Analysis of the complex sound power in the near field of spherical loudspeaker arrays

Accepted Manuscript Analysis of the complex sound power in the near field of spherical loudspeaker arrays A.M. Pasqual

PII:

S0022-460X(19)30313-X

DOI:

https://doi.org/10.1016/j.jsv.2019.05.044

Reference:

YJSVI 14781

To appear in:

Journal of Sound and Vibration

Received Date: 17 December 2018 Revised Date:

7 May 2019

Accepted Date: 20 May 2019

Please cite this article as: A.M. Pasqual, Analysis of the complex sound power in the near field of spherical loudspeaker arrays, Journal of Sound and Vibration (2019), doi: https://doi.org/10.1016/ j.jsv.2019.05.044. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Analysis of the complex sound power in the near field of spherical loudspeaker arrays A. M. Pasquala,∗ a

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Technological Institute of Aeronautics, Division of Mechanical Engineering Pra¸ca Mal. Eduardo Gomes 50, 12228-900, S˜ao Jos´e dos Campos/SP, Brazil.

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Abstract

Over the past decade, compact spherical loudspeaker arrays have emerged as directivity controlled

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electroacoustic sources. The usual approach to control such a device is to provide it with a finite set of pre-programmed directivities corresponding to spherical harmonics or to the acoustic radiation modes (ARMs) of the array. However, these orthogonal bases are not suited for the acoustic control in the near field, which requires a number of spherical harmonics larger than the number of loudspeakers, and modes that take into account the reactive part of the complex sound power, not only the active part as is the case for the conventional ARMs. This paper presents a theoret-

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ical and numerical study of alternative ARM approaches applied to spherical loudspeaker arrays that consider both components of the sound power, with focus on the arrangements following the regularity of the five Platonic solids. For such arrays, it is shown that the active, reactive, and

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the so-called generalized ARMs form exactly the same basis, which does not depend either on frequency, source-receptor distance or on the vibration pattern and shape of the individual loudspeakers, provided that these are identical and axisymmetric. Closed-form expressions for the

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modal active and reactive efficiencies are derived and evaluated for the Platonic solid loudspeakers as a function of frequency and source-receptor distance. In addition, an application example in local active noise control is presented, in which a spherical loudspeaker array is used as a secondary source aiming at producing a zone of quiet in its near field. Keywords: Spherical loudspeaker array, Spherical harmonics, Reactive sound power, Active noise control, Acoustic radiation mode, Platonic solid

Preprint submitted to Journal of Sound and Vibration

May 21, 2019

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1. Introduction Compact spherical arrays of independent loudspeakers operating at the same frequency range have been extensively investigated as directivity controlled sound sources [1–19]. Such devices

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have found applications in electroacoustic music [1, 18], reproduction of radiation patterns of musical instruments [6], room acoustics measurements [15, 19], active noise control [7, 12], and psychoacoustic research [17]. Arrays ranging from four [20] to 120 loudspeakers [4] mounted on a spherical (or nearly spherical, such as polyhedra) cabinet have been built, most of them made up

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of electrodynamic transducers.

A spherical loudspeaker array is usually provided with a set of pre-programmed basic direc-

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tivities corresponding to spherical harmonic functions, which can be linearly combined to render more complex patterns. Indeed, for a small cabinet radius compared to the acoustic wavelength, the source is able to produce far-field radiation patterns that match pure spherical harmonics up to a maximum order that depends on the number of loudspeakers [3, 4, 10]. Accordingly, the controllable degrees of freedom (DOFs) of the source are related to spherical harmonics in a

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straighforward manner. However, as frequency increases or near-field synthesis is sought, the so-formed basic directivites will no longer radiate pure spherical harmonics. Instead, an intricate combination of them up to higher orders will take place — which has been referred to as spatial aliasing in the audio literature — and thus the control strategy based on these functions will

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fail. Besides, although spherical loudspeaker arrays are able to synthesize spherical harmonics in the low-frequency range, the associated radiation efficiencies drop rapidly as frequency decreases

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and/or the spherical harmonic order increases [11, 21], so that an attempt to achieve a desired sound power might overload the transducers. An alternative approach based on the acoustic radiation modes (ARMs) can be used to tackle these issues. The ARMs of a vibrating body are an orthogonal set of surface velocity distributions that radiate sound energy independently and are ranked according to their radiation efficiencies. They can be obtained through the eigenvalue analysis of a radiation operator. Such an approach to deal with radiation problems is particularly useful in active structural acoustic control insofar ∗

Corresponding author. Tel.: +55 (12)3947-5900. Email address: [email protected] (A. M. Pasqual)

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as it allows a reduced number of sensors and actuators [22, 23]. Although the ARMs were introduced in the early 1990s [24–26], their use in sound field synthesis and control by loudspeaker arrays has been explored only in a few studies [8, 11, 27]. In these applications, the ARMs are

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expressed as vectors in terms of the controllable DOFs, whose number generally equals the number of loudspeakers, as it will be assumed in this paper. In the ARM approach, each vector is a pre-programmed basic pattern. Unlike the spherical harmonic approach, any radiation pattern the array is able to produce can be expressed as a linear combination of its ARMs, which always

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span a finite dimensional subspace. In addition, since these modes are ranked according to their radiation efficiencies, low-efficient ARMs can be discarded to avoid overloading the transducers.

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Traditionally, only the active component of the sound power is considered to evaluate the ARMs, and thus we will refer to such modes as active ARMs, and to the associated efficiencies as active radiation efficiencies. The active ARMs form a suitable basis to represent the source radiation in the far field. However, in the near field, the reactive component of the sound power is as important as the active power, so that it cannot be neglected. We will refer to the modes and efficiencies evaluated considering only the reactive power as reactive ARMs and reactive ra-

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diation efficiencies, respectively. A formal definition of these terms will be given in Section 2.2. The modal approach of the sound radiation has already been extensively investigated by many researchers, but there are only few works that take the two components (active and reactive) of

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the complex sound power into account. Chen and Ginsberg [28] and Chen [29] were probably the first researchers to propose a theoretical framework to deal with this problem. Later, Maury and co-workers [30–32] investigated the active and reactive ARMs of some planar and spherical

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radiators, motivated by the active structural acoustic control of large-scale structures and/or in the low-frequency range, when the control domain is in the source near field. Recently, the active and reactive ARMs of planar loudspeaker arrays have been applied to the synthesis of sound fields at the vicinity of flat surfaces [27]. This paper investigates the complex sound power produced by spherical loudspeaker arrays, especially in the source near field, where the reactive power plays an important role. The active and reactive ARMs, as well as their radiation efficiencies, are numerically evaluated from an acoustic model based on the solution of the Helmholtz’s equation in spherical coordinates. In 3

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addition, an application example in active control of sound is presented, namely, the generation of a zone of quiet in the near field of a spherical loudspeaker array, which is used as a secondary source. We focus on the most common loudspeaker layouts, those corresponding to the five con-

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vex regular polyhedra (Platonic solids): tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron. These layouts, which we will refer to as Platonic solid loudspeakers (PSL), constrain the maximum number of controllable DOFs to four, six, eight, twelve, and twenty, respectively. The paper is organized as follows. Section 2 presents a sound radiation model for spherical

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loudspeaker arrays, as well as theoretical developments concerning the active and reactive ARMs. Next, the model is applied to the PSLs made up of identical axisymmetric loudspeakers in Sec-

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tion 3. Then, numerical results for the active and reactive efficiencies of the PSLs are presented in Section 4, where the loudspeaker membranes are modeled as spherical caps. Section 5 presents an application example in local active control of sound. Finally, the main conclusions are summarized in Section 6. 2. Theoretical developments

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This section describes a frequency-domain model for the sound radiation from a spherical loudspeaker array in a non-dissipative fluid medium under free-field conditions. In addition, different

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eigenvalue analysis of the radiation operator are presented in a general framework. Throughout √ this paper, a harmonic time dependence of the form e−jωt is assumed, where j ≡ −1, ω is the angular frequency, and t is time.

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2.1. Acoustic model and complex sound power The sound radiation from spherical bodies can be naturally expressed in spherical coordinates, (r, θ, φ), where r ≥ 0, 0 ≤ θ ≤ π, and 0 ≤ φ < 2π are, respectively, the radial coordinate, the polar angle, and the azimuth angle. The point r = 0 is at the geometrical center of the spherical radiator. The solution of the homogeneous Helmholtz’s equation in spherical coordinates is given by a series involving spherical Hankel functions of the first kind, hn (·), and spherical harmonics,

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Ynm (·), defined here following Arfken and Weber [33] as s (2n + 1) (n − m)! m Ynm (θ, φ) ≡ (−1)m P (cos θ)ejmφ , 4π (n + m)! n

(1)

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where 0 ≤ n < ∞ and −n ≤ m ≤ n are integers, and Pmn (·) is the associated Legendre function of the first kind.

