Elastic wave band gaps and single scattering

Solid State Communications, Vol. 105, No. 5. pp. 327-332, 1998 @ 1998 Published by Elsevicr Science Ltd Printed in Great Britain. All rights reserved 0038-1098198 S19.00 + .OO PII: s0038-1098(~10

ELASTIC WAVE BAND GAPS AND SINGLE SCATTERING A. D. Klironomos and E. N. Economou Research Center of Crete, FORTH, PO. Box 1527,711lO Heraklio and Department of Physics, University of Crete, Crete (Received 25 July 1997; accepted 11 September 1997 by l? H. Dederichs)

The scattering cross section of an elastic plane wave incident at right angle to a circular cylinder was calculated. The frequency dependence of this single scattering cross section is connected to the existence (or not) of spectral gaps for elastic waves propagating in a periodic array of parallel circular cylinders. The present work confirms the picture of propagation along two distinct channels: one employing mainly the host material and the other exploiting the resonance modes of each cylinder. @ 1998 Published by Elsevier Science Ltd Keywords: D. acoustic properties, D. elasticity.

1. INTRODUCTION There is a growing interest in classical wave propagation in periodic or random media [l-4]. The experimental demonstration [5] of the existence of spectral gaps in the so called photonic “crystals” has opened up a new area of applications and has contributed to a better understanding of wave propagation in periodic media [6]. Recently the field has been extended to the study [6-l 1] of acoustic and elastic waves in composite periodic or disordered materials. Elastic (EL) waves, with their full vector character and their three independent parameters per component (mass density, p, longitudinal, cl and transverse , ct, sound velocities) offer richer physics and further opportunities for applications. Exploiting the mass contrast between the two components of the composite (besides the velocity contrast) structures have been found exhibiting full spectral gaps (for all directions of propagation) [8,12,13]. For spherical inclusions in a host material the existence of full gaps has been connected [6] to the following simple picture: There are two channels for propagation. One is using mainly the host material; the other is employing the resonance states (which appear as peaks in the scattering cross section by a single spherical inclusion). This second channel is created by coherent jumping from resonance state to neighboring resonance states in analogy with the linear combi-

nation of atomic orbitals (otherwise called tight binding approximation) in the electronic band structure. According to this picture the appearance of full spectral gaps in periodic composite materials (or of localized bands in disordered ones) requires the blocking of both channels. This in turn implies that gaps should appear (if at all) at frequencies: (a) far away from resonances (so that the resonance channel to be inoperative); (b) such that the propagation along the host material is inhibited. The latter frequency regions are those at which the single scattering cross section is large and is due entirely (or almost entirely) to a rigid (or empty) inclusion, i.e. one which does not allow the wave to enter to its interior. Thus the conclusion is the following: Whenever the single inclusion scattering cross section exhibits well separated resonances with a strong background in between due to a rigid (or empty) inclusion, spectral gaps may appear in this (in between) frequency region [6]. In the present work we have tested this simplyfying and physically attractive picture by comparing the frequency dependence of the single cylinder cross section with band structure results in periodic arrays.

2. RESULTS AND DISCUSSION The cylinder has a circular cross section with its axis parallel to the z axis passing through the origin. We

327

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ELASTIC WAVE BAND GAPS AND SINGLE SCATTERING

consider an EL plane wave propagating in the x direction. Two kinds of polarization are considered for the incident wave: i) Longitudinal (L-scattering), with the polarization in the x direction, ii) Transverse (Tscattering), with the polarization in the y direction. A polarization in the z direction is not considered as this particular wave is essentially decoupled from the other two polarizations and behaves as a scalar wave. The details of the derivation and the solution of the vector wave equation for an isotropic and homogeneous medium are presented in [14,15]. The incident, scattered and the wave inside the cylinder are expanded in terms of the cylindrical Bessel functions, appropriate for this particular problem. The next step is the application of the boundary conditions on the surface of the cylinder and the calculation of the dimensionless scattering cross section in terms of the longitudinal and transverse scattered wave coefficients. The results are remarkably simple:

as = -

2-

(414

%*

{(1

“=oEn Ct1 =

IanI’ + IhI*

where (31and & are the scattering cross sections (divided by the diameter d and unit of length along the z axis) for longitudinal and transverse incident wave respectively, kl and q1 are the longitudinal and transverse wave numbers for the host, cl, and ct, are the longitudinal and transverse velocities for the host, a, and b, are the longitudinal and transverse scattered wave coefficients and en is the Neumann factor: E, = 1 for n = 0, ln = 2 otherwise. The above results hold for the most general case where the host as well as the cylinder are solid elastic media. Other combinations (i.e. solid cylinder in liquid host, liquid cylinder in liquid host, etc.) are special cases of the above. In order to check the physical origin of the background between two consecutive resonances, we subtract from the scattering amplitude the corresponding scattering amplitude due to a rigid (or empty) cylinder [6] depending on whether the density of the inclusion is larger (or smaller) than that of the host. The infinite series for the scattering cross sections are approximated by finite sums, by keeping the first 20-25 terms. For the most part of the frequency range this is more than adequate. The band structure results for periodic media are obtained by solving the elastic wave equation for a locally isotropic medium (p, h are the Lame elastic coefficients):

