Active control of acoustic scattering from a passively optimised

spherical shell

Stephen Elliott 1 , Mihai Orita, Erika Quaranta and Jordan Cheer Institute of Sound and Vibration Research, University of Southampton Southampton, SO17 1BJ, UK

ABSTRACT

At low frequencies, the sound power scattered from a spherical shell can be minimised by de- signing the material properties and thickness so that its mass and compressibility are the same as that of the displaced fluid. The scattered power is then dominated at higher frequencies by that due to the resonances of the structural modes of the shell, particularly the ovalling mode. The peaks in the scattered power due to structural resonances can be reduced somewhat by material damping but are more effectively attenuated with active control using structural actuators as sec- ondary sources. Of particular interest are structural actuators and sensors that are distributed over the surface of the sphere, rather than just acting at single points. Simulations are presented of the scattered sound power of such a shell when subject to feedforward control, which assumes knowledge of both the incident and scattered acoustic sound fields, and structural feedback con- trol, which only assumes that the velocity on the surface of the sphere can be measured .

1. INTRODUCTION

The scattering of sound is important in a number of applications, such as binaural sound reproduc- tion, where the physical presence of the head plays an important role in the perceived sound [1], and in acoustic cloaking of objects [2], which is important in scenarios involving acoustic detection. The sound scattered from a body surrounded by a fluid can be calculated numerically, using finite elements or boundary elements for example, or analytically if the body has a simple shape, such as a sphere [3,4,5].

Bobrovnitskii [6,7] introduced an impedance-based approach to the analysis of sound scattering by assuming that the surface of the scattering body was divided into a large number of discrete ele- ments, which are assumed to be small compared with a wavelength in the surrounding fluid. If the pressure and velocity over the surface are instead expressed in terms of a modal expansion, an en- tirely analogous analysis of scattering can be formulated. Assuming tonal excitation proportional to 𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 , the vectors of complex total modal pressures and total modal velocities on the surface of the scattering body are denoted 𝐩𝐩 𝑡𝑡 and 𝐯𝐯 𝑡𝑡, where 𝐯𝐯 𝑡𝑡 is measured normal and outward with respect to the surface. Each of these vectors is made up of contributions from the sound field incident on the scat- tering body, and contributions from the scattered sound field, so that 𝐩𝐩 𝑡𝑡 can be written as 𝐩𝐩 𝑖𝑖 plus 𝐩𝐩 𝑠𝑠 and 𝐯 𝐯 𝑡 𝑡 as 𝐯 𝐯 𝑖 𝑖 plus 𝐯 𝐯 𝑠 𝑠 . Three input impedance matrices are then defined, which are the in-vacuo struc- tural impedance matrix of the scattering body, 𝐙𝐙 𝐵𝐵 , the impedance matrix of the internal volume of

1 sje@isvr.soton.ac.uk

‘inter.a 21-24 AUGUST SCOTTISH EVENT CAMPUS ? O? ? GLASGOW

the scattering body if filled with the surrounding fluid, 𝐙𝐙 𝐼𝐼 , and the outward radiation impedance ma- trix into the surrounding fluid, 𝐙𝐙 𝑅𝑅 , so that

𝐩𝐩 𝑡𝑡 = −𝐙𝐙 𝐵𝐵 ∙𝐯𝐯 𝑡𝑡 (1) 𝐩𝐩 𝑖𝑖 = −𝐙𝐙 𝐼𝐼 ∙𝐯𝐯 𝑖𝑖 , (2) 𝐩𝐩 𝑠𝑠 = 𝐙𝐙 𝑅𝑅 ∙𝐯𝐯 𝑠𝑠 . (3) Using simple manipulations of the defining equations (1), (2) and (3), the vector of scattered sur- face pressures, 𝐩𝐩 𝑠𝑠 , can be expressed in terms of the vector of incident surface pressures, 𝐩𝐩 𝑖𝑖 , as [6,7]

