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Analogue Transformation

Acoustics

Covariant Methodology in

Transformational Acoustics

Final Report Authors: Alberto Favaro1, Martin W. McCall1 and Paul Kinsler1, Luzi Bergamin2, Sante Carloni3

Affiliation: 1Imperial College London, 2KBP GmbH, Fliederweg 10, 3007 Bern, Switzerland, 3Advanced Concepts Team

ESTEC, PPC-PF, Keplerlaan 1, Postbus 299, 2200 AG Noordwijk, The Netherlands Date: 11th September 2012 Contacts: Martin McCall Tel: +44(0)20 7594 7749 Fax: +44(0)20 7594 7749 e-mail: [email protected]

Leopold Summerer (Technical Officer) Tel: +31(0)715654192 Fax: +31(0)715658018 e-mail: [email protected]

Available on the ACT website http://www.esa.int/act

Ariadna ID: 11-1301 Ariadna study type: Small

Contract Number: 4000104573/11/NL/KML

Contents

1 Introduction 5

2 Linearised Acoustics Theory 7

3 Equations of Motion for Linearized Acoustics 83.1 Equation of Motion for the Displacement . . . . . . . . . . . 83.2 Equation of Motion for the Pressure . . . . . . . . . . . . . . 93.3 Equation of Motion for the Stress Tensor . . . . . . . . . . . 9

4 Linearized Acoustics with Anisotropic Mass 10

5 Pentamode Media 11

6 Inertial Transformation Acoustics 126.1 Basic Transformation Acoustics Theorem . . . . . . . . . . . 14

7 Pentamode Acoustic Transformation 157.1 Pentamode-to-Pentamode Transformation Acoustics Theorems 17

8 Application: Cylindrical Acoustic Cloak 198.1 Inertial Cloaking Limit . . . . . . . . . . . . . . . . . . . . . . 218.2 Isotropic Mass Density Limit . . . . . . . . . . . . . . . . . . 22

9 Transformation acoustics beyond pentamodes 23

10 Geometric (‘Ray’) Transformation Acoustics 2410.1 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . . 2410.2 Conformal Metric Transformation . . . . . . . . . . . . . . . . 2510.3 Direct Transformation of Geodesics . . . . . . . . . . . . . . . 2610.4 Combining Conformal-Transformation with Direct Transfor-

mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.5 Example: Acoustic Ray Cloak . . . . . . . . . . . . . . . . . . 28

11 Simple Covariant Acoustics 3011.1 Vector Calculus Description . . . . . . . . . . . . . . . . . . . 3011.2 Tensor Description . . . . . . . . . . . . . . . . . . . . . . . . 3111.3 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . 3211.4 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.5 Pentamode Acoustics . . . . . . . . . . . . . . . . . . . . . . . 3411.6 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 36

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11.7 An example: the spatial carpet cloak . . . . . . . . . . . . . . 3711.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A Anisotropic Mass Density: on the Correct Index Positioningof the Mass Tensor 42

B Push-forward (ϕ∗) and Pull-back (ϕ∗) Notation, 44

C Proof of Inertial Cloaking Theorem 44

D Proof of Theorem 2 (Section 7.1) 46

3

Analogue Transformational Acoustics: An

Alternative Theoretical Framework for Acoustic

Metamaterials

or

Covariant Methodology in Transformational

Acoustics

Alberto Favaro, Martin W. McCall and Paul KinslerDepartment of Physics, The Blackett Laboratory

Imperial College of Science, Technology and MedicinePrince Consort Road, London SW7 2AZ, UK

Luzi BergaminKBP GmbH, Fliederweg 10, 3007 Bern, Switzerland.

Sante CarloniAdvanced Concepts Team

ESTEC, Postbus 299, 2200 AG Noordwijk, The Netherlands

11th September 2012

Abstract

The work of Ariadne study reference 11-1301 is described. Focuswas given to developing a formal theory of transformation acousticsthat analogued the exact theory of transformation electromagnetics.This was developed both for simple isotropic media and for so-calledpentamode acoustic media. In addition a generalised ray-based trans-formation theory that provided a simple, but highly flexible, recipefor making transformation-based devices, such as an acoustic cloak.Finally, an analogue acoustic theory that mimics the structure of co-variant electromagnetism is discussed.

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1 Introduction

The programme of actualizing point deformations in media [1],[2] has openedup many avenues of research, particularly in electromagnetics. Conceptually,the idea is beautiful: think of a deformation that maps points in space tonew points in space1. For a given domain of physics (electromagnetism,acoustics, etc.), enquire as to how the relevant medium properties shouldbe modified (anisotropy and inhomogeneity) so as to reposition the fieldquantities in question (e.g. electromagnetic fields) to ones located at thenew points. If this question is answered, then we have a recipe for ‘designer’deformation, the most celebrated example of which is the electromagneticcloak [3].

For electromagnetism, the prescription is known to be exact, i.e. all prop-erties of Maxwell’s equations, including both near- and far-field attributes,are inherited by the transformed fields, irrespective of wavelength, obstaclesizes and shapes. Whatever the technological challenges in producing the re-quired electromagnetic medium, if it can be made, then Maxwell’s equationsare mapped exactly onto the new domain. The linear rays of geometricaloptics in a homogeneous medium are then mapped precisely to the curvedpaths defined by the mapping.

Why the programme is so outstandingly successful in electromagnetism,and not in other areas of physics, is still a matter of debate and ongoing re-search. The fact that Maxwell’s equations admit a covariant, or co-ordinateindependent formulation is undoubtedly significant. Notably, the most con-certed mathematical studies of transformation electromagnetics has beencarried out by Thompson et al [4, 5]. Acoustics, the focus of this currentreport, has been the most explored area for transformation theory besideselectromagnetism. Milton et al [6] have shown that in general an anisotropicmass density is required for elastodynamic equations that are the naturalcounterparts of Maxwell’s equations to remain invariant under spatial trans-formation. However, the equations Milton uses as acoustic counterparts ofMaxwell are posed in terms of ordinary, rather than covariant, derivatives.As a consequence, Milton’s equations of motion are not inherently forminvariant and thus do not admit the full range of transformation-type so-lutions that are possible in electromagnetism. One of our key objectivesin this report is to provide a properly covariant framework for acoustics.In two-dimensions, the acoustic equations in a fluid are isomorphic to the

1We draw a distinction between deformations that reposition points in a space, tocoordinate transformations that merely relabel points in the space. Although deformationscan be related to a coordinate relabelling, the converse is not true in general.

5

single polarization Maxwell equations in a way that also preserves bound-ary conditions as was shown by Cummer et al [7]. The form invariance ofHelmholz equations under general curvilinear transformations was studiedby Greenleaf et al [8] who showed in particular that the scalar wave equationfor the pressure in an inviscid (i.e. non-viscous) medium can be cast as

1√|g|

∂

∂xi

(√|g|gij ∂p

∂xj

)=∂2p

∂t2, (1)

where gij is a metric induced by the proposed transformation. Acousticcloaking in inviscid media for which the mass density is anisotropic andthe bulk modulus varies, have become known as Inertial Cloaks [9]. Norrisshowed that inertial cloaking requires the cloaking medium to have a locallyinfinite mass density. If instead more general anisotropic media are consid-ered in which the bulk modulus is also anisotropic, then the infinite massconstraint can be avoided [10]. The simplest such media are known as penta-mode media which represents a limiting class of anisotropic media in whichfive modes (hence ‘penta’) of shear combine with one mode of stress andstrain. Generalizing further, Norris also showed how utilizing ‘pentamode-to-pentamode’ transformation media allowed the resultant medium to beinertially isotropic.

Another reason why transformation electromagnetics is so natural is thatelectromagnetism is pre-metric [11]. By contrast acoustics is not pre-metric,and another of our aims in this work was to examine carefully how the metricshould be interpreted in any transformational acoustics scheme (cf. Section6, point 2).

We endevoured in this project to examine the above literature carefullyand attempt to formulate transformation acoustics more formally, focus-ing specifically on purely spatial transformations. Several approaches wereadopted and described in the sections below. We commence in Section 2by summarizing linearized acoustics theory from the most general covari-ant perspective. Then, in Section 3 we derive covariant second order waveequations for the fluid displacement, pressure and stress tensor. In Section 4we generalize the linearized theory to include anisotropic mass and therebyobtain generalized second order equations for the pressure and the stress.These appear to be quite new. Likewise, we derive the relevant second or-der equations for pentamode media in Section 5. In Section 6 we return tofirst order equations that retain the conceptual clarity of the original equa-tions set out in sections 3 and 4, and derive a key theorem, which, for thefirst time, gives a precise covariant recipe for inertial acoustic cloaking, in

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which the elasticity tensor takes its simplest form. In Section 7 we extendthe theory to pentamodes and show in particular, via another key theorem(Theorem 2 of Section 7.1), how the technical requirement that the tensordefining the pentamode medium (Sij) be divergence free can be manufac-tured. In Section 8 we apply the developed formalism to a specific and novelexample in acoustic cloaking, giving the explicit medium cloaking recipes forpentamode media in the two limits of inertial cloaking and for isotropic massdensity. Some commentary of how the formalism might be extended beyondpentamodes is provided in Section 9.

In Section 10 we take a different tack in which we attempt to produce ageometric transformation theory, i.e. a method that retains only directions ofenergy transport (i.e. rays), and then defines how the rate of transport mustbe modified to produce a given ray deformation. Although such an approachis necessarily imperfect, it did yield a very general recipe (Equation (127))for transformation media that can be applied across multiple disciplines,including acoustics. An explicit cloaking recipe is discussed.

