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Classical vibrational modes in phononic lattices:theory and experiment

Mihail SigalasI, Manvir S. KushwahaII, Eleftherios N. EconomouIII, Maria KafesakiIII, Ioannis E. Psarobas*, IV

and Walter SteurerV

I Communication and Optics Lab, Agilent Labs, Agilent Technologies, 3500 Deer Creek Road, MS26M9, Palo Alto, California 94304, USAII Institute of Physics, University of Puebla, P.O. Box J-45, Puebla 72570, MexicoIII FORTH, Institute of Electronic Structure and Laser, P.O. Box 1527, 71110 Heraklion, Crete, GreeceIV Section of Solid State Physics, The University of Athens, Panepistimioupolis, 15784 Athens, GreeceV Laboratory of Crystallography, Department of Materials, ETH Zurich and MNF, University of Zurich, 8093 Zurich, Switzerland

Received May 31, 2005; accepted June 9, 2005

Phononic crystal / Sonic bandgap materials /Classical spectral gap materials /Vibrations-lattice dynamics / Elastic waves

Abstract. We present a review, through selected illustra-tive examples, of the physics of classical vibrationalmodes in phononic lattices, which elaborates on the theo-ry, the formalism, the methods, and mainly on the numer-ical and experimental results related to phononic crystals.Most of the topics addressed here, are written in a self-consistent way and they can be read as independent indi-vidual parts.

Part I: Theory-formalism

1. Introduction

In recent years, propagation of classical waves [electro-magnetic (EM) or elastic waves] in composite materialswith dielectric or, respectively, elastic properties which areperiodic functions of the position, with a period compar-able to the wavelength of the corresponding field, hasbeen the object of considerable attention [1][4]. Thesematerials, photonic and phononic crystals, respectively,whether they exist naturally or are artificially fabricated,exhibit a rich variety of physical properties of interest tofundamental and applied research. There are striking ana-logies between the propagation of electrons in ordinarycrystals and EM/elastic waves in photonic/phononic crys-tals (see Table 1), so that a great variety of multiple-scat-tering (MS) methods as well as other traditional methodsoriginally developed for electronic-structure calculationshave been transferred to the field of photonic and phono-nic crystals. The reader may consult the work of Modinoset al. [5], where a theory of electron, EM, and elasticwave propagation in systems consisting of nonoverlappingscatterers in a host medium is presented. The theory pro-

vides a framework for a unified description of wave propa-gation in three-dimensional periodic structures, finite slabsof layered structures, and systems with impurities: isolatedimpurities, impurity aggregates, or randomly distributedimpurities.

Phononic crystals, like photonic crystals, attracted a lotof interest among researchers mainly because of the possi-bility of frequency regions, known as absolute phononicgaps, over which there can be no propagation of elasticwaves in the crystal, whatever the direction of propaga-tion. At the beginning of this review we shall describebriefly the methods presently available for the calculationof the frequency band-structure of phononic crystals. Theplane-wave method (PW) [6, 7] seems the most forwardand, in conjuction with a supercell approach, can alsotreat defects [8] and finite slabs [9]. However, it has con-vergence problems, especially in the case of fluid-solidcomposites [12]. A MS method emanating from the tradi-tional Korringa-Kohn-Rostoker (KKR) method [10, 11]developed for the calculation of the electronic structure ofsolids appears to be numerically more efficient [1214].The above methods apply to infinite phononic crystalsmade of non-dispersive lossless materials. In reality, how-ever, one usually measures the transmittance or reflectanceof finite slabs, which may be dispersive or dissipative. Thewell-known finite-difference-time domain (FDTD) method[1517] gives the transmission, reflection, and absorptioncoefficients of elastic waves incident on a finite slab of aphononic crystal, but it gets computationally cumbersomefor 3D systems [18]. Psarobas et al. [19] proposed an on-shell layer-MS method for the calculation of the complexband structure of 3D phononic crystals and of the trans-mittance/reflectance of slabs of the same. The method issimilar to the layer-KKR (LKKR) method of low-energyelectron diffraction (LEED) [20] and electron emission[21]. Liu et al. [13] proposed independently a methodalong the same lines for the transmittance/reflectance of aslab. Also a variational method was developed bySanchez-Dehesa et al. [2224]. Finally, over the years,certain combinations of the above methods appeared (see

Z. Kristallogr. 220 (2005) 765809 765# by Oldenbourg Wissenschaftsverlag, Munchen

* Correspondence author (e-mail: [email protected])

e.g. Ref. [25]), and well established numerical methods,such as the finite element method and the transfer matrixmethod have been used to address specific problems (e.g.,a more complex geometry).

