PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Tonal active control of the power scattered by locally-reacting
spheres using a small number of radiators near the surface
Mihai ORITA1; Stephen ELLIOTT2; Jordan CHEER3
1, 2, 3

Institute of Sound and Vibration Research, United Kingdom

ABSTRACTA theoretical investigation is performed where spherical harmonic series expansions are used to describe theprimary and secondary pressure fields in the problem of actively controlling sound scattering around spheres.
The study focuses on the scenario of point-monopoles radiators in the vicinity of a sphere with a uniform,
locally-reacting and real-valued surface impedance. A control method based on minimizing the contributionsof the spherical harmonic components to the scattered sound power is proposed. Under the establishedcircumstances, any radiated sound power is found to be dominated by a few spherical harmonic terms of low
degree in the region of large relative wavelength. As this wavelength decreases, numerous terms of higher
degree become significant, thus increasingly more sources are required to achieve any suppression. A study
case is performed where one point-monopole is placed at infinite distance to form an incident plane-wave.
The scattering due to this is controlled with up to four point-monopoles on or near the surface of the sphere.
Progressively better attenuation is achieved at large relative wavelength when increasing the number ofcontrol sources and the behaviour can be described using asymptotes of the spherical harmonic coefficients.
Keywords: Active Control, Sound Scattering, Spherical Harmonics

1. INTRODUCTION
Scattering represents the behaviour generated in the medium surrounding an obstacle when the
combination of travelling waves forming a disturbance impinges on it. This behaviour is the total
result of reflection, absorption and diffraction. In the case of acoustic waves, the understanding andcontrol of scattering is important for a number of applications. One such example is binaural soundreproduction where the physical presence of the head plays a major role on the perceived sound [1].
Another example is acoustic cloaking, which has its purpose in situations where it is complicated toachieve desirable sound properties of certain elements. Structural components such as walls, columns,
rafters and balconies have an impact on the soundscape within a building [2]. The ability toacoustically hide these once constructed could potentially simplify some of the challenges indesigning spaces within concert halls, schools and hospitals that require specific acoustic properties.
In general, traditional passive absorption has drawbacks when it comes to performance and covered
frequency range, which has sparked research interest into alternatives such as active control [3, 4].
When it comes to actively controlling sound scattering, there are currently two fundamental strategies.
The first is a theoretical approach based on surrounding the obstacle with continuous layers of sensors
and actuators in order to achieve perfect cancellation [5, 6]. The second method is rooted in practicalrealization and relies on supressing the scattered field, for a given frequency, by minimizing its
radiated power at a finite number of error sensor positions with a finite number of control actuators[7-9]. In some of these works, the sound fields are modelled as series of orthogonal basis functions, i.e.
expansions into uncoupled components that can be individually studied and manipulated. However,this quality can be further exploited when deriving the optimal strengths of the control sources.
The current paper proposes and theoretically investigates an alternative to the second fundamental
strategy described above. This is done in the rudimentary case of an absorptive model for a sphericalscatterer, surrounded by point-monopole sources, and consists of minimizing the contributions to thescattered sound power of the spherical harmonic components rather than the pressure at error points.
1

Mihai.Orita@soton.ac.uk
S.J.Elliott@soton.ac.uk
3
J.Cheer@soton.ac.uk
269

2. THEORETICAL BACKGROUND
2.1 Spherical Harmonic Representations for Sound Scattering from Spheres
Let there be a fluid that is stationary, isentropic, inviscid and a perfect gas. The motion of acoustic
waves propagating through this medium is governed by the linearized wave equation in the time
domain. Given a system of spherical polar coordinates, as defined in [10], the equation can be solved
to yield the general solution for the complex-valued acoustic pressure at location ሺ‫ݎ‬ǡ ߠǡ ߮ሻ [10, 11]
௡
ஶ
ሺԹሻ
ሺԹሻ
‫݌‬ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍ ܷ௡ǡ௠ ݆௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻ ൅ ෍ ෍ ܸ௡ǡ௠ ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻǡ௡ୀ଴ ௠ୀି௡௡ୀ଴ ௠ୀି௡
ஶ௡

(1)

at a single angular frequency ߱. ݇ represents the wavenumber associated with ߱. A factor of ݁ ௝ఠ௧ isomitted in (1) for convenience of notation, and it represents the chosen time-frequency convention.
Expression (1) is composed of an interior solution, the first double summation, and an exterior one,the second double summation, which are distinct through their different regions of validity [10].
݆௡ ሺ݇‫ݎ‬ሻ and ݄௡ ሺ݇‫ݎ‬ሻ are the spherical Bessel function of the first kind and, respectively, the sph erical
Hankel function of the second kind, as detailed in [10]. They describe the variation of the acoustic fieldሺԹሻwith radial distance. The basis functions ܻ௡ǡ௠ ሺߠǡ ߮ሻ are the real-valued spherical harmonics [11]ሺԹሻ(2)ܻ௡ǡ௠ ሺߠǡ ߮ሻ ൌ ඥߝ௠ ȀሺͶߨሻඥሺʹ݊ ൅ ͳሻሺ݊ െ ݉ሻǨȀሺ݊ ൅ ݉ሻǨ ܲ௡௠ ሺ  ߠሻ  ௠ ሺ݉߮ሻǡ௠
where ߝ௠ ൌ ʹ for ȁ݉ȁ ൒ ͳ and ߝ଴ ൌ ͳ , while  ିȁ௠ȁ ൌ  and  ȁ௠ȁ ൌ  . ܲ௡ ሺ‫ݔ‬ሻ are theassociated Legendre functions; however, they are defined as in [10], rather than in [11], to include theሺԹሻሺെͳሻ௠ factor corresponding to the Condon-Shortley phase. ܻ௡ǡ௠ ሺߠǡ ߮ሻ can be interpreted as 3D
patterns of angular variation around a given coordinate system and they can be visualized in [12]. ܷ௡ǡ௠and ܸ௡ǡ௠ are complex-valued coefficients which uniquely define a particular sound field and arederived based on the circumstances of the given problem (e.g. boundary conditions on obstacles).
When it comes to the exterior solution in particular, the acoustic power it radiates is [10, 11]
ஶ