Finally, in Section 11 we present some analysis in which first order acous-tic equations are tied as closely as possible to their electromagnetic coun-terparts. The formalism is applied specifically to the design of an acousticcarpet cloak.

2 Linearised Acoustics Theory

We here summarize the linear theory of elasticity, expressed in an arbitraryspatial coordinate system. This form invariance is essential in discussingtransformation acoustics.Firstly we have the equation of motion for a mass element of the fluid:

Dkσkm = ρgmnu

n , (2)

(see [13] §2). Here Dk is a spatial covariant derivative, σ km is the stress

tensor (i.e. Tm = σ kmnk is the stress (= force per unit area) vector acting

through a unit area specified by the normal nk), ρ is the (isotropic) massdensity, gmn is the spatial metric (expressed in arbitrary coordinates), andu represents the acceleration of the mass element in question.

Next, we have the strain tensor given by

enk =1

2(Dkuk +Dnuk) = D(nuk) = D(ngk)mu

m , (3)

(see [13] §3). It represents the deformation of the line element that occurswhen the body is deformed from an initial configuration S to a deformed

7

configuration S, i.e.ds2 − ds2 = 2enkdx

ndxk . (4)

In the linearized theory we assume that stress and strain are linearly related,i.e.

σ ji = C jkl

i ekl . (5)

(see [13] §4). For any linear medium one can decompose the stress tensoras:

σ ji =

(σ ji −

1

3σ kk δ

ji

)+

1

3σ kk δ

ji , (6)

= (σ ji + pδ j

i )− pδ ji , (7)

= 6σ ji − pδ

ji , (8)

where p := −13σ

ii and 6σ j

i := σ ji −

13σ

kk δ

ji = σ j

i + pδ ji (see [14] §2). Note

that the pressure is related in a trivial way to the trace of the stress, whereas

6σ ji is the trace-free part of the stress (i.e. the shear stresses). If a body

undergoes hydrostatic compression a uniform, isotropic pressure p, acts onevery unit surface area of the body, directed normally inwards. In that casethe shear stresses vanish and σ j

i = −pδ ji .

3 Equations of Motion for Linearized Acoustics

We now proceed to derive equations of motion for a linearized medium. Wedo this in various ways:

1. by deriving a set of coupled equations for the displacement ui.

2. by deriving an equation of motion for the (isotropic) pressure p coupledto the stresses

3. by deriving an equation of motion for the stress tensor σ ji coupled to

the pressure.

3.1 Equation of Motion for the Displacement

Combining Eq. (2) with Eqs. (5) and (3)

DkCkijm Digjau

a = ρgmnun . (9)

8

Now compatibility between the metric and the covariant derivative2 requiresthat Digjk = 0 so that the last equation can be written as

DkCkijm gjaDiu

a = ρgmnun . (10)

This equation is three coupled equations for the displacement ui. Althoughbuilding a successful transformation acoustics scheme based on Eq. (10) istechnically challenging, the compactness and generality of Eq. (10) makethis idea very tempting for future investigation.

3.2 Equation of Motion for the Pressure

Combining Eq. (2) with Eq. (8)

Dj6σ ji −Dip = ρgiku

k . (11)

Combining p = −13σ

ii with Eq. (5) and Eq. (3) we obtain

p = −1

3C ikli ekl = −1

3C ikli Dkglmu

m . (12)

Now since C jkli , Di and gij are time independent

p = −1

3C ikli Dkglmu

m . (13)

Combining Eq. (11) with Eq. (13)

p =1

3C ikli Dk

[1

ρDlp

]− 1

3C ikli Dk

[1

ρDm6σ m

l

]. (14)

Note the last term couples the equation of motion for pressure to the shearstresses.

3.3 Equation of Motion for the Stress Tensor

Similarly, we can obtain equations of motion for the shear stresses of anylinearized medium. Combining Eq. (8) with Eq. (5) and Eq. (3) yields

6σ ji = C jkl

i ekl −1

3C mklm eklδ

ji , (15)

= C jkli Dkglnu

n − 1

3δ ji

(C mklm Dkglnu

n), (16)

2This is how a unique covariant derivative is determined from a metric gij .

9

whence¨6σ ji = C jkl

i Dkglnun − 1

3δ ji

(C mklm Dkglnu

n). (17)

Combining the last equation with Eq. (11)

¨6σ ji = C jkl

i Dk

[1

ρDm6σ m

l −1

ρDlp

]−1

3δ ji C

nkln Dk

[1

ρDm6σ m

l −1

ρDlp

]= C jkl

i Dk

[1

ρDm6σ m

l

]− 1

3δ ji C

nkln Dk

[1

ρDm6σ m

l

]−C jkl

i Dk

[1

ρDlp

]+

1

3δ ji C

nkln Dk

[1

ρDlp

]. (18)

Note that the last two terms couple the equation of motion for the stressesto the pressure.

4 Linearized Acoustics with Anisotropic Mass

It was previously mentioned that the idea of building a transformation acous-tics scheme based on Eq. (10) is tempting. However, Eq. (10) is somewhatincomplete, in that it does not include an anisotropic mass. One is thus mo-tivated to ask: what is the equivalent of Eq. (10) with an anisotropic mass?One technical point is how the mass density tensor should be properly in-dexed. This issue is dealt with in detail in Appendix A, and we here assumethe form of the mass density should be written as ρij . With anisotropicmass Eq. (2) generalizes to

Dkσkm = gmnρ

nlul . (19)

We assume that there exists(ρ−1)ij

such that(ρ−1)ijρjk = δik. Omitting

algebraic details, the three equations of motion analogous to Eqs. (10), (14)and (18) are straightforwardly derived as

Dj

[C jmni gmp

]Dnu

p = gikρklul , (20)

p =1

3C ikli Dk

[glm

(ρ−1)mpgpq]Dqp

−1

3C ikli Dk

[glm

(ρ−1)mpgpq]Dr6σ r

q , (21)

10

and

¨6σ ji = C jkl

i Dk

[gln(ρ−1)npgpq]Dr6σ r

q

−1

3δ ji C

mklm Dk

[gln(ρ−1)npgpq]Dr6σ r

q

−C jkli Dk

[gln(ρ−1)npgpq]Dqp

+1

3δ ji C

mklm Dk

[gln(ρ−1)npgpq]Dqp . (22)

Relative to the extant literature [9, 13, 14], these equations appear to benew in their use of anisotropic mass.

5 Pentamode Media

A pentamode medium [9] is characterized by the following form for theelasticity tensor

C jkli = λgimS

mjSkl , (23)

where Sij is symmetric and invertible. An equation for the pressure in termsof the strain is obtained by combining p = −1

3σii with σ j

i = C jkli ekl (Eq.

(5)) and ekl = D(kgl)mum (Eq. (3)) to obtain

p = −1

3K(gimS

im)

(gpqSqrDru

p) . (24)

From this we can obtain the shear stress as

6σ ji = σ j

i + pδ ji ,

= KgimSmjSklDkglmu

m − 1

3K(gimS

im)

(gpqSqrDru

p) δ ji ,

= K (gpqSqrDru

p)

[gikS

kj − 1

3(gmnS

mn) δ ji

]. (25)

We now introduce the so-called ‘pseudo-pressure’ [9]

p = −K (gpqSqrDru

p) , (26)

and proceed to express p, 6σ ji and σ j

i in terms of p as

p =1

3p (gmnS

mn) , (27)

6σ ji = −p

[gikS

kj − 1

3(gmnS

mn) δ ji

], (28)

σ ji = −p

[gikS

kj]. (29)

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On the right hand sides of Eqs. (27), (28) and (27), the only time de-pendent quantity is the pseudo-pressure. Accordingly, the pressure, shearstresses and stress tensor consist of a time-independent geometric struc-ture modulated by a time-dependent scalar, namely the pseudo-pressure.This observation is related to the fact that pentamodes are well-suited totransformation acoustics. Now, combining Eqs. (19) and (29) we obtainstraightforwardly

Dj

[−pgimSmj

]= gikρ

klul . (30)

After multiplying through by(ρ−1)pqgqi and utilizing the fact that the co-

variant derivative commutes with the metric we obtain

up = −(ρ−1)pmDn [pSmn] . (31)

Differentiating Eq. (26) twice with respect to time and substituting for up

from Eq. (31) yields

¨p = KgpqSqrDr

(ρ−1)pmDn [pSmn] . (32)

If Sij is assumed to be divergence free, i.e. DiSij = 0 then the preceeding

equation is straightforwardly manipulated to

¨p = KDi

[Sijgjk

(ρ−1)klSlm

]Dmp . (33)

6 Inertial Transformation Acoustics

In this section we aim to provide a general recipe for so-called ‘inertial trans-formation acoustics’, i.e. a transformational scheme that maps a shear-freemedium with anisotropic inertial mass to another similar medium. Thereforethe only constitutive parameters altered by the transformation scheme arethe bulk modulus (λ), and the (anisotropic) mass tensor ρij . In contrast toother authors ([6], [8], [9]) who derive transformation algorithms for secondorder wave equations, in this section we retain first order equations. Thischoice provides better conceptual clarity, and keeps the methodology closerto transformation optics. There are two requirements imposed to achievethe recipe for transformation acoustics:

1. a simple conversion from covariant to ordinary derivatives.

2. we need to have (effective) constitutive quantities where the metriccan be hidden after it has been transformed.

