Active acoustic cloaking of cylindrical shells in low Mach number flow

The vibro-acoustic response of a two-dimensional cylindrical shell in low Mach number flow is herein derived. The analytical model takes into account the structural elasticity and coupling of the shell vibration with its interior and exterior acoustic fields in the presence of a moving fluid. The cylindrical shell is modelled using Donnell-Mushtari theory. Taylor transformations are employed to transfer the convected wave equation into the ordinary wave equation which was then solved using scattering theory. Three excitation cases corresponding to a plane wave, an external monopole source and a radial point force applied directly to the shell are considered. Shell circumferential resonances and interior acoustic resonances are identified. Two active control strategies are then applied to acoustically cloak the cylindrical shell at its acoustic and structural resonances. The first control approach employs acoustic control sources in the exterior fluid domain. In the second approach, control forces are applied to directly excite the elastic shell, whereby the structural response is actively modified to manipulate the scattered and radiated acoustic fields arising from plane wave excitation of the shell. Results show that the second approach is superior in terms of both reduced control effort and cloaking of the global exterior domain. For both control approaches, the performance of the active cloak is shown to deteriorate if the convected flow field is not accounted for in the control process. © 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Acoustic cloaking has received significant interest in recent years with potential applications for stealth, masking, and reduction of anthropogenic noise. Passive cloaking approaches are well established, namely using a coordinate transformation approach wherein acoustic metamaterials are designed to render a region acoustically invisible. Metamaterial cloaks have been designed with functionally graded density, commonly referred to as inertial cloaks [1], or by jointly varying the metamaterial stiffness and density to guide the acoustic field around a cloaked region [2]. Multilayered cloaks using isotropic and metamaterial layers have also been proposed [3,4]. Popa et al. [5,6] applied the coordinate transformation method for the design of an acoustic ground cloak. Alternative passive methods using impedance matching skins with subwavelength inclusions, or through scattering cancellation using a specially designed isotropic elastic layer, have been explored [7,8]. Various active cloaking techniques have been proposed that can be broadly categorised as either exterior cloaking or interior cloaking. Exterior cloaking, in which control sources are located outside of the cloaked region, was achieved using multipole sources to cloak a body from an incident field [9e12]. Interior cloaking, in which control sources are located within the cloaked region, has been demonstrated using piezoelectric coatings [13e16], or a discrete number of secondary sources to cancel the scattered field [17e20].
The aforementioned studies on the design of passive cloaks ignore the Doppler effect associated with convected flow of the surrounding acoustic medium. For stealth purposes in aeronautical and maritime applications, cloaking of a body in a moving fluid is required. Cloaking in mean flow was first demonstrated with a passive acoustic ground cloak in a moving fluid medium, which was theoretically constructed using a space-time transformation method [21]. Ryoo and Jeon [22] expanded on the work by Huang et al. [23], taking into account the effects of fluid compressibility and flow non-uniformity. Iemma and Palma [24] presented a convective correction for metamaterial cloaks originally designed for a stationary acoustic medium, which was found to effectively cloak a 2D cylinder from monopole excitation in both stationary and low Mach number flow. An increasing Mach number yielded diminished cloaking performance attributed to the low Mach number formulation of the governing equations in the cloak design. Huang et al. [25] proposed an optimisation routine to cancel turbulent noise from high Reynolds number flow over a cylinder by varying the metamaterial properties of the cloak using the method of steepest descent. More recently, they designed an acoustic cloak for an aerofoil in turbulent flow that was insensitive to frequency variations [26]. As an alternative to passive cloaking in mean flow, acoustic cloaking of a 2D rigid cylinder in low Mach number flow using an active control approach was recently proposed [27]. Failure to account for potential flow was shown to result in adverse constructive interference that enhanced the original acoustic field.
The current paper extends the recent study by Eggler et al. [27] on acoustic cloaking of a rigid cylinder, to consider an elastic cylindrical shell in low Mach number flow. A number of studies have investigated the vibrational responses of an elastic structure in convected flow, although to the best of the authors' knowledge, there is very little literature on the acoustic responses of an elastic structure in a moving fluid. Dynamic responses have been analytically studied for a cylindrical shell in cross flow [28] and axial flow [29]. Amabili and Garziera [30] reported a decrease in the eigenfrequencies of a cylindrical shell in axial flow with increasing flow speed. Taylor [31] pioneered a transformation method to derive the acoustic field arising from a vibrating surface in low Mach number potential flow. The method was then employed to model acoustic generation by pulsating and vibrating elastic spheres in homentropic potential flow.