௡

ͳ
ଶ
ܹൌ
෍ ෍ หܸ௡ǡ௠ ห Ǥ
ଶ
ʹߩܿ݇

(3)

௡ୀ଴ ௠ୀି௡

Each spherical harmonic component ሺ݊ǡ ݉ሻ contributes independently to the overall radiation and,thus, they are uncoupled from each other. This is a direct result of the basis functions being orthogonal
and leads to theoretical and practical benefits, as will become apparent later in the study.
The acoustic scattering from a uniform, locally-reacting impedance sphere can be described interms spherical harmonic components by applying the boundary condition(4)݆݇‫݌‬ሺ‫ݎ‬ǡ ߠǡ ߮ሻ െ ߞ߲‫݌‬ሺ‫ݎ‬ǡ ߠǡ ߮ሻȀ߲‫ ݎ‬ൌ Ͳǡ
to the general solution in (1). ߞ ൌ ܼȀሺߩܿሻ is the surface impedance of the sphere normalised by thecharacteristic acoustic impedance of the medium exterior to it. This model represents an impedance
that is manifested at each point of the boundary in the outgoing normal direction, independent of its
value at all other points. Furthermore, it is considered that ߞ ‫ א‬Թஹ଴. A negative real part translates into
the sphere creating additional energy, which is not generally encountered in the natural world.
In the case of a monochromatic point-monopole, when placed in free-field at ሺ‫ݎ‬Ԣǡ ߠԢǡ ߮Ԣሻ while
having a volumetric strength ‫ݍ‬Ԣ, it generates a sound pressure that is expanded as [10, 11]
ஶ

௡
ሺԹሻ

ଶ

ሺԹሻ

(5)

‫݌‬௜ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ‫ݍ‬Ԣ݇ ߩܿ ෍ ෍ ݆௡ ሺ݇‫ݎ‬ழ ሻ݄௡ ሺ݇‫ݎ‬வ ሻܻ௡ǡ௠ ሺߠԢǡ ߮Ԣሻܻ௡ǡ௠ ሺߠǡ ߮ሻǡ௡ୀ଴ ௠ୀି௡

where ‫ݎ‬ழ is equal to ሺ‫ݎ‬ǡ ‫ݎ‬Ԣሻ and ‫ݎ‬வ is equal to ሺ‫ݎ‬ǡ ‫ݎ‬Ԣሻ. Due to linearity, the total pressure field‫݌‬௧ resulting from the interaction between the point-monopole and a spherical obstacle of radius ܽ,
centred at the origin, is the superposition between the incident field (5) and a scattered field ‫݌‬௦ taking
the form of the exterior solution from (1). At ‫ ݎ‬ൌ ܽ, applying (4) to the total field leads toஶ௡‫ ۓ‬෍ ෍ ‫ ݍ‬ᇱ ݇ ଶߩܿൣ݆ ሺ݇‫ ݎ‬ᇱ ሻ െ ܴ ݄ ሺ݇‫ ݎ‬ᇱ ሻ൧ܻ ሺԹሻ ሺߠԢǡ ߮Ԣሻ݄ ሺ݇‫ݎ‬ሻܻ ሺԹሻ ሺߠǡ ߮ሻ ǡ ‫ ݎ‬൐ ‫ ݎ‬ᇱ
௡
ஶǡ௡ ௡
௡
௡ǡ௠
௡ǡ௠
ۖ
‫݌‬௧ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ௡ୀ଴ ௠ୀି௡
௡
ஶ

‫۔‬
ሺԹሻ
ሺԹሻ
ۖ ෍ ෍ ‫ ݍ‬ᇱ ݇ ଶߩܿൣ݆௡ ሺ݇‫ݎ‬ሻ െ ܴஶǡ௡ ݄௡ ሺ݇‫ݎ‬ሻ൧ ݄௡ ሺ݇‫ݎ‬Ԣሻܻ௡ǡ௠ ሺߠԢǡ ߮Ԣሻܻ௡ǡ௠ ሺߠǡ ߮ሻǡ ‫ ݎ‬൏ ‫ ݎ‬ᇱ ‫ ە‬௡ୀ଴ ௠ୀି௡and, by extension, to the scattered field

70

ǡ

(6)

௡

ஶ

ሺԹሻ

ሺԹሻ

‫݌‬௦ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍ െ‫ݍ‬ᇱ ݇ ଶ ߩܴܿஶǡ௡ ݄௡ ሺ݇‫ ݎ‬ᇱ ሻܻ௡ǡ௠ ሺߠԢǡ ߮Ԣሻ ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻǡ(7)

௡ୀ଴ ௠ୀି௡
ܴஶǡ௡ ൌ ሾ݆௡ ሺ݇ܽሻ ൅ ݆ߞ݆௡ᇱ ሺ݇ܽሻሿȀሾ݄௡ ሺ݇ܽሻ ൅ ݆ߞ݄௡ᇱ ሺ݇ܽሻሿ

where. ݆௡ᇱ ሺ݇ܽሻ and ݄௡ᇱ ሺ݇ܽሻ are the first
derivatives in respect to the dimensionless quantity ݇ܽ, which describes the wavelength of soundrelative to the size of the scatterer as an alternative to frequency. The scattered field (7) is present inboth radial regions of (6) and is an exterior solution. The acoustic field generated directly by the
point-monopole is converging towards the origin between ‫ݎ‬Ԣ and ܽ, where it has the form of aninterior solution, while diverging away from the origin after ‫ݎ‬Ԣ, where it is an exterior solution.
Allowing ‫ݎ‬Ԣ ՜ λ in (6) imposes the first branch and results in the scattering of a plane-wave ofmagnitude ܲ௜ arriving from angles ሺߠԢǡ ߮Ԣሻ, where the total and scattered pressure fields are [10, 11]
ஶ

௡

ሺԹሻ

ሺԹሻ

‫݌‬௧ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍ Ͷߨܲ௜ ݆ ௡ ൣ݆௡ ሺ݇‫ݎ‬ሻ െ ܴஶǡ௡ ݄௡ ሺ݇‫ݎ‬ሻ൧ ܻ௡ǡ௠ ሺߠԢǡ ߮Ԣሻܻ௡ǡ௠ ሺߠǡ ߮ሻǡ(8)