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We start with the force density fi given by

fi = gijρjku

k , (34)

and the constitutive laws

C jkli = λδ j

i gkl , (35)

C jkli ekl = σ j

i , (36)

and finally the differential equations

Diσij = fj , (37)

eij =1

2

(Digjku

k +Djgikuk), (38)

but avoid in the following using second covariant derivatives.Substituting Eq. (36) into (37)

Di

[C iklj ekl

]= fj , (39)

and then using Eq. (35) we obtain

Di

[λδ i

j

(gklekl

)]= fj . (40)

Finally substituting Eq. (34) into the above equation yields

Di [λ (gpqepq)] = gijρjku

k . (41)

Now the strain equation, Eq. (38) can be decomposed into a traceless partand an isotropic part:

eij = gim

[/emj + (gpqepq) δ

mj

], (42)

= gim/emj + (gpqepq) gij . (43)

This decomposition of the strain tensor is quite crucial, since it facilitates,as we show subsequently, a simple conversion from covariant to ordinaryderivatives.3 Note that the traceless part, /eij does not couple with Eqs.(34), (35), (36) and (37). Furthermore,

gpqepq = gpq1

2(Dpgqmu

m +Dqgprur) = Dru

r , (44)

3This decomposition of eij is irreducible under GL(4,R).

13

the last equality arising because of the covariant compatibility of the metric.Using the above, we express the differential equation Eq. (41) and the strainequation Eq. (38) in the following way

Di (λξ) = gijρjku

k , (45)

ξ = Diui . (46)

Eqs. (45) and (46) constitute four equations in four unknowns. Using wellknown properties of the covariant derivative, these last two equations canbe written in terms of ordinary derivatives as

∂i (λξ) = gijρjku

k , (47)

ξ =1√

det gij

∂

∂xp

[√det giju

p], (48)

which again is 4 equations in four unknowns (i.e. three unknown uα andone unknown ξ). We are now ready to state the basic “inertial cloaking”version of transformation acoustics (cf. [9]). We do this via a theorem.

6.1 Basic Transformation Acoustics Theorem

Theorem: If up and ξ are solutions of Eqs. (47) and (48), then

ξ′ =

√det gij√det g′ij

ξ , (49)

and

u′p =


up , (50)

are solutions of∂i(λ′ξ′)

= g′ijρ′jku′k , (51)

ξ′ =1√

det g′ij

∂

∂xp

(√det g′iju

′p), (52)

where

λ′ =

√det g′ij√det gij

λ , (53)

14

and

ρ′ij =


g′imgmnρnj . (54)

Note the following definitions (see Appendix B):

ξ = ϕ∗ξ , ui = (ϕ∗)

ij u

j , λ = (ϕ∗λ) , ρmn = (ϕ∗ρ)mn , gij = (ϕ∗)mi (ϕ∗)

njgmn ,

(55)where g′ij is a spatial metric. Stating the theorem in words: under a trans-formation solutions of the elastodynamic equations are mapped to solutionsof the elastodynamic equations at the price of redefining the constitutive pa-rameters. Note how the metric is ‘hidden’ in Eq. (54) after transformation.The proof of the theorem is given in Appendix C.

Lemma: p′ = −13σ′ ii obeys p′ = p.

Proof: p′ = −13λ′δ ii g′kle′kl = −3

3λ′ξ′ = −λ′ξ′ = −λξ = p.

7 Pentamode Acoustic Transformation

The previous section presented the basic transformation acoustics theoremfor a trivial medium in which the elasticity tensor took its simplest form asgiven by Eq. (35). Here we extend the theorem to pentamode media wherethe elasticity tensor is given by Eq. (23). Note that since the trivial mediumCijkl = λgijgkl (which is equivalent to Eq. (35)) is also a pentamode, thetheory of this section encompasses the transformation (trivial medium) →(pentamode medium) as a sub-case. For clarity we restate all the startingequations for linearised elasticity (i.e. small displacement) in a pentamodematerial as

C jkli = λgimS

mjSkl , (56)

σ ji = C jkl

i ekl , (57)

fi = gijρjku

k , (58)

Djσji = fi , (59)

eij =1

2

(Digjku

k +Djgikuk). (60)

To these we add a crucial additional assumption, namely that

DiSij = 0 . (61)

15

It is emphasised again, that the subsequent development is restricted to firstorder equations, and no attempt is made to generate wave equations thatwill involve second spatial (covariant) derivatives. Combining Eqs. (56)-(59)yields

Dj

[λgimS

mj(Sklekl

)]= gipρ

pquq . (62)

The strain of Eq. (60) can be decomposed with respect to Sij as

eij = Sim

[/emj + (Spqepq) δ

mj

]= Sim/e

mj + (Spqepq)Sij , (63)

where we note that the trace-free part of the strain /eij , does not couple toEq. (57). Note, in Eq. (63) (and subsequently) the symbol Sim refers tothe imth component of the inverse of Sjk. For a pentamode medium onlythe component of the strain given by ξ := Sijeij couples to Eq. (57). Thedecomposition given by Eq. (63) is actually quite crucial as it permits usto develop equations in which the covariant derivative is easily related tothe partial derivatives. In turn, this allows transformation acoustics to beperformed successfully. Now, by using Eq. (60), we can easily show byrelabelling indices and the symmetry of Spq, that the term in parenthesesin the last equation can be written as

Spqepq = SpqDp (gqrur) . (64)

Thus, in a manner analogous to Eqs. (45) and (46) we arrive at

Dj

[λξgimS

mj]

= gipρpquq , (65)

ξ := Sijeij = SijDi

(gjku

i), (66)

which is four equations in four unknowns. If we now make use of the addi-tional assumption Eq. (61), and the invertibility of Sij , then the above twoequations become

Di (λξ) = Sijρjku

k , (67)

ξ = Di

(Sijgjku

k), (68)

or, with further use of the invertibility of Sij

Di (λξ) = SijρjkgklSlmw

m , (69)

ξ = Diwi , (70)

16

where wi := Sijgjkuk. These two sets of first-order equations encode the

dynamics of pentamode materials in two slightly distinct ways. As before,both sets (i.e. Eqs. (67)-(68) and (69) and (70)) may be expressed in termsof ordinary derivatives:

∂i (λξ) = Sijρjku

k , (71)

ξ =1√

det gpq∂i

[√det gpqS

ijgjkuk], (72)

and

∂i (λξ) = SijρjkgklSlmw

m , (73)

ξ =1√

det gpq∂i

[√det gpqw

i]. (74)

Before stating the relevant pentamode transformation acoustics theorems,we note that the assumption of vanishing divergence of Sij (i.e. DiS

ij =0) must hold true for the tensor Sij both before and after transformationacoustics is applied. This leads to an additional constraint that in practicemay be quite hard to satisfy, particularly if the transformational schemeis applied to Eqs. (71) and (72). By contrast, the set (73) and (74) hasadditional flexibility in the choice of the transformed tensor S′ij which can

be exploited to manufacture DiS′ij = 0. Hence in the next section we give

two transformational acoustic theorems, one for set (71) and (72) and onefor set (73) and (74).

Another point concerns the quantity ξ. It is related to Norris’ pseudopressure, p [9] via p = −λξ, and can be called the ‘trace’ of the strain withrespect to Sij .

7.1 Pentamode-to-Pentamode Transformation Acoustics The-orems

We now state the transformation acoustics theorem analogous to that givenin Section 6.1:

Theorem 1

If ξ and uk are solutions of Eqs. (71) and (72), then

ξ′ =

√det gpq√det g′pq

ξ and u′i =


ui (75)

17

are solutions of

∂i(λ′ξ′)

= S′ijρ′jku′k , (76)

ξ′ =1√

det g′pq∂i

[√det g′pqS

′ijg′jku′k], (77)

where

S′ij := Simgmng′nj , (78)

ρ′jk :=

√det g′pq√det gpq

S′jmSmnρnk , (79)

λ′ :=


λ . (80)

Here (see Appendix B for notation) ξ = ϕ∗ξ and ui = (ϕ∗u)i aretransformed variables; Sij = (ϕ∗S)ij , ρij = (ϕ∗ρ)ij and λ = ϕ∗λ

are transformed constitutive parameters; gij = (ϕ∗g)ij is thetransformed metric and g′ij is some other metric.

To give the theorem in words, the primed quantities are quantities obtainedafter:

transformation + reinterpretation as a new medium

The proof of the theorem is entirely analogous to that given in AppendixC. Attention is also drawn in particular to Eq. (78) which is effectively arequirement on the new S′ij . In order for a transformational scheme basedon Eqs. (75)-(80) to work it is also necessary for DiS

′ij = 0, i.e. the primedSij must be covariantly conserved. Satisfying this requirement turns outto be quite problematic. However, this difficulty may be circumvented byinstead basing the transformational acoustics scheme on the equation set(73)-(74). We now give the second transformational acoustic theorem basedon the equation set (73) and (74).

Theorem 2

If ξ and wk are solutions of Eqs. (73) and (74), then

ξ′ =


ξ and w′i =


wi (81)

18

are solutions of

∂i(λ′ξ′)

= S′ijρ′jkg′klS′lmw

′m , (82)

ξ′ =1√

det g′pq∂i

[√det g′pqw

′i], (83)

provided

S′ijρ′jkg′klS′lm =


SiaρabgbcScm , (84)

and

λ′ =


λ . (85)

Here quantities denoted with an overbar are the transformed quantities,whereas those denoted with a prime are quantities actually implemented inthe final medium. The quantity g′pq is an arbitrary metric, usually set to beequal to gpq, the metric of Euclidean space.

Note that if Sij = gij and S′ij = Sij , the above transformation schemereplicates that of Norris [9], taking a simple shear-free material as being aspecial case of a pentamode. Our development, in contrast to Norris, hasbeen consistently first order.

When we consider a specific application of transformational acoustics inSection 8, we will exploit the additional flexibility referred to above and useTheorem 2. Since exploitation of Theorem 2 is more fruitful than Theorem1, we give a detailed proof in Appendix D.