𝐩𝐩 𝑠𝑠 = (𝐘𝐘 𝑅𝑅 + 𝐘𝐘 𝐵𝐵 ) −1 ∙(𝐘𝐘 𝐼𝐼 −𝐘𝐘 𝐵𝐵 ) ∙𝐩𝐩 𝒊𝒊 = 𝐑𝐑∙𝐩𝐩 𝒊𝒊 , (4) where the admittance matrices 𝐘𝐘 𝐵𝐵 , 𝐘𝐘 𝑅𝑅 and 𝐘𝐘 𝐼𝐼 are the inverses of the impedance matrices 𝐙𝐙 𝐵𝐵 , 𝐙𝐙 𝑅𝑅 and 𝐙𝐙 𝐼𝐼 , assuming that these matrices are non-singular. Under the conditions of linearity and reciprocity, these matrices are also symmetric, and when all the processes involved are passive, the real parts of the matrices are positive definite and so all of their associated impulse responses are causal. It is im- portant to note that despite being formulated in terms of the in-vacuo structural response of the body, the loading of the fluid on the structure, as well as the sound scattering, are all accounted for in equation (4).

These impedance matrices are fully populated in the original formulation using elemental radia- tors, but each could be diagonalised by choosing a model expansion involving either the structural modes of the body, for 𝐙𝐙 𝐵𝐵 , the interior acoustic modes of the space, for 𝐙𝐙 𝐼𝐼 , or the radiation modes, for 𝐙𝐙 𝑅𝑅 . The eigenvectors of any of the three impedance matrices could thus potentially be used to define this modal expansion. For the particular case of the scattering from a thin, uniform, empty spherical shell in an infinite fluid, however, an expansion in terms of spherical harmonics diago- nalises all three impedance matrices. This expansion is truncated here to 𝑛𝑛= 𝑁𝑁’ terms, so that the pressure and velocity on the surface of the sphere are

𝑝𝑝(𝜃𝜃, 𝜑𝜑) = ∑ ∑ 𝑝𝑝(𝑛𝑛, 𝑚𝑚) 𝑌𝑌 𝑛𝑛 𝑚𝑚 (𝜃𝜃, 𝜑𝜑) 𝑛𝑛𝑚𝑚=−𝑛𝑛 𝑁𝑁 𝑛𝑛=0 , (5) 𝑣𝑣(𝜃𝜃, 𝜑𝜑) = ∑ ∑ 𝑣𝑣(𝑛𝑛, 𝑚𝑚)𝑌𝑌 𝑛𝑛 𝑚𝑚 (𝜃𝜃, 𝜑𝜑), 𝑛𝑛𝑚𝑚=−𝑛𝑛 𝑁𝑁 𝑛𝑛=0 (6) where 𝑌𝑌 𝑛𝑛 𝑚𝑚 (𝜃𝜃, 𝜑𝜑) is the complex spherical harmonic of index (𝑛𝑛, 𝑚𝑚) and 𝑝𝑝(𝑛𝑛, 𝑚𝑚) and 𝑣𝑣(𝑛𝑛, 𝑚𝑚) denote the modal amplitudes. The vectors of 𝑁𝑁= (𝑁𝑁’ + 1) 2 modal pressures and velocities are then de- fined as

𝐩𝐩= [𝑝𝑝(0,0), 𝑝𝑝(1, −1), 𝑝𝑝(1,0), 𝑝𝑝(1,1) … 𝑝𝑝(𝑁𝑁, −𝑁𝑁) … 𝑝𝑝(𝑁𝑁, 0) … 𝑝𝑝(𝑁𝑁, 𝑁𝑁)] 𝑇𝑇 , (7)

𝐯𝐯= [𝑣𝑣(0,0), 𝑣𝑣(1, −1), 𝑣𝑣(1,0), 𝑣𝑣(1,1) … 𝑣𝑣(𝑁𝑁, −𝑁𝑁) … 𝑣𝑣(𝑁𝑁, 0) … 𝑣𝑣(𝑁𝑁, 𝑁𝑁)] 𝑇𝑇 . (8) In the case of a uniform spherical shell, all three impedance matrices are only dependant on the index 𝑛𝑛 , and the diagonal elements associated with the 𝑛𝑛 -th terms of the two acoustic impedance matrices can then be written as [8]

𝑍𝑍 𝐼𝐼 (𝑛𝑛) = 𝑗𝑗𝑗𝑗𝑗𝑗 𝑗 𝑗 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 ) 𝑗 𝑗 𝑛 𝑛 ′ (𝑘 𝑘 𝑘 𝑘 ) , (9)

