Comment on "Design of acoustic devices with isotropic material via conformal transformation" [Appl. Phys. Lett. 97, 044101 (2010)]

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by  Andrew N. Norris

Comment on “Design of acoustic devices with isotropic material via
conformal transformation” [Appl. Phys. Lett. 97, 044101 (2010)]
Andrew N. Norris1
Mechanical and Aerospace Engineering, Rutgers University, Piscataway NJ 08854
(Dated: 9 January 2012)
The paper presentsincorrectformulasforthe density andbulk modulus under a conformaltransformationof
coordinates. The fault lies with an improper assumption of constant acoustic impedance.
The wave equation for an acoustic medium with uni- for constant c > 0. These are the correct expressions
form density ρ0 and bulk modulus λ0 is, assuming time for the bulk modulus and density in the transformed
2 dependence e−iωt, medium.
1 The general theory of transformation acoustics allows
0 ∇·∇p+ω2(ρ0/λ0)p=0. (1) for the possibility of non-unique expressions for the ma-
2 terial parameters in the transformed domain2. The ma-

n Renetal.1demonstratedthatundertheconformaltrans- terial descriptions range from what is known as inertial
a formation xi → xi′, z(= x1 +ix2) → ζ(z), eq. (1) be- fluids, withanisotropicdensityandscalarbulk modulus,
J
comes, for p′(xi′)=p(xi), topentamodebehaviorwithelasticconstitutiveresponse
6
andgenerallyanisotropicdensity,plusaspectrumofpos-
] ∇′·∇′p′+ω2(ρ′/λ′)p′ =0, (2) sibilitiesinbetween3. Intheinertialfluidlimitthetrans-
h formedbulk modulus anddensitytensorare(eq. (2.8)of
p Ref. 2)
- where
ath λ′/ρ′ =|ζ′(z)|2λ0/ρ0. (3) λ′ =λ0detF, ρρρ′ =ρ0(detF)(FFT)−1, (9)
m where Fi′i = ∂xi′/∂xi. At the other extreme, the pen-
[ The authors1 then imposed the constraint that the tamode material is defined by a density tensor ρρρ and a

1 impedance in the transformed domain remains un- fourth orderelasticity tensor with components Cijkl (eq.
v changed, i.e. (4.3) of Ref. 2)
0
7 λ′ρ′ =λ0ρ0, (4) Cijkl =λ′SijSkl, ρρρ=Sρρρ′S, (10)
4
1 where λ′,ρρρ′ are given by (9) and S=ST is positive def-
. which together with (3) implies the identities (eq. (6) inite satisfying divS = 0, but otherwise arbitrary. For
1 of1)
the special case of a conformal transformation we have
0
2 detF = |ζ′(z)|2, FFT = |ζ′(z)|2I where I is the iden-
′ ′ ′ ′
1 ρ =ρ0/|ζ (z)|, λ =λ0|ζ (z)|. (5) tity. Then eq. (9) reduces to (8), apart from the factor
: c which could be incorporated into F, and taking S = I
v
i The expressions (5) are not correct as we explain next. eq. (10) also reduces to (8) since Cijkl = λ′δijδkl cor-
X ′
Theequationsofmotionforanacousticfluidofdensity responds to the acoustic fluid of bulk modulus λ. In
r ρ′ and bulk modulus λ′ are conclusion,whenthetransformationisconformalthepa-
a
rameters are isotropic, with both the inertial and the
−ω2ρ′u′ =−∇′p′, p′ =−λ′∇′·u′. (6) pentamodal models yielding ρ′ and λ′ given by eq. (8).
Eliminating the displacement u′ yields
ACKNOWLEDGMENTS
′ ′ ′ ′ ′ 2 ′ ′ ′
ρ∇ ·(1/ρ)∇p +ω (ρ/λ)p =0. (7)
Thanks to N. Gokhale and A. J. Nagy for comments.
Comparisonofeqs.(2)and(7)impliesthattheformeris 1C. Ren, Z. Xiang, and Z. Cen, “Design of acoustic devices with
thereducedequationforthepressureinanacousticfluid isotropic material via conformal transformation,” Appl. Phys.
only if ρ′ is constant. That is, eqs. (2) and (3) represent Lett.,97,044101+ (2010).
2A. N. Norris, “Acoustic cloaking theory,” Proc. R. Soc. A, 464,
the equation of motion of an acoustic medium with
2411(2008).
3A. N. Norris, “Acoustic metafluids,” J. Acoust. Soc. Am., 125,
ρ′ =cρ0, λ′ =cλ0|ζ′(z)|2, (8) 839(2009).