8 Application: Cylindrical Acoustic Cloak

We now give a specific application of the above formalism to acoustic cloak-ing, illustrating the use of the second theorem given in the last section. Wetake the radial transformation to be

r =(r2 + a2

)1/2,

θ = θ . (86)

The origin point r = 0 is mapped to the circle r = a, and for large r we haver ≈ r. Note that this improves on the ‘standard’ cloaking transformation

r =

(b− ab

)r + a ,

θ = θ . (87)

19

which, although extensively used in the literature right from the inceptionof transformation optics [1], fails to satisfy r → r as r →∞.

In order to proceed we need to first express the underlying Euclideanmetric gij = δij in cylindrical polar coordinates

r =(x2 + y2 + z2

)1/2, θ = tan−1

(yx

), (88)

with the z-coordinate unaltered. Using gi′j′ = ∂xi∂xi′

∂xj∂xj′

gij , we find the polar

representation grr = 1, gθθ = r2, gzz = 1, i.e. gi′j′ = diag(1, r2, 1

). The

inverse is given by gi′j′ = diag

(1, r−2, 1

). For convenience we now drop the

primes so that gij represents the untransformed Euclidean metric expressedin cylindrical coordinates.

In order to implement the transformation scheme we need to calculatethe push-forward quantities (ϕ∗g

−1)ij , (ϕ∗g)ij , and (ϕ∗λ), where ϕ is themapping given by Eqs. (86), and λ is the bulk modulus of the originalhomogeneous medium denoted by λ = κ, where κ is a constant. We find

(ϕ∗g−1)ij = diag

[(r2 − a2)/r2, (r2 − a2)−1, 1

], (89)

(ϕ∗g)ij = diag[r2/(r2 − a2), (r2 − a2), 1

], (90)

(ϕ∗λ) = κ . (91)

We are now ready to apply the transformational scheme of Theorem2. We make the following assumptions concerning the initial (i.e. untrans-formed) medium:

1. The initial medium is a simple fluid, i.e.

Cjkli = λδ ji g

kl . (92)

2. The initial medium is characterized by an isotropic mass density, i.e.

ρij = ρδij . (93)

3. The initial medium is characterized by a constant bulk modulus, i.e.

λ = κ . (94)

4. The initial medium is characterized by a constant mass density, i.e.

ρ = M = constant . (95)

20

After application of Theorem 2 we find that Eq. (84) becomes

S′ijρ′jkg′klS′lm =


Mgim , (96)

λ′ =


λ , (97)

or, explicitly for the transformation of Eqs. (86) (now dropping the tilde),

S′ijρ′jkg′klS′lm =√

det g′pq√det gpq

M

[diag

(r2

(r2 − a2)1/2,(r2 − a2)3/2

r,(r2 − a2)1/2

r

)], (98)

λ′ =


(r2 − a2)1/2

rκ . (99)

We now focus on the possibilities for the medium designed to mimic thetransformation. One natural assumption is that the medium exists, likeits untransformed antecedent, in Euclidean space, so that g′ij = gij =

diag(1, r2, 1) and g′ij = gij = diag(1, r−2, 1). Eqs. (98) and (99) thenbecome

S′ijρ′jkgklS′lm = M

[diag

(r

(r2 − a2)1/2,(r2 − a2)3/2

r,(r2 − a2)1/2

r

)],

(100)

λ′ =(r2 − a2)1/2

rκ . (101)

Evidently, there are various possibilities for satisfying Eq. (100). We con-sider just two.

8.1 Inertial Cloaking Limit

In this caseS′ij = gij = diag(1, r−2, 1) , (102)

and we have for r > a

ρ′ij = M

[diag

(r

(r2 − a2)1/2,(r2 − a2)3/2

r3,(r2 − a2)1/2

r

)], (103)

21

λ′ =(r2 − a2)1/2

rκ . (104)

In this case the requirement that DiS′ij = 0 is automatically satisfied by

the covariant compatibility condition Digjk = 0 which associates the metric

gij with its unique covariant derivative Di.

8.2 Isotropic Mass Density Limit

In this case we set ρ′ij = M ′δij and obtain for r > a

S′ij =

(M

M ′

)1/2[

diag

(r1/2

(r2 − a2)1/4, r1/2(r2 − a2)3/4,

(r2 − a2)1/4

r1/2

)],

(105)

λ′ =(r2 − a2)1/2

rκ . (106)

Now the requirement DiSij = 0 sets a condition on the radial mass density

M ′(r). Explicitly, Sij must satisfy

∂iSij + ΓiaiS

aj + ΓjaiSai = 0 , (107)

where the only non-zero connection coefficients in cylindrical polar coordi-nates are Γrθθ = −r and Γθrθ = r−1. After inverting Sij in Eq. (105) andsubstituting in the above equation, we find that M ′(r) must satisfy(

r2 − a2)1/4

r1/2∂

∂r

[ln

(M ′

M

)]+

∂

∂r

(r2 − a2

)1/4r1/2

+

(r2 − a2

)1/4r3/2

− r1/2

(r2 − a2)3/4= 0 . (108)

The solution isM ′

M= const.

(1− a2

r2

)1/2

, (109)

where const. is an integration constant. This mass distribution, togetherwith the bulk modulus distribution (which for this case is still given by Eq.(104)) ensures that acoustic disturbances are bent around the region r < a,and, being mass-isotropic, should be easier to achieve than the previousinertial limit case.

22

9 Transformation acoustics beyond pentamodes

As explained in Section 6, to achieve a valid transformation acoustics schemeit is essential that the equations of motion can be easily cast in terms of or-dinary derivatives, as opposed to covariant ones. When the transformationacoustics scheme takes its moves from a pentamode material (Section 7),the effortless conversion from covariant to ordinary derivatives is made pos-sible by the identification of a crucial scalar, namely the pseudo pressurep. As an alternative, one can make use of the parameter ξ, given that thisquantity is a proxy for p. For more general materials, the determinationof a single scalar, such as the pseudo pressure, becomes insufficient so asto characterise the behaviour of sound waves. In particular, shear effectsmay occur that are truly tensorial in nature. An important consequence fol-lows: the fundamental equations for more general materials do not manifesta simple conversion from covariant to ordinary derivatives, unless suitablymanipulated. Ergo, developing a transformation acoustics scheme that goes“beyond pentamodes” is a significant challenge. One solution may comefrom the identification of multiple scalars, describing the numerous featuresof sound propagation in complex media. To this end, a decomposition ofthe stress-strain tensor Cijkl put forward in the article [15] by Milton andCherkaev is employed:

Cijkl =6∑I=1

λI SijI S

klI = λ1S

ij1 S

kl1 + · · ·+ λ6S

ij6 S

kl6 . (110)

The scalars λI may be regarded as bulk moduli. In addition, the tensorsSijI are not necessarily full-rank. A pentamode material is defined by theproperty that five of the parameters λI vanish. Contracting Cijkl with thestrain eij allows one to formulate the stress tensor, with one index raised,as

σij = −6∑I=1

pISijI = −

(p1S

ij1 + · · ·+ p6S

ij6

). (111)

Each of the scalars(p1, . . . , p6

):=(−λ1Sij1 eij , . . . ,−λ6S

ij6 eij

)can be in-

terpreted as a pseudo pressure. As anticipated, the identification of thesemultiple parameters is the key for constructing a transformation acousticsscheme that is applicable beyond pentamodes. In fact, subject to the con-dition that

DkSklI = 0 for all I, (112)

23

it is possible to identify six coupled wave equations, governing the evolutionof the scalars pI . Notably, the covariant derivatives appearing in such equa-tions of motion are easily converted into ordinary ones. As a result, one candevise a transformation acoustics scheme for media of very general nature.However, it should be recognised that the constrain of Eq. (112), whichmust be imposed on all SI , is very restrictive. Thus the scheme outlinedabove can only be claimed to a be a first step towards the TransformationAcoustics of a non-scalar wave equation. Second, numerous transformationacoustics endeavors that are otherwise productive fail to comply with therequirement (112) for all I.

10 Geometric (‘Ray’) Transformation Acoustics

In this section we formulate a transformation theory that operates at thelevel of geometrical acoustics, i.e. the goal of an exact field transformationtheory (cf. foregoing sections) is compromised in favour of a ray-based theorythat seeks to define material properties that can morph rays in a prescribedor designed way. The key idea is to regard the rays as geodesics with respectto a metric. A ‘speed’ is defined at each point on a given geodesic accordingto the magnitude of the tangent vector at that point. A desired morphingof the rays will be shown to relate to a single parameter n(r), given bythe ratio of the speed along the unperturbed geodesics, to the speed alongthe perturbed geodesics. Such a refractive index can then be related tosimple medium parameters such as the contrast in bulk modulus and massdensity. This transformational strategy appears to be quite generic. Anyray theory (e.g. geometrical optics, geometrical acoustics, etc.) admits aparameter equivalent to the refractive index of geometrical optics, so thatrelating spatial deformations to the required index distribution (and therebymaterial parameters) gives us access to e.g. an acoustic cloak or black hole,a refractive-index based cloak or concentrator for optics, etc. One just has toprescribe the desired deformation, and the following theory will then deliverthe required index distribution. The price of such generality is that any suchtransformation device is necessarily imperfect, since the very index contrastused to create the device induces scattering.