𝑍𝑍 𝑅𝑅 (𝑛𝑛) = −𝑗𝑗𝑗𝑗𝑗𝑗 ℎ 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 )

ℎ 𝑛 𝑛 ′ (𝑘 𝑘 𝑘 𝑘 ) , (10)

where 𝜌𝜌 and 𝑐𝑐 are the density and speed of sound in the surrounding fluid, 𝑗𝑗 𝑛𝑛 (𝑘𝑘𝑘𝑘) and ℎ 𝑛𝑛 (𝑘𝑘𝑘𝑘) are 𝑛𝑛 -th order spherical Bessel function of the first kind and spherical Hankel function of the second kind and the prime superscript denotes derivation in respect to the normalized frequency, 𝑘𝑘𝑘𝑘 , where k is the acoustic wavenumber in the surrounding fluid and a is the radius of the sphere. The modal impedance of the spherical shell, is given by [4]

2 )(𝛺 𝛺 2 −𝛺 𝛺 𝑛 𝑛 2

2 ) 𝛺 𝛺 2 −(1 + 𝛽 𝛽 2 )(𝑣 𝑣 + 𝜆 𝜆 𝑛 𝑛 −1) , (11)

(𝛺 𝛺 2 −𝛺 𝛺 𝑛 𝑛 1

𝑍𝑍 𝐵𝐵 (𝑛𝑛) = 𝑗𝑗 𝜌 𝜌 𝑠 𝑠 𝑐 𝑐 𝑝

ℎ 𝑎

𝛺

Figure 1: The normalised scattered power, in water, for spherical shells made of steel and of al- loy, as described in the text, compared with that of a rigid sphere. where 𝜌𝜌 𝑠𝑠 is the density of the shell, 𝑐𝑐 𝑝𝑝 2 = 𝐸𝐸 𝑠𝑠 /[𝜌𝜌 𝑠𝑠 (1 −𝑣𝑣 2 )] with 𝐸𝐸 𝑠𝑠 and 𝑣𝑣 being the Young`s modu- lus and Poisson`s ratio of the shell, ℎ is the shell thickness, 𝑎𝑎 is the shell radius, 𝛽𝛽 2 is ℎ 2 /(12𝑎𝑎 2 ) , 𝜆𝜆 𝑛𝑛 is 𝑛𝑛(𝑛𝑛+ 1) , and 𝛺𝛺 is 𝜔𝜔𝜔𝜔/𝑐𝑐 𝑝𝑝 with 𝜔𝜔= 𝑘𝑘𝑘𝑘 , which can also be written as 𝛺𝛺= ൫𝑐𝑐/𝑐𝑐 𝑝𝑝 ൯𝑘𝑘𝑘𝑘 . The val- ues of 𝛺𝛺 𝑛𝑛1 ⬚ and 𝛺𝛺 𝑛𝑛2 ⬚ are the in-vacuo natural frequencies of vibration of the shell and are given by the solutions to the equation

𝛺𝛺 4 −[1 + 3𝜈𝜈+ 𝜆𝜆 𝑛𝑛 −𝛽𝛽 2 (1 −𝑣𝑣−𝜆𝜆 𝑛𝑛 2 −𝑣𝑣𝜆𝜆 𝑛𝑛 )]𝛺𝛺 2 + +(𝜆𝜆 𝑛𝑛 −2)(1 −𝑣𝑣 2 ) + 𝛽𝛽 2 [𝜆𝜆 𝑛𝑛 3 −4𝜆𝜆 𝑛𝑛 2 + 𝜆𝜆 𝑛𝑛 (5 −𝑣𝑣 2 ) −2(1 −𝑣𝑣 2 )] = 0 , (12)

It is convenient to write the impedance of the shell in normalised form as

𝑍𝑍 𝐵𝐵 (𝑛𝑛) = 𝜌𝜌𝜌𝜌𝜁𝜁 𝑛𝑛 (𝑘𝑘𝑘𝑘), (13) where 𝜁𝜁 𝐵𝐵 (𝑛𝑛) is the in-vacuo impedance of the shell normalized by the characteristic acoustic im- pedance of the fluid, 𝜌𝜌𝜌𝜌 .