௡ୀ଴ ௠ୀି௡
௡
ஶ
ሺԹሻ

ሺԹሻ

‫݌‬௦ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍ െͶߨܲ௜ ݆ ௡ ܴஶǡ௡ ܻ௡ǡ௠ ሺߠԢǡ ߮Ԣሻ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻǤ(9)௡ୀ଴ ௠ୀି௡

Furthermore, allowing ‫ݎ‬Ԣ ൌ ܽ in (6) also imposes the first branch and yields the radiated pressureof a point-monopole with source strength ‫ݍ‬Ԣ, situated on a spherical baffle at location ሺߠԢǡ ߮Ԣሻ [10, 11]ஶ௡‫݌‬௥ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍௡ୀ଴ ௠ୀି௡

‫ݍ‬ԢߩܿߞሺԹሻሺԹሻܻ ሺߠԢǡ ߮Ԣሻ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻǤܽଶ ݄௡ ሺ݇ܽሻ ൅ ݆ߞ݄௡ᇱ ሺ݇ܽሻ ௡ǡ௠

(10)

2.2 Tonal Active Control of Scattering Based on Spherical Harmonic Coefficients
When it comes to the problem of actively supressing the scattering around the sphere of radius ܽ,
let the primary field be the superposition of scattered fields due to ‫ ܨ‬incident point-monopole
disturbances, situated external to the obstacle. Each of these fields can take any of the forms expressedin (7), (9) or (10). In the same region of space, let there also be a secondary field composed of ‫ܩ‬point-monopole control sources, distinct from the previous set. Each of these fields can take the forms
expressed in (6) or (10). It is chosen that the control system attenuates the radiated sound power of the
primary, a measure that is invariant with space outside of the obstacle. When determining the soundpower before and after control on a virtual sphere that encloses the ‫ ܩ‬sources, all possible forms ofacoustic field in the control problem are purely radiations, i.e. behave as exterior solutions.
The above tactic leads to the considered primary and secondary fields being
ሺ௙ሻ

ሺԹሻ

݂ ‫ א‬ሼͳǡ ʹ ǥ ‫ܨ‬ሽǡ

(11)

ሺԹሻ

݃ ‫ א‬ሼͳǡ ʹ ǥ ‫ܩ‬ሽǡ(12)‫݌‬ௗ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍ ݀௡ǡ௠ ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻ ǡ
௙ ௡ǡ௠
ሺ௚ሻ

‫݌‬஻ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ෍ ‫ݍ‬௚ ‫ܤ‬௡ǡ௠ ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻ ǡ
௚ ௡ǡ௠

ሺ௙ሻ

where the double summations are denoted in a compact way. The possible forms taken by ݀௡ǡ௠ andሺ௚ሻ‫ܤ‬௡ǡ௠ are shown in Table 1 for different regimes of ‫ݎ‬Ԣ, where ܴ௡ ሺ‫ݎ‬Ԣሻ ൌ ሺ݇ܽሻଶ ൣ݆௡ ሺ݇‫ ݎ‬ᇱ ሻ െ ܴஶǡ௡ ݄௡ ሺ݇‫ ݎ‬ᇱ ሻ൧
and ܴ௡ ሺܽሻ ൌ ሺ݇ܽሻଶൣ݆௡ ሺ݇ܽሻ െ ܴஶǡ௡ ݄௡ ሺ݇ܽሻ൧ ൌ ߞȀሾ݄௡ ሺ݇ܽሻ ൅ ݆ߞ݄௡ᇱ ሺ݇ܽሻሿ. The field after control is
ሺԹሻ

ሺ௙ሻ

ሺ௚ሻ

ሺԹሻ

‫݌‬௘ ሺ‫ݎ‬ǡ ߠǡ ߮ሻ ൌ ෍ ݁࢔ǡ௠ ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻ ൌ ෍ ቌ෍ ݀௡ǡ௠ ൅ ෍ ‫ݍ‬௚ ‫ܤ‬௡ǡ௠ ቍ ݄௡ ሺ݇‫ݎ‬ሻܻ௡ǡ௠ ሺߠǡ ߮ሻ
௡ǡ௠

௡ǡ௠

௙

(13)

௚

and, from (3), radiates a sound power of
ଶ

ͳ
ͳ
ଶ
ሺ௙ሻ
ሺ௚ሻ
ܹ௘ ൌ
෍ห݁࢔ǡ௠ ห ൌ෍ ቮ෍ ݀௡ǡ௠ ൅ ෍ ‫ݍ‬௚ ‫ܤ‬௡ǡ௠ ቮ
ଶ
ଶ
ʹߩܿ݇
ʹߩܿ݇
௡ǡ௠

௡ǡ௠

௙

(14)

௚

ܹ௘ is a ݃-dimensional, continuous, quadratic function with the source strengths ‫ݍ‬௚ as variables.
Active control is meant to find the set of ‫ݍ‬௚ for which this function admits a minimum, if any. In
practice, manipulating (13) and (14) requires the truncation of the infinite summation over ݊, turningthem into numerical results. Let ܰ ൒ Ͳ be the truncation and let the spherical harmonic terms ሺ݊ǡ ݉ሻ

71

Table 1 – Possible forms of spherical harmonic coefficients in primary and secondary
ሺ௙ሻ

ሺ௚ሻ

݀௡ǡ௠

‫ܤ‬௡ǡ௠
ሺԹሻሺԹሻ‫ݎ‬Ԣ ൌ ܽ‫ݍ‬௙ ߩܿܽିଶ ܴ௡ ሺܽሻܻ௡ǡ௠ ൫ߠ௙ ǡ ߮௙ ൯ߩܿܽିଶ ܴ௡ ሺܽሻܻ௡ǡ௠ ൫ߠ௚ ǡ ߮௚ ൯‫ݎ‬Ԣ ൐ ܽെ‫ݍ‬௙ ݇ ଶ ߩܴܿஶǡ௡ ݄௡ ሺ݇‫ ݎ‬ᇱ ሻܻ௡ǡ௠ ൫ߠ௙ ǡ ߮௙ ൯‫ݎ‬Ԣ ՜ λെͶߨܲ௙ ݆ ௡ ܴஶǡ௡ ܻ௡ǡ௠ ൫ߠ௙ ǡ ߮௙ ൯