a2ui I a -at2 = pipj-$

ad

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+

&Lg +gm I

The periodic functions ~1,h and p-’ are expanded in Fourier series and the solution is then obtained by means of the Bloch theorem. The relevant theory is presented in [16]. In Figs l-4 we present our results. Every figure consists of three panels. The first is the scattering cross section for the single scatterer, the second is the rigid or empty cylinder scattering cross section [17] while the third panel is the total scattering cross section after the subtraction of the amplitude of the rigid (or empty) cylinder contribution. We compare with the corresponding band structure results of refs [lo] and [18]. The position of the gap obtained in refs [lo] or [18] is denoted by the full arrows while the single arrows indicate the position of “flat” bands (peaks in the density of states). Figure 1 shows the scattering results for a medium of water cylinder (p = 1025 kgr/m3, ci = 1531 m/s) immersed in mercury (p = 13500 kgr/m3, ci = 1450 m/s) host. The periodic composite exhibits an extremely large gap [IO] between the first and second band, AU/& = 0.901 for a square lattice and a filling fraction f = 0.34, Ace/ii, = 0.984 for a hexagonal lattice and a filling fraction of f = 0.27. The single scattering cross sections show an almost ideal case, in which the resonances are very well separated by a region of strong scattering due almost entirely to the empty cylinder contribution. Notice the similar situation for the 3.7 I (kid) I 7.0 range which indicates that another gap may appear. This is indeed the case, a complete band gap, although not as wide as the previous one, appears in the band structure between the fourth and fifth bands [lo]. Careful1 examination of the previous figure shows that this is a most favorable case for the creation of a wide gap. The resonances appear not only well separated but also isolated, a condition not met in any of the following cases. In addition, in what follows, both L and T-scattering profiles exist, as we are dealing with solid scatterers in a solid host rather than liquid scatterers in a liquid host. Figures 2 and 3 illustrate the single scattering cross sections for lead (p = 11357 kgr/m3, ci = 2158 m/s, ct = 860 m/s) and steel (p = 7800 kgr/m3, q = 5940 m/s, ct = 3220 m/s) cylinders in epoxy (p = 1180 kgr/m3, ci = 2540 m/s, ct = 1160 m/s) host. For the case of the steel scatterers a complete gap appears with Au)/& = 0.44 for a square lattice and a filling fraction f = 0.4 [I 81. For the case of the lead scatterers a complete band gap appears with Au/ii, = 0.37 for a square lattice and a filling fraction f = 0.283 [18].

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ELASTIC WAVE BAND GAPS AND SINGLE SCATTERING

329

1

8.0

(a)

(c) -

8.0 i

1 6.0

6.0

l-lr-4

o.o/’ 0.0

2.0

4.0

6.0

V

4.0

4.0

2.0

2.0

0.0 0.0

2.0

(k,d)

4.0

6.0

II

:

0.0

0.0

2.0

(Q-l)

4.0

6.0

(k,d)

Fig. 1. Dimensionless scattering cross sections for water (a), empty (b) cylinders embedded in mercury host. Panel (c) results by subtracting the amplitudes of (a) and (b). For the arrows see text.

Pb in Epoxy (longitudinal incident wave)

s

-= : z g 0

10.0

10.0

10.0

8.0

8.0

8.0

6.0

6.0

6.0

4.0

4.0

(4

_

2.0 0.0 0.0

1.0

2.0

0.0

3.0

1.0

2.0

3.0

(k,d)

(k,d)

Pb in Epoxy (transverse incident wave)

(a) -

10.0

s

10.0

8.0

8.0 -

6.0 4.0

(4

L--.-! 10.0

8.0

6.0 4.0

2.0

0.0

0.0

1.0

(k,d)

2.0

3.0

0.0

1 .o

(k,d)

2.0

3.0

L

0.0

1.0

2.0

3.0

(k,d)

Fig. 2. Dimensionless scattering cross sections for lead (a), rigid (b) cylinder embedded in epoxy host. Panel (c) results by subtracting the amplitudes of (a) and (b). For the arrows see text.