The diagonal elements of the modal form of equation (4) then leads directly to an expression for the ratio between the 𝑛𝑛 -th spherical harmonic component of the scattered pressure and that of the incident pressure [15,5]

1 10° 10?

𝑗 𝑗 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 ) + 𝑗 𝑗 𝜁 𝜁 𝐵 𝐵 (𝑛 𝑛 )𝑗 𝑗 𝑛 𝑛 ′ (𝑘 𝑘 𝑘 𝑘 ) ℎ 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 ) + 𝑗 𝑗 𝜁 𝜁 𝐵 𝐵 (𝑛 𝑛 )ℎ 𝑛 𝑛 ′ (𝑘 𝑘 𝑘 𝑘 ) . (14)

𝒑 𝒑 𝒔 𝒔 (𝑛 𝑛 ) 𝒑 𝒑 𝒊 𝒊 (𝑛 𝑛 ) = 𝑌 𝑌 𝐼 𝐼 (𝑛 𝑛 ) −𝑌 𝑌 𝐵 𝐵 (𝑛 𝑛 )

𝑌 𝑌 𝑅 𝑅 (𝑛 𝑛 ) + 𝑌 𝑌 𝐵 𝐵 (𝑛 𝑛 ) = − ℎ 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 )

𝑗 𝑗 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 )

Using this formulation, the normalised sound power of a spherical shell in response to an inci- dent plane-wave, which is proportional to the spatial integral of the mean square scattered pressure in the far-field, can be calculated as [5]

= 4 (𝑘 𝑘 𝑘 𝑘 ) 2 ෍(2𝑛𝑛+ 1) ቤ 𝑗 𝑗 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 ) + 𝑗 𝑗 𝜁 𝜁 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 )𝑗 𝑗 𝑛 𝑛 ′ (𝑘 𝑘 𝑘 𝑘 ) ℎ 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 ) + 𝑗 𝑗 𝜁 𝜁 𝑛 𝑛 (𝑘 𝑘 𝑘 𝑘 )ℎ 𝑛 𝑛 ′ (𝑘 𝑘 𝑘 𝑘 ) ቤ

Π 𝒔𝒔 = 𝑊 𝑊 𝑠 𝑊 𝑊 𝑖

2 𝑁𝑁

, ⬚ ⬚ (15)

𝑛𝑛=0

Figure 1 shows the normalised scattered power in water for two spherical shells of different ma- terials and thicknesses, as a function of the normalised frequency 𝑘𝑘𝑘𝑘 , together with that for a rigid sphere. At low frequencies the normalised power scattered by the steel shell, 𝜌𝜌 𝑠𝑠 = 7,700 kg/m 3 , 𝐸𝐸 𝑠𝑠 = 190 GPa , 𝜈𝜈= 0.28 , ℎ/𝑎𝑎= 2.3% ( , is proportional to 𝑘𝑘𝑘𝑘) 4 , which is the same proportionality as the rigid shell. The properties of the “alloy” shell, 𝜌𝜌 𝑠𝑠 = 10,000 kg/m 3 , 𝐸𝐸 𝑠𝑠 = 72.9GPa , 𝜈𝜈= 0.28 , ℎ/𝑎𝑎= 3.33% , have been chosen so that its low-frequency stiffness and mass match that of a sphere of water, so that the low frequency scattering due to the 𝑛𝑛= 0 and 𝑛𝑛= 1 spherical harmon- ( ics is suppressed [10,11]. The normalised scattered power is then proportional to 𝑘𝑘𝑘𝑘) 8 at low fre- quencies and hence much smaller than a rigid sphere. Although the alloy shell is modelled here as a thin, uniform, empty shell with appropriate mechanical properties, it might be realised in practice by a metal shell of a suitable thickness, surrounded by a soft coating with a similar density to the fluid, but with a thickness chosen to give the required overall stiffness [12]. In both cases shown in Figure 1 the scattered power at higher frequencies, from around ka=1 to 3, is dominated by peaks due to the structural resonances of the fluid-loaded shell, particularly corresponding to the n=2, ovalling, n=3 and n=4 vibration modes.