ሺԹሻ

ሺԹሻ

ߩܿܽିଶ ܴ௡ ሺ‫ݎ‬Ԣሻܻ௡ǡ௠ ൫ߠ௚ ǡ ߮௚ ൯

ሺԹሻ

--

be arranged in the order: ሺͲǡͲሻ, ሺͳǡ െͳሻ, ሺͳǡͲሻ, ሺͳǡͳሻ … ሺܰǡ െܰሻ … ሺܰǡ Ͳሻ… ሺܰǡ ܰሻ. This leads to a
total of ‫ ܮ‬ൌ ሺܰ ൅ ͳሻଶ components present in the series, of which ʹ݊ ൅ ͳ are of the same degree ݊.
The version of (13) truncated to ݊ ൌ ܰ can be arranged into the matrix equation(15)‫ ܍‬ൌ ‫ ܌‬൅ ۰‫ܙ‬Ǥ
The ‫ ܮ‬ൈ ͳ column vector ‫ ܌‬contains the components of the primary and is expressed as
୘
ሺ௙ሻ

ሺ௙ሻ

ሺ௙ሻ

ሺ௙ሻ

ሺ௙ሻ

ሺ௙ሻ

ሺ௙ሻ

‫ ܌‬ൌ ቎෍ ݀଴ǡ଴ ǡ ෍ ݀ଵǡିଵ ǡ ෍ ݀ଵǡ଴ ǡ ෍ ݀ଵǡଵ ǥ෍ ݀ேǡିே ǥ෍ ݀ேǡ଴ ǥ෍ ݀ேǡே ቏ Ǥ
௙

௙

௙

௙

௙

௙

(16)

௙

In the ‫ ܮ‬ൈ ‫ ܩ‬matrix ۰, the ݃-th column contains the corresponding components in the secondary forthe ݃-th control source and is given by
ሺ௚ሻ

ሺ௚ሻ

ሺ௚ሻ

ሺ௚ሻ

ሺ௚ሻ

ሺ௚ሻ

ሺ௚ሻ ୘

۰‫כ‬ǡ௚ ൌ ቂ‫ܤ‬଴ǡ଴ ǡ ‫ܤ‬ଵǡିଵ ǡ ‫ܤ‬ଵǡ଴ ǡ ‫ܤ‬ଵǡଵ ǥ‫ܤ‬ேǡିே ǥ‫ܤ‬ேǡ଴ ǥ‫ܤ‬ேǡே ቃ Ǥ

(17)

The ‫ ܩ‬ൈ ͳ column vector ‫ ܙ‬is governed by the constraint of the control problem and is equal to୘(18)‫ ܙ‬ൌ ൣ‫ݍ‬ଵ ǡ ‫ݍ‬ଶ ǡ ‫ݍ‬ଷ ǥ ‫ݍ‬௚ ൧ ǤThe ‫ ܮ‬ൈ ͳ column vector ‫ ܍‬contains the components after control is applied and is written as୘
(19)‫ ܍‬ൌ ൣ݁଴ǡ଴ ǡ ݁ଵǡିଵ ǡ ݁ଵǡ଴ ǡ ݁ଵǡଵ ǥ݁ேǡିே ǥ݁ேǡ଴ ǥ݁ேǡே ൧ Ǥ
Relation (15) forms a linear system of equations that is an established topic in the area of active
control [3, 4]. The system emerges when using the traditional strategy of minimizing the means squarepressures or particle velocities at a finite set of sensor positions, given a finite set of seco ndary sources.
If (15) is overdetermined, i.e. ‫ ܩ‬൏ ‫ܮ‬, and ۰ ୌ ۰ is positive-definite, then source strengths given by‫ܙ‬௢௣௧ ൌ െሾ۰ୌ ۰ሿିଵ۰ୌ ‫܌‬
(20)
form the unique, optimal set that produces a minimum, if it exists, in the cost functionଶ‫ ܬ‬ൌ ܹ௘ ൌ ‫܍‬ୌ ‫ ܍‬ൌ ෍ห݁௡ǡ௠ ห 

(21)

௡ǡ௠

at a single value of ݇ܽ. (21) is the truncated version of (14) without the factor of ͳȀሺʹߩܿ݇ ଶ ሻ. Thisfactor is a constant relative to ݇ܽ and does not affect the derivation of ‫ܙ‬௢௣௧ from (20); however, it
does scale the minimum achievable value of ܹ௘ . It can be concluded that minimizing either the meansquared pressures at sensing locations or the mean squared spherical harmonic coefficients of thepressure leads to the same analytical result for the scattered power after control. The two methods
differ when it comes to inner and cross coupling of primary and secondary. For a grid of pressure
sensors, the elements in ‫ ܌‬and, respectively, in each column of B are inherently interconnected, thus
limiting the optimization. Spherical harmonic coefficients are uncoupled and do not exhibit this.
The amount of attenuation achievable after the optimization depends on which elements of ‫ ܌‬aredominant and on how well the corresponding elements of each row in ۰ can couple into them. The actof altering or completely cancelling a specific spherical harmonic component in (15) establishes one
constraint on the set of source strengths, thus creating residuals in the other non -zero components. The
effect is referred to as spill-over [3, 4] and is unavoidable in the current model. This is because allforms of the primary and secondary described in Table 1 have at least a countably infinite set of
non-zero spherical harmonic components, regardless of point-monopole location in the region ‫ ݎ‬൒ ܽ.
Imposing specific behaviours on ‫ ܩ‬distinct components requires at least ‫ ܩ‬distinct sources.
While the problem is not ill-conditioned, the optimization balances between manipulating specificcomponents and generating residuals, such that the passive situation is not made worse after control.
However, (20) is a numerical result and cannot directly describe how each optimal variable is chosen.