For the sake of notation compactness, the spherical harmonics are combined into a column vector y(·), so that yi (·) ≡ Ynm (·), with i = n2 + n + m + 1 for linear indexing of the harmonics. Also,

Z

π

Z

0

2π

y(θ)∗ y(θ)T sin θdθdφ = I,

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unit sphere [33], one has

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θ and φ are grouped in a vector θ. Thus, since the spherical harmonics are orthonormal over the

0

(2)

where the asterisk denotes the complex conjugation, the superscript T indicates the transpose, and I is the identity matrix.

Consider a compact array made up of L identical axisymmetric loudspeakers mounted on a rigid spherical cabinet of radius rs . Because of the axial symmetry, the acoustic field produced by

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each individual loudspeaker is a function of r and θ(l) only, where 0 ≤ θ(l) ≤ π is the angle between the position vector of the field point and the symmetry axis of the lth loudspeaker. Therefore, only the spherical harmonics with m = 0 must be retained to express the acoustic variables in this local coordinate system. The acoustic pressure field, p(l) , and the radial acoustic velocity, v(l) , produced

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by the lth loudspeaker are given by [34]

and

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p(l) (r, θ(l) ) = ρcul

∞ X

An hn (kr)Yn0 (θ(l) )

(3)

Vn (r)Yn0 (θ(l) ),

(4)

n=0

v (r, θ ) = ul (l)

(l)

∞ X n=0

where r ≥ rs , ρ is the medium density, c is the sound speed, k = ω/c is the wavenumber, ul is a reference velocity considered here as the radial velocity at (r, θ(l) ) = (rs , 0), An are coefficients related to the loudspeaker vibration pattern, and Vn (r) = −jAn h0n (kr). The prime in h0n indicates the first derivative. 5

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Since Yn0 (θ(l) ) =

√

(2n + 1)/(4π)Pn (cos θ(l) ), Eqs. (3) and (4) are expansions in series of Legen-

(5)

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dre polynomials. Therefore, Vn can be obtained as [33] r Z 2n + 1 1 (l) (2n + 1) ul Vn (r) = v (r, θ(l) )Pn (cos θ(l) )d(cos θ(l) ). 4π 2 −1

Eqs. (3) and (4) express the acoustic field produced by the lth loudspeaker in local coordinates, θ(l) . To superimpose the acoustic fields produced by all the loudspeakers, it is usefull to express p(l) and v(l) in a global coordinate system, θ. Let θl ≡ (θl , φl ) be the polar and azimuth angles that

It can be shown that p(l) and v(l) can be rewritten as [11]

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define the orientation of the symmetry axis of the lth loudspeaker in the global coordinate system.

(6)

v(l) (r, θ) = −jul (y(θl ))H H0 (kr) y(θ),

(7)

and

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p(l) (r, θ) = ρcul (y(θl ))H H(kr) y(θ)

where the superscript H denotes the complex conjugate transpose. H(·) and H0 (·) are diagonal

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matrices whose elements are given by

r

Hii (kr) = An

and

r

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Hii0 (kr)

= An

4π hn (kr), 2n + 1

(8)

4π 0 h (kr), 2n + 1 n

(9)

respectively. Note that i is the same index used to group the spherical harmonics. Since −n ≤ m ≤

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n, H(·) and H0 (·) have 2n + 1 repeated entries for each n. Let u be a column vector containing the reference velocities of the L loudspeakers, and Y be a matrix that contains y(θl ) as columns. By superimposing the acoustic fields produced by the L identical loudspeakers, the sound pressure and the radial acoustic velocity generated by the spherical array become [11] p(r, θ) = ρc uT YH H(kr) y(θ)

(10)

v(r, θ) = −juT YH H0 (kr) y(θ) = −jy(θ)T H0 (kr)T Y∗ u.

(11)

and 6

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Now, let r¯ ≥ r s be the radius of a concentric spherical surface enclosing the spherical sound

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source. The complex acoustic power produced by the loudspeaker array is Z π Z 2π 1 ∗ 2 pv r¯ sin θdθdφ. Π= 2 0 0 Substitution of Eqs.(2), (10), and (11) into (12) yields Π = uT Bu∗ ,

B≡j

 ρc¯r2  H Y H(k¯r) H0 (k¯r)H Y 2

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where

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is an L × L complex matrix.

(12)

(13)

(14)

The angle between the symmetry axis of the lth and the l0 th loudspeakers, χll0 , can be related to Ynm (θl ) and Ynm (θl0 ) by the addition theorem for spherical harmonics [33], n 4π X m ∗ m Y (θl ) Yn (θl0 ). Pn (cos χll0 ) = 2n + 1 m=−n n

(15)

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An inspection of Eqs.(8), (14), and (15) reveals that the ll0 th element of B is given by ∞ Bll0 1 X = 2 |An |2 bn (k¯r)Pn (cos χll0 ), ρc 2k n=0

(16)

where bn (k¯r) ≡ j(k¯r)2 hn (k¯r)hn (k¯r)∗ . This equation shows that BT = B. Therefore, Eq.(13) can be Π = uH Bu.

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rewritten as

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0

(17)

The complex sound power can be decomposed into its active and reactive parts, so that n o n o Π = Re uH Bu + jIm uH Bu = uH Re {B} u + juH Im {B} u.

(18)

n o The expression Re jx2 hn (x)h0n (x)∗ = 1 can be derived from the Wronskian relation hn (x)h0n (x)∗ − h0n (x)hn (x)∗ = −2jx−2 given by Abramowitz and Stegun [35], so that Re {bn } = 1. Hence, inspection of Eq.(16) reveals that the ll0 th element of Re {B} is ∞ ρc X |An |2 Pn (cos χll0 ). Re {Bll0 } = 2 2k n=0

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(19)

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This equation shows that Re {B} is an L × L real symmetric matrix that couples the active power produced by the individual loudspeakers. In addition, comparison of Eqs.(18) and (19) shows that the active power, Re {Π}, does not depend on r¯, as expected.

Im {Bll0 } =

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Also, Eq.(16) shows that the ll0 th element of Im {B} is ∞ ρc X |An |2 Im {bn (k¯r)} Pn (cos χll0 ). 2k2 n=0

(20)

Im {B} is an L×L real symmetric matrix that couples the reactive power produced by the individual

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loudspeakers. Unlike Re {bn }, Im {bn } depends on n and k¯r. Therefore, the reactive power, Im {Π}, depends on r¯, as expected. Note that, for large arguments, one has hn (x) ≈ (−j)n+1 ejx /x and

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h0n (x) ≈ (−j)n ejx /x [34]. Therefore, for large k¯r, it can be shown that bn ≈ 1, so that Im {bn } ≈ 0, and thus Im {B} ≈ 0, showing that the reactive power is negligible for large distances, as expected. 2.2. Active and reactive radiation modes and efficiencies

The radiation efficiency, σa , of an arbitrary radiator is commonly defined as [21, 26, 36] Re {Π}

, ρcS |vn (xs )|2

(21)

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σa ≡

where S is the area of the vibrating surface S, vn is the velocity normal to S, xs ∈ S is a point on R the vibrating surface of the body, and h·i ≡ (2S )−1 S (·)dxs is a spatial mean operator. In this paper,

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the subscript “a” is used in σa to indicate that only the active sound power is considered. To take into account the reactive power, let us extend the concept of radiation efficiency and

where

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propose a complex radiation efficiency defined as σ≡

Π  = σa − jσr , ρcS |vn (xs )|2

σr ≡ −

Im {Π} 

ρcS |vn (xs )|2

(22)

(23)

will be called reactive radiation efficiency, as opposed to the active radiation efficiency, σa . The chosen sign convention aims at obtaining positive values for σr since Im {Π} < 0 for free-field conditions and a harmonic dependence of the form e−jωt . Let ϑ be the argument of σ, i.e., tan ϑ = 8

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−σr /σa . Following the AC circuit analogy described by Chen [29], cos ϑ will be referred to as the power factor when σr is evaluated at the source surface. For a spherical loudspeaker array, the total net vibrating surface is S = S1 ∪ S2 ∪ . . . SL , where

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Sl is the vibrating surface of the lth loudspeaker. Note that Sl ∩ Sl0 = ∅ for l , l0 , otherwise the individual loudspeakers will overlap. The lth element of u is the normal velocity at the geometrical center of Sl . The use of a velocity vector with only L elements does not represent a constraint on the allowable vibration patterns of the loudspeaker membranes, it just means that there are only

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L controllable DOFs through the voice-coil voltages. Now, let us require the loudspeakers to be identical and symmetric around the axis defined by the centers of the spherical cabinet and the

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loudspeaker membrane. The velocity distribution normal to Sl can be written as ul ξ(xs ), where ξ(xs ), with xs ∈ Sl , defines an axisymmetric vibration pattern on the loudspeaker membrane. Accordingly, it can be shown that [11] D

where

E ¯ H u, |vn (xs )|2 = ξu

Z

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1 ξ¯ ≡ 2S

(24)

|ξ(xs )|2 dxs .

