acoustic cloaks for plates and cylinders

7/30/2019 Acoustic Cloaks for Plates and Cylinders

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Acoustic Cloaks for Plates and Cylinders

Beena Mary John

Rajagiri School of Engineering and Technology, Kakkanad, Kochi 682039

D. D. Ebenezer

Naval Physical and Oceanographic Laboratory, Thrikkakara, Kochi 682021

Abstract : A method is presented to determine the properties of thin, lossy, homogeneous

coatings that can acoustically cloak plates and cylinders. Flat panels and cylinders are often

used as basic building blocks in mathematical models of submarines and are, therefore, of

interest. The frequency-dependent sound speed in the coating required to reduce the

broadband backscattered pressure to zero is determined by using analytical models.

Numerical results indicate that materials with the required sound speed can be developed.

The sensitivity of the reflection coefficient to the complex sound speed is also presented to

illustrate the effect of variations in sound speed that will occur during production.

Keywords: Echo, backscatter, multilayer, internal loss

1. Introduction

Rendering a submarine invisible to interrogating acoustic waves would constitute a

gigantic leap in stealth technology. An ideal acoustic cloak conceals a submarine by causing

the total pressure field to be the same as that due to the incident field alone; without echoes or

shadows [1]. Cai [2] presents multilayer elastic cloaks for cylinders that, unlike the cloak in

Ref. 1, do not have any material singularity, but perform very well in the low frequency

region where there are no resonances. Cheng et al [3] present a cloak that is effective against

waves with different wave-front shapes.

In this paper, a method is presented to design broadband acoustic cloaks for

underwater plates and cylinders such that the reflected and backscattered pressures,

respectively, are zero. Flat panels and cylinders are often used as basic building blocks in

mathematical models of submarines and are, therefore, of interest. The method is used to

determine the frequency-dependent properties of a homogeneous cloak for a flat plate – and it

is likely that such materials can be developed. The effect of a small difference between the

actual and ideal material properties on the effectiveness of the cloak is also studied. A model

of a cylinder coated with a thin lossy layer is used to show that coatings without losses can

amplify echoes whereas designed ones can reduce the backscattered pressure to zero.



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2. Acoustic Cloaks for Plates

Consider an infinite, flat, metal plate with a lossy homogeneous coating immersed in

infinite fluid as shown in Fig. 1. The objective is to determine the speed of sound in the

coating that would make the reflected pressure zero when a broadband wave is normally

incident. The thickness and density of the metal plate and the speed of sound in it, and the

thickness and density of the coating are known. Losses in the coating are modelled using

complex Lame’s coefficients and the speed of sound is therefore complex.

The pressures due to the incident,

reflected, and transmitted waves are expressed

as )(

11xkt j

i e AP−

=ω , )(

11xkt j

r eBP+

=ω , and

)(

4

1 xk t j

t e AP

−

=ω

respectively, where 11

/ck ω =

is the acoustic wave number in water, ω is the

angular frequency, 1c is the speed of sound in

water, and t denotes time. The normal stresses

in the coating and the metal plate are expressed as )(

2

)(

2222 xkt jxkt j

eBe AT+−

+=ω ω and

)(

3

)(

3333 xkt jxkt j

eBe AT+−

+=ω ω , respectively, where

22/ ck ω = and

33/ ck ω = . The continuity

conditions for stress and particle velocity at the first, second, and third interfaces are used to

determine the coefficients 2 A , 3 A , 2 B , 3 B , and 4 A when 1 A is known. Plates with several

coatings are analyzed using a transfer matrix approach [4].

The reflection coefficient 11 / ABR = is a function of the density and the sound speed in

each layer and the thicknesses of the metal and coating layers. It is assumed that all these

properties except the speed of sound in the coating, 2c , are known. The complex value of 2c

at which R becomes zero is determined. The method used to find 2c is presented in a later

section.

3. Acoustic Cloaks for Cylinders

Consider the solid infinite circular cylinder shown in Fig. 2. It has a thin coating. A

plane acoustic wave travelling along the direction perpendicular to the axis of the cylinder is

incident on it. The backscattered acoustic wave is of interest and is determined using a model

of a multilayer cylinder [5]. The effect of the material properties of the coating on the

backscattered pressure is studied. The analysis can be extended to study coated thin

cylindrical shells.

Fig.1. An infinite coated plate; and

incident, reflected, and transmitted plane

acoustic waves.