10.1 Geodesic Equation

A covariant derivative provides a mechanism for parallel transportation,allowing vectors at different locations to be compared. Insisting that agiven vector v, be parallel transported with respect to itself results in the

24

geodesic equation:

viDivj = vi∂iv

j + Γjkivkvi = 0 . (113)

Alternatively, if time t affinely parameterizes the curve for which v = d/dt =vi∂i is tangent, then vi = dxi/dt, and

d2xi

dt2+ Γjki

dxk

dt

dxi

dt= 0 . (114)

A standard result from differential geometry is that a metric induces a uniquecovariant derivative for which the connection coefficients are given by

Γijk =1

2gim (∂kgjm + ∂jgkm − ∂mgjk) . (115)

10.2 Conformal Metric Transformation

The cartesian components of the Euclidean metric of flat three-dimensionalspace are just δij , so that according to Eq. (115) all connection coefficientsvanish, leaving

d2xi

dt2= 0 . (116)

This integrates up toxj(t) = xj(0) + vj0t , (117)

i.e. the familiar straight lines of Euclidean geometry. Also, according toNewtonian mechanics, Eq. (117) is the trajectory of a free particle withvelocity vj0. We now take the Euclidean metric and conformally transformit according to (in Cartesian coordinates)

gij = n2δij . (118)

The transformed metric generates a new covariant derivative for which theconnection coefficients are determined from Eq. (115). In Cartesian coordi-nates we have

Γijk =1

2δim (δmj∂k + δmk∂i − δjk∂m) [lnn] . (119)

The new covariant derivative produces new geodesics that satisfy

d2xi

dt2+ Γjki

dxk

dt

dxi

dt= 0 . (120)

25

Equation (119) tells us how the new geodesics arise from the old. The firsttwo terms on the right hand side of Eq. (119) correspond to a reparameter-ization of the original geodesics from time t to a dilated time, T (t). This isseen by transforming Eq. (113) with Γijk = 0 to one in which a ‘bad’ clockT (t) is used:

d2xi

dt2= T

d

dT

(Tdxi

dT

)= T

[Td2xi

dT 2+(∂j T

) dxidT

dxj

dT

]= 0 , (121)

or

d2xi

dT 2+ ∂j

(ln T

) dxidT

dxj

dT= 0 , (122)

or

d2xi

dT 2+

1

2δim (δmj∂k + δmk∂i)

(ln T

) dxjdT

dxk

dT= 0 . (123)

Thus we see that indeed the time reparameterization generates the first twoterms on the right of Eq. (119) provided T = n.

The final term of Eq. (119) is the most interesting. It corresponds tobending the original geodesic, when there is a gradient of n = T as we movebetween neighbouring geodesics. When ∂xn = 0, for example, geodesicsparallel to the x-axis in the Euclidean plane deform to ones satisfying

d2x

dt2+ 2 [∂y (lnn)]

dx

dt

dy

dt= 0 , (124)

d2y

dt2+ [∂y (lnn)]

[(dy

dt

)2

−(dx

dt

)2]

= 0 , (125)

Clearly, the solutions to Eqs. (124)-(125) will describe a geodesic that de-viates and curves away from the x-axis as illustrated in Fig. 1 We thusconclude that conformal transformation of the spatial metric has two ef-fects: to reparameterise the geodesics so that n can be interpreted as arefractive index, and to deform the geodesics in a potentially useful way.

10.3 Direct Transformation of Geodesics

The previous subsection showed how geodesics are morphed by the param-eter n(r). Here we note that without reference to n(r), any coordinatemorphing, i.e. x(p)→ x[ϕ(p)] transforms the connection coefficients to (see[16], p.36)

Γlmn = (ϕ∗)li (ϕ∗)jm (ϕ∗)kn Γijk + (ϕ∗)

lj,k (ϕ∗)jm (ϕ∗)kn , (126)

where (ϕ∗)il = ∂xi/∂xl, and (ϕ∗)il (ϕ∗)lj = δij (see Appendix B).

26

t=-2 t=-1 t=0 t=1 t=2k=-2

k=-1

k=0

k=1

k=2

t=-2 t=-1 t=0t=1

t=2

x

y

Figure 1: Illustration of the action of −12δimδjk (∂m lnn) (cf. Eq. (119)) in

bending geodesics.

10.4 Combining Conformal-Transformation with Direct Trans-formation

The key result is now obtained by combining Eq. (115) with Eq. (126) toyield

Γlmn = (ϕ∗)li (ϕ∗)jm (ϕ∗)kn Γijk + (ϕ∗)

lj,k (ϕ∗)jm (ϕ∗)kn

=1

2gla (∂mgan + ∂ngam − ∂agmn) , (127)

where gij is determined from its Cartesian form given in Eq. (118). Thislinks some desired deformation (ϕ : p→ ϕ(p)) of the geodesics of a homoge-neous medium to the index distribution n(r), of an inhomogeneous mediumrequired to achieve that deformation.

27

10.5 Example: Acoustic Ray Cloak

As an application of Eq. (127) we consider again the 2-D cloaking trans-formation of Section Eq. (86) 8. In a polar coordinate basis, the non-zeroconnection coefficients for Euclidean space are given by

Γrθθ = −r , Γθrθ = Γθθr = r−1 . (128)

Also, from Eq. (86), we note

(ϕ∗)rr =∂r

∂r=

r

(r2 − a2)1/2, (ϕ∗)

rr,r =

a2

r3. (129)

The Cartesian metric gij = n2δij becomes gmn = n2diag(1, r2) in polarcoordinates, so that the non-zero connection coefficients are

Γrrr = ∂r (lnn) , Γrθθ = −r [1 + r∂r (lnn)] , Γθrθ = r−1 + ∂r (lnn) . (130)

Eq. (127) for Γrrr then turns out to be

d (lnn)

dr=

a2

r(r2 − a2), (131)

which readily integrates up to

n = n0

(1− a2

r2

)1/2

, (132)

where n0 is the index of the initial homogeneous medium. Note that theindex becomes zero at r = a reflecting the fact that transport aroundthe edge of the cloak must correspond to the null transit time across thepre-transformed origin. There is no causality violation as we are dealingwith steady-state. Therefore to acoustically cloak an object immersed in amedium characterized by mass density ρ0 and bulk modulus λ0, the mediummust be locally modified so that(ρ

λ

)=

(ρ0λ0

)(1− a2

r2

), (133)

where the tilde has been dropped. The resultant cloaking function is il-lustrated in Fig. 2. The optical index cloak defined by Eq. (132) might

28

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x

y

Figure 2: Acoustic ray cloaking function produced by the medium distribu-tion described by Eq. (133). The red tick marks on each geodesic indicatetime.

be achieved by surrounding the object to be cloaked with a semiconduc-tor4 with graded free-carrier density, N(r). The refractive index is thenapproximately described by

n =

(1−

ω2p

ω2

)1/2

, (134)

where ω is the angular frequency of the incident light, and ωp =(Ne2/mε0

)1/2is the plasma frequency determined from the free-carrier density, N . Equa-tion (134) bears a striking similarity to Eq. (132). Provided always thatω > ωp, the refractive index is real and less than unity. For an acousticcloak, the initial homogeneous medium cannot be vacuum, and the requiredradial profile of the contrast in mass density and/or bulk modulus is deter-mined from Eq. (133). Note also from Eq. (132) and Fig. 2 that the localphase velocity must become infinite at the cloak boundary r = a. Again thisis unsurprising as the mapping of Eq. (86) takes the origin point, crossed inzero time by the original geodesics, to the points on the circle r = a.

4Note the structure must be band-gap engineered to prevent radial diffusion of the freecarrier density.

29

11 Simple Covariant Acoustics

Here we present a simple but general description of scalar acoustics ina covarient-compatible form similar to tensor descriptions of electromag-netism. As noted previously (see Section 6) we use first order equations tomodel our wave mechanics [17], so that a pair of them are needed (in concertwith constitutive or state equations) are needed to generate wave behaviour.If desired, the first order equations can be substituted inside one anotherto give a familiar second order form, but in fact the first order formulationis less restrictive. It is also more straightforward to apply a transformationscheme to the first order system.

A typical description of linearized acoustic waves [18] uses an an isotropicpressure field p, and fluid velocity field v, linked by a bulk modulus λ anda density ρ. Here we extend the theretical description to explicitly allowfor a momentum density field V and a dimensionless scalar field P ; butfor now choose to neglect the role of material velocity. In this way we canwrite down a simplified pseudo-acoustic (p-acoustic) form that is equivalentto a theory describing linear (i.e. perturbative) acoustic waves on a sta-tionary background medium. The advantage for our purposes, however, isto emphasize both the similarities when compared to the electromagneticdescription, which also uses two pairs of related quantities. Because we canuse the same mathematical skeleton, transformation optics (T-optics) andtransformation acoustics (T-acoustics) then are merged to become the sameprocedure. Only the construction of a particular transformation device (T-device) will differ between the two, being dependent on the expression ofoptics or acoustics within the overall machinery.

Moreover, p-acoustics is a more general model than the comparable tra-ditional acoustic one, which typically reduces the equations back down to pand v – or even just a second order wave equation in p.

11.1 Vector Calculus Description

To make wave equations in the way we want, we need to link pairs of quan-tities by spatio-temporal differential equations, as well as by constitutiverelations. Note that whilst a time derivative leaves a scalar as a scalar,and a vector as a vector, a spatial derivative can either promote a scalarto a vector, using the gradient, or demote a vector to a scalar, using thedivergence5.

5Another alternative is the curl operation, which takes a vector to a vector, as seen inEM.

30

Since the divergence ∇ · demotes the velocity field v to the change in adimensionless scalar P , and the gradient operation ∇ promotes the scalar pto a temporal rate of change of vector momentum density V , the p-acousticsystem is

∂tP (r, t) = ∇ ·v(r, t) +QP (r, t), (135)

∂tV (r, t) = ∇ p(r, t) +QV (r, t), (136)

where QP is a dimensionless source term for P , andQV a momentum densitysource term for V .