In this work, active acoustic cloaking of an elastic cylindrical shell in convected flow is presented. The active control methodology depends on prediction of the vibro-acoustic responses of the elastic shell arising from excitation due to the primary incident field and the control sources. To this end, we utilise Taylor's transformation method to derive the structural and acoustic responses of an elastic cylindrical shell excited by acoustic sources or localised mechanical forces in a moving fluid. The cylindrical shell is modelled using standard shell theory, in which the shell displacements are coupled with the interior and exterior acoustic fields. Taylor transformations are then applied to the interior and exterior acoustic fields to reduce the convected wave equation to the standard Helmholtz equation, which is subsequently solved using scattering theory for an elastic shell. The vibro-acoustic responses of the cylindrical shell with and without the presence of flow are compared. Two active control approaches are then employed to acoustically cloak the shell at its structural and acoustic resonances. The first control approach called active noise cloaking employs acoustic control sources in the exterior fluid domain uniformly distributed around the cylindrical shell. In the second approach, termed active structural acoustic cloaking, control forces are applied to directly excite the elastic shell. The structural response is actively modified to manipulate the scattered and radiated acoustic fields arising from plane wave excitation of the shell. The effectiveness of the two active control approaches are compared. Further, the performance of the acoustic cloaks with and without accounting for convective flow is evaluated.

Elastic cylindrical shell
A 2D elastic cylindrical shell is located in a flow of free stream velocity U ∞ as shown in Fig. 1. Donnell-Mushtari shell theory with a FlüggeeByrneeLur'ye modifying operator to account for the midplane displacements and the curvature of the shell is employed. The equations of motion in the radial and tangential directions are respectively given by [32] vvðq; tÞ vq where w and v are the radial and tangential shell displacements, respectively, a is the mid-thickness shell radius, b ¼ is the non-dimensional shell thickness parameter, h is the shell thickness, and E, n, r are respectively the Young's modulus, Poisson's ratio and density of the shell.
The equation of motion for the shell radial displacement is modified to consider a localised point force as well as the acoustic pressure loading on the interior and exterior surfaces of the shell. Assuming time-harmonic separable solutions for w and v such that wðq; tÞ ¼ wðqÞe Àiut and vðq; tÞ ¼ vðqÞe Àiut , where i ¼ ffiffiffiffiffiffi ffi À1 p and u is the angular frequency, the equations of motion in the radial and tangential directions become [33] D d 4 wðqÞ þ Db 2 d dq vðqÞ ¼ p int ða; qÞ À p ext ða; qÞ þ f i dðq À q i Þ; (3) G ¼ Eh=ða 2 ð1 Àn 2 ÞÞ and D ¼ G=b 2 are shell stiffness parameters. p int ðr; qÞ is the interior acoustic pressure. The exterior acoustic pressure p ext ðr; qÞ comprises both the incident acoustic pressure due to an acoustic source, p inc ðr; qÞ, and the combined scattered and radiated pressure arising from acoustic excitation of the shell, p sc=rad ðr; qÞ. That is, for acoustic excitation, p ext ðr; qÞ ¼ p inc ðr; qÞ þ p sc=rad ðr; qÞ. For structural excitation of the shell, the exterior acoustic pressure comprises only the structure-borne acoustic pressure, p rad ðr;qÞ, such that p ext ðr;qÞ ¼ p rad ðr;qÞ. f i and q i denote the amplitude and angular location of the i th localised point force, respectively, and dðq Àq i Þ is the Dirac delta function. The shell radial and tangential displacements are expressed in terms of their Fourier coefficients, w n and v n , by summing over the modal order n as follows [32] wðqÞ ¼ X ∞ n¼À∞ w n e inq ; vðqÞ Three different excitation cases are herein considered corresponding to an incident plane wave travelling in the direction of the flow, a monopole source located in the exterior acoustic domain and a radial point force applied directly to the shell, as shown in Fig. 1. Assuming time harmonic excitation, the incident acoustic pressure fields in the absence of flow for plane wave and monopole source excitation are respectively given by [34] where R is the distance from the monopole source to the field point x ¼ ðr;qÞ, J n is the Bessel function of the first kind of order n, H 0 is the zeroth order Hankel function of the first kind, ðÞ 0 is the derivative with respect to the argument and k a is the acoustic wavenumber. The incident field coefficient for a plane wave is a n ¼ i n J 0 n ðk a aÞ [34]. Equations (7) and (8) must be modified to account for convection effects as outlined in Sections 2.3 and 2.4, respectively.