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ELASTIC WAVE BAND GAPS AND SINGLE SCATTERING

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Steel in Epoxy (longitudinal incident wave)

(a)

8.0

(b)

8.0

6.0

6.0

0.0 0.0

2.0

4.0

6.0

8.0

8.0

0.0 I0.0

2.0

(k,d)

4.0

6.0

8.0

0.0

2.0

(W)

4.0

6.0

8.0

WV

Steel in Epoxy (transverse incident wave)

(a)

8.0

0-4

8.0

s ‘E 6.0

I’

8.0 -

r1 I r

” 77I@I

6.0

6.0

Q 2 4.0 2 0 2.0

4.0 2.0

J 0.0

2.0

4.0

6.0

8.0

0.0

l-----J 2.0 4.0

0.0

:I

I

0.0 6.0

8.0

0.0

2.0

4.0

6.0

8.0

W,d) Fig. 3. Dimensionless scattering cross sections for steel (a), rigid (b) cylinder embedded in epoxy host. Panel (c) results by subtracting the amplitudes of (a) and (b). For the arrows see text. Again, the single scattering results are in excellent agreement with the band gap results; furthermore they correctly identify the most preferable case of the two. The case of the steel cylinders in epoxy host exhibits the well separated, with strong background in between, resonances necessary for gap creation. However, a noticeable fact is that in both cases, especially for the T-scattering profile for the lead cylinders in epoxy host, additional resonances appear, resulting in a narrower gap. In Fig. 4 the case of epoxy scatterers in lead host is shown. As it can be inferred from the figure, this is a less favorable case for wide gap creation because the background is not due entirely to the empty cylinder as it can be seen-from panels (c) where a non-zero background remains. The band structure results show a gap with Am/(?, = 0.22 for a filling fraction f =

0.739 [18].

In conclusion, the comparison of the band structure results of refs [IO, 181 with our present single scattering data, supports our picture of two distinct main channels of propagation and clearly confirms our proposal that spectral gaps appear in frequency regions between two well separated resonances in the single inclusion scattering cross section with a strong background in between (due to rigid or empty cylinder). Acknowledgements-We would like to acknowledge support by EU grants TMR-CT-96-0042, TMRCT96-0640, CHRX-CT93-0136 and GSRT-Hellas grant 15774-5. A. D. K. would like to thank M. Kafesaki for helpful discussions.

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ELASTIC WAVE BAND GAPS AND SINGLE SCATTERING

331

Epoxy in Pb 3.5

(longitudinal incident wave) 3.5

3.0

3.0

g 2.5

2.5

2.5

‘Z 2.0 2 2 1.5 S? ” I.0

2.0

2.0

1.5

1.5

1.0

1 .o

0.5

0.5

0.5

0.0

tl

0.0

2.0

4.0

6.0

0.0

--

02

L I

0.0

I

2.0

(k,d)

4.0

6.0

0.0

/L/h IIII

0.0

2.0

4.0

6.0

WV

t&d) Epoxy in Pb (transverse incident wave) 3.5’

3.5

(a)

3.0

r

3.0

(W

3.5

g 2.5

2.5

.$ 2.0

2.0

z2

1.5

1.5

5

1.0

1 .o

1.0

0.5

0.5

0.5 0.0

:;

t!

0.0

0.0 2.0

4.0

6.0

ll

0.0

2.0

(k,d)

4.0 (k,d)

(c)

3.0 2.5 2.0 1.5

6.0

0.0 t 0.0

2.0

4.0

6.0

(V)

Fig. 4. Dimensionless scattering cross sections for epoxy (a), empty (b) cylinder embedded in lead host. Panel (c)-results by subtracting the amplitudes of (a) and (b). For the arrows see text. REFERENCES

1.-Soukoulis, C.M., ed., Photonic Band Gaps and Localization. Plenum, New York, 1993. 2. Sheng, P., Introduction to Wave Scattering, Locakzation and Mesoscopic Phenomena. Academic Press, London, 1995. 3. Soukoulis, C.M., ed., Photonic Band Gap Materials. Kluwer, Dordrecht, 1996. 4. Joannopoulos, J.D., Meade, R.D. and Winn, J.N., Photonic Crystals. Princeton Univ. Press, Princeton, 1995. 5. Yablonovitch, E., Gmitter, T.J. and Leung, K.M., Phys. Rev. Lett., 67, 1991, 2295. _ 6. Kafesaki, M. and Economou, E.N., Phys. Rev. B, 52, 1995, 13317. 7. Sigalas, M.M. and Economou, E.N., 1 Sound Vi-

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15. Stratton, IA., Electromagnetic Theory. McGrawHill, New York, 1941. 16. Economou, E.N. and Sigalas, M.M., J Acoust. Sot. Am., 95, 1994, 1734. 17. The low frequency strong peak in the cross section by an empty cylinder (panel (b) in Fig. 1) is due to isotropic scattering. For the cases of Figs. 2 and 3,

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the low frequency strong peak in the cross section by a rigid cylinder (panels (b)) is due to the dipole term. Although it is not clear in the figures, u - 0 for w - 0 as it should. 18. Sigalas, M.M., 1 Acoust. Sot. Am., 101, 1997, 1256.