(25)

Sl

Substitution of Eqs.(18), (24), and (25) into (21) and (23) leads to

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and

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σa (u) =

uH Re {C} u , uH u

(26)

uH Im {C} u , uH u

(27)

σr (u) = −

¯ −1 B is an L × L complex symmetric matrix. By referring to respectively, in which C ≡ (ρcS ξ) Eq.(16), the ll0 th element of C is Cll0 =

∞ 1 X |An |2 bn (k¯r)Pn (cos χll0 ). 2 ¯ 2S ξk n=0

(28)

Note that σa and σr are in the form of the Rayleigh quotient in Eqs.(26) and (27). Therefore, the eigenvectors of Re {C} and Im {C} are points of maximum, minimum, or saddle points of σa (u) and σr (u), respectively (see, e.g., Ref. [37]). Since these are real symmetric matrices, they have 9

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L real orthogonal eigenvectors. The eigenvectors of Re {C} will be denoted by ψa1 , ψa2 , . . . , ψaL , corresponding to the real eigenvalues σa1 , σa2 , . . . , σaL , where σal ≡ σa (ψal ). These eigenvectors are the active ARMs and the eigenvalues are their corresponding active radiation efficiency coeffi-

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cients. Similarly, the eigenvectors of Im {C} will be denoted by ψrl , with 1 ≤ l ≤ L, corresponding to the real eigenvalues σrl , where σrl ≡ σr (ψrl ). These eigenvectors are the reactive ARMs and the eigenvalues are their corresponding reactive radiation efficiency coefficients.

The active and reactive ARMs are alternative basis that span a finite dimension subspace on

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which any vibration pattern the array is able to produce can be projected with no approximation error. Therefore, if Ψa and Ψr are L × L modal matrices whose columns contain the active and

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reactive radiation modes, respectively, one may write

or

u = Ψa αa ,

(29)

u = Ψr αr ,

(30)

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where αa and αr are L × 1 vectors of modal contributions.

The active ARMs radiate sound energy independently to the far field and allow ranking the expansion terms by their active radiation efficiencies. However, for near-field radiation, the reactive power must also be taken into account, so that the reactive ARMs might become a better basis

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insofar as they allow ranking the expansion terms by their reactive radiation efficiencies. The modal representations in Eqs.(29) and (30) take into account only the active or the reactive

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sound power, respectively. Chen [29] describes a different approach that combines the real and imaginary parts of the sound power by using the ratio between them, η≡−

Re {Π} 1 σa =− = . Im {Π} tan ϑ σr

(31)

Substitution of Eqs.(26) and (27) into (31) yields η(u) = −

uH Re {C} u , uH Im {C} u

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(32)

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which is in the form of the generalized Rayleigh quotient. Therefore, the solution of the generalized eigenvalue problem Re {C} ψg = −η Im {C} ψg

(33)

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leads to L eigenvectors ψg1 , ψg2 , . . . , ψgL corresponding to eigenvalues η1 , η2 , . . . , ηL , where ηl ≡ η(ψgl ). These eigenvectors will be referred to as generalized ARMs, which can be arranged in an

Eqs.(29) and (30): u = Ψg αg , where αg is an L × 1 vector of modal contributions.

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L × L modal matrix, Ψg . Finally, the following decomposition can be used as an alternative to

(34)

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¯ and An , the shape and vibration pattern of the individual axisymBecause C depends on S , ξ, metric loudspeakers must be known to evaluate Cll0 in Eq. (28). Indeed, these parameters are required to evaluate the eigenvalues of Re {C} and Im {C}, as it will be shown in Section 4, where they will be explicitly given for some case studies. However, the eigenvectors of Re {C} and Im {C}

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for PSLs do not depend on such parameters, as it will be shown in Section 3.1. 3. Platonic solid loudspeakers

The model presented in Section 2 holds for any arrangement of identical axisymmetric loudspeakers mounted on a rigid spherical cabinet. In the following, such model is applied to a specific

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set of loudspeaker arrays, namely, the five PSLs.

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3.1. Active, reactive, and generalized ARMs As discussed in Section 2.2, the active and reactive ARMs and their corresponding radiation efficiencies can be obtained through the eigendecomposition of the real and imaginary parts of the matrix C given in Eq. (28), respectively. Note that the angles χll0 are the only parameters that can cause any two elements of C to be different, and thus they completely determine this matrix structure. In particular, although C is also a function of frequency, r¯, and the vibration pattern and shape of the individual loudspeakers, its structure does not depend on these parameters. Because of the high symmetry, the Platonic solids have just a small number Q of distinct values for χll0 . 11

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Table 1 presents these angles, where χi (with 1 ≤ i ≤ Q) is the ith value of χll0 . Accordingly, C possesses only Q distinct entries, which will be referred to as ci . Therefore, Eq. (28) can be ∞ 1 X |An |2 bn (k¯r)Pn (cos χi ). ci = 2 ¯ 2S ξk n=0

(35)

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rewritten as

Table 1: Values of Q and χi (rad) for the five Platonic solids [11]. Q

χ1

χ2

χ3

χ4

χ5

χ6

Tetrahedron

2

0

1.910633

–

–

–

–

Hexahedron

3

0

π/2

π

–

–

–

Octahedron

4

0

1.230959

1.910633

π

–

–

Dodecahedron

4

0

1.107149

2.034444

π

–

–

Icosahedron

6

0

0.729728

1.230959

1.910633

2.411865

π

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Platonic solid

Fig. 1 displays the structure of C (which is the same of Re {C} and Im {C}) for the five PSLs, where the shades of gray correspond to the index i, and the indexes l and l0 generate a square grid of side L according to the numbering convention given in Appendix A for the polyhedron

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faces. These matrices are bisymmetric, i.e., symmetric about both of their main diagonals. In a paper about the active ARMs of the PSLs, Pasqual and Martin [11] make use of this property to demonstrate that the eigenvectors of real matrices presenting the structures depicted in Fig. 1 do

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not depend on the values of the matrix individual elements. Accordingly, they conclude that the PSLs possess active ARMs that do not depend either on frequency or on the vibration pattern and shape of the individual loudspeakers, provided that these are identical and axisymmetric. Since

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Re {C} and Im {C} are real matrices with the same structure, they must therefore have the same eigenvectors, so that an active ARM is also a reactive ARM for PSLs. In addition, an active or reactive ARM is also a generalized ARM, since these modes emerge from a generalized eigenvalue problem involving matrices that possess the same eigenvectors. Hence, the columns of the modal matrices can be arranged and normalized such that Ψa = Ψr = Ψg ≡ Ψ. These matrices are explicitly given in Appendix A for the five PSLs, which were evaluated as explained in Ref. [11], and sorted in ascending order of spatial variation. The ARMs of most vibroacoustic sources depend on frequency, albeit they present a “nesting” 12

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Hexahedron

Octahedron

Dodecahedron

Icosahedron

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Tetrahedron

Figure 1: Bisymmetric structure of the matrices C, Re {C}, and Im {C} for the five PSLs.