These are not yet wave equations, however, since the two differentialequations are independent. To link them, and create a wave theory, wedefine constitutive relations. In the most general case these are

P = λ−1p−α · v (137)

V = βp+ ρv, (138)

where λ is a scalar bulk modulus relating the scalar p to the scalar P , and ρis a matrix of densities relating the velocity field v to the momentum densityV . In addition there may also be unconventional couplings between the ve-locity field v and dimensionless P parameterized by a vector α, and anotherbetween the pressure field p and the momentum density parameterized bya vector β.

11.2 Tensor Description

Whilst the vector calculus description above is accessible to a wide audience,we can gain considerable benefits by re-expressing it in a tensor form. Thisoffers similar advantages to the re-expression of EM’s Maxwell’s equations intensor form. For our purposes, the main advantage is that a transformationacoustics based on this prescription follows the same mathematical steps asfor transformation optics. That is, one mathematical methodology can beused for both T-acoustics and T-optics, there is no need for separate andincompatible approaches.

So, just as for EM, we can now embed the field components P,v intoa tensor Fαβ, and the p,V fields into a tensor density Gαβ; as well aspopulating a constitutive tensor χ with ρ, λ,α,β appropriately. The p-acoustic wave mechanics will now look the same as EM, being

∂αFαβ = Jβ, ∂βG

αβ = Kα . (139)

31

with constitutive relations

Gαβ = χαβγδFγδ, (140)

and source terms Jα,Kα; but the internal structure – how the tensors arepopulated – will differ.

To see how the vector calculus quantities populate the tensor description,we utilise the Cartesian (t, x, y, z) coordinates. Then the tensor fields Fαβ

and Gαβ can be written out as matrices

Fαβ =

P 0 0 0−vx 0 0 0−vy 0 0 0−vz 0 0 0

, Gαβ =

0 −Vx −Vy −Vz0 p 0 00 0 p 00 0 0 p

, (141)

together with the source terms J0 = QP andK = (Ki) = Qν . In particular,note the repeated appearance of the single quantity p in three elements ofGαβ.

It is instructive to compare these p-acoustic field tensors with their EMcounterparts, which are antisymmetric, have no repeated elements, and areonly zero on the diagonal. In order to make the merger between acousticsand EM a seamless as possible, we use an unconventional choice of fieldtensor pair; i.e. the usual F but the dual of G:

Fαβ =

0 −cBx −cBy −cBzcBx 0 Ez −EycBy −Ez 0 ExcBz Ey −Ex 0

, (142)

Gαβ =

0 Dx Dy Dz

−Dx 0 cHz −cHy

−Dy −cHz 0 cHx

−Dz cHy −cHx 0

. (143)

One consequence of this, but not further discussed here, is that the matrixrepresentation of the EM constitutive tensor becomes (block) off-diagonal.

11.3 Constitutive Relations

As with EM, the constitutive tensor has four indices, making a rewriting asa matrix problematic; so we follow the same approach as EM and compactit: we stack the non-zero elements of the Fαβ to make a vector FA, anddo likewise for Gαβ to make a vector GB, However, because of the repeated

32

element p in the Gαβ tensor, the situation is less straighforward than forEM.Explict form: The most explict Compaction contains redundant informa-tion, but will reveal post transformation inconsistencies most clearly. To dothis we choose to compact the G tensor and preserve independently each ofthe diagonal elements Gxx, Gyy, and Gzz – even though each is simply thesame quantity p. This helps us identify transformations or deformations ofthe coordinates that give rise to material properties incompatible with thevector model; i.e. where Gxx, Gyy and Gzz are not all identical. Thus theFA and GB column vectors are[

FA]

= [P, vx, vy, vz]T , (144)[

GB]

= [p, p, p, Vx, Vy, Vz]T . (145)

The matrix representation of χαβγδ is not square, but still has the upperindex denoting columns, and the lower index denoting rows. It has a matrixconstitutive relation given by[

GA]

=[χAB

] [FB], (146)

Gxx

Gyy

Gzz

Gxt

Gyt

Gzt

=

λ αx αy αzλ αx αy αzλ αx αy αzβx ρxx ρxy ρxzβx ρyx ρyy ρyzβx ρzx ρzy ρzz

F tt

F tx

F ty

F tz

. (147)

At this point we might hope to generalize away from our original vectormodel and replace the scalar λ with a vector, and the vector α into a matrix;but this would emphatically not be equivalent to the original p-acousticequations in eqns. (135), (136). Further, if we were to try to design a T-device by transforming eqn. (147), whatever transforming we pick wouldhave to preserve the repeated rows of

[χAB

]intact.

Abbreviated form: We can choose to use the fact that the elements Gxx,Gyy and Gzz are all just p, and so write a more compact vector for G usingonly (e.g.) Gxx. We now have a 4×4 constitutive matrix, with field columnvectors GA, FB, [

GA]

= [p, Vx, Vy, Vz]T (148)[

FB]

= [P, vx, vy, vz]T . (149)

33

The matrix representation of χαβγδ has the upper index denoting columns,and the lower index denoting rows, so that the matrix constitutive relationis [

GA]

=[χAB

] [FB]

(150)

Gcc

Gxt

Gyt

Gzt

=

λ αx αy αzβx ρxx ρxy ρxzβx ρyx ρyy ρyzβx ρzx ρzy ρzz

F tt

F tx

F ty

F tz

. (151)

Note that we must remember that the Gcc element can be freely substitutedby Gxx, Gyy or Gzz; this acts as an implicit constraint on allowed transfor-mations, which are only allowed to transform x, y, and z equivalently. Thusany allowed non-isotropic wave behaviour has to derive from α, β, or ρ, andnot from a transformation.

11.4 Covariance

This implementation of acoustic waves (p-acoustics) is written down in acovariant way; thus the mathematical description is intrinsically covariant –the form of the equations remains invariant under Lorentz transformations.However, note that although the mathematics is covariant, the underlyingphysics of acoustics is not, since the constitutive parameters λ, ρ are notframe independent, unlike the ε0 and µ0 in EM. Nevertheless, the covariantdescription mimics (as far as possible) the tensor representation of Maxwell’sequations, which emphasises the similarity of the transformation mechanicalmachinery used to design T-devices in either T-acoustics or T-optics.

11.5 Pentamode Acoustics

Stress/shear acoustic waves can be modelled using the “pentamode mate-rial” model [9], based on 6-vectors for stress σ and strain ε, linked in aconstitutive relation by a 6x6 matrix C of elastic moduli. This theory re-duces to an orientation vector s and two scalars that scale it to get stressand strain vectors. As a result, for small amplitude disturbances, the modelcan be reduced to a form suggestively similar to p-acoustics, but with theaddition of an extra matrix S encoding the anisotropic material parameters.

Thus, just as the p-acoustics vector calculus eqns. (135) and (136) canbe inferred from traditional acoustics models, similar forms can be generated

34

for pentamode acoustics. The relevant vector calculus re-representation ofNorris’s pentamode wave equations are

∂tV = −∇ ·Sp , (152)

∂tP = −S : ∇v , (153)

with

V = ρv + βp , (154)

P = λ−1p+α · v , (155)

where the pseudo-pressure p is derived from a negative single stress termas p = −λTr(Sε). Note that this is a more general model than Norris’spentamode description, and is not an approximation of it.

In an indexed notation, these differential and constitutive equations are

∂tVi = −∇k Skip , (156)

∂tP = −Ski∇i vi , (157)

with

Vi = ρjivj + βip , (158)

P = λ−1p+αjvj . (159)

Taking S to be a constant, i.e. that these equations describe waves ina uniform medium. Since S should have an inverse S−1 = T, we can pre-multiply the PM model’s ∂tVi equations,

T il∂tVi = −T il∇k Skip , (160)

∂tTilVi = −∇k T ilSkip , (161)

∂tVl = −∇k δklp , (162)

∂tVl = −∇l p , (163)

leaving us with a differential equation for V with identical structure to thebasic p-acoustics ∂tV equation; with

Vl = T ilVi , Vi = S li vl . (164)

Likewise the PM model’s ∂tP equation can be adapted,

∂tP = −Ski∇i vk (165)

∂tπ = −∇i Skivk (166)

= −∇i vi , (167)

35

and again this leaves us with a differential equation for P with identicalstructure as the basic p-acoustics ∂tP equation, with

vi = Skivk; vk = T ik vi . (168)

So we see that the two vector wave properties v and V have been trans-formed. This affects the vector constitutive relation as follows:

Vi = ρ ji vj + βip , (169)

T ilVi = T ilρji T

mk vm + T ilβip , (170)

Vi = ρ mi vm + βip , (171)

with ρ mi = T ilρ

ji T

mk , (172)

and βi = T il βl . (173)

It also affects the scalar constitutive relation,

P = λ−1p+αivi , (174)

= λ−1p+ αiT ji vj , (175)

= λ−1p+ αj vj , (176)

with αj = T jiαi . (177)

Thus the transformed PM-acoustics equations are

∂tV = −∇ p , (178)

∂tP = −∇ ·v , (179)

V = ρv + βp , (180)

P = λ−1p+ α · v . (181)

Thus the PM acoustic equations codify the same wave behaviour as theordinary p-acoustic ones, albeit with the matrix S encoding a skew betweenthe physical velocity field v and the effective one v; and betwwen the physicalmomentum density and the effective one V . Further, the density matrix ρ,and cross-couplings α and β also have to be transformed to match.

