## Abstract

The effect of a non-zero shear modulus on two-dimensional acoustic cloaking based on the transformation technique is investigated. Using the method of multiple scales, approximate solutions are found to the elastic equations with anisotropic density when the ratio of the shear modulus to the bulk modulus is small. These solutions indicate that a non-zero shear modulus causes the cloaking effect to become limited to a band of frequency, which becomes wider as the shear modulus is reduced. Resonances associated with shear waves are seen in the tangential component of displacement but do not affect the scattering to first order in the asymptotic expansions. No finite solutions exist for the case when the transformation shrinks the cloaked object to zero size (perfect cloaking).

## 1. Introduction

An acoustic cloak is material that surrounds an object and makes it invisible to incident sound. It is a generalization to sonar of the electromagnetic cloak invented by Pendry *et al.* (2006) using the idea of transformation optics. By exploiting a particular invariance in the form of the wave equation under a coordinate transformation, they showed that an object surrounded with a material with particular dielectric and magnetic properties would be equivalent to free space. Such materials are typically metamaterials in that they have properties that are usually difficult to obtain in nature but can be achieved by tailoring the sub-wavelength structure of the material.

The transformation technique also works for the acoustic wave equation and acoustic cloaking solutions have been found in two (Cummer & Schurig 2007) and three (Chen & Chan 2007; Cummer *et al.* 2008) dimensions. Cloaks have also been found for linear surface waves in a fluid (Farhat *et al.* 2008*a*). The required materials are metafluids (Norris 2008, 2009, 2010) and typically fall into two categories: one is termed inertial cloaking and utilizes a metafluid with an isotropic bulk modulus but anisotropic density. The other class consists of pentamode materials (Milton & Cherkaev 1995; Norris 2008) that have (in general) both anisotropic density and anisotropic modulus; however, the stress is still characterized by a scalar function (‘pressure’), so they remain in some sense ‘fluid-like’. Inertial cloaks can, in principle, be approximated using layers of isotropic fluids (Torrent & Sánchez-Dehesa 2008). The properties of these layers would have to be tailored by the use of scatterers, or a combination of rigid plates and inclusions (Pendry & Li 2008), at the microscale to achieve the required effective properties.

Although an ideal acoustic cloak is composed purely of metafluids, the practicalities of manufacturing a material loaded with local scatterers suggest that a solid variant should be considered. It has been shown that the invariance of the wave equation that allows the transformation technique to work fails for the elastic equations (Milton *et al.* 2006), though cloaks can be found for the special case of anti-plane shear waves (Farhat *et al.* 2008*b*; Guenneau *et al.* 2010). At first glance, this would seem to indicate that acoustic cloaking using solid, elastic metamaterials is impossible; however; a recent numerical study using a finite-element simulation by Urzhumov *et al.* (2010) showed that a cloaking effect could still be observed using elastic materials with a Poisson’s ratio *ν*>0.49 at the frequency used in the simulation.

In this paper, the effect of a non-zero shear modulus on the cylindrical (two-dimensional) cloaking solution of Cummer & Schurig (2007) is examined using singular perturbation theory. It turns out to be convenient to consider a slight generalization of the usual cloaking solutions (first introduced by Urzhumov *et al.* 2010) where the cloak shrinks the object isotropically from size *a* to apparent size *a*′, the traditional case of *a*′=0 then being referred to here as *perfect* cloaking. The bulk modulus and density are taken to have the generalized form of the cloaking solution (Cummer & Schurig 2007) and—since the metamaterial will presumably have been achieved using scattering on the microscale that is likely to mix bulk and shear processes—the shear modulus is taken to have the same dependence on position as the bulk modulus. The resulting generalized elastic equations are then solved approximately using the method of multiple scales (Smith 1975) for the case where the background shear modulus is ‘small’.

The results obtained here are broadly consistent with those of Urzhumov *et al.* (2010); however, the asymptotic results also show that:

— When the shear modulus is non-zero, there is no solution for perfect cloaking.

— There are a number of resonant frequencies of the tangential component of displacement associated with shear waves; however, they have no effect on the far-field acoustic scattering to first order.

— Although cloaking can be achieved over a range of frequencies when the shear modulus is small compared with the bulk modulus, the cloaking effect is predicted to break down at both low and high frequencies.

In §2, the generalized elastic equations are formulated and an asymptotic solution for the displacements in the cloaking layer found using singular perturbation theory. This solution is then used to calculate the scattering in §3 and the effect of shear at low frequencies examined. The theory is then extended to high frequency in §4 and conclusions summarized in §5.

## 2. The generalized elastic equations

The starting point for the study is a simple generalization of the inhomogeneous elastic equations to anisotropic density. By analogy with the metafluid case (Norris 2008, 2009), they are given in an arbitrary coordinate system by
2.1Here, the Einstein summation convention has been assumed for repeated indices and the covariant derivative denoted ∇_{i}. Covariant components have raised indices (*u*^{i}) and contravariant components lowered indices (*u*_{i}).