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property [38] that has been used to obtain frequency-independent spatial filters in active structural acoustic control applications [23]. Besides the PSLs, only few radiators are known to possess

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ARMs that are strictly frequency-independent, such as the sphere, whose ARMs are the spherical harmonic functions [21, 24]. It is worth mentioning that spherical loudspeaker arrays with layouts other than Platonic lead to ARMs that are mostly frequency-dependent [39]. 3.2. Closed-form expressions for the modal radiation efficiencies

Unlike their eigenvectors, real matrices with the structures displayed in Fig. 1 have eigenvalues

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that depend on their entries. Therefore, the σal and σrl associated to the lth ARM of a PSL are functions of the frequency and the vibration pattern and shape of the individual loudspeakers. In addition, Im {C} changes also with r¯, and so does σrl . For such a set of real matrices, Pasqual and Martin [11] present exact closed-form expressions for their eigenvalues given as linear combina-

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tions of the matrix elements. Hence, substitution of Re {ci } and −Im {ci } into these expressions leads to σal and σrl , respectively. Moreover, since we can always make ψal = ψrl ≡ ψl for PSLs,

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the complex radiation efficiencies, σl ≡ σal − jσrl , can be directly obtained by substituting ci in the closed-form expressions presented in Ref. [11]. This leads to Table 2, in which the lth ARM is the lth column of the Ψ presented in Appendix A. It is commonly found that the ARMs are split into a finite number of subsets, each containing ARMs with the same radiation efficiency, and thus forming a radiation group [21]. The number of orthogonal eigenvectors in each subset is equal to the multiplicity of the corresponding eigenvalue. As a consequence, any linear combination of ARMs in a radiation group is also an ARM with the same radiation efficiency. For the PSLs, Table 2 shows that the tetrahedron, hexahedron, 13

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Table 2: Complex radiation efficiencies associated to the ARMs of the PSLs (see Pasqual and Martin [11]). Octahedron

Dodecahedron

Icosahedron

c1 +3c2

c1 +4c2 +c3

c1 +3(c2 +c3 )+c4

c1 +5(c2 +c3 )+c4

c1 +3(c2 +2c3 +2c4 +c5 )+c6

c1 −c2

c1 −c3

c1 +c2 −c3 −c4

√ c1 + 5(c2 −c3 )−c4

√ c1 + 5(c2 −c5 )+2(c3 −c4 )−c6

–

c1 −2c2 +c3

c1 −c2 −c3 +c4

c1 −c2 −c3 +c4

c1 +c2 −2(c3 +c4 )+c5 +c6

–

–

c1 −c2 −c3 +c4

c1 −c2 −c3 +c4

c1 −3(c2 −c3 )−c4

c1 −c2 −c3 +c4

–

–

c1 +c2 −2(c3 +c4 )+c5 +c6

[Y3m ]

[Y2m ]

[Y2m ]

–

–

–

c1 −c2 −c3 +c4

c1 +c2 −2(c3 +c4 )+c5 +c6

–

–

–

√ c1 − 5(c2 −c3 )−c4

c1 −3(c3 −c4 )−c6

–

–

–

–

c1 −3(c3 −c4 )−c6

2 to 4 [Y1m ] 5 to 6 [Y2m ] 7 [Y2m ] 8 [Y2m or Y3m ] 9 [Y2m ] 10 to 12 [Y3m ] 13

14 to 16 [Y3m ]

–

17 to 20 –

–

–

√ c1 − 5(c2 −c5 )+2(c3 −c4 )−c6

–

–

–

c1 −2(c2 +c5 )+c3 +c4 +c6

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[Y4m ]

c1 +c2 −2(c3 +c4 )+c5 +c6

–

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[Y3m ]

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[Y00 ]

Hexahedron

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1

Tetrahedron

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ARM

octahedron, dodecahedron, and icosahedron have two, three, four, four, and six radiation groups,

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respectively (note that this number is equal to Q). The eigenvalues in the first row have unit multiplicity, so that the ARM # 1 is the only mode in its radiation group, regardless of the PSL. This ARM is given by u1 = u2 = · · · = uL , which is the desired vibration pattern for the “omnidirectional” loudspeakers widely used in room acoustics. Indeed, for low krs values, the far-field radiation pattern of a PSL operating in its ARM # 1 is approximately the spherical harmonic of order zero (Y00 ) [40], i.e., uniform over a sphere of radius r¯ >> rs . In addition to the ARM # 1, only the ARM # 8 of the octahedron corresponds to an eigenvalue with unit multiplicity. For low krs , the far-field radiation pattern produced by an octahedral source operating in this ARM is approx14

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imately a linear combination of the third-order spherical harmonics (Y3m , with −3 ≤ m ≤ 3) [11]. The remaining ARMs belong to radiation groups made up of two or more orthogonal eigenvectors. Any linear combination of ARMs in a given radiation group leads to far-field radiation patterns

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that can be nearly expressed as linear combinations of spherical harmonics of a given order n, provided that krs is low enough [8, 10, 11]. The first column of Table 2 indicates in square brackets the order of the spherical harmonics associated to each radiation group.

The discussion presented in this section applies to PSLs regardless of the shape and vibration

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pattern of the individual axisymmetric loudspeakers, i.e., regardless of ξ¯ and An . However, these parameters must be known to evaluate ci in Eq. (35) and σl in Table 2. The infinite series in Eq. (35)

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must be truncated for numerical implementation. Although this might affect the evaluation of σl , it has no effects on Ψ since the truncation does not change the structure of C. In the next section, the loudspeaker diaphragm is modeled as a spherical cap to obtain numerical results for the active and reactive radiation efficiencies.

4. Numerical results for the spherical cap model applied to PSLs

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The diaphragm of a loudspeaker mounted on a rigid sphere can be modeled as a convex spherical cap [9, 41]. Therefore, vn is a radial velocity, dxs = rs2 sin θdθdφ and S = 2πrs2 (1 − cos θ0 )L, where θ0 is the aperture angle of the cap. Note that θ0 cannot be larger than an upper value to avoid

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overlapping of the caps in the array. For the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron, this upper limit is 54.7◦ , 45.0◦ , 35.2◦ , 31.7◦ , and 20.9◦ , respectively. Such max-

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imum values were used to obtain the simulation results presented in this section. Three different axisymmetric vibration patterns of the individual caps were considered, which are described in Section 4.1. Then, the corresponding radiation efficiencies are presented in the remaining sections.

4.1. Axisymmetric vibration patterns of the individual caps The numerical simulations were carried out for the following axisymmetric vibration patterns of the caps: constant radial velocity (labeled as “A”), constant axial velocity (labeled as “B”), and a smooth radial velocity distribution (labeled as “C”). These velocity distributions are summarized 15

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in Table 3 and illustrated in Fig. 2 for the icosahedron (θ0 = 20.9◦ ). Because these profiles are axisymmetric, they can be mathematically expressed as a function of a single independent variable,

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θ(l) . In addition, Eq. (25) becomes Z θ0 Z 2π Z θ0 1 1 (l) 2 2 (l) (l) ξ¯ = |ξ(θ )| rs sin θ dθ dφ = |ξ(θ(l) )|2 sin θ(l) dθ(l) . 2S 0 2L(1 − cos θ0 ) 0 0

(36)

The profile A is the most usual in the spherical loudspeaker array literature and can be expressed by ξ = 1 for 0 ≤ θ(l) ≤ θ0 . Therefore, Eq. (36) leads to ξ¯ = 1/(2L). The profile B

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is a piston-like motion, i.e., the cap moves as a rigid body in the direction of the cap symmetry axis. To achieve this, the radial component of the cap velocity must change along θ(l) so that ξ¯ = (1 − cos3 θ0 )/[6L(1 − cos θ0 )].

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ξ(θ(l) ) = cos θ(l) for 0 ≤ θ(l) ≤ θ0 [42]. Substitution of this expression into Eq. (36) leads to

The profiles A and B present a velocity discontinuity at the interface between the cap border and the rigid spherical cabinet (i.e., at θ(l) = θ0 ), which is a non-realistic assumption. Therefore, a third vibration pattern (profile C) that does not have velocity discontinuities is considered here, which is given by ξ(θ(l) ) = (cos θ(l) − cos θ0 )/(1 − cos θ0 ) for 0 ≤ θ(l) ≤ θ0 . Substitution of this

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expression into Eq. (36) leads to ξ¯ = 1/(6L).

Table 3: Axisymmetric radial velocity distributions of the spherical caps. ξ(θ(l) ) : 0 ≤ θ(l) ≤ θ0

ξ¯

A: constant radial velocity

ξ=1

B: constant axial velocity

ξ(θ(l) ) = cos θ(l)

1 ξ¯ = 2L 1 − cos3 θ0 ξ¯ = 6L(1 − cos θ0 ) 1 ξ¯ = 6L

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Profile

ξ(θ(l) ) =

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C: smooth radial velocity distribution

cos θ(l) − cos θ0 1 − cos θ0

The coefficients An can be evaluated from Vn (rs ), An = jVn (rs )/h0n (krs ), as described in Section 2.1. Because v(l) (rs , θ(l) ) = ul ξ(θ(l) ) for 0 ≤ θ(l) ≤ θ0 , Eq. (5) evaluated at r = rs becomes Z θ0 p Vn (rs ) = π(2n + 1) ξ(θ(l) )Pn (cos θ(l) ) sin θ(l) dθ(l) . (37) 0

This integral can be analytically solved for the three vibration patterns considered here. For the

16

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Figure 2: Radial velocity distributions along a vibrating spherical cap mounted on a rigid spherical cabinet with

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θ0 = 20.9◦ : (a) constant radial velocity, (b) constant axial velocity, and (c) smooth radial velocity distribution. The solid lines indicate the exact profile, whereas the dashed lines correspond to truncated Legendre polynomial series.