11.6 Transformations

The transformations used in designing T-devices are usually expressed ascoordinate transformations between an original set α with a known solutionof the wave equations (usually a flat space solution), and a new set α′ in

36

which the solution is distorted to form the T-device, such as one with acentral hole for use as a cloaking T-device. We can write down the Jacobianof the transformation between α and α as

Tα′

α =

[∂α′

∂α

], (182)

so that the field density G and field F transform as

Gα′β′

= {det(T )}−1 Tα′α T

β′

β Gαβ, Fα

′β′= Tα

′α T

β′

β Fαβ, (183)

and the rules of tensor calculus ensure that both the differential equationand constitutive relation works in any coordinate system:

∂αFαβ = Jβ, ∂βG

αβ = Kα ⇔ ∂α′Fα′β′

= Jβ′, ∂β′Gα

′β′= Kα′

(184)

Gαβ = χαβγδ Fγδ ⇔ Gα

′β′= χα

′β′

γ′δ′ Fγ′δ′ , (185)

where

χα′β′

γ′δ′ =1

det(T )Tα

′α T

β′

β χαβγδ T

γγ′T

δδ′ . (186)

Of course there is an important caveat here: although the transformed

constitutive relation and tensor χα′β′

γ′δ′ are mathematically valid, the new

χα′β′

γ′δ′ might not have a form allowed by the physical model specified in theoriginal theory – i.e. the physical χ is not a tensor, despite the mathematicalnotation. We will see how this happens in practise in the example thatfollows.

11.7 An example: the spatial carpet cloak

It is instructive to consider an example showing how a particular T-devicemight be implemented. Consider a carpet cloak, a means of concealing abump in a planar reflective surface. Here this lies flat on the y − z plane atx = 0, and reflects waves coming from the negative x half-space. We decideto transform a localized region of space reaching no higher than x = −Habove the plane and no further than y = ±Λ sideways. This is done tocreate a triangular cloaked region h high that is undetectable to any exteriorobserver, as shown on fig. 3.

37

x

y

h

H

Figure 3: The (a) spatial carpet cloak. Continuous lines show the actualray trajectories, dotted lines the apparent (illusiory) ray trajectories.

The deformation that achieves these T-devices is applied (only) insidethe shaded regions on fig. 3, and is as follows:

t′ = t, y′ = y, z′ = z, (187)

x′ = Rx− Λ− y. sgn(y)

Λh = Rx+ h− rsy, (188)

where R = (H − h)/H is the ratio of actual depth to apparent depth, andr = h/Λ, and s = sgn(y).

The effect of this deformation from eqn. (188) is given by the tensorLα

′α , which specifies the differential relationships between the primed and

un-primed coordinates, where α ∈ {t, x, y, z} and α′ ∈ {t′, x′, y′, z′}. If welet columns span the upper indices, and rows the lower, then in matrix formwe have

[Lα

′α

]=

[∂α′

∂α

]=

1 0 0 00 R 0 00 −rs 1 00 0 0 1

, (189)

and note that det(Lα′α ) = R.

For p-acoustic transformations, we need two compactified transformationmatrices, with correctly arranged ordering. This is less straightforward thanin EM, where rows and columns are indexed by the same list of coordinatepairs. To transform Gαβ, compactly indexed by cc, xt, yt, zt, we need [NA′

A ][GA

′]

=[NA′A

] [GA]

(190)Gc′c′

Gx′t′

Gy′t′

Gz′t′

=

{R2, 1

}0 0 0

0 R −rs 00 0 1 00 0 0 1

GccGxtGytGzt

. (191)

38

Note that the top left corner entry has two entries, i.e. R2 if the cc indexstands for xx, but 1 if yy or zz. This is the first explicit sign that alltypes of T-device are not possible with p-acoustics, and that in most casesrestrictions will be imposed by the scalar nature of p-acoustics itself. Forthe moment we will continue with the ambiguity, so that we can determinehow to resolve this at the final step when all consequences of the T-devicedesign will be visible.

To transform Fαβ, compactly indexed by tt, tx, ty, tz, we need [MA′A ][

FA′]

=[MA′A

] [FA]

(192)Ft′t′

Ft′x′

Ft′y′

Ft′z′

=

1 0 0 00 R −rs 00 0 1 00 0 0 1

FttFtxFtyFtz

, (193)

The compact constitutive matrix transforms like,

[χA′

B′ ] =[NA′

A ][χAB][MB′B ]T

det(L), (194)

where a transpose operation has to be applied to the M matrix to ensure itrepresents the tensor summation correctly.

Hence

[χA′

B′ ] =[NA′

A ][χAB][MB′B ]T

det(L)=

{λR, λ/R} 0 0 0

0 ρ(R2 + r2)/R −rsρ/R 00 −rsρ/R ρ/R 00 0 0 ρ/R

.(195)

Just as for the N transforation matrix, we can see an ambiguity in constitu-tive matrix, i.e. in the modulus λ′ which must be both λR and λ/R. Thiskind of problem is intrinsic to any T-device design for a scalar wave thatscales different directions differently.

One point to note is that although the waves will propagate differentlydepending on our choice for λ′, this is just a scalar multiple applied to theanisotropic properties specified by ρ′, i.e. the 3x3 lower-right submatrix ofχ′ dependent on the reference density ρ. This means that although the raytrajectories will be correctly preserved in our (necessarily imperfect) carpetcloak, the wave speeds will not. Further, we cannot perfectly impedancematch across the free-space to cloak interface, leading to reflections.

39

Only with non-trivial acoustic waves can we rescue this situation, atleast to a degree; e.g. with pentamode waves we might restrict ourselvesto cloaking in the 2D xy plane with the wave oscillations at right anglesalong z. We can achieve this means of a careful choice of the matrix S

defining the stress-strain orientation, thus isolating the wave dynamics fromthe ambiguity in the constitutive matrix.

Alternatively, we could design a corner-cloak, the cloaking counter partto a corner cube reflector, but this is a rather trivial device, with each ofx, y, z being scaled by the same amount with no cross-coupling, i.e. c′ =Rc+ h.

11.8 Discussion

The above description makes two important points. First, that acoustics andEM can use a unified description in which a single transformation theorycan be defined without reference to either acoustics or optics specifically –although the implementation of any given T-device will be type dependent.To clarify this statement, for acoustics, which is not a priori pre-metric,there seem to be essentially three ways of defining a transformation scheme:

1. Instead of the exact theory, take an approximation, such as the RayAcoustics of Section 10.

2. Despite the fact that the equations of the exact acoustic theory arenot premetric, transform the metric dependent equations and try tofind a transformation scheme at least for certain specific cases. Thisis what was done in Sections 6-8.

3. Since the equations of an exact acoustics theory are not premetric, wecan aim to construct a new model which is premetric, as is done inthis section. While this model is capable of being adapted directlyfor describing many types of acoustic wave in practical circumstances,it is important to note that it could only be mapped onto an exactmetric-dependent acoustics in specific cases.

Secondly, the pentamode theory as finally applied by Norris as an exampleof an acoustic wave for which more general T-acoustic devices might bepossible is not so – it is mathematically identical to ordinary p-acoustics.The similarity is guaranteed by the symmetry (and hence invertibility) ofthe stress-strain orientation matrix S.

It is worth noting that there are other useful properties that follow fromthis p-acoustic wave theory. We can construct flux vectors analogous to the

40

Poynting vector of EM, which are simple products of one choice of scalarquantity (i.e. p or P ) and one choice of vector quantity (i.e. v or V ). Just asfor EM, there are four possibilities, each with their own continuity equation.Also, scalar wave potential(s) can be defined, with wave quantities beingextracted by a time derivative or a gradient operation; as can vector wavepotential(s), with wave quantities being extracted by a divergence operationor a time derivative.

As a final note, consider the linearized acoustics with anisotropic massas discussed at the beginning of this report. The wave theory there is basedon eqns. (19), (3), (5), i.e.

Dkσkm = gmnρ

nlul,

enk =1

2(Dkuk +Dnuk) ,

σ ji = C jkl

i ekl.

Now the usual linearized acoustics situation involves a bulk backgroundmedium with spatially and temporally constant parameters. This allowsus to shift the time derivatives to define a material velocity field vk and amomentum density Vm, and to redefine the density using the metric severaltimes to raise and lower indices, to get

vk = ∂tuk, (196)

Vm = gmnρnl∂tu

l = ρmlvl = ¯ρ k

mvk. (197)

These allow us to rewrite that wave theory in a manner similar to thatused here for both p-acoustics, PM acoustics, and even EM; i.e. as pairs ofquantities linked by constitutive relations, and cross-linked by differentialequations. Taking the time derivative of the enk equation, inverting the σ j

i

equation using a T such that C jkli T ijkl = 1, we can achieve this. We find

that the wave theory can be written using the differential equations

∂tVm = Dkσkm ,

∂tenk =1

2(Dk +Dn) vk, (198)

with the constitutive equations

Vm = ¯ρ kmvk (199)

ekl = T ijklσji . (200)

41

This theory interlinks a pair two-index quantities σ km , enk to a pair of single-

index ones Vm, vk; we might choose to express it as pairs of matrices andvectors. A new feature is the non-standard spatial derivative part in the∂tenk equation, which here is a sum of two parts.

This demonstrates that the type of approach used in this section is valideven for treating acoustic wave theory of the kind discussed in the earlierpart of this report.