The Lamé constants, *λ* and *μ*, can be expressed in terms of the engineering bulk modulus, *K*, and shear modulus, *G*, by the relations
2.2thus, *λ* becomes the bulk modulus as the shear modulus tends to zero. The aim is to find position-dependent material properties so that a rigid cylinder of radius *a* embedded in an external fluid and surrounded by a layer of the material with outer radius *b* looks, under the coordinate transformation from polar coordinates (*r*,*ϕ*,*z*) to new (scaled) coordinates (*r*′,*ϕ*,*z*), like a smaller cylinder of radius *a*′ surrounded by a layer of the external fluid of thickness *b*−*a*′ (figure 1). The required transformation is *r*′=*Ar*−*B*, where
2.3

When the shear modulus is zero, it is a straightforward matter (Cummer & Schurig 2007; Norris 2008) to show that
2.4and
2.5where *λ*_{0} is the bulk modulus of the surrounding fluid and *ρ*_{0} is its density. For the elastic case, the shear modulus is assumed to have the same position dependence as the bulk modulus,
2.6Here, the constants *λ*_{0} and *μ*_{0} will be referred to as the background moduli and the associated wave speeds, and , as the background longitudinal and shear wave speeds.

If it is assumed that the scattering will be from an incident plane wave normal to the axis of the cylinder, the problem is two dimensional and the elastic equations in the (*r*,*ϕ*) coordinate system become
2.7aand
2.7bIt is assumed in the following that *u*^{i} has time dependence which will be dropped throughout.

To solve these equations, first a scaled displacement vector is defined by
2.8then the coordinate transformation *r*′=*Ar*−*B* is performed. The scaled displacement vector now has components *V* ^{r′}=*AV* ^{r} and *V* ^{ϕ′}≡*V* ^{ϕ} in the new (*r*′,*ϕ*′) coordinate system. Since the transformation left the polar angle unchanged, *V* ^{i′}(*r*′,*ϕ*) must be periodic in *ϕ* with period 2*π* and thus can be expanded in the Fourier series
2.9

Using this in equation (*a*,*b*), and defining *z*=*r*′/*B*, leads to the system of equations
2.10aand
2.10bwith and *k*_{l}=*ω*/*c*_{l}. It can be seen that when *ε*=0, these are just the equations for a longitudinal wave in an isotropic medium, which is to be expected since the sequence of transformations and scalings above are simply those that were used in the transformation technique to derive the bulk properties in the first place.

When *n*=0, the equations for and are uncoupled. The equation for has regular singular points at *z*=0 and −1; series solutions in the different regions can be obtained using standard techniques. The equation for , however, has an irregular singular point and one expects at best to be able to get asymptotic expansions for the solution (Olver 1974). When *n*≠0, equations (*a*,*b*) form a coupled system; thus, attention is now turned to approximate techniques. Since the shear modulus is typically several orders of magnitude smaller than the bulk modulus for elastomeric materials, it seems reasonable to look for approximate solutions for *ε*≪1. Use of normal secular perturbation theory, however, only gives two independent general solutions to the leading-order equations, whereas four would be expected on physical grounds. Problems of this type are termed singular and can be solved using the method of multiple scales (Smith 1975).

The basis of the method is to introduce a new scale, *X*, by
2.11where *h*(*z*) is a (yet to be determined) monotonically increasing function of *z*. It will prove convenient to choose *h* so that *X*=0 when *z*=*a*′/*B*. The scales *z* and *X* are then treated as independent variables and derivatives are calculated using the chain rule
2.12Here, the prime denotes the derivative with respect to the argument.

Using the ansatz
2.13aand
2.13bin the system (*a*,*b*) and equating each power of *ε* to zero leads to a hierarchy of equations that can be solved for each order, subject to a set of consistency criteria that ensures non-uniform terms in the expansion are zero. In particular, at second order (with comma denoting partial derivative), the relation
2.14is obtained, indicating that *h*(*z*) should be chosen so that
2.15It is then found that
2.16
2.17
2.18and
2.19with (‘*’ denotes complex conjugate)
2.20and
2.21

The Green function *G*(*x*,*x*_{0}) is chosen to be the solution to
2.22which satisfies Neuman boundary conditions at *x*=*x*_{a}=*k*_{l}*a*′ (*r*′=*a*′) and *x*=*x*_{b}=*k*_{l}*b* (*r*′=*b*). Thus,
2.23and
2.24with
2.25The constants *A*_{0}, *A*_{1}, *B*_{0}, etc., are then fixed by the boundary conditions.