(38)

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profiles A, B, and C, it can be shown, respectively, that (cf. [42])   p π      2n+1 [Pn−1 (cos θ0 ) − Pn+1 (cos θ0 )], if n > 0 , VnA (rs ) =   √     π(1 − cos θ0 ), if n = 0

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and

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 ( ) !  n   p π n+1    [Pn (cos θ0 )−Pn+2 (cos θ0 )]+ [Pn−2 (cos θ0 )−Pn (cos θ0 )] , if n > 1   2n+1  2n+3 ! 2n−1      1 − cos3 θ0 √ VnB (rs ) =  3π , if n = 1 ,   3 !      √ 1 − cos2 θ0    , if n = 0   π 2 (39) ! ! 1 cos θ0 VnC (rs ) = VnB (rs ) − VnA (rs ). 1 − cos θ0 1 − cos θ0

(40)

Substitution of Eq. (38), (39), or (40) in Eq. (4) leads to the Legendre polynomial expansion of ξ for the corresponding vibration pattern. In this work, the infinite series in Eq. (4) was truncated to n ≤ 100 for the profiles A and B, and to n ≤ 30 for the profile C. Fig. 2 shows these truncated Legendre polynomial series as dashed lines. The profiles A and B present the Gibbs phenomenon due to the jump discontinuity at θ(l) = θ0 and require a larger number of terms compared with profile C. 17

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Finally, the expressions developed here for ξ¯ and An can be used in Eq. (35) — truncated to the order mentioned above — to obtain ci . The numerical results are presented in the following sections.

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4.2. Active radiation efficiency

For the five PSLs and the velocity profile A, Fig. 3 shows the active radiation efficiencies corresponding to the ARMs of the loudspeaker arrays. These results have already been presented in

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previous papers [10, 11], but are included here for completeness. For low krs values, the radiation groups have clearly distinct efficiencies, which increase monotonically with krs at different rates depending on the ARM. As krs gets higher, the radiation groups become progressively closer, and

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finally merge into a single radiation group. For tetrahedra and icosahedra, this occurs approximately for krs > 2 and krs > 5, respectively, taking a difference of 1 dB as a criterion. These non-dimensional numbers are within the audio frequency range for typical cabinet sizes. At such high frequencies, the spherical array radiates very efficiently to the far field, regardless of its sur-

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face velocity distribution.

Figure 3: Modal active radiation efficiencies for the Platonic solid loudspeakers with velocity profile A: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

18

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Figure 4: Modal active radiation efficiencies for the Platonic solid loudspeakers with velocity profile B: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

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Figs. 4 and 5 show the modal active radiation efficiencies for the velocity profiles B and C, respectively. Note that the velocity profiles A and B present virtually the same σa curves. The profile-C and profile-A (or B) curves present a subtle difference, which does not exceed 1.5 dB for the whole krs range considered. This indicates that σa is not appreciably affected by the vibration

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pattern of the individual caps. The amplitude and phase relations between caps, as well as krs , are

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the main parameters that affect the active radiation efficiency of a spherical loudspeaker array. 4.3. Reactive radiation efficiency Figs. 6, 7, and 8 show the reactive radiation efficiencies corresponding to the ARMs of the five PSLs for the velocity profiles A, B, and C, respectively, which were evaluated at the source surface, i.e., r = rs . Note that Figs. 6 and 7 present virtually the same σr curves (the difference is within ±1 dB for the considered krs range), so that the profiles A and B lead almost to the same reactive radiation efficiency. On the other hand, the difference between the profile-C and profileA (or B) curves is more pronounced, which is within ±4 dB for the considered krs range. This 19

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Figure 5: Modal active radiation efficiencies for the Platonic solid loudspeakers with velocity profile C: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

the source near field.

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indicates that σr is significantly affected by the vibration pattern of the individual caps, at least in Unlike σa , σr does not increase monotonically with krs , but presents a maximum point instead that is related to the near- to far-field transition, as it will be shown in Section 4.4. Also, the

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σr difference between any two radiation groups for a given krs and velocity profile is no more than 6 dB approximately, which is negligible compared with the σa differences for low krs values.

AC C

This can be explained by considering that the acoustic field is very intricate at the source surface, regardless of the phase and amplitude relations between the individual loudspeakers, so that the reactive powers associated to distinct radiation groups are not as different as the active powers. As observed for σa , the reactive radiation groups also degenerate into a single group for high krs values, with a constant decay rate of 3 dB/octave, i.e., σr ∝ (krs )−1 . This occurs approximately for krs > 30 for all PSLs. As shown in Figs. 6 and 7 (profiles A and B), some σr curves present oscillations in the high krs range, which does not occur for the profile-C curves (Fig. 8). These oscillations arise due to the jump discontinuity (Gibbs phenomenon) of the profiles A and B. 20

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Figure 6: Modal reactive radiation efficiencies for the Platonic solid loudspeakers with r = rs and velocity profile A:

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(a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

Figure 7: Modal reactive radiation efficiencies for the Platonic solid loudspeakers with r = rs and velocity profile B: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

21

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Figure 8: Modal reactive radiation efficiencies for the Platonic solid loudspeakers with r = rs and velocity profile C: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

Because it does not take place in a real loudspeaker, such oscillations can be considered as non-

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physical modeling artifacts. It is interesting to note that, in the low krs range, σr ∝ krs for all ARMs and PSLs, which is substantially smaller than the increase rate of σa . The reactive radiation efficiency decays with distance. Fig. 9 shows σr as a function of r/rs

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for the ARMs of dodecahedral and icosahedral sources, with krs = 0.1 and krs = 1, and the velocity profile A. The corresponding σa values are indicated by thin horizontal lines. The reactive

AC C

efficiency drops with distance in a rate that depends on frequency, ARM order, and distance. The decay rate is larger for low krs , high-order ARMs, and in the near field. Indeed, as r/rs increases within the near-field region, the relative contribution of the high-order modes to the reactive power gradually decreases. The ARM # 1 presents a constant decay rate of 3 dB per doubling of distance, i.e., σr ∝ 1/r. If r/rs is large enough, the remaining ARMs also exhibit such a behavior, which can be used to identify the source far field. This 1/r dependence has been described for the far-field radiation from beams and plates [31], as well as from simple spherical sources [32]. The transition from the near to the far field occurs at large distances for low krs and high order ARMs. Except 22

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for the ARM # 1, Fig. 9 indicates that the transition can be identified by noting that σa > σr in the far field, regardless of the radiation group and loudspeaker layout. Note that, although we chose the dodecahedron and icosahedron with the velocity profile A and only two krs values to illustrate

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profiles lead to similar curves, so that they are not presented here.

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the effects of distance on the reactive efficiency, the remaining PSLs and different krs and velocity

Figure 9: Modal reactive radiation efficiencies for the PSLs as a function of r/rs for the velocity profile A: (a) dodecahedron with krs = 0.1, (b) dodecahedron with krs = 1, (c) icosahedron with krs = 0.1, and (d) icosahedron with

AC C

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krs = 1. The thin horizontal lines indicate the active radiation efficiencies.

4.4. Relation between the active and reactive radiation efficiencies For low krs values, σa and σr decrease as the radiation group order increases, i.e., ARMs associated to low spatial variations radiate more efficiently than those corresponding to high spatial variations. Therefore, if the source is to be used only to generate high sound power levels, regardless of active or reactive, it should be driven in its first ARM. On the other hand, some applications can benefit from a low η (≡ σa /σr ), such as in local active noise control, as it will be shown in Section 5. Fig. 10 shows σa /σr curves as a function of krs for the ARMs of the PSLs with r = rs 23

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and velocity profile C, which correspond to the solution of the generalized eigenvalue problem expressed in Eq.(33). Note that high order ARMs lead to smaller σa /σr ratios. Indeed, the ARM # 1 yields the largest σa /σr for all PSLs, and thus to the largest power factor, cos ϑ. The results for

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the velocity profiles A and B are similar and will not be presented here.