In this review, we present briefly the principles of theMS and the FDTD methods. The rest of the main methodsappear in Part II through certain illustrative examples.More detailed presentations can be found in other contri-butions to this special issue. In addition, Part II also in-cludes an extended collection of numerical results on pho-nonic crystals, where the underlying physics is discussed.Finally, Part III is solely devoted to various experimentalobservations and results on the field.

2. Multiple scattering method

2.1 Introduction

Although the PW method is a very powerful tool for thecalculation of the dispersion relations of acoustic and elas-tic waves, it fails in some important cases: It can not cal-culate accurately the dispersion relation of mixed (fluid/solid) composites, it also fails in cases where the contrastin the material parameters between scatterers and host isvery high (due to the fact that very large-step functionsneed an extremely large number of Fourier components inorder to be reproduced accurately).

These difficulties are overcome within the MS method,an approach based on the well-known in the electronicband-structure community Korringa-Kohn-Rostoker (KKR)theory [10, 11, 2628]. The success of this theory in bothelectronic and electromagnetic [2931] band-structure cal-culations was a strong motivation for its application in theacoustic/elastic problem as well. Moreover, in addition toits capability to calculate dispersion for mixed composites[12] and for high contrast composites [32], the MS meth-od is capable to calculate transmission through finite slabsof those composites, both periodic and random; thus it isa valuable tool in the acoustic/elastic problem.

In what follows we will describe in detail the methodand its application to both band structure and transmissioncalculations, restricting ourselves to three-dimensional per-

iodic or random systems consisting of solid or fluid scat-terers in a fluid host, i.e. to the scalar version of the meth-od. For the full vector version (including all possiblecombinations of hosts and scatterers, bulk and layer-KKR)see Refs. [19, 33] and [13, 34] as well.

2.2 Description of the method

2.2.1 Main equations

One begins with the observation that in a system of manyscatterers, either periodic or random, the incident wave ateach scatterer is the sum of the scattered waves by all theother scatterers (plus the external field, if present). Thisidea is used for the determination of the total field and,through it, for the calculation of either the transmissioncoefficient or the dispersion relation.

The application of the method starts by writing the totalpressure field pr in the fluid host of the system as [12]

pr p0r Pnpscn r ; 1

where p0r is the external field and pscn r the scatteredfield by the scatterer at the position Rn. This scatteredfield, pscn r, can be written as a sum of elementary spheri-cal waves,

pscn r pscr RnP

lmbnlmhlkojr Rnj Ylmr Rn : 2

(hl jl inl is the spherical Hankel function of the firstkind and order l [35], and ko w=co, with co the wavevelocity in the host material; Ylm are the spherical harmo-nics [36, 37]). Thus, the determination of the total field ofEq. (1) is reduced to the calculation of the coefficients bnlm.The determination of bnlm is done by writing the incidentfield at the scatterer at the position Rn as

pincn r p0r Pp 6 n

pscp r : 3

Writing pincn r as a sum of elementary spherical waves,pincn r pincr Rn

Plm

anlmjlkojr Rnj Ylmr Rn ; 4

766 M. Sigalas, M. S. Kushwaha, E. N. Economou et al.

Table 1. Band-structure-related properties of three periodic systems. (After Kushwaha et al. [1]).

Property Electronic crystal Photonic crystal Phononic crystal

Materials Crystalline solid(natural or grown)

Constructed of at least 2dielectric materials

Constructed of at least 2elastic materials

Parameters Atomic numbers er; mr(E/M permitivities & permeabilities)

rr; clr; ctr(Mass densities, sound speeds)

Lattice constant 15 A (microscopic) 0.1 mm1 cm (macroscopic) &1 mm (macroscopic)Waves De Broglie (electrons) w EM (photons) (E, H) Vibrational or sound (phonons) u

Polarization Spin "; # Transverser D 0r E 6 0

Coupled shear-compressionalr u 6 0;r u 6 0

Diff. equation h2=2m r2 Vr w ih @tw r er E e=c2 @2t E see Table footnoteaFree particle limit W h2k2=2m (parabolic) w ck= ep (linear) w cl; t k (linear)Spectral region Radio waves, microwaves, optical, x-rays Microwaves, optical w . 1 GHz

a: rc2trr ur r rc2l u 2r u u r rrc2t rrc2t r u r @2t u:

and substituting this expression and Eq. (2) in Eq. (3), oneobtainsP

lmanlmjlkojr Rnj Ylmr Rn

p0r Pp 6 n

Plm

bplmhlkojr Rpj Ylmr Rp : 5

The coefficients bnlm are proportional to anlm,

bnlm tnl anlm ) anlm tnl 1bnlm ; 6where the proportionality coefficients tnl ( 1=1 iwnl with wnl Im tnl 1 real) can be found by solving asingle scattering problem [12, 38, 39]. Moreover, the sphe-rical functions of (5) centered at Rp can be transformed tofunctions centered at Rn,