2. ACTIVE FEEDFORWARD CONTROL

A frequency domain feedforward control formulation can be used to calculate the optimal perfor- mance of an array of secondary forces in minimising the scattered power, assuming knowledge of the incident and scattered fields. This allows evaluation of the best possible performance with a given number of secondary sources, without having to be concerned with the sensing of the refer- ence or of the error signals, or with the implementation of a practical controller, and so can be used as the first step in a hierarchical design approach for active control [9].

The shell is assumed to be controlled with 𝐿 𝐿 internal forces distributed over a spherical cap of angle θ 0 , each of which has a magnitude 𝑓𝑓 𝑙 𝑙 acting at (𝜃 𝜃 𝑙 𝑙 , 𝜑 𝜑 𝑙 𝑙 ) that generates a modal pressure of

𝑝𝑝 𝑙𝑙 (𝑛𝑛, 𝑚𝑚) = − ቈ 𝑍 𝑍 𝐵 𝐵 (𝑛 𝑛 ) 𝑍 𝑍 𝐵 𝐵 (𝑛 𝑛 ) + 𝑍 𝑍 𝑅 𝑅 (𝑛 𝑛 ) ቉ 𝑓 𝑓 𝑙

𝑎 𝑎 2 ቈ P 𝑛 𝑛 −1 ( cos θ 0 ) − P 𝑛 𝑛 +1 ( cos θ 0 )

(2𝑛 𝑛 + 1)(1 −cos θ 0 ) ቉ 𝑌𝑌 ഥ 𝑛𝑛 𝑚𝑚 (𝜃𝜃 𝑙𝑙 , 𝜑𝜑 𝑙𝑙 ) , (16)

where the overbar denotes complex conjugation. The scattered modal pressure after control with 𝐿𝐿 secondary point-forces is thus

𝑝𝑝 𝑠𝑠𝑠𝑠 (𝑛𝑛, 𝑚𝑚) = 𝑝𝑝 𝑠𝑠 (𝑛𝑛, 𝑚𝑚) −ቈ 𝑍 𝑍 𝐵 𝐵 (𝑛 𝑛 ) 𝑍 𝑍 𝐵 𝐵 (𝑛 𝑛 ) + 𝑍 𝑍 𝑅 𝑅 (𝑛 𝑛 ) ቉෍ 𝑓 𝑓 𝑙

𝑎 𝑎 2 ቈ P 𝑛 𝑛 −1 ( cos θ 0 ) − P 𝑛 𝑛 +1 ( cos θ 0 ) (2𝑛 𝑛 + 1)(1 −cos θ 0 ) ቉𝑌𝑌 ത 𝑛𝑛 𝑚𝑚 (𝜃𝜃 𝑙𝑙 , 𝜑𝜑 𝑙𝑙 )

𝐿𝐿

, (17)

𝑙𝑙=1

which can be written in vector form as

𝐩𝐩 𝑠𝑠𝑠𝑠 = 𝐩𝐩 𝑠𝑠 −𝐁𝐁∙𝐩𝐩෥ 𝑐𝑐 , (18)

𝑎 𝑎 2 [ 𝑓𝑓 1 , 𝑓𝑓 2 , 𝑓𝑓 3 … 𝑓𝑓 𝐿𝐿 ] 𝑇𝑇 is the vector of 𝐿𝐿 control forces, which has a tilde to denote these are discrete rather than modal pressures, and 𝐁𝐁 is equal to 𝐙 𝐙 𝐵 𝐵 [𝒁 𝒁 𝑩 𝑩 + 𝒁 𝒁 𝑹 𝑹 ] − 1 times the N by L matrix 𝐒𝐒 , where the 𝑙𝑙 -th column of 𝐒𝐒 ቂ has elements as P 𝑛 𝑛 −1 (cos θ 0 ) − P 𝑛 𝑛 +1 (cos θ 0 )

where 𝐩𝐩෥ 𝑐𝑐 = 1

(2𝑛 𝑛 +1)(1−cos θ 0 ) ቃ𝑌𝑌 ത 𝑛𝑛 𝑚𝑚 (𝜃𝜃 𝑙𝑙 , 𝜑𝜑 𝑙𝑙 ) .