Transformation of the exterior and interior acoustic potential
The exterior and interior acoustic fields are evaluated in terms of the acoustic potential 4 a ðx; tÞ using the linearised potential flow formulation of the convected wave equation which is given by where 4 v ðxÞ is the velocity potential governing the mean flow of the convected field. This equation is valid for isentropic, inviscid and irrotational flow and neglects terms of OðM 2 Þ where M ¼ U ∞ =c f is the Mach number and c f is the speed of sound in the fluid. As such, our analysis is limited to low Mach number potential flow (M 2 ≪1). Since the physical system has two distinct acoustic fields, Eq. (9) must be solved separately for both the exterior acoustic potential 4 a;ext ðx; tÞ and the interior acoustic potential 4 a;int ðx;tÞ. The reference spatial and temporal coordinates ðx; tÞ are mapped to the transformed coordinates ðX; TÞ according to the Taylor transformations given by [37] X Note, x ¼ ðr; qÞ and X ¼ ðb r; b qÞ respectively denote the coordinates in the original and transformed cylindrical coordinate systems for a given field point. Employing these transformations reduces Eq. (9) to the Helmholtz equation in the transformed coordinates, which can then be solved using standard scattering theory. However, the acoustic-structure interaction between the elastic structure and the convected acoustic field complicates the solution for the scattered and structurally radiated sound field. Taylor [31] proposed that the acoustic effects of a vibrating surface in potential flow defined by Gðx; tÞ ¼ GðxÞþ sðx; tÞ ¼ 0 could be captured using the following boundary condition where GðxÞ ¼ 0 defines the mean position of the surface and sðx; tÞ is the time-varying component of the surface position. To simplify this boundary condition, we define G and s by restricting our analysis to an elastic cylindrical shell in plane strain, which results in G ¼ r À a and sðq; tÞ ¼ À wðqÞe Àiut . The velocity potential for a 2D cylinder in cylindrical coordinates is defined by [39] 4 v ðxÞ ¼ U ∞ r þ a 2 r cos q: (12) For the case of a finite cylindrical shell in potential flow, a numerical solution to the Laplace equation for the velocity potential would be required. Substituting the expressions for G, s and 4 v into the boundary condition given by Eq. (11) and simplifying the resulting expression yields the boundary condition for the exterior acoustic potential as v vr 4 a;ext ðcos q þ in sin qÞ w n e inq e Àiut : Recognising that the interior acoustic domain is stationary (i.e. U ∞ ¼ 0Þ, the boundary condition for the interior acoustic potential is given by v vr The solution to the equations of motion given by Eqs. (3) and (4) is herein derived for the case of plane wave excitation. 4 a;ext ðx; tÞ and 4 a;int ðx; tÞ are first expressed as their transformed equivalents, denoted by b 4 a;ext ðX; TÞ and b 4 a;int ðX; TÞ, respectively. As Eq. (9) reduces to the Helmholtz equation under the Taylor transformations, b 4 a;ext and b 4 a;int are solutions to the exterior and interior Helmholtz equations respectively, and can be expressed in their general forms as follows The first term on the right hand side of Eq. (15) corresponds to the transformed incident acoustic potential b 4 a;inc ðX; TÞ and the second term corresponds to the transformed scattered and radiated acoustic potential b 4 a;sc=rad ðX; TÞ, where H n is the Hankel function of the first kind of order n. b a n denotes the coefficients describing the transformed incident acoustic potential, b b n represents the coefficients for the transformed scattered/radiated acoustic potential, and b c n are the coefficients for the transformed interior acoustic potential. The boundary conditions in the transformed domain are obtained by applying Taylor's transformations to Eqs. (13) and (14) and substituting the expressions for the transformed acoustic potentials given by Eqs. (15) and (16), respectively. A system of algebraic equations for the unknown coefficients can be obtained by considering orthogonality of the Fourier series basis e in b q . First considering the exterior acoustic potential, applying Taylor's transformations to Eq. (13), substituting Eq. (15) for the transformed exterior acoustic potential, multiplying by e Àim b q and then integrating between 0 and 2p, results in an infinite set of integrals without a closed form solution. The infinite set of integrals can be represented by the matrix where b a is a column vector containing the elements b a n , b b is a column vector containing the elements b b n , and w is a column vector containing the shell radial Fourier coefficients w n . F is a matrix containing the elements F mn given by the following expression The elements F mn are evaluated numerically via the trapezoidal method using 2n subintervals. Now considering the interior acoustic potential, applying Taylor's transformations to Eq. (14) and substituting Eq. (16) yields the interior boundary condition in the transformed domain. Multiplying by e Àim b q and integrating between 0 and 2p results in b c n ¼ À ic f w n , thus directly relating the transformed interior acoustic potential to the shell radial displacement.