Figure 10: Ratio between the modal active and reactive radiation efficiencies for the Platonic solid loudspeakers with

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r = rs and velocity profile C: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron, and (e) icosahedron.

Except for the ARM # 1, which presents a very short near field, a comparison between Figs. 8 and 10 shows that the maximum point of a σr curve takes place at a krs value that leads to σr ≈ σa .

AC C

This corresponds to the transition region between the near to the far field, which is therefore related to the maximum of σr when evaluated at r = rs . Hence, the near- to far-field transition will occur at the very vicinity of the source if krs > arg maxkrs σr . An ideal pulsating sphere presents σa = cos2 ϑ for all krs . Chen [29] argued that this relation roughly holds for different source shapes and vibration patterns, and provided several examples of radiators that corroborate this hypothesis. To investigate this behavior for PSLs, parametric curves of σa (krs ) versus cos ϑ(krs ) are presented in Fig. 11, where 0.1 ≤ krs ≤ 100. Only the velocity profile A is considered since it leads to the same vibration pattern as the pulsating sphere over the 24

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Figure 11: Parametric curves showing modal active radiation efficiencies and power factors for the Platonic solid loudspeakers, 0.1 ≤ krs ≤ 100, and velocity profile A: (a) tetrahedron, (b) hexahedron, (c) octahedron, (d) dodecahedron,

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and (e) icosahedron. The dotted line corresponds to a parabola, σa = cos2 ϑ.

cap surface. The reference parabola, σa = cos2 ϑ, is indicated as a dotted line. Except for the octahedron and icosahedron, the ARM # 1 curves are virtually parabolic. Because the ARM # 1 approximates the behavior of a pulsating sphere, this is an expected result. Note that the ratio

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between the total area of the spherical caps and the area of a sphere with radius rs is S /(4πrs2 ) = (1 − cos θ0 )L/2. The substitution of the simulation parameters into this equation leads to the

AC C

following area ratios for the tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron, respectively: 0.844, 0.879, 0.731, 0.895, and 0.658. Although the icosahedron presents the largest number of faces, it has the smallest area ratio, followed by the octahedron. Therefore, these Platonic layouts present a larger deviation from the pulsating sphere compared with tetrahedral, hexahedral, and dodecahedral sources, and thus a larger deviation from the parabolic trend. In addition, the parabolic behavior degrades as the ARM order increases, so that the hypothesis under study does not hold for the last radiation groups of PSLs, especially for spherical arrays made up of many loudspeakers. Although the curves for the velocity profiles B and C are not presented 25

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here, it can be shown that they lead to a still larger deviation from a parabola than the profile A. 5. Application example: local active control of sound

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This section illustrates the concepts developed throughout this paper by applying them to a simulated problem of local active control of sound. A PSL is used as a secondary source aiming at producing a quiet zone in its near field.

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5.1. Problem formulation

The primary pressure field, ppri (r, θ), is the superposition of the pressure field impinging on the

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PSL and the pressure field scattered by the PSL when it is switched off. In this case, the secondary source is modeled as a rigid sphere of radius rs . When the loudspeaker array is switched on, the superposition of ppri (r, θ) and the secondary field, p(r, θ), leads to the total pressure field, ptot (r, θ), i.e., ptot (r, θ) = ppri (r, θ) + p(r, θ). Substitution of Eq. (10) into this equation yields ptot (r, θ) = ppri (r, θ) + ρcy(θ)T H(kr)Y∗ u.

(41)

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Let (rq , θq ) (with rq ≥ rs ) be the spherical coordinates of the qth point in the sound field, where 1 ≤ q ≤ M and M is the number of field points (simulated error microphones). Hence, Eq. (41) can be written as

(42)

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ptot = ppri + Du,

where ptot and ppri are M ×1 vectors containing ptot (rq , θq ) and ppri (rq , θq ) as elements, respectively;

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D ≡ ρcYH(kr)Y∗ is an M × L matrix, and Y is an M-row matrix that contains y(θq )T as rows. A straightforward manner to achieve a quiet zone at the region defined by the M field points is to let M ≥ L and evaluate u that minimizes kppri + Duk2 , where k · k is the 2-norm. The optimal solution of this unconstrained least-squares problem is uopt = −D† ppri , where D† is the pseudoinverse of D. However, the so-obtained sound attenuation may take place at the expense of a substantial increase of the sound levels outside the quiet zone [7]. To tackle this issue, we propose to reduce the solution space such that the feasible u must be in a subspace spanned by L˜ ˜ α, ˜ is an L × L˜ reduced modal matrix, which ˜ where Ψ ARMs, with L˜ ≤ L. Accordingly, u = Ψ 26

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˜ is an L˜ × 1 vector of modal contributions. Therefore, contains selected ARMs as columns, and α the optimization problem becomes ˜ αk ˜ 2. min kppri + DΨ

(43)

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˜ α

˜ † ppri , and thus uopt = Ψ ˜α ˜ opt = −(DΨ) ˜ opt . The sound amplification outside The optimal solution is α ˜ be made up of ARMs with low σa /σr ratios. the quiet zone can be mitigated by letting Ψ

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5.2. Simulation example

A numerical example investigated by Rafaely [7] is revisited here using the ARM approach and the PSL radiation model presented in the previous sections. The considered primary field is a

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unit-amplitude plane wave traveling in the negative y-axis direction and scattered by the secondary source. The primary pressure field was evaluated using the equations given by Rafaely [7, 43]. An icosahedral secondary source of radius rs = 0.1 m, θ0 = 20.9◦ , and velocity profile C (see Table 3) is used to obtain a quiet zone in a region in the xy plane defined by 0.2 ≤ r ≤ 0.5 m, θ = 90◦ , and −30◦ ≤ φ ≤ 30◦ . We simulated M = 48 error microphones placed on a polar grid with six and

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eight points in the r and φ directions, respectively. Fig. 12 illustrates the secondary source and the

AC C

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orientation of the xyz axes, in which the arrows indicate the symmetry axes of the caps.

Figure 12: Icosahedral array used as secondary source in the active sound control simulations. The arrows indicate the symmetry axes of the caps.

27

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Fig. 13 shows contour plots of the primary pressure field on the xy plane for 250 Hz and 500 Hz. This field is composed of the incident plane wave and the scattering due to the PSL, which is modeled as a rigid sphere. The grayscale indicates |ppri | in decibels, where the reference

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value is the acoustic pressure of the incident plane wave. Note that, for 250 Hz, the scattering is virtually negligible since the acoustic wavelength is much larger than the secondary source radius.

(a)

0.75

(b)

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Figure 13: Primary pressure field in decibels: (a) 250 Hz, and (b) 500 Hz.

Figs. 14 and 15 show contour plots of the sound attenuation (or amplification) when the secondary source is switched on for 250 Hz (krs = 0.46) and 500 Hz (krs = 0.92), respectively. Note

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that the radiation groups of the secondary source are not merged into a single group in this frequency range (see Figs. 5-e, 8-e, and 10-e). The grayscale indicates |ptot |/|ppri | in decibels. The ˜ = Ψ, optimal cap velocities were evaluated as described in Section 5.1 for three cases: (a) Ψ

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˜ made up of the last three radiation groups i.e., unconstrained u (all ARMs considered); (b) Ψ ˜ made up of the first three radiation groups (ARMs # 1 to 9). (ARMs # 10 to 20); and (c) Ψ For both frequencies, the unconstrained optimal solution leads to the largest attenuation in the control zone. This is an expected result since the secondary source has twenty effective DOFs in this case, whereas it has eleven and nine effective DOFs in the considered constrained solutions, plots (b) and (c), respectively. However, the unconstrained solution produces a very large sound amplification outside the quiet zone. The sound level is increased by more than 30 dB at 500 Hz. On the other hand, the sound amplification in the source far field is reduced to less than 10 dB if 28

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Figure 14: Acoustic pressure attenuation (or amplification) achieved by an icosahedral secondary source for 250 Hz:

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(a) ARMs # 1 to 20, (b) ARMs # 10 to 20, and (c) ARMs # 1 to 9.

the loudspeaker array is constrained to operate with the ARMs # 10 to 20, while still keeping a significant quiet zone, as shown in Figs. 14-b and 15-b. This can be explained by the low σa /σr ratios of these ARMs in the considered frequency range. Indeed, Figs. 14-c and 15-c show that the sound amplification is reduced by a substantially smaller amount if ARMs with higher σa /σr

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-10

-10

x (m)

-0.75 -0.75 -0.5 -0.25

-0.5

0

0.75

20 10

-0.25

-10

0.5

0

-10

0.25

0

20

20

0

0

-10

-0.25

-0.5

-0.75 -0.75 -0.5 -0.25

0.25 -10

-0.5

0

-10

0

-0.25

20

-20

0

y (m)

0.25

-20

y (m)

0

-10 0 10

0.25

0.5

0

0.5

30 10

0

0.5

(c) 10

0

0.75

0

(b)

TE D

(a)

0.75

10

ratios are used instead.