The scattered power after control is given by

𝑊𝑊 𝑠𝑠𝑠𝑠 = 𝑎 𝑎 2

2 Re {𝐩𝐩 𝑠𝑠𝑠𝑠 H ∙𝐯𝐯 𝑠𝑠𝑠𝑠 } = 𝑎 𝑎 2

2 𝐩𝐩 𝑠𝑠𝑠𝑠 H ∙ Re {𝐘𝐘 𝑅𝑅 } ∙𝐩𝐩 𝑠𝑠𝑠𝑠 , (19)

since 𝐯𝐯 𝑠𝑠𝑠𝑠 is given by 𝐘𝐘 𝑅𝑅 ∙𝐩𝐩 𝑠𝑠𝑠𝑠 , which is a quadratic function of 𝐩𝐩෥ 𝑐𝑐 . The scattered pressure can thus be minimised by the vector of control forces given by [9]

(opt) = [𝐁𝐁 H ∙ Re {𝐘𝐘 𝑅𝑅 } ∙𝐁𝐁 ] −1 ∙𝐁𝐁 H ∙ Re {𝐘𝐘 𝑅𝑅 } ∙𝐑𝐑∙𝐩𝐩 𝑖𝑖 , (20)

𝐩𝐩෥ 𝑐𝑐

where equation (4) has been used to relate 𝐩𝐩 𝑠𝑠 to 𝐩𝐩 𝑖𝑖 . Figure 2 shows, on an enlarged scale, the re- sults of minimising the scattered sound power of the alloy shell using either one distributed second- ary force, with θ 0 = 𝜋𝜋/10, facing the incident wave, or two distributed secondary forces, facing to- wards and away from the incident wave. Significant control of the scattered sound power can be achieved at the peaks due to the structural resonances.

Figure 2: Optimal results of using active feedforward control with one or two distributed force actuators to minimise the scattering from the alloy shell in water.

wo! 10° 10" 107 10 ke 0

3. ACTIVE FEEDBACK CONTROL

The effect of active feedback control on scattering can be calculated by initially considering the consequences of feedback control on the in-vacuo response of the shell. Figure 3(a) shows the phys- ical arrangement in which the signals from 𝐾𝐾 discrete velocity sensors is fed to 𝐿𝐿 internal point force actuators via a feedback controller matrix, 𝐇𝐇 . The effect of such a controller on the modal ve- locity in response to a general modal pressure excitation is shown in Figure 3(b).

In the absence of control and using equation (1), the vector of modal velocities in response to a general vector of modal pressures is

𝐯𝐯= −𝐘𝐘 𝐵𝐵 ∙𝐩𝐩 , (21) In the presence of active control with an array of internal point-force sources, as in Section 2, the modal pressures acting on the shell are modified, although in this in-vacuo case there is no fluid loading so that the matrix B in equation (18) reduces to S , so that

𝐯𝐯= −𝐘𝐘 𝐵𝐵 ∙(𝐩𝐩−𝐒𝐒∙𝐩𝐩෥ 𝑐𝑐 ), (22) For the feedback arrangement shown in Figure 3(a), the secondary forces are due to feedback from the velocities at the 𝐾𝐾 sensor positions, 𝐯𝐯෤ 𝑐𝑐 , which also has a tilde to denote discrete rather than a modal velocities, via the feedback controller 𝐇𝐇 , so that

𝐩𝐩෥ 𝑐𝑐 = 𝐇𝐇∙𝐯𝐯෤ 𝑐𝑐 , (23) where 𝐯𝐯෤ 𝑐𝑐 = 𝐓𝐓∙𝐯𝐯 , (24) 𝐯𝐯 being the vector of modal velocities, and 𝐓𝐓 is a 𝐾𝐾 by 𝑁𝑁 transformation matrix, which, from equa- tion (6), has elements of the form 𝑌𝑌 𝑛𝑛 𝑚𝑚 (𝜃𝜃 𝑘𝑘 , 𝜑𝜑 𝑘𝑘 ) . Thus,