It is assumed that the cloaked cylinder is rigid; thus (letting , )
2.26Setting and noting that *u*^{r}=(*r*′/*r*)*V* ^{r′}, the first boundary condition at *r*=*b* () is
2.27From this, and the first condition of equation (2.26),
2.28is immediately obtained.

Since the surrounding medium is a fluid, the final boundary condition is that the shear stress, *σ*_{rϕ}, should be zero at *r*=*b*. Thus,
2.29Rewriting in terms of *z* and *X*, and using the ansatz (2.13), leads to
2.30at *r*=*r*′=*b*. This relation must hold at each order in *ε*. Use of these boundary conditions then gives
2.31and the constants for *V* ^{1} and *U*^{1} are given in appendix A.

When *n*=0, *U*(*z*,*X*) is unchanged (and plays no role in the scattering solution), however, the radial displacement is independent of the scale *X* and has the form
2.32The leading-order term is
2.33and *V* ^{1} is simply
2.34where is now the Green function of equation (2.22) that satisfies Dirichlet boundary conditions at *x*=*k*_{l}*a*′ and *x*=*k*_{l}*b*. The constants and are easily determined and are given by
2.35

## 3. The scattering solution

The expressions (2.16)–(2.19) give an asymptotic approximation for the displacements in the cloaking material, in the transformed coordinate system, when the radial displacement of the outer surface () is known. In this section, the scattering from the cloaked cylinder is calculated and the physical meaning of these expressions examined.

First note from equation (2.31) that both *B*_{0} and *C*_{0} diverge as 1/*a*′^{2} as *a*′→0. This arose from the need to satisfy *σ*_{rϕ}=0 at the interface between the cloaking material and the external fluid. If *μ*_{0}=0, this condition is satisfied automatically and *B*_{0} and *C*_{0} can be taken to be zero. It can be seen that *V* (*z*,*X*) is then simply the solution for longitudinal waves in a fluid layer and perfect cloaking can be obtained. When *μ*_{0} is non-zero, however, *B*_{0} and *C*_{0} must ensure that equation (2.29) holds and there is no finite solution as *a*′→0. Thus, the presence of a shear modulus, no matter how small, means that there is no finite solution for the perfect cloaking case.

It can also be seen from equation (2.31) that *B*_{0} and *C*_{0} are undetermined when *x*_{b}=*π*/2+*mπ*. This is a resonance of the tangential component of the displacement at leading order. From equation (2.15),
3.1and since the wave number based on the background shear modulus, *k*_{s}, is given by *k*_{s}=*k*_{l}/*ε*^{1/2}, *X* is the characteristic length associated with shear waves. There are thus shear wave resonances at frequencies
3.2however, it will be seen that they have no effect on the scattering to first order at low frequency.

To calculate the scattering, the acoustic pressure in the external fluid is expanded into an incident plane wave and scattered cylindrical waves:
3.3and *p*_{s} has a Fourier expansion in terms of Hankel functions given by
3.4The fluid has been taken to have sound velocity *c* and wave number *k*=*ω*/*c*.

Continuity of stress and radial displacement are then imposed at the interface of the cloak and external fluid. Since the Fourier components of the radial stress are given by
3.5and
3.6(the Fourier components of *p*(*b*,*ϕ*)), it can be shown that
3.7where the input impedance for order *n* has the asymptotic form
3.8with
3.9
3.10and
3.11for *n*≠0. When *n*=0, the asymptotic form of the input impedance is
3.12where is simply equation (3.9) evaluated at *n*=0 and
3.13with
3.14and
3.15

If the external fluid has characteristic impedance *Z*_{f}=*ρ*_{f}*c*, the Fourier components of the scattered wave are given by
3.16Of particular interest is the total scattering cross section, which is the integral over all angles of the squared amplitude of the far-field pressure (Morse & Feshbach 1953). In the plots that follow, the dimensions of the cylinder and the cloak have been chosen to match the numerical results of Urzhumov *et al.* (2010), i.e. *a*=0.2 m and *b*=0.4 m. The external fluid is air, with speed of sound *c*=343 m s^{−1}, and the cloak is taken to have *c*_{l}=*c* and *a*′=0.1 *a*. It is convenient to define *δ*=*ε*^{1/2}=*c*_{s}/*c*_{l}.

Figure 2 shows a plot of the total scattering cross section against *kb* when *δ*=0.0001. Also shown are the total scattering cross sections for rigid cylinders of radius *a* and *a*′. As *δ*→0, the input impedance (3.8) tends to *Z*_{0}, the input impedance of a fluid layer of thickness *b*−*a*′ surrounding a rigid cylinder of radius *a*′, and for much of the frequency range, the cloak behaves as expected. At low frequency, however, the *O*(*ε*^{1/2}) term in equation (3.8) becomes significant and the solution deviates from the fluid-cloaking solution. This is perhaps to be expected: when shear is present, the transformation to the (*r*′,*ϕ*′) coordinates no longer removes all the positional dependence of the material properties and the average velocity over the layer at low frequency will not be *c*_{l}. At low frequency, it is this average velocity that will determine the behaviour of the layer, and there is a corresponding thickness resonance. As *δ*→0, this thickness resonance (and anti-resonance) shifts down to zero frequency and the scattering follows the smaller cylinder curve above it.