-30 0

0.25

0.5

0.75

x (m)

Figure 15: Acoustic pressure attenuation (or amplification) achieved by an icosahedral secondary source for 500 Hz: (a) ARMs # 1 to 20, (b) ARMs # 10 to 20, and (c) ARMs # 1 to 9.

In a real active sound control system, the power of the loudspeaker signals must be limited to avoid saturation. Thus, it is worth knowing how the solution norm, kuopt k, is affected by the ˜ considered in optimization constraints. Table 4 shows kuopt k in decibels for the frequencies and Ψ the simulations. The value of kuopt k for 250 Hz and the unconstrained optimization problem was 29

ACCEPTED MANUSCRIPT

taken as the reference value. Note that, for the same primary field, a huge reduction in the signal power is obtained by only constraining the source to operate in a few radiation groups. Such a desirable result can be understood by realizing that the disposal of some ARMs is a regularization

RI PT

technique, which improves the conditioning of the matrix involved in the optimization problem. Indeed, for the example under study, the condition number of D (or DΨ) is ∼ 107 , whereas the ˜ is ∼ 103 if the ARMs # 10-20 are retained, and ∼ 104 for the ARMs # 1-9. condition number of DΨ

ARMs # 1-20

ARMs # 10-20

ARMs # 1-9

250 Hz

0 dB

−29.3 dB

−49.7 dB

500 Hz

0.3 dB

6. Conclusion

M AN U

Frequency

SC

˜ Table 4: Norm of the optimal cap velocities, kuopt k, in decibels for different frequencies and Ψ.

−43.0 dB

−47.0 dB

This paper presented a theoretical and numerical analysis of the complex sound power pro-

TE D

duced by compact spherical loudspeaker arrays, with focus on the Platonic layouts and the source near field. A symmetric matrix that couples the complex sound power generated by the individual loudspeakers was derived from the solution of the homogeneous Helmholtz’s equation in spherical

EP

coordinates. The adopted model holds for arrays made up of identical axisymmetric loudspeakers mounted on a rigid spherical cabinet. The matrices obtained for the five PSLs were subjected to eigenvalue analyses to derive the active, reactive, and generalized ARMs, as well as closed-form

AC C

expressions for the corresponding radiation efficiencies. The individual loudspeakers were modeled as convex baffled spherical caps to numerically evaluate the modal efficiencies, which were investigated as a function of the non-dimensional frequency and distance, krs and r/rs , respectively. Three different vibration patterns on the surface of the individual caps were considered: constant radial velocity, constant axial velocity, and a smooth radial velocity distribution. In addition, the ARM approach was applied to an active sound control problem, where an icosahedral loudspeaker array was used as a secondary source aiming at producing a zone of quiet in its near field. 30

ACCEPTED MANUSCRIPT

We found that the active, reactive, and generalized ARMs form strictly the same basis for a PSL, which is explicitly given in the appendix. In addition, they do not depend on krs , r/rs , and the vibration pattern and shape of the individual axisymmetric loudspeakers, as it had been already

RI PT

demonstrated for the active ARMs of PSLs in a previous study. These results are due to the highly symmetrical distribution of loudspeakers in Platonic layouts, leading to matrices with bisymmetric structures.

The evaluation of the eigenvalues (σa , σr , and σa /σr ) revealed that, in the low krs range, the

SC

ARM # 1 is the most efficient mode, both in active and reactive power. However, it presents the largest σa /σr ratio, so that high-order modes are preferable in applications that require a given

M AN U

amount of reactive power with a minimum active power. For low krs and r = rs , we observed that σr ∝ krs . As krs increases, σr attains a maximum value, and then decreases so that σr ∝ (krs )−1 for krs > 30 approximately. In addition, it was observed that σr ∝ 1/r in the far field, which agrees with recent findings for different radiators. The r/rs range in which the transition from the near to the far field takes place depends on krs and the ARM order, occurring at larger r/rs as krs decreases and the ARM order increases. These observations hold for the three different vibration patterns

TE D

that were imposed on the caps. In fact, the changes in the cap vibration pattern led to variations in σr of less than 4 dB, whereas these variations did not exceed 1.5 dB for σa . The number of loudspeakers, their distribution on the sphere, the amplitude and phase relations between them,

EP

the frequency, and the cabinet radius have a greater effect on the complex radiation efficiency than the vibration pattern of each individual loudspeaker. The claimed parabolic relation between the active radiation efficiency and the power factor,

AC C

σa = cos2 ϑ, was also investigated. For the constant-radial-velocity vibration pattern and the ARM # 1, the parabolic behavior was verified for the tetrahedron, hexahedron, and dodecahedron. However, it was not observed for the remaining PSLs, which presented a deviation from a parabola, especially the icosahedron. As the ARM order increases, the relation between σa and cos ϑ cannot be described by a parabola, even as a rough approximation. Therefore, we conclude that the parabolic behavior is not a general trend, but holds only for some surface velocity patterns and radiator shapes. The application example in local active sound control showed that the generalized ARMs of a 31

ACCEPTED MANUSCRIPT

multi-channel secondary source is a worthy alternative as a basis to represent the control signals. The disposal of ARMs that have high σa /σr ratios might lead to a great reduction in the sound amplification outside the control zone, as well as in the power of the loudspeaker signals, while

RI PT

still keeping a significant zone of quiet nearby the error microphones. Appendix A. Explicit ARMs of Platonic solid loudspeakers

This appendix explicitly presents the L × L modal matrices, Ψ, for the five PSLs, which were

SC

evaluated as described in a previous paper [11]. The ARMs (columns of Ψ) are sorted in ascending order of spatial variation, so that the ARM # 1 is given by Ψ11 = Ψ21 = · · · = ΨL1 . The rows of

M AN U

Ψ correspond to the individual loudspeakers of the PSL, which are labeled as shown in Fig. A.16. This numbering convention must be used to produce the bisymmetric matrices displayed in Fig. 1.

2 3

2

1 3

4

2

TE D

1

5

1

4

4

4

3

6

2

7

1 3

7

8

EP

4

5

AC C

10

4

13

6

12

3

1

2

7

14

5

2

4

9

12

1

2 7

6

2 6

4

9

3

8 15 20 11

7

8

1 5

5

19 18 13 6

5

3

6

10

1 3

1

10

17

2

1

9

12

11

10

16

Figure A.16: Face numbering convention for the five convex regular polyhedra: tetrahedron (top left), hexahedron (top right), octahedron (center left), dodecahedron (center right), and icosahedron (bottom).

32

ACCEPTED MANUSCRIPT

Appendix A.1. Tetrahedron

(A.1)

  0.4082 0.7071  0.0000 0.0000 −0.2887 0.5000     0.4082 0.0000 0.7071 0.0000 −0.2887 −0.5000     0.0000 0.7071 0.5774 0.0000  0.4082 0.0000  Ψ =   0.4082 0.0000 0.0000 −0.7071 0.5774 0.0000     0.4082 0.0000 −0.7071 0.0000 −0.2887 −0.5000     0.4082 −0.7071 0.0000 0.0000 −0.2887 0.5000

(A.2)

RI PT

  0.5000 0.5000  0.5000 0.5000     0.5000 0.5000 −0.5000 −0.5000   Ψ =  0.5000 −0.5000 −0.5000 0.5000      0.5000 −0.5000 0.5000 −0.5000

M AN U

SC

Appendix A.2. Hexahedron

0.3536 0.3536 0.3536

0.3536

0.3536

0.3536

−0.3536

0.3536

0.3536

0.3536

0.3536

−0.3536 −0.3536 0.3536

−0.3536 −0.3536

0.3536

−0.3536 −0.3536

−0.3536

EP

−0.3536

AC C

 0.3536   0.3536   0.3536   0.3536 Ψ =  0.3536   0.3536  0.3536   0.3536

TE D

Appendix A.3. Octahedron

−0.3536

0.3536

0.3536

0.3536

−0.3536 −0.3536 −0.3536 0.3536

0.3536

−0.3536

0.3536

0.3536

−0.3536

−0.3536 −0.3536 −0.3536 −0.3536 0.3536

−0.3536 −0.3536 −0.3536

−0.3536

0.3536

0.3536

0.3536

−0.3536

0.3536

 0.3536   −0.3536  −0.3536   −0.3536   0.3536    0.3536    0.3536    −0.3536

(A.3)