hlkojr RpjYlmr RpP

l0m0jl0 kojr Rnj Yl0m0 r Rn ghl0m0lmRp Rn

for jr Rnj < jRp Rnj ; 7where

gRl0m0lmD

Pl

1l0 l l=2 4pCl0m0; lm; lmm0RlkD

Ylmm0 D ; R j or h 8and Cl0m0; lm; lm are the Gaunt numbers [12, 36]. Substitut-ing (6) and (7) in (5), interchanging the l0; m0 with thel; m in the r.h.s and writing the external field, p0, also asa sum of spherical waves with center at Rn,

p0r Plm

a0nlm jlkojr Rnj Ylmr Rn ; 9

we obtain (after few algebraic manipulations)Pl0m0

Pptpl0 1 dll0dmm0dpn ghlml0m0 Rp Rn 1 dpn

bpl0m0 a0nlm : 10Using Eq. (10), which constitutes a linear algebraic sys-

tem, one can calculate the coefficients bnlm, and throughthem the total field and the transmission coefficient, interms of the coefficients a0nlm of the incident wave.

For the calculation of the dispersion relation of an infi-nite periodic system, i.e. for the eigenmodes of the systemin the absence of any external field, the coefficients a0nlm ofEq. (10) are set equal to zero, the coefficients bplm of thedifferent lattice sites are connected through Blochs theo-rem, i.e. bplm eik Rp Rnbnlm, and one seeks the frequen-cies for which the determinant of the homogeneous sys-temP

l0m0tpl0 1 dll0dmm0

Ppeik Rp Rn

ghlml0m0 Rp Rn 1 dpnibnl0m0 0 11

vanishes. The calculation of this determinant requires thetruncation of the summation over l0m0 and the calculationof the infinite sum

Ppeik Rp Rnghlml0m0 Rp Rn. The

way to perform this calculation, since the sum is not con-vergent, is through a technique known as Ewalds summa-

tion [12, 28, 40], which transforms the non-convergent totwo equivalent convergent sums. The Ewalds summationprocedure as it is applied in our case is described in detailin Ref. [12].

2.2.2 Generalized transmission coefficient

The transmission coefficient, T , for an acoustic wavetransmitted through a periodic or random finite systemis given by the transmitted energy flux (in the far fieldregime), normalized by the incident energy flux,T jJj=jJ0j. The energy flux vector, J, is given by

Ji Re sij Re _uuj w Im s*ijuj

2; 12

where uj are the components of the displacement vector,u, and sij are the stress tensor components [41]. The lastpart in the r.h.s. of Eq. (12) is obtained for a time depen-dence of the form u; s / e iwt.

For fluids sij pdij and u rp=rw2; thus,Eq. (12) becomes

J 12rw

Im p*rp : 13

For an incident wave of the form

p0r hlkor Ylmr ; 14and in the far field (r !1), we have

p0r il 1 1ko

Ylmr eikor

r f 0r^r e

ikor

r

and J0r 1

2rw

kor2jf 0r^rj2 : 15

Under the presence of the sample (in the far field),

pr p0r Pnpscn r f 0r^r f scr^r

eikor

r

) Jr 12rw

kor2jf 0r^r f scr^rj2 : 16

Thus

qT 1 fscr^rf 0r^r

2 : 17

The scattering amplitude f sc can be calculated usingPnpscn r f scr^r eikor=r and the expansions of the

pscn r in the far field. Manipulating these expansions prop-erly, one can find either the expression

f scr^r Pn

Plm

Pl0m0

bnlmgjl0m0lmRn

il0 1ko

Yl0m0 r 18

or its alternative,

f scr^r Pn

Plm

bnlmil 1

koYlmr Rn eikojRnj cos qn ;

19with qn the angle between r and Rn.

Classical vibrational modes in phononic lattices theory and experiment 767

3. Finite difference time domain method

3.1 Introduction

The FDTD method presented here is based on the discreti-zation of the full elastic time dependent wave equationthrough a finite difference scheme. Both the time and thespace derivatives are approximated by finite differencesand the field at a given time point is calculated throughthe field at the previous points. Thus, one can obtain thefield as a function of time at any point of a slab. Thefrequency dependence of the field is obtained through fastFourier transform of the time results.

The FDTD method, while it was well known in theacoustics and seismology communities [42][45], had notbeen applied until only few years ago in the study of pho-nonic crystals. Here the most important advantages of themethod are that: a) it can give the field at any point insideand outside a sample, at every time; b) it can give thefield in both frequency and time domains; c) its resultscan be directly compared with the experimental data, sincethe method calculates the transmission through finite sam-ples; d) it can be applied in systems with arbitrary materi-al combination (e.g. solids in fluids or fluids in solids); e)it can be applied in periodic systems as well as in systemswith arbitrary configuration of the scatterers, giving thusthe possibility to study defect states, waveguides, randomsystems etc. These important advantages of the methodhave been already exploited extensively in the field ofelectromagnetic wave band gap materials (photonic crys-tals) [46][51]. Here we will present the method as it isapplied in elastic two-dimensional systems, i.e. systems ofcylinders embedded in a host material, for transmissioncoefficient calculations.