C o a t i n g

P l a t e



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The pressure in the incident wave is

expressed in cylindrical coordinates using

Bessel functions of the first kind [6]. The

solutions to the equations of motion of the

solid cylinder and the coating are determined

using the method of separation of variables

[7]. In the azimuthal direction, a Fourier

series expansion with orthogonal terms is

used. In the radial direction, Bessel functions

are used. The scattered acoustic pressure is also expressed in terms of a Fourier-Bessel series.

Each orthogonal component has two coefficients in the solid cylinder, four coefficients in the

coating, and one coefficient in water.

The seven coefficients in the solutions are determined using seven continuity

conditions at the interfaces: at the water-coating interface, the normal stress is equal to the

pressure, the shear stress on the surface of the coating is zero, and the radial displacement is

continuous; at the coating-cylinder interface, the normal and shear stresses, and radial and

tangential displacements are continuous. The coefficients can be used to determine the

pressure in any direction. However, here, only the back-scattered pressure is computed

because this is of interest when a monostatic sonar interrogates the object. Losses in the

coating are modelled using complex Lame’s coefficients in this case also.

4. Results and discussion

Numerical results are presented to illustrate cloaks that render plates and cylinders

invisible to monostatic sonars. Results are presented for a steel plate of thickness 6 mm,

density 7700 kg/m3, Young’s modulus 195 GPa, and Poisson ratio 0.3. The thickness of the

coating is 40 mm for the plate and the density of the coating is 1200 kg/m3.

The reflection coefficient, at 10 kHz, for a coated flat plate is shown in Fig. 3 as a

function of the real and imaginary parts of the speed of sound in the coating. It is seen that R

has several local minima that are not nearly zero. However, it is seen from Fig. 3 that R is

approximately zero at the global minimum where 2c = 710+j100 m/s. R does not change very

rapidly in this neighborhood indicating that small deviations from the ideal2c will not result

in a large change in R . The change is much more rapid at local minima. After visually

finding the approximate global minimum using the 3D plot, the search for the ideal sound speed is continued by using 2D plots. This is easier than a fully automatic search.

Fig. 2. A solid elastic cylinder with a thin

acoustic coating. A plane wave travelling

along the x axis is incident on it.



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2D graphs are shown in Fig. 4. Each line is for a constant integer value of )Im( 2c

ranging from 97 to 101. The five lines are nearly the same and cannot be distinguished. The

ideal speed of sound for the cloak is the one corresponding to R =0. If such a sound speed

does not exist, the sound speed at which R is minimum is the ideal sound speed.

The sensitivity of R to the sound speed, in the neighborhood of the ideal sound

speed, is illustrated using a contour plot in Fig. 5. Contour lines are shown for R = 0.02,

0.05, 0.1, 0.2, and 0.3. It is seen that R < 0.1 (-20 dB) in a region that is approximately

elliptical. This indicates that the cloak will be effective even if the actual sound speed is a

little different from the ideal sound speed either due to inability to achieve the ideal sound

speed or due to variation during production.

The feasibility of using an acoustic cloak to conceal a cylinder at high normalized

frequencies is illustrated by presenting results for rigid and Titanium cylinders of radius 1 m.

Fig. 3. R as a function of real and

imaginary parts of 2c at 10 kHz.

Fig. 4. Reflection coefficient at 10 kHz. Five

lines at integer values of )Im( 2c ranging

from 97 to 101 are shown.

Table I. Ideal sound speed in an acoustic

cloak for a steel plate.

Frequency(kHz)

Ideal sound speed (m/s) Young’s

Modulus(MPa)Re

(c)Im(c)

2 158 0.75 1.75 +0.01j

4 305.5 8.7 6.53+0.37j

6 450 28.7 14.14+1.81j

8 584.5 60 23.7+4.92j

10 710 99 34.65+9.85j

Fig. 5. Contour plot of the amplitude of thereflection coefficient at 10 kHz. Contour

lines are shown at R = 0.02, 0.05, 0.1, 0.2,

and 0.3.

600 650 700 750 800 8500

0.1

0.2

0.3

0.4

0.5

Re(C2)

R e f l e c t i o n C o e f f i c i e n t

0.02

0.05

0.1

0.2

0.3 0.3

Re (c2)

I m

( c 2

)

600 650 700 750 800 85050

100

150



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The thickness of the coating is 50 mm. The Young’s modulus, Poisson’s ratio, and density of

Ti are 110 GPa, 0.3, and 4533 kg/m3, respectively. The Young’s modulus, Poisson’s ratio,

and density of the rubber coating are 0.11(1-j0.1) GN/m2, 0.48(1-j0.1), and 910 kg/m3,

respectively.