72

Table 2 – Asymptotic behavior relative to ݇ܽ of ܴ௡ ሺ‫ݎ‬ఛᇱ ሻ factors in spherical harmonic coefficients݇ܽ ‫ͳ ا‬ǡ ݊ ൌ Ͳܴ௡ ሺܽሻሺ݇ܽሻଶܴ௡ ሺ‫ݎ‬Ԣሻሺ݇ܽሻଶ ൤ͳ െ

ܴஶǡ௡

ሺ݇ܽሻଶ

ߞ
݆݇ܽ ൅ ߞܽ ͳ െ ݆ߞ݇ܽȀ͵൨‫ݎ‬Ԣ ͳ ൅ ߞȀሺ݆݇ܽሻͳ െ ݆ߞ݇ܽȀ͵݆݇ܽ ൅ ߞ݇ܽ ‫݊ ا‬ǡ ݊ ൒ ͳ݇ܽ ՜ λʹ௡ିଵ ሺ݊ െ ͳሻǨߞሺ݇ܽሻ௡ାଶ
ሺʹ݊ െ ͳሻǨ݆݇ܽ ൅ ߞሺ݊ ൅ ͳሻߞ݆݇ܽ ௡ାଵ ݁ ି௝௞௔ ሺͳ ൅ ߞሻܽ ଶ௡ାଵ
ሺ݇ܽሻଶ ሺ݇‫ݎ‬Ԣሻ௡݇ܽ ൅ ݆ߞ݊൤ͳ െ ݆ ቀ ቁ൨ሺʹ݊ ൅ ͳሻԥ‫ݎ‬Ԣ݆݇ܽ ൅ ߞሺ݊ ൅ ͳሻ݁ ି௝௞ሺ௥ ି௔ሻ ࣠ሺ݇ܽሻଶ݊ߨቈ ቀ݇‫ݎ‬Ԣ െ ቁ െ቉ሺͳ ൅ ߞሻ
ʹሺ݇‫ݎ‬Ԣሻʹ௡ ݊Ǩ ଶ ሺ݇ܽሻଶ௡ାଵ݇ܽ ൅ ݆ߞ݊൤൨ሺʹ݊ሻǨ ሺʹ݊ ൅ ͳሻ ݆݇ܽ ൅ ߞሺ݊ ൅ ͳሻ݆࣠ ௡ାଵ ݁ ି௝௞௔ ሺͳ ൅ ߞሻ

ᇲ

2.3 Method for Assessing the Performance of Active Control
To interpret the outcome of (20), the spherical harmonic coefficients after control are compared toanalytical predictions. These require studying the behaviour of the coefficients in the primary andሺ௙ሻ
ሺ௚ሻsecondary. The forms of ݀௡ǡ௠ and ‫ܤ‬௡ǡ௠ from Table 1 share a general structure of three factorsሺԹሻ(22)߬௡ǡ௠ ൌ Ձఛǡ௡ ܴ௡ ሺ‫ݎ‬ఛᇱ ሻܻ௡ǡ௠ ሺߠఛ ǡ ߮ఛ ሻǤ
Relative to ݇ܽ, Ձఛǡ௡ is either a real or purely imaginary constant that gives the units of ߬௡ǡ௠ . ܴ௡ ሺ‫ݎ‬ఛᇱ ሻ
is a dimensionless, complex-valued quantity. It solely governs the variation with ݇ܽ of ߬௡ǡ௠ and,hence, that of the contribution to the radiated sound power, before and after control, from the sphericalሺԹሻ
harmonic terms of degree ݊. ܻ௡ǡ௠ ሺߠఛ ǡ ߮ఛ ሻ is a real-valued, dimensionless constant relative to ݇ܽ.
When using the general form (22) into (3), the spherical harmonics closure relation reduces (3) to
ͳ
ܽଶ
ଶ
ଶ
ܹఛ ൌ
෍ห߬
ห
ൌ
෍ሺʹ݊ ൅ ͳሻหՁఛǡ௡ ห ȁܴ௡ ሺ‫ݎ‬ఛᇱ ሻȁଶ Ȁሺ݇ܽሻଶ ൌ ෍ ܹఛǡ௡ ǡ(23)௡ǡ௠ʹߩܿ݇ ଶ
ͺߨߩܿ
௡ǡ௠

௡

௡

where ܹఛǡ௡ is the combined contribution to the radiated power of components with degree ݊.
In (23), หՁఛǡ௡ ห does not depend on ݊, given the results from Table 1. The asymptotic behaviour ofthe ܴ௡ ሺ‫ݎ‬ఛᇱ ሻ factors when ݇ܽ becomes very small or very large can be established, as seen in [13, 14].
Choosing the appropriate expansion for the spherical Bessel and Hankel functions that composeܴ௡ ሺ‫ݎ‬ఛᇱ ሻ and then considering the leading order term, yields the results from Table 2, where ࣠ ൌሺ݇ܽ െ ݊ߨȀʹሻ ൅ ݆ߞ ሺ݇ܽ െ ݊ߨȀʹሻ. Computing these results for different values of ݊ and using
them to analyse ܹఛǡ௡ reveals that, in the region of ݇ܽ ‫ͳ ا‬, the contribution of a lower ݊ component
is larger than that of a higher ݊ one while, generally, also varying as a lower power of ݇ܽ. This leads
to the radiated power being dominated by the first few spherical harmonic components. Furthermore,at large ݇ܽ, ܹఛ is dominated by a number of spherical harmonic components of high ݊.
As ݇ܽ increases, achieving considerable suppression of the scattered power implies more control
sources, such that each component that becomes significant can be addressed. Thus, when using asmall number of sources, the predicting the asymptotes when ݇ܽ ‫ ͳ ا‬is of interest in the analysis.