Appendix A.4. Dodecahedron   Ψ   11 Ψ12   Ψ =   , Ψ Ψ 21 22

33

(A.4)

ACCEPTED MANUSCRIPT

where

Ψ12

0.6455

0.2236

0.4472

0.0000

−0.1291

0.2236

0.1382

0.4253

−0.1291

0.2629

−0.1291

0.2236 −0.3618

0.2236 −0.3618 −0.2629 −0.1291 0.2236

0.1382

 0.0000    0.4472    −0.3618  ,  0.1382   0.1382    −0.3618

−0.4253 −0.1291

 −0.1291 −0.3618   −0.2236 0.3618 0.2629 −0.1291 0.1382    −0.2236 0.3618 −0.2629 −0.1291 0.1382   ,  −0.2236 −0.1382 −0.4253 −0.1291 −0.3618  −0.2236 −0.4472 0.0000 −0.1291 0.4472    −0.5000 0.0000 0.0000 0.6455 0.0000 0.4253

SC

−0.2236 −0.1382

(A.5)

RI PT

0.0000

(A.6)

    0.0000 0.0000 0.0000 0.0000 0.0000 0.5000      0.4472 0.0000 0.0000 0.4472 0.0000 −0.2236     0.1382 0.4253 −0.2629 −0.3618 −0.2629 −0.2236    ,  =  −0.3618 0.2629 0.4253 0.1382 0.4253 −0.2236    −0.3618 −0.2629 −0.4253 0.1382 −0.4253 −0.2236     0.1382 −0.4253 0.2629 −0.3618 0.2629 −0.2236

(A.7)

EP

and

0.0000

TE D

Ψ21

 0.2887   0.2887   0.2887 =  0.2887   0.2887   0.2887

0.5000

M AN U

Ψ11

 0.2887   0.2887   0.2887 =  0.2887   0.2887   0.2887

AC C

    0.1382 −0.4253 0.2629 0.3618 −0.2629 0.2236     −0.3618 −0.2629 −0.4253 −0.1382 0.4253 0.2236      0.4253 −0.1382 −0.4253 0.2236  −0.3618 0.2629  . =    0.1382 0.4253 −0.2629 0.3618 0.2629 0.2236       0.4472 0.0000 0.0000 −0.4472 0.0000 0.2236     0.0000 0.0000 0.0000 0.0000 0.0000 −0.5000

Ψ22

(A.8)

Appendix A.5. Icosahedron   Ψ  Ψ 12   11  , Ψ =   Ψ 21 Ψ22 34

(A.9)

ACCEPTED MANUSCRIPT

where

Ψ12

 −0.0214    0.1950  −0.1991   −0.2238   0.2493  =  −0.4444    0.2205    0.0288  −0.0502   0.2453

0.0980 −0.1174 0.2119

0.0727

0.2398

0.2954 −0.1350 0.1688

0.3078

0.1482

0.1825 −0.0200 −0.1431

0.3078

0.2194 −0.0845 0.1011

0.1821

−0.0727 0.2068 −0.3193 0.3807 −0.1865 −0.3078 0.2272

0.0606 −0.3468 −0.1645

−0.0727 0.0204

0.3799

0.3078 −0.1278 0.1973

0.0024

0.1001

0.0139 −0.4941

−0.3078 0.0126

M AN U

0.3078 −0.0126 −0.2348 −0.2433 0.1177

 0.1181 −0.0324 0.2432 −0.3360   −0.0419 −0.2968 0.3185 −0.0063  −0.4024 −0.1698 −0.1117 0.3794   −0.3877 0.2786 0.0348 −0.0063  0.0402 0.4464 0.0757 −0.0307  , (A.10) 0.1198 −0.1820 −0.1510 0.0370    0.2424 −0.0945 0.1870 −0.0433   0.0271 −0.1517 −0.4650 0.0126    −0.0254 0.0021 0.0708 −0.3486  0.3098 0.2002 −0.2023 0.3423 

RI PT

−0.0727 0.3676

0.2075

SC

0.3550 −0.1368 0.3644

0.2348 −0.2433 0.1177

0.3098

−0.3078 0.1278 −0.1973 0.0139 −0.4941 −0.0254 0.0727 −0.0204 −0.3799 0.0024

0.1001

0.0271

0.3078 −0.2272 −0.0606 −0.3468 −0.1645 0.2424 0.0727 −0.2068 0.3193

0.1011

0.1821

TE D

−0.3078 −0.2194 0.0845

0.3807 −0.1865 0.1198

0.0402

−0.3078 −0.1482 −0.1825 −0.0200 −0.1431 −0.3877 −0.0727 −0.2398 −0.2954 −0.1350 0.1688 −0.4024 0.0727 −0.3676 −0.0980 −0.1174 0.2119 −0.0419 −0.0727 −0.3550 0.1368

0.3644

0.2075

EP

Ψ21

 0.2236   0.2236  0.2236   0.2236  0.2236  =  0.2236   0.2236  0.2236   0.2236  0.2236

0.0727

0.1181

 0.2002 −0.2023 −0.3423   0.0021 0.0708 0.3486   −0.1517 −0.4650 −0.0126  −0.0945 0.1870 0.0433   −0.1820 −0.1510 −0.0370  , (A.11) 0.4464 0.0757 0.0307    0.2786 0.0348 0.0063    −0.1698 −0.1117 −0.3794   −0.2968 0.3185 0.0063   −0.0324 0.2432 0.3360 

−0.2366 −0.1750 −0.1474 0.3411 −0.1091 −0.3709 0.1687 −0.2449 0.3193 −0.1257 −0.3595 0.0703

AC C

Ψ11

 0.2236   0.2236  0.2236   0.2236   0.2236 =  0.2236   0.2236  0.2236   0.2236  0.2236

−0.1236 0.0343 0.3716

0.2482

0.2131 −0.0676 −0.3807

0.1776 −0.2384 −0.0119 0.2004

0.1084 −0.3785 0.0092

0.0115 −0.0323 0.1935 −0.1478 0.3011 0.1407

0.1801

0.2852

0.3456

0.0817 −0.2524 −0.3176 −0.1817

0.2334 −0.2870 0.2617 −0.2555 −0.1275 0.4221

0.3601

0.0817

0.0160

0.1351

0.1067 −0.2859 0.0822

0.1903 −0.0434 −0.2340 0.3341

−0.1267 −0.4277 −0.0510 −0.0467 0.3811

0.0631 −0.0157 −0.1288

−0.1099 0.2527

0.0945

0.3364

0.0573

−0.1350 0.0666 −0.0593 0.2209

0.1832

0.4187 −0.1173

0.3125 −0.2209 0.1168 −0.1702

35

 0.1651    −0.0714  −0.2006  0.0493   0.0575   , (A.12) −0.3165   0.1782    0.4233    0.0445   −0.3295

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and

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 0.1350 −0.0666 0.0593 −0.2209 −0.3125 −0.2209 0.1168 −0.1702 −0.3295   0.1099 −0.2527 −0.3364 −0.0573 −0.1832 0.0945 0.4187 −0.1173 0.0445   0.1267 0.4277 0.0510 0.0467 −0.3811 0.0631 −0.0157 −0.1288 0.4233   −0.3601 −0.1407 −0.1801 −0.2852 −0.1903 −0.0434 −0.2340 0.3341 0.1782   −0.0115 0.0323 −0.1935 0.1478 −0.3011 0.1067 −0.2859 0.0822 −0.3165  . (A.13) −0.2334 0.2871 −0.2617 0.2555 0.1275 0.4221 0.0160 0.1351 0.0575    −0.3716 −0.1084 0.3785 −0.0092 −0.0817 −0.2524 −0.3176 −0.1817 0.0493    0.1236 −0.0343 −0.2482 −0.1776 0.2384 −0.0119 0.2004 0.3456 −0.2006   0.2449 −0.3193 0.1257 0.3595 −0.0703 0.2131 −0.0676 −0.3807 −0.0714  0.2366 0.1750 0.1474 −0.3411 0.1091 −0.3709 0.1687 0.0817 0.1651 

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Ψ22

 −0.2453    0.0502  −0.0288   −0.2205    0.4444 =  −0.2493    0.2238   0.1991   −0.1950   0.0214

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