A Anisotropic Mass Density: on the Correct In-dex Positioning of the Mass Tensor

Consider a 4-dimensional setting. Conservation of particle (baryon) numberreads

dJ = 0 , (201)

where J is the baryon current density 3-form, and it is a twisted quantity.Note that in Eq. (201), the symbol d represents the exterior derivative. Nowimpose the condition that

ncJ = 0 , (202)

where n is the tangent vector to the baryons’ trajectories (i.e. n is the4-velocity of the baryons). Note that in Eq. (A), the symbol c representsthe interior product.

nαJαβγ = 0 . (203)

Note that in Eq. (203), the greek indices run over the four dimensions ofspacetime. Now define

Jα :=1

3!εαβγδJβγδ ⇒ Jαβγ = εαβγδJ

δ , (204)

where εαβγδ is the Levi-Civita symbol. The quantity Jα is therefore a twistedvector density of weight +1. Then

nαεαβγδJδ = 0 ⇔ εαβγδn

γ Jδ = 0 . (205)

Contract with 12εµναβ as follows:

1

2εµναβ εαβγδn

γ Jδ = 0 . (206)

Recall that 12εµναβ εαβγδ = δµνγδ , and obtain

δµνγδ nγ Jδ = 0 ⇔ n[µ J ν] = 0 ⇔ Jν = ηnν , (207)

42

where η is a twisted scalar density of weight +1, and represents the densityof baryons (see [12]). In 3-dimensions, the density of mass is a twisted 3-form. Thus, in 4-dimensions, there must exist a current density 3-form K,which is twisted. Hence the mass tensor is a linear map relating K and J .More precisely, in the non-relativistic limit in which mass does not dependon speed,

K = m(J) . (208)

Since J is dimensionless, and K has units of mass, then [m] = [mass]. Also,m is untwisted. In tensor-index notation

Kαβγ =1

3!m µνλαβγ Jµνλ , (209)

where m µνλαβγ is a true tensor. Note that:

m µνλ[αβγ] = m µνλ

αβγ , (210)

andm

[µνλ]αβγ = m µνλ

αβγ . (211)

Define:

Kα :=1

3!εαβγδKβγδ ⇒ Kαβγ = εαβγδK

δ . (212)

Evidently,

εαβγδKδ =

1

3!m µνλαβγ εµνλτ J

τ . (213)

Contract through by 13!ε

αβγ :

1

3!εθαβγ εαβγδK

δ =

(1

3!

)2

εθαβγm µνλαβγ εµνλτ J

τ , (214)

so thatKθ = mθ

τ Jτ , (215)

where mθτ is a true

[11

]tensor, defined via

mµν :=

(1

3!

)2

εµαβγm ηθταβγ εηθτν . (216)

AlsoKµ = mµ

ν Jν = ηmµ

νnν , (217)

or [ηm0

0 00 ηmi

j

]= ηmµ

ν . (218)

The term on the bottom right of this equation is what Norris [9] calls√gρij .

43

B Push-forward (ϕ∗) and Pull-back (ϕ∗) Notation,

Central to any transformation scheme is the mapping of one manifold M ,to another manifold N , i.e. ϕ : M → N . For various technical reasons, weassume that the mapping is a diffeomorphism, so that ϕ is invertible, andM ∼ N (i.e. M is diffeomorphic to N). Let xi represent coordinates inM and xj coordinates in N . For a given vector field u in M , ϕ induces avector field u = ϕ∗u in N (the push-forward of u) via the standard rules forcoordinate transformation:

(ϕ∗u)j = (ϕ∗)jiui , (219)

where (ϕ∗)ji = ∂xj/∂xi, and the summation convention has been used.

Likewise, since ϕ is a diffeomorphism, the pull-back ϕ∗w of a vector w ∈ Nmay be defined via

(ϕ∗w)i = (ϕ∗)ijwj , (220)

where (ϕ∗)ij = ∂xi/∂xj , and (ϕ∗)ik(ϕ∗)kj = δij . In this notation gmn =

(ϕ∗g)mn are the pushed forward components of a (2, 0) tensor whose com-ponents in M are gij . They are given by

(ϕ∗g)mn = (ϕ∗)mi (ϕ∗)

nj g

ij . (221)

For a mixed second rank tensor6 in M whose components are given by ρij ,the corresponding pushed forward components in N are ρmn = (ϕ∗ρ)mn, andare given by

(ϕ∗ρ)mn = (ϕ∗)mi (ϕ∗)jn ρ

ij . (222)

Finally7, a function ξ on M may be pushed forward to ξ = ϕ∗ξ on N via

ϕ∗ξ [ϕ(p)] = ξ(p) , (223)

where p ∈M .

C Proof of Inertial Cloaking Theorem

This appendix gives the proof of the theorem stated in Section 6.

6For a mixed tensor the distinction between push-forward and pull-pack is blurred, andso the following definition should be considered a formal one which is only defined whenϕ is a diffeomorphism.

7Again, this definition relies on the invertibility of ϕ.

44

Part 1:

λ′ξ′ =


ξ


= ξλ .

Part 2:

g′ijρ′jku′k = g′ij


g′jmgmnρnku′k =


ginρnk


ük

= gij ρjkük.

Henceg′ijρ

′jku′k = gij ρ

jkük.

Part 3: On account of parts 1 and 2

∂i(λ′ξ′)

= g′ijρ′jku′k ⇒ ∂i

(λξ)

= gij ρjkük.

Part 4:

ξ′ =1√

det g′ij

∂

∂xp

(√det g′iju

′p)

Hence √det gij√det g′ij

ξ =1√

det g′ij

∂

∂xp

(√det g′ij


up

),

ξ =1√

det gij

∂

∂xp

(√det gij u

p).

Part 5:

∂i(λ′ξ′)

= g′ijρ′jku′k ⇒ ∂i

(λξ)

= gij ρjkük ⇒ ∂i (λξ) = gijρ

jku

k ,

and

ξ′ =1√

det g′ij

∂

∂xp

(√det g′iju

′p)

⇒ ξ =1√

det gij

∂

∂xp

(√det gij u

p)

⇒ ξ =1√

det gij

∂

∂xp

(√det giju

p), QED.

45

D Proof of Theorem 2 (Section 7.1)

Here we give the proof of the second theorem given in Section 7.1 (cf. Eqs.(81)-(85)). As previously, we divide the proof into several steps.

Step 1

λ′ξ′ =


λ


ξ = λξ . (224)

Step 2

S′ijρ′jkg′klS′lmw

′m =


SiaρabgbcScm


¨wm, (225)

Step 3

Combining Step 1 and Step 2 implies that if

∂i(λ′ξ′)S′ijρ

′jkg′klS′lmw

′m , (226)

then

∂i(λξ)Sij ρ

jkgklSlm ¨w

m. (227)

Step 4

If

ξ′ =1√

det g′pq∂i

(√det g′pqw

′i), (228)

then √det gpq√det g′pq

ξ =1√

det g′pq∂i

(√det g′pq


wi

). (229)

Hence

ξ =1√

det gpq∂i

(√det gpqw

i). (230)

Step 5

46

Combining the results of Step 3 and Step 4 yields

∂i(λ′ξ′)

= S′ijρ′jkg′klS′lmw

′m , (231)

⇔ ∂i(λξ)

= Sij ρjkgklSlm ¨w

m, (232)

⇔ ∂i (λξ) = SijρjkgklSlmw

m , (233)

and

ξ′ =1√

det g′pq∂i

(√det g′pqw

′i), (234)

⇔ ξ =1√

det gpq∂i

(√det gpqw

i), (235)

⇔ ξ =1√

det gpq∂i

(√det gpqw

i). (236)

QED.

References

[1] J.B. Pendry, D. Schurig and D.R.Smith, “Controlling electromagneticfields”, Science, 312, 1780-1782, (2006).

[2] U. Leonhardt, “Optical conformal mapping”, Science, 312, 1777-1780,(2006).

[3] D. Schurig, J.J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A.F. Starr and D. R. Smith, Science, 314, 977-980, (2006).

[4] R. T. Thompson, S. A. Cummer and J. Frauendiener, “A completely co-variant approach to transformation optics”, J. Opt., 13, 024008, (2011).

[5] R. T. Thompson, S. A. Cummer and J. Frauendiener, J. Opt., “Gener-alized transformation optics of linear materials”, 13, 055105, (2011).

[6] G. W. Milton GW, M. Briane and J. R. Willis, “On cloaking for elastic-ity and physical equations with a transformation invariant form”, NewJ. Phys. 8, 248, (2006).

[7] S. A. Cummer and D. Schurig. “One path to acoustic cloaking”, NewJournal of Physics, 9, 45, (2007).

47

[8] A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, “Full-Wave Invisi-bility of Active Devices at All Frequencies”, it Commun. Math. Phys.,275, 749789, (2007).

[9] A. N. Norris, “Acoustic cloaking theory”, Proc. Roy. Soc. A, 464, 2411-2434, (2008).

[10] A. N. Norris, “Acoustic metafluids”, J. Acoust. Soc. Am., 125, 839-849,(2009).

[11] F. Hehl and Y. Obukhov, Foundations of Classical Electrodynamics, vol.33 of Progress in Mathematical Physics. Birkh’auser, Boston, 2003.

[12] A. Yavari, Journal of Mathematical Physics, 49, 022901, (2008).

[13] Amenzade, Y., Theory of Elasticity, MIR publishers, (1979).

[14] L.D. Landau and E.M. Lifschitz, The classical theory of fields, Volume2, Butterworth-Heinemann, (1975).

[15] Milton, G.W. and Cherkaev, A.V., “Which elasticity tensors are realiz-able?”, Journal of Engineering Materials and Technology, 117, 483–493,(1995).

[16] E. J. Post, Formal Structure of Electromagnetics, Dover, (1997).

[17] R. Geroch, Partial Differential Equations of Physics, Proceedings ofthe Forty-Sixth Scottish Summer School in Physics, eds. G. S. Hall; J.R. Pulham, 1996.

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