As *δ* is increased, more structure becomes evident. Figure 3 shows the total scattering cross section when *δ*=0.01 and figure 4 that for *δ*=0.1. The study of Urzhumov *et al.* (2010) was at 500 Hz, corresponding to *kb*=3.7, and reasonable cloaking is still observed at this frequency with reductions in scattering cross section of between 10 and 20 dB. Since *δ*=0.1 corresponds to a Poisson’s ratio, *ν*, of 0.49, these results are consistent with the previous study. Away from this region of frequency, though, cloaking is less marked and at low frequency, the scattering is actually increased: the addition of the cloak has doubled the size of the cylinder and the Fourier impedances do not need to deviate greatly from *Z*_{0} before the effect of this interface begins to be seen. As *δ* (*ε*^{1/2}) increases, higher order terms can no longer be neglected, so the scattering calculated using equation (3.8) is not expected to be accurate; however, these results do indicate that the impedance is no longer close to that for a fluid layer of wave velocity *c*_{l}, so cloaking is no longer expected.

## 4. The high-frequency behaviour

When calculating the far-field pressure using the Fourier expansion (3.4), the behaviour of the Hankel functions in *a*_{n} usually means that only terms up to *n*≃*kb* contribute to the sum. For *n*=*O*(1), the expression (3.8) is a valid asymptotic expansion; however, examination of the expression for shows that the first-order term is actually *O*(*ε*^{1/2}*n*^{2}), thus it is to be expected that the previous results are only valid for .

To examine the large *n* behaviour, set *n*=*N*/*ε*^{1/4} in equations (*a*,*b*) and introduce a new scale,
4.1The radial and tangential components of displacement are then expanded using
4.2aand
4.2band the method of multiple scales applied as before. The shear scale *X* remains unchanged and is given by equation (3.1). The new scale, *Y* , is found to be
4.3To first order, the solution is then given by
4.4
4.5
4.6and
4.7with
4.8
4.9and
4.10

The boundary conditions are applied as before, leading to (setting , and ) 4.11 4.12and 4.13and the constants for the next-order term are, for completeness, given in appendix A.

These expressions can be used to calculate an asymptotic expression for the input impedance, as before. It is given by
4.14Thus, it can be seen that the higher angular orders actually start to look like a perfectly reflecting cylinder as *ε*→0 (for *N*=*O*(1)). Low angular orders are still expected to be close to a cloaking solution; thus, the net result depends on the overall contribution of each order to the scattering. Figure 5 shows the result of using equation (4.14) for when *δ*=0.001. Since *ε*^{1/4}≃0.03, the asymptotic results would be expected to be a reasonable approximation. Figure 5 indicates that the large *n* terms gradually dominate the scattering and the cloaking effect is completely destroyed when for this model.

## 5. Conclusions

The current study has looked at a generalization of the cylindrical acoustic cloaking solution of Cummer & Schurig (2007) to include shear and non-perfect cloaking. The resulting system has been solved asymptotically using the method of multiple scales. These solutions exhibit a cloaking effect for scaled frequencies, *kb*, of order one and (Poisson’s ratio ), which is consistent with the numerical results of Urzhumov *et al.* (2010). A number of resonant frequencies associated with the tangential component of the displacement in the cloak are predicted but do not affect the scattering to first order. Deviations from perfect cylindrical symmetry could mean that these resonances will influence the scattering in practice.

The asymptotic solutions also indicate that significant deviations from a cloaked state are expected at low frequency and the cloaking effect will be completely destroyed at high frequencies, . The presence of shear in this model thus means that a cloaking effect can only be achieved over a range of frequencies. The lower the shear modulus is compared with the bulk modulus, the wider this range is. A non-zero shear modulus also means that perfect cloaking cannot be achieved, but this is not a significant limitation in practice.

## Acknowledgements

The authors would like to thank I. Youngs and L. Raynes for support and useful discussions. © Crown Copyright 2010. This work was funded by the Ministry of Defence and is published with the permission of DSTL on behalf of the controller of HMSO.

## Appendix A. The constants at first order

In this appendix, the constants for first order, determined by the boundary conditions, are given. When *n*=*O*(1),
A1
A2
A3and
A4and has been split into two parts by
A5

When *n*=*O*(1/*ε*^{1/4}),
A6
A7
A8and
A9

- Received December 10, 2010.
- Accepted January 21, 2011.

- This journal is © 2011 The Royal Society