loudspeakers per wavelength.

Figure 6 – Simulated and analytical sound field at 500 Hz
Considering the suggested number of loudspeakers per wavelength, the maximum frequency thatcan be solved by the models would be frep=c/2DL : 930 Hz, 1250 Hz and 1560 Hz respectively.

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Figure 7 – Simulated and analytical sound field at 1000 Hz

Figure 8 – Simulated and analytical sound field by an extended imaginary part =1.0 and Af=33240

Fig. 6 and Fig. 7 show the sound radiated by the models and the exact analytical values. At 500 Hz(Fig. 6), the three models are able to reproduce accurately the sound radiation of the complex
monopole since the relative errors are smaller than 10%. At 1000 Hz (Fig. 7), only the model with 32
loudspeakers reproduces well the radiation pattern of the monopole with an error  C=4.9%.
If the singularity ring of the complex monopole must be always contained inside the model, the sizeof the model limits the amplitude of the imaginary part of the monopole. If the simulation is performed
for bigger imaginary parts:  =0.5 and  =1.0, large errors (C ) for all frequencies are obtained. The
accuracy of the simulation can be increased by shifting the position of the control points by a factor Affrom their original positions (Af=1) so that the singularity ring remains inside the surface bounded by
the control points (Fig. 8).

6. CONCLUSIONSThe present work shows that the numerical realization of the radiation pattern of complex
multipoles with a loudspeaker array is feasible. Accurate simulations can be achieved when the surfaceof the radiator is equipped by at least two loudspeakers per wavelength. Additionally, the controlpoints must enclose the singularity ring. Highly focused radiation patterns are possible even at low
frequencies, but at a certain distance from the radiator, since large imaginary parts are required toproduce big singularity rings.

ACKNOWLEDGEMENTS
This work was supported by the German Research Association (DFG) in the frame of the Research
Project “Theory and Application of Acoustic Multipoles with Complex Singularities”.

REFERENCES
1. A.N. Norris, Complex point-source representation of real point sources and the Gaussian beamsummation method, J. Opt. Soc. Am. A 3(12), 1986.
2. G. Pavić, A. Lindberg, Airborne sound characterisation by a dummy source approach, J. Sound &Vibration, Vol 392, pp. 91-112, 2016.
3. S.Y. Shin, L.B. Felsen, Gaussian beams by multipoles with complex source points. J. Opt. Soc. Am.
1977; 67(5): 699-700.

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