3. STUDY CASE: ACTIVE CONTROL OF SINGLE PLANE-WAVE SCATTERING
3.1 Defining the Active Control Problem
The specific example presented in this paper consists of controlling the scattering of a single
monochromatic plane-wave as it insonifies the impedance sphere. The plane-wave has a magnitudeܲ௜ ൌ ͳ and arrives from the incidence angles ሺߠ௜ ǡ ߮௜ ሻ. The generated scattered field represents theሺԹሻprimary and is expressed as (9), with coefficients given by ݀௡ǡ௠ ൌ െͶߨܲ௜ ݆ ௡ ܴஶǡ௡ ܻ௡ǡ௠ ሺߠ௜ ǡ ߮௜ ሻ. The
plane-wave is chosen to propagate parallel to the yz-plane, in the direction of the z-axis from െλ to
λ. This corresponds to ሺߠ௜ ǡ ߮௜ ሻ ൌ ሺߨǡ ߨȀʹሻ and, from (2), results in only the components of order݉ ൌ Ͳ being activated in the primary. Furthermore, three spheres of surface impedances ߞ ൌ ͲǤͲͳ ,ߞ ൌ ͳ and ߞ ൌ ͳͲͲ are exemplified. The first and the last cases approximate a soft sphere of ߞ ൌ Ͳand, respectively, a hard sphere of ߞ ൌ λ. This was done to maintain consistency, as a secondary
source on the surface of a soft obstacle cannot radiate. The case of ߞ ൌ ͳ represents the impedance ofthe sphere matched to that of the surrounding medium. Air at normal atmospheric conditions is used in
all results.

73

The scattered power due to the plane-wave, ܹௗ , is expressed using the coefficients ݀௡ǡ௠ in (3).
This is normalized by the power associated with the plane-wave, ܹ௜ ൌ ߨܽଶ ܲ௜ଶ Ȁሺʹߩܿሻ, then plotted asȫௗ ൌ ܹௗ Ȁܹ௜ versus ݇ܽ for the three cases of impedance in the graphs of Figure 1. A truncation
degree of ܰ ൌ ͳͲͲ was used alongside a range of ͳͲͲͳ logarithmically-spaced points between݇ܽ ൌ ͲǤͳ and ݇ܽ ൌ ͳͲͲ. Simulations have shown that, at a given large value of ݇ܽ, the number ofspherical harmonic degrees ݊ that contribute significantly to this scattered power is equal to ݇ܽ.
The asymptotic behaviour of ȫௗ when ݇ܽ ‫ ͳ ا‬is dictated by the most dominant sphericalharmonic terms in that region, which can be established from Table 2. T he ߞ ൌ ͲǤͲͳ and ߞ ൌ ͳspheres are both dominated by the ሺ݊ǡ ݉ሻ ൌ ሺͲǡͲሻ term, which leads to their normalized powers
having asymptotes of Ͷ and, respectively, Ͷሺ݇ܽሻଶ . The ߞ ൌ ͳͲͲ sphere is dominated by bothሺ݊ǡ ݉ሻ ൌ ሺͲǡͲሻ and ሺ݊ǡ ݉ሻ ൌ ሺͳǡͲሻ terms, which contribute to the scattered power as Ͷሺ݇ܽሻସȀͻ and,
respectively, ͵ሺ݇ܽሻସ Ȁͻ. This leads to an asymptote of ȫௗ equal to ͹ሺ݇ܽሻସ Ȁͻ, as described in [15].
In the following paragraphs, results of active control are shown for the four different arrangementsof point-monopole sources depicted in the diagram of Figure 1. To have corresponding sphericalharmonic components activated in both primary and secondary, the sources are placed on the z-axis.
3.2 Control Results when Using a Small Number of Surface Radiators
A secondary field composed of point-monopoles on the scattering surface has the coefficientsሺԹሻሺ௚ሻ‫ܤ‬௡ǡ௠ ൌ ߩܿܽିଶ ܴ௡ ሺܽሻܻ௡ǡ௠ ൫ߠ௚ ǡ ߮௚ ൯. Simulations show that the power radiated by such a field follows thesame truncation rule as the primary, where ܰ ൌ ݇ܽ terms are required at a given large value of ݇ܽ.
Choosing the same truncation degree, ܰ ൌ ͳͲͲ, for both primary and secondary, the minimization(20) was computed over the ܰ ൅ ͳ contributing spherical harmonic components of ݉ ൌ Ͳ, at eachͳͲͲͳ logarithmically-spaced points between ݇ܽ ൌ ͲǤͳ and ݇ܽ ൌ ͳͲͲ. For a single control source
placed at ሺߠଵ ǡ ߮ଵ ሻ ൌ ሺͲǡͲሻ, it is assumed that the minimization attempts to perfectly cancel theሺ݊ǡ ݉ሻ ൌ ሺͲǡͲሻ term, which is the largest for all three cases of ߞ. Then, the optimal source strength is‫ݍ‬ଵ ൌ െ݀଴ǡ଴ Ȁ‫ܤ‬଴ǡ଴ ൌ Ͷߨܲ௜ ܽଶ ሺߩܿሻିଵ ܴஶǡ଴ Ȁܴ଴ ሺܽሻǡ(24)
and, after control, the ሺ݊ǡ ݉ሻ ൌ ሺͳǡͲሻ term is(25)݁ଵǡ଴ ൌ ݀ଵǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଵǡ଴ Ȁ‫ܤ‬଴ǡ଴ ൌ ʹξ͵ߨܲ௜ ൣെ݆ܴஶǡଵ ൅ ܴஶǡ଴ ܴଵ ሺܽሻȀܴ଴ ሺܽሻ൧ǤThe formulas from Table 2 are used in (24) and (25) to calculate the asymptotic behaviour after control
when ݇ܽ ‫ͳ ا‬. The hierarchy between contributions to the power is maintained, ignoring the cancelled
component. The ሺ݊ǡ ݉ሻ ൌ ሺͳǡͲሻ term varies as the lowest power of ݇ܽ and, thus, is the new dominantcomponent. This translates into the normalized scattered power after control, ȫ௘ ൌ ܹ௘ Ȁܹ௜ , having
asymptotes of ͳʹሺ݇ܽሻଶ, Ͷሺ݇ܽሻସ Ȁ͵ and ͵ሺ݇ܽሻସȀͻ for ߞ ൌ ͲǤͲͳ, ߞ ൌ ͳ and, respectively, ߞ ൌ ͳͲͲ.