Solution for the exterior and interior acoustic pressure
The exterior and interior acoustic responses of the cylindrical shell in potential flow can now be solved using the shell equations of motion. The acoustic pressure is related to the acoustic potential using Bernoulli's equation as follows Taylor's transformations are applied to Eq. (18), neglecting terms of OðM 2 Þ, to relate the transformed exterior and interior acoustic pressures respectively denoted by b p ext ðX; TÞ and b p int ðX; TÞ, to the corresponding transformed acoustic potentials given by Eqs. (15) and (16). The resulting expressions are evaluated at In the final step, Taylor's transformations are applied to Eqs. (3) and (4) to describe the shell's equations of motion in the transformed domain. Substituting Eqs. (19) and (20) into the transformed equations of motion and solving simultaneously with b b ¼ Fw À b a and b c n ¼ Àic f w n yields the following solution for the vector w of radial displacement coefficients w ¼ ðB þ DFÞ À1 ððD À CÞ b a Þ: B is a diagonal matrix with elements given by and C and D are the matrices with elements given by which are evaluated numerically via the trapezoidal method using 2n subintervals, as per Eq. (17). The elements of b a are defined by b a n ¼ a n =iur f such that b 4 a;inc describes an unconvected incident field when  (26) in sin q b a n J n ðk a rÞ J t and z n capture the effect of the radial and tangential components of potential flow on the acoustic pressure at any field point in the exterior domain, respectively.

Monopole and force excitation
For monopole source excitation, a modification to the solution process must be implemented. The transformed exterior acoustic potential given by Eq. (15) where b R is the distance between the monopole source and the field point X ¼ ðb r; b qÞ in the transformed domain. Equation (19) is still used at the boundary, however, the incident field due to the monopole source must first be expressed in the cylinder's local coordinates. This is achieved with Graf's addition theorem [34], yielding Thus, for monopole source excitation b a n ¼ ð1 =iur f ÞðÀ1Þ n H n ðk a bÞJ 0 n ðk a aÞe Àina . The procedure to find the coefficients b b n , b c n , and w n remains the same as described previously. However, Eqs. (25)e(28) must be modified to account for phase changes associated with the effect of the fluid flow between the location of the monopole source and the field point [27,37]. For monopole excitation, the exterior and interior acoustic pressures in the untransformed domain are expressed as (32) wðqÞ ¼ X N j¼1 w j ðqÞ: When the cylindrical shell is directly excited by a radial point force, p ext ¼ p rad in Eq. (3) and the solution procedure follows that of plane wave excitation. In this case, the radial shell displacement is given by where the matrix elements for F, B, D are given by Eqs. (17), (22) and (24), respectively. f is a column vector containing the elements f n ¼ f i e Àinqi =2p. When N radial point forces are employed, the elements within the column vector for f become

Physical system results
An infinitely long elastic cylindrical shell of mean radius a ¼ 0. For the frequency range considered in Fig. 2, the five resonances correspond to the first three acoustic spinning resonances, the first acoustic axisymmetric resonance, and the first structural resonance corresponding to the zeroth shell circumferential mode. Acoustic resonances were identified and categorised according to the interior field patterns described in Ref. [40]. Structural resonances were identified by the shell circumferential displacement pattern. The broad peak in Fig. 2 around 420 Hz corresponding to the first structural resonance is due to high radiation damping for the radially axisymmetric mode of the cylindrical shell [33,41]. Table 1 lists the frequencies for the first three acoustic spinning, acoustic axisymmetric and circumferential resonances of the cylindrical shell in the absence of potential flow (M ¼ 0) and in convected flow (M ¼ 0:12). The first three structural resonances correspond to the breathing (radially axisymmetric), bending and ovalling shell circumferential modes. The frequencies of the acoustic resonances are unaffected by the convected flow field. The presence of flow can yield a reduction in the resonance frequencies of the structural modes, as observed previously by Amabili and Garziera [30]. Fig. 3 presents contour plots of the acoustic pressure fields in the absence of potential flow (left column) and in the presence of a convected flow field of M ¼ 0:12 (right column) for the first acoustic spinning resonance of the cylindrical shell due to excitation by an incident plane wave (Fig. 3(a)), a monopole source