The pressure backscattered by a rigid cylinder, at a distance of 500 m from the

cylinder, is shown in Fig. 8 for three cases: without coating, coated with lossless layer, and

coated with lossy layer. At large distances from the cylinder, the pressure varies inversely

with square root of radial distance. It is seen that the backscattered pressure is nearly

independent of frequency if frequency > 1 kHz. Using a coating without losses causes rapid

oscillations and higher backscattered pressure in several narrow frequency bands. However,

there are no oscillations in the 3.5 – 8.5 kHz band and the backscattered pressure is very

nearly the same as that from the rigid cylinder. When a layer with loss is used and frequency

> 500 Hz, the backscattered pressure is less than that when there is no coating. There are no

oscillations at frequency > 3 kHz. There is a minimum at about 7.2 kHz and the coating is a

cloak at this frequency.

The backscattered pressure is shown in Fig. 9 for a Ti cylinder. At high frequencies,

the pressure oscillates about the pressure scattered by a rigid cylinder. In contrast to the rigid

Fig. 6. Re(c2) corresponding to R = 0, 0.1 and

0.2. Im(c2) = ideal value at each frequency.Fig. 7. Im(c2) corresponding to R = 0, 0.1

and 0.2. Re(c2) = ideal value at each

frequency.

Fig. 8. Magnitude of pressure backscattered

by a rigid cylinder of radius 1 m.

Fig. 9. Magnitude of pressure backscattered

by a titanium cylinder of radius 1 m

1 2 3 4 5 6 7 8 9 1 00

100

200

300

400

500

600

700

800

Frequency(kHz)

R e ( C 2

)

Reflection Coefficient = 0

Reflection Coefficient = 0.1


1 2 3 4 5 6 7 8 9 1 00

20

40

60

80

100

Frequency(kHz)

I m ( C 2

)

Reflection Coefficient = 0



0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

Frequency(kHz)

B a c k s c a t t e r e d p r e s s u r e ( P a )

Rigid cylinder without coating

Rigid cylinder with rubber coating(losses not included)

Rigid cylinder with rubber coating(losses included)

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

Frequency(kHz)

B a c k s c a t t e r e d p r e s s u r e ( P a )

Ti cylinder without coating

Ti cylinder with coating (losses not included)

Ti cylinder with coating (losses included)



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cylinder case, there are several oscillations even at high frequencies. A thin coating without

loss does not significantly affect the backscattered pressure. However, when there is loss in

the coating, oscillations are still present but the pressure is considerably reduced. The coating

is nearly an ideal cloak at approximately 7 kHz.

It is feasible to compute the frequency-dependent characteristics of the ideal coating

by using the method used for plates.

5. Conclusions

Analytical models are used to show that it is feasible to acoustically cloak plates and

cylinders such that the backscattered pressure is zero. Models of multilayered flat plates and

cylinders with lossy coatings, based on exact equations of motion, are used to compute the

backscattered pressure at high normalized frequencies. It is shown that the frequency-dependent complex sound speed in the coating can be suitably designed to achieve zero

reflection. The effect of a small difference between the ideal and actual sound speed on the

reflected pressure is also presented to assist in setting production tolerance.

Acknowledgement

Facilities to do the work provided by Director, NPOL are gratefully acknowledged.

References

1. S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New Journal of Physics

9, pp 1-8 (2007).

2. L.-W. Cai, “Optimizing imperfect cloaks to perfection,” J. Acoustical Soc. America.132,

2923- 2931 (2012)

3. Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multi layered structured acoustic cloak

with homogeneous isotropic materials,” Applied Phy. Let. 92, 151913 (2008).

4. D. D. Ebenezer and Pushpa Abraham, “Effect of multilayer baffles and domes on

hydrophone response,” J. Acoustical Soc. America. 99, 1883-1893 (1996).

5. J. S. Sastry and M. L. Munjal, “Response of a multi-layered infinite cylinder to two-

dimensional pressure excitation by means of transfer matrices,” J. Sound and Vibration,

209, 123-142 (1998).

6. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York,1968).

7. M. C. Junger and D. Feit, Sound, Structures, and their Interaction, (MIT Press,

Cambridge, MA, 1972).

8. L. E. Kinsler, A. R. Frey, A. B. Coppens and J. V. Sanders, Fundamentals of Acoustics (John Willey & Sons,1972), 4th ed.

acoustic cloaks for plates and cylinders

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