Figure 1 – Normalized scattered power due to plane-wave before and after control, as a function of ݇ܽ74

A second point-monopole is now placed on the surface of the sphere at ሺߠଶǡ ߮ଶ ሻ ൌ ሺߨǡ ߨȀʹሻ and all
the computational circumstances are maintained. It is assumed the minimization attempts to perfectly
supress both the ሺ݊ǡ ݉ሻ ൌ ሺͲǡͲሻ and ሺ݊ǡ ݉ሻ ൌ ሺͳǡͲሻ terms, which are the largest two for all three casesof ߞ. Due to the parity of the spherical harmonic functions, the linear combinationsሺଶሻ
(26)‫ݍ‬ଵ ൅ ‫ݍ‬ଶ ൌ െ݀଴ǡ଴ Ȁ‫ܤ‬଴ǡ଴ ൌ Ͷߨܲ௜ ܽଶ ሺߩܿሻିଵ ܴஶǡ଴ Ȁܴ଴ ሺܽሻǡሺଶሻ
(27)‫ݍ‬ଶ െ ‫ݍ‬ଵ ൌ െ݀ଵǡ଴ Ȁ‫ܤ‬ଵǡ଴ ൌ Ͷߨ݆ܲ௜ ܽଶ ሺߩܿሻିଵ ܴஶǡଵ Ȁܴଵ ሺܽሻǡ
of the two optimal source strengths govern (15). After control, the ሺ݊ǡ ݉ሻ ൌ ሺʹǡͲሻ term is expressed as
ሺଶሻ
ሺଶሻ
(28)
݁ଶǡ଴ ൌ ݀ଶǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ Ȁ‫ܤ‬଴ǡ଴ ൌ ʹξͷߨܲ௜ ൣܴஶǡଶ ൅ ܴஶǡ଴ ܴଶ ሺܽሻȀܴ଴ ሺܽሻ൧and is now the dominant component in the ݇ܽ ‫ ͳ ا‬region, yielding asymptotes of ȫ௘ equal toʹͲሺ݇ܽሻସȀͻ, ʹͲሺ݇ܽሻ଺ Ȁͺͳ and Ͷሺ݇ܽሻ଼ ȀͶͲͷ for ߞ ൌ ͲǤͲͳ, ߞ ൌ ͳ and, respectively, ߞ ൌ ͳͲͲ.
For both control arrangements, the predicted asymptotes, seen as dashed lines in Figure 1, are good
representations of the computed outcomes of (20), seen as solid coloured lines in Figure 1. Overall, thereduction of ȫ௘ has improved when going from one to two sources. For ߞ ൌ ͳͲͲ, considerably better
attenuation is achieved with two sources as both significant terms in the primary are addressed.

3.3 Control Results when Using a Small Number of Surface and Near-Surface Radiators
To control more than the first two spherical harmonic components, extra variables need to beሺ௚ሻ
introduced in (15). However, new sources have to be placed such that their coefficients ‫ܤ‬௡ǡ௠ are
ሺଵሻ

ሺଶሻ

considerably different from ‫ܤ‬௡ǡ௠ and ‫ܤ‬௡ǡ௠ used previously. Otherwise, the control generates as muchattenuation of the power as for the previous two sources and manifests ill-conditioning when inverting۰ ୌ ۰. Therefore, extra point-monopoles are placed away from the surface of the scatterer at a distanceሺ௚ሻof ‫ ݎ‬ᇱ ൌ ʹܽ. In this way, the secondary has ‫ܤ‬௡ǡ௠ coefficients that can have either of the two forms from
Table 1. Simulations show that the power radiated by such a secondary requires a truncation degree ofܰ ൌ ݇‫ݎ‬Ԣ, at a given large value of ݇ܽ, where ‫ݎ‬Ԣ corresponds to the source furthest away.
In the previous arrangement of two control sources, a third point-monopole is now added atሺ‫ݎ‬ଷᇱ ǡ ߠଷ ǡ ߮ଷ ሻ ൌ ሺʹܽǡ ߨǡ ߨȀʹሻ. (20) is computed under the same circumstances, but the higher truncation
degree corresponding to the secondary, ܰ ൌ ʹͲͲ, is chosen rather than that of the primary to avoid
aliasing. This mismatch introduces terms of high degree ݊ that are zero in the primary but non-zero in
the secondary. Attenuation is achieved only in the ݇ܽ ‫ ͳ ا‬region, which is dominated by terms of low
degree ݊ and, hence, is minimally affected by the mismatch. Assuming that the spherical harmoniccomponents of the first three degrees ݊ are perfectly cancelled leads to the optimal source strengthsሺଶሻሺଶሻ ሺଷሻሺଶሻ ሺଷሻ

ሺଶሻሺଷሻ ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଶሻ

(29)

ሺଶሻ

ሺଶሻ ሺଷሻሺଶሻ ሺଷሻ

ሺଶሻሺଷሻ ሺଶሻሺଶሻ ሺଷሻሺଶሻ

(30)‫ݍ‬ଵ ൅ ‫ݍ‬ଶ ൌ െ݀଴ǡ଴ Ȁ‫ܤ‬଴ǡ଴ െ ቀ݀ଶǡ଴ ‫ܤ‬଴ǡ଴ ‫ܤ‬଴ǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ ቁ Ȁ ቀ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ ‫ܤ‬଴ǡ଴ െ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ ቁǡ‫ݍ‬ଶ െ ‫ݍ‬ଵ ൌ െ݀ଵǡ଴ Ȁ‫ܤ‬ଵǡ଴ െ ቀ݀ଶǡ଴ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଵǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ ‫ܤ‬ଵǡ଴ ቁ Ȁ ቀ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଵǡ଴ െ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ ‫ܤ‬ଵǡ଴ ቁǡ
ሺଶሻ

ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଶሻ

(31)

ሺଷሻ

‫ݍ‬ଷ ൌ ቀ݀ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁ Ȁ ቀ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁǡthat govern the individual equations of (15). After control, the ሺ݊ǡ ݉ሻ ൌ ሺ͵ǡͲሻ term is expressed as
ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଷሻ

ሺଶሻ

ሺଶሻ

ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଶሻ

ሺଷሻ

݁ଷǡ଴ ൌ ݀ଷǡ଴ െ ݀ଵǡ଴ ‫ܤ‬ଷǡ଴ Ȁ‫ܤ‬ଵǡ଴ ൅ ቀ‫ܤ‬ଷǡ଴ െ ‫ܤ‬ଵǡ଴ Ȁ‫ܤ‬ଵǡ଴ ቁ ቀ݀ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁ Ȁ ቀ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁ(32)

and now dominates in the ݇ܽ ‫ ͳ ا‬region, yielding asymptotes of ȫ௘ equal to ͲǤͶͻͺͻሺ݇ܽሻ଺ ,ͲǤͲ͵ͻͶሺ݇ܽሻ଼ and ͲǤͲͲͲͻሺ݇ܽሻ଼ for ߞ ൌ ͲǤͲͳ, ߞ ൌ ͳ and, respectively, ߞ ൌ ͳͲͲ.
Lastly, a fourth source is now placed at ሺ‫ݎ‬ସᇱ ǡ ߠସ ǡ ߮ସሻ ൌ ሺʹܽǡ ͲǡͲሻ. The previous computational context
is maintained. Assuming that the components of first four degrees ݊ are perfectly cancelled yields
ሺଷሻ

ሺଷሻ

ሺଶሻ ሺଷሻ

ሺଶሻ

ሺଷሻ

(33)

ሺଷሻ

ሺଷሻ

ሺଶሻ ሺଷሻ

ሺଶሻ

ሺଷሻ

(34)

ሺଶሻ

ሺଶሻ

ሺଶሻ ሺଷሻ

ሺଶሻ

ሺଷሻ

(35)

ሺଶሻ

ሺଶሻ

ሺଶሻ ሺଷሻ

ሺଶሻ

ሺଷሻ

(36)

‫ݍ‬ଵ ൅ ‫ݍ‬ଶ ൌ ቀ݀ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁ Ȁ ቀ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ െ ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ ቁǡ‫ݍ‬ଶ െ ‫ݍ‬ଵ ൌ ቀ݀ଷǡ଴ ‫ܤ‬ଵǡ଴ െ ݀ଵǡ଴ ‫ܤ‬ଷǡ଴ ቁ Ȁ ቀ‫ܤ‬ଵǡ଴ ‫ܤ‬ଷǡ଴ െ ‫ܤ‬ଷǡ଴ ‫ܤ‬ଵǡ଴ ቁǡ‫ݍ‬ଷ ൅ ‫ݍ‬ସ ൌ ቀ݀ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ݀଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁ Ȁ ቀ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ െ ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ ቁǡ‫ݍ‬ଷ െ ‫ݍ‬ସ ൌ ቀ݀ଷǡ଴ ‫ܤ‬ଵǡ଴ െ ݀ଵǡ଴ ‫ܤ‬ଷǡ଴ ቁ Ȁ ቀ‫ܤ‬ଷǡ଴ ‫ܤ‬ଵǡ଴ െ ‫ܤ‬ଵǡ଴ ‫ܤ‬ଷǡ଴ ቁǡwhich govern the individual equations in (15). After control, the ሺ݊ǡ ݉ሻ ൌ ሺͶǡͲሻ term becomes

75

ሺଷሻ

ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଶሻ ሺଷሻሺଷሻ ሺଶሻ

ሺଶሻ

ሺଷሻ

ሺଶሻ

ሺଷሻ

݁ସǡ଴ ൌ ݀ସǡ଴ ൅ ቂ݀଴ǡ଴ ቀ‫ܤ‬ସǡ଴ ‫ܤ‬ଶǡ଴ െ ‫ܤ‬ସǡ଴ ‫ܤ‬ଶǡ଴ ቁ ൅ ݀ଶǡ଴ ቀ‫ܤ‬ସǡ଴ ‫ܤ‬଴ǡ଴ െ ‫ܤ‬ସǡ଴ ‫ܤ‬଴ǡ଴ ቁቃ Ȁ ቀ‫ܤ‬଴ǡ଴ ‫ܤ‬ଶǡ଴ െ ‫ܤ‬ଶǡ଴ ‫ܤ‬଴ǡ଴ ቁ(37)

and now dominates in the ݇ܽ ‫ ͳ ا‬region, yielding asymptotes of ȫ௘ equal to ͲǤͲͶ͵ͳሺ݇ܽሻ଼ ,ͲǤͲͲʹͳሺ݇ܽሻଵ଴ and ͲǤͲͲͲͲ͹ሺ݇ܽሻଵଶ for ߞ ൌ ͲǤͲͳ, ߞ ൌ ͳ and, respectively, ߞ ൌ ͳͲͲ.
For the current two control arrangements, the predicted asymptotes, seen as dashed lines in Figure
1, are also good representations of the computed outcomes of (20), seen as solid coloured lines in
Figure 1. Going from two to three to four sources progressively improves the obtained suppression at
low ݇ܽ. The amount is different between cases of ߞ, as their corresponding asymptotes are dissimilar.

4. CONCLUSIONS
A theoretical investigation has been realized where spherical harmonic expansions were used todescribe the primary and secondary fields, as well as the minimization method, in the problem ofactively controlling the sound scattering from a sphere. Simulations were performed for truncatedversions of these expansions, in the case of a single plane-wave arriving onto a uniform,
locally-reacting sphere. A small number of point-monopoles on and near the surface of the obstacle
were ideally placed relative to the known incidence. In the Rayleigh region, the spherical harmoniccoefficients in both primary and secondary form a hierarchy of ascending powers of ݇ܽ withincreasing ݊. Thus, progressively better attenuation of the scattered power was achieved when adding
a few more sources, as more of the first dominant components in the hierarchy can be controlled. Thisbehavior was analytically predicted using the asymptotes of the spherical harmonic coefficients.
In terms of future work, the presented modelling and control strategies can be extended to scenarios
beyond that of a single plane-wave. Also, characterizing the effects on control performance of using alarge number of sources and of solely moving the sources away from the obstacle can be further
explored. In terms of experimental validation, a necessary first step is to devise a practical active
control system that can sense and manipulate a 3D sound field as spherical harmonic components.

ACKNOWLEDGEMENTS
This work is supported by the Defense Science and Technology Laboratory, United Kingdom.

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