𝐯𝐯= −𝐘𝐘 𝐵𝐵 ( 𝐩𝐩−𝐒𝐒∙𝐇𝐇∙𝐓𝐓∙𝐯𝐯) , (25) and so,

𝐯𝐯= −[𝐈𝐈+ 𝐘𝐘 𝐵𝐵 ∙𝐒𝐒∙𝐇𝐇∙𝐓𝐓] −1 ∙𝐘𝐘 𝐵𝐵 ∙𝐩𝐩 . (26)

(a) (b) Figure 3: Feedback control using secondary point-force actuators driven by the measured veloci- ties at discrete sensors on the surface of the body (a), and the block diagram of the equivalent modal feedback system (b).

xed =

The modal pressures can this be expressed as

(𝑐𝑐𝑐𝑐) ∙𝐯𝐯, (27) where 𝐙𝐙 𝐵𝐵 (𝑐𝑐𝑐𝑐) is the overall modal impedance for the shell with closed-loop feedback control, which can be written as

𝐩𝐩= −𝐙𝐙 𝐵𝐵 ∙[𝐈𝐈+ 𝐘𝐘 𝐵𝐵 ∙𝐒𝐒∙𝐇𝐇∙𝐓𝐓] ∙𝐯𝐯= −𝐙𝐙 𝐵𝐵

(𝑐𝑐𝑐𝑐) = 𝐙𝐙 𝐵𝐵 + 𝐒𝐒∙𝐇𝐇∙𝐓𝐓 . (28) Although 𝐙𝐙 𝐵𝐵 is diagonal for a spherical harmonic expansion, the matrix 𝐒𝐒∙𝐇𝐇∙𝐓𝐓 is generally not

𝐙𝐙 𝐵𝐵

diagonal. Nevertheless, the matrix ቂ𝐙𝐙 𝐵𝐵 (𝑐𝑐𝑐𝑐) ቃ −1 can now be used instead of 𝐘𝐘 𝐵𝐵 in equation (4) to give the vector of scattered modal pressures after feedback control in terms of the vector of incident modal pressures, and, hence, the scattered power after control can be calculated using equation (19).

In the absence of an incident field, the matrix of “plant” responses, 𝐆𝐆 𝑐𝑐 , between the point-force actuators and the discrete velocity sensors is

𝐯𝐯෤ 𝑐𝑐 = 𝐓𝐓∙[𝐙𝐙 𝐵𝐵 + 𝐙𝐙 𝑅𝑅 ] −1 ∙𝐒𝐒∙𝐩𝐩෥ 𝑐𝑐 = 𝐆𝐆 𝑐𝑐 ∙𝐩𝐩෥ 𝑐𝑐 , (29) since the force actuators have to overcome both the impedance of the shell and the radiation imped- ance. If there are the same number of actuators as sensors and they are collocated, then 𝐒𝐒 is equal to 𝐓𝐓 H so

𝐆𝐆 𝑐𝑐 = 𝐓𝐓∙[𝐙𝐙 𝐵𝐵 + 𝐙𝐙 𝑅𝑅 ] −1 ∙𝐓𝐓 𝐇𝐇 , (30) which is entirely passive. The stability of the closed-loop system is thus guaranteed provided the feedback gain matrix, 𝐇𝐇 , is also passive [13,14, 16] and is unconditionally stable for direct velocity feedback.

Several designs of feedback controller are possible. The simplest is decentralised local velocity feedback, for which 𝐇𝐇 is equal to 𝛾𝛾𝑰𝑰 , where 𝛾𝛾 is the gain of each local feedback loop. Figure 4 shows the effect of decentralised velocity feedback on the scattered power from the alloy shell in Figure 1 with one or two distributed force actuators, as above, and collocated distributed velocity sensors. The feedback gain, 𝛾𝛾 , is chosen to give a reasonable compromise between supressing the original lightly damped structural resonances and not exciting higher-order resonances by pinning the structure [13, 14]. This feedback controller is able to effectively supress the very lightly damped resonances between about 𝑘𝑘𝑘𝑘 equal to 1 and 3 .