located at ðb; aÞ ¼ ð2a; 0Þ (Fig. 3(b)), and a radial        point force at q i ¼ p (Fig. 3(c)). Scattering both upstream and downstream of the shell is amplified by the presence of flow due to the Doppler effect, whereby expansion and contraction of the acoustic wavelength distorting the response can be observed. Fig. 4 shows similar results to that of Fig. 3, in which contour plots of the acoustic pressure for the third acoustic spinning resonance due to each excitation case, in the absence of potential flow (left column) and at M ¼ 0:12 (right column), are shown. Enhanced scattering of the exterior acoustic field both upstream and downstream of the cylinder can be observed with increasing frequency. Similar contour plots of the acoustic pressure due to each excitation case are presented in Figs. 5 and 6 for the first acoustic axisymmetric resonance and first structural resonance, in the absence of potential flow (left column) and at M ¼ 0:12 (right column). For a cylindrical shell of finite length, the presence of axial modes would result in a significantly higher number of structural resonances. Diffraction around the finite cylinder edges would yield greater scattering effects.

Cloaking methodology
We apply two active control strategies, namely active noise cloaking (ANCL) that utilises acoustic control sources and active structural acoustic cloaking (ASACL) that employs structural forces to actively modify the radiated sound field [18]. For ANCL, N monopole control sources and L microphone error sensors were equispaced in circumferential arrays around the cylinder at radial distances of 2a and 3a, respectively. For ASACL, N point forces were applied to directly excite the shell, also uniformly distributed around the shell.
The cloaking algorithm is set up to minimise the acoustic field given by e ¼ p sc=rad þ Zq, where p sc=rad corresponds to the combined scattered and radiated exterior acoustic pressure at the error sensor locations arising from excitation of the cylindrical shell by an incident plane wave, and Zq denotes the acoustic field generated by the control sources. Z is a matrix of acoustic transfer functions at the error sensor locations arising from each of the control sources, and q is a vector of control source amplitudes to be optimised. The optimisation process consists of minimising a quadratic cost function given by J ¼ e H e based on a feedforward adaptive least-mean-square algorithm [42]. The resultant optimal control source amplitudes are evaluated by solving vJ=vq ¼ 0 and are obtained as where H denotes the Hermitian transpose. The current active control strategy requires a priori knowledge of the incident field as well the ability to directly measure the scattered and radiated acoustic pressure arising from the incident plane wave. There currently exist limited strategies to identify the scattered and radiated acoustic field separately from an incident acoustic field, for example, see Ref. [19,20,43,44]. A procedure outlined previously was used to determine the number of control sources and error sensors such that a percentage error function given by [18] D ¼ jjp inc j À jp cl jj jp inc j Â 100% (42) gives a predefined percentage error of D < 1%. In Eq. (42), p inc is a vector of incident acoustic pressures at the error sensors and p cl denotes the cloaked acoustic pressure at the error sensors. Table 2 summarises the control configurations employed for ANCL and ASACL at the first and third acoustic and structural resonances of the cylindrical shell in a convected flow field of M ¼ 0:12. For each control configuration, the number of error sensors is greater than the number of control sources, thus ensuring that the least-mean-squares problem is overdetermined and yielding a unique solution provided by Eq. (41). The required number of control sources and error sensors to maintain D < 1% increases with increasing frequency due to the reduction in the acoustic wavelength, as discussed previously [18]. The first and third acoustic and structural resonances of the cylindrical shell under incident plane wave excitation in a convected flow field of M ¼ 0:12 are herein examined for the cloaking process. Active acoustic cloaking using either ANCL (left column) or ASACL (right column) is presented in Fig. 7 for the first acoustic spinning resonance ( Fig. 7(a)), the first acoustic Table 2 Cloaked resonance frequencies with number of control sources ðNÞ and error sensors ðLÞ for ANCL and ASACL in a convected flow field of M ¼ 0:12.      