In the limiting case, where it is assumed that there are as many actuators and sensors as there are modes, 𝐿𝐿= 𝐾𝐾= 𝑁𝑁 , and assuming that the transducers are collocated and positioned so as to actuate and sense all of the modes such that 𝐓𝐓 H ∙𝐓𝐓 is a good approximation to the identity matrix, then per- fect control of all the modes would be possible. In this case, the closed-loop impedance of the shell could, in principal, be set equal to the input impedance of the volume of the scattering body filled with fluid, 𝐙𝐙 𝐼𝐼 , so that the scattering would be completely suppressed. Setting equation (28) equal to 𝐙𝐙 𝐼𝐼 in this case leads to the feedback controller would having the form [6,7]

𝐇𝐇= 𝐓𝐓∙(𝐙𝐙 𝐼𝐼 −𝐙𝐙 𝐵𝐵 ) ∙𝐓𝐓 H

(31)

The stability of this feedback controller is by no means guaranteed, however, and was found not to be stable in the cases considered here.

Figure 4: Normalised scattered sound power from the alloy shell after active feedback control with one or two distributed force actuators and collocated distributed velocity sensors using decen- tralised velocity feedback.

4. CONCLUSIONS

A modal formulation for acoustic scattering from a flexible body is reviewed. This general theory is shown to take a particularly simple form in the case of scattering from a thin, uniform, empty spher- ical shell surrounded by an infinite fluid, since the internal and external acoustic modes, and the structural modes, are then all spherical harmonics. Examples of scattering are calculated for spheri- cal shells of different materials, and it is shown that the scattering due to both the 𝑛𝑛= 0 and the 𝑛𝑛= 1 spherical harmonics can be suppressed in a “alloy” shell with a suitable choice of the shell’s material properties.

This formulation is then used to calculate the effect of active feedforward or feedback control on the scattering, using distributed forces acting on the flexible spherical shell as secondary actuators. The effect of both forms of active control in reducing the scattering from the composite shell is mainly to due to the suppression of a few lightly damped structural resonances.

5. ACKNOWLEDGEMENTS

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

6. REFERENCES

1. Duda, R.O. and Martens, W. L. Range dependence of the response of a spherical head model, The Journal of the Acoustical Society of America, 104 (5), 3048–4058, (1998).

2. Nelson, P.A. and Elliott, S.J. Active Control of Sound, Academic Press, London, UK (1991).

3. Rayleigh, W.S. Investigation of the disturbance produced by a spherical obstacle on the waves of sound, Proceedings of the London Mathematical Society, s1-4 (1), 253-283 (1871).

4. Junger, M. C. and Feit, D. Sound, structures and their interactions, 2nd edition, MIT press, Mary-land, Massachusetts (1986).

5. Elliott, S. J., Orita, M. and Cheer, J. Active Control of the Sound Power Scattered by a Locally- Reacting Sphere, The Journal of the Acoustical Society of America, 147, 1851-1862, (2020).

6. Bobrovnitskii, Y.I. A new solution to the problem of an acoustical transparent body, Acoustical Physics, 50 (6), 647–650, (2004).

7. Bobrovnitskii, Y.I. A new impedance-based approach to analysis and control of sound scatter- ing, Journal of Sound and Vibration, 297 (3-5), 743–760, (2006).

8. Williams, E. G. Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography, Aca- demic Press, London (1999).

9. Elliott, S.J. Signal processing for active control, Academic press, London (2001).

10. Alu, A. and Engheta, N. Achieving transparency with plasmonic and metamaterials coatings, Physical Review E, 72 (1), 16623 (2005).

11. Guild, M.D., Alu A. and Haberman M.R. Cancellation of acoustic scattering from an elastic sphere, The Journal of the Acoustical Society of America, 129 (3), 1355-1365, (2011).

12. Rohde C.A. et al. Experimental demonstration of underwater acoustic scattering cancellation. Sci-entific Reports, 5 (3), 13175 (2015).

13. Preumont, A. Vibration Control of Active Structures: An introduction, 3rd Edition, Kluwer Aca-demic Publishers, Dordrecht (1999).

14. Elliott, S.J.et al. Active control with multiple local feedback loops, The Journal of the Acousti- cal Society of America, 111 (2), 908-915 (2002).

15. Junger, M.C. Sound scattering by thin elastic shells, The Journal of the Acoustical Society of America, 24 (4), 366-373 (1952).

16. Meirovitch, L. Dynamics and control of structures , John Wiley, New York, (1990).