axisymmetric resonance (Fig. 7(b)) and the structural breathing resonance (Fig. 7(c)). Using ANCL, the resultant controlled field resembles that of the convected incident field beyond the control source perimeter. Using ASACL, the cloaked field immediately takes effect beyond the control force perimeter corresponding to the surface of the shell. Similar results to Fig. 7 are included in Fig. 8 for the third acoustic and structural resonances, whereby a cloaked acoustic field beyond the circumferential array of control sources is achieved using ANCL, whereas the entire exterior acoustic field beyond the surface of the shell is cloaked using ASACL. The acoustic performance of the cloaking process when the convected flow field is not accounted for is shown in Figs. 9e11 at both the first and third acoustic and structural resonances. Figs. 9(a)-11(a) present the uncontrolled acoustic pressure for a plane wave impinging on the elastic cylindrical shell (also known as the primary acoustic field). Figs. 9(b)-11(b) present the resultant field when cloaking is attempted using ANCL neglecting convective effects. Figs. 9(c)-11(c) present the resultant field when cloaking is attempted using ASACL neglecting convective effects. Comparison of Figs. 9(b)-11(b) with Figs. 9(c)-11(c) reveals that when using ANCL, any departure in the cloaking process yields significantly greater deviation of the resultant field from that of the desired field compared to using ASACL.
An error analysis of the active cloaking results is presented by examining the acoustic directivity, corresponding to the absolute acoustic pressure evaluated at a specific radial distance for all field points along the circumference, as well as the real acoustic pressure as a function of distance from the cylinder centre. Figs. 12(a)e14(a) present the acoustic directivity at r ¼ 3a for the incident field, the actively cloaked field when accounting for flow, and the controlled field when neglecting flow, using ANCL (left column) and ASACL (right column). Both control strategies exhibit identical directivities of the cloaked and incident fields when accounting for convection, but the resultant directivity significantly deviates from that of the incident field when neglecting convection. This effect is more pronounced using ANCL than using ASACL. This is attributed to the fact that since ASACL employs mechanical excitation, the Doppler effect is only applied to the radiated acoustic field. However, for ANCL which employs acoustic excitation, the Doppler effect is enhanced due to the additional contributions from the incident component of the acoustic control sources as well as the scattered and radiated response. In Figs. 12(b)-14(b), the same results are compared as a function of distance from the cylinder centre along q ¼ 0. Effective cloaking is observed from the control source perimeter of 2a for ANCL (left column), and from the shell surface for ASACL (right column). Significantly greater deviation of the resultant field from that of the desired field is observed using ANCL compared with ASACL, when the effect of flow is not taken into consideration.

Conclusions
Acoustic cloaking of an elastic structure in a moving fluid using active control methods has been presented. The vibroacoustic responses of an elastic cylindrical shell in potential flow under plane wave, monopole source and radial point force excitation were derived using Taylor's transformations and Donnell-Mushtari shell theory. The shell equations of motion and boundary conditions on the interior and exterior shell surfaces were solved in the transformed domain. Scattering theory was applied to resolve the exterior and interior acoustic field coefficients for the different acoustic and structural excitation cases. Results for the first three acoustic spinning resonances, acoustic axisymmetric resonances and structural resonances were presented in the absence and presence of convected flow. Two active control approaches were implemented to acoustically cloak the elastic cylindrical shell in a moving fluid. In the first control approach using ANCL, acoustic control sources in the exterior fluid domain were uniformly distributed in a circumferential array around the cylindrical shell. In the second control approach using ASACL, point control forces were uniformly distributed to directly excite the elastic shell. ASACL was shown to have several distinct advantages over ANCL. First, the control effort was reduced as the number of required control inputs was lower. Second, using ASACL the cloaked exterior field took effect from the shell surface whereas using ANCL, the acoustic domain between the shell and acoustic control sources was uncloaked. Further, using ANCL, any departure in the cloaking process yielded significantly greater deviation of the resultant field from that of the desired field compared to using ASACL.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.