3.2 Description of the method

The elastic wave equation in isotropic inhomogeneousmedia is [41],

@2ui@t2

1r

@sij@xj

; 20

where sij lr ulldij 2mr uij and uij @ui=@xj @uj=@xi=2 (in Cartesian coordinates). In the above ex-pressions ui is the ith component of the displacement vec-tor, ur, sij are the stress tensor components and uij thestrain tensor components; lr and mr are the so-calledLame coefficients of the medium [41] and rr is themass density. The l, m and r are connected with the wavevelocities in a medium through the relations m rc2t andl rc2l 2rc2t , where cl and ct are, respectively, the ve-locity of the longitudinal and the transverse component ofthe wave. In a multicomponent system the l, m and r arediscontinuous functions of the position, r.

As was mentioned above, here we consider systemsconsisting of infinitely long cylinders embedded in ahomogeneous material. A cross section of such a system(periodic) is shown in Fig. 1. We consider the z axis to beparallel to the axis of the cylinders and propagation in thexy plane. As the system has translational symmetryalong z-direction, the parameters lr, mr and rr do

not depend on the coordinate z, and the wave equation forthe z component is decoupled from the equations for the xand the y component. The equations for the x and the ycomponent can be written as

@2ux@t2

1r

@sxx@x

@sxy@y

;

@2uy@t2

1r

@sxy@x

@syy@y

;

21

where

sxx l 2m @ux@x

l @uy@y

;

syy l 2m @uy@y

l @ux@x

;

sxy m @ux@y

@uy@x

:

22

The above Eqs., (21) and (22), constitute the basis forthe implementation of the FDTD in 2D systems. For thisimplementation the computational domain is divided intoimax jmax subdomains (grids), with dimensions Dx; Dy,and the displacement vector components are discretizedaccording to

ui; j; k ui Dx; j Dy; k Dt ; x; y ; 23with 1 i imax, 1 j jmax and k 0.

In the Eqs. (21) and (22), the derivatives are ap-proximated in both space and time with finite differences[46]. For the space derivatives, central differences areused:

@u@x

i; j; k

Dx0ui; j; k ui 1=2; j; k ui 1=2; j; k=Dx ;

24@u@y

i; j; k

Dy0ui; j; k

ui; j 1=2; k ui; j 1=2; k=Dy :For the time derivatives, a combination of forward andbackward differences are used:

@2u@t2

i; j; k

DtDtui; j; k ; 25

where

Dtui; j; k ui; j; k 1 ui; j; k=Dt ;Dtui; j; k ui; j; k ui; j; k 1=Dt ;

26

and x; y:From the equation for ux of (21), expanding around

i; j; k and following the procedure described above, one


x

yFig. 1. The computational cell.

obtains

uxi; j; k 1 2uxi; j; k uxi; j; k 1

D2t

ri; j Dx sxxi1=2; j; k sxxi 1=2; j; k

D2t

ri; j Dy sxyi; j1=2; k sxyi; j 1=2; k :

27Similarly, from the equation for uy of (21), expandingaround i 1=2; j 1=2; k, one finds

uyi 1=2; j 1=2; k 1 2uyi 1=2; j 1=2; k uyi 1=2; j 1=2; k 1

D2t

ri 1=2; j 1=2 Dx

(

sxyi 1; j 1=2; k sxyi; j 1=2; k)

D2t

ri 1=2; j 1=2 Dy

(

syyi 1=2; j 1; k syyi 1=2; j; k):

28The sxx, sxy, syy at the time k Dt are functions of the

displacement vector components at the same time, k Dt,and are used for the updating of the fields for the nexttime. They are also discretized through Eq. (24) and theirexpressions after the discretization are given in 3.3.

The discretization presented in the previous equationsinsures second order accurate central differencing for thespace derivatives. This has as a result, however, the fieldcomponents ux and uy to be located at different spacepoints, i; j for the ux and i 1=2; j 1=2 for the uy.

Using the procedure described above, the componentsux and uy at the time step k 1 are calculated throughtheir values at the step k. For insuring stability of the cal-culation the stability criterion used is [46]

Dt 0:5=c1=Dx2 1=Dy2

p; 29

where the velocity c is the highest among the sound velo-cities of the components of the composite.

To treat periodic systems, one can use Blochs (peri-odic) boundary conditions [ur R exp ik R ur(R: lattice vector)] at the boundaries of the computationalsystem along the propagation direction (at the i 1 andi imax of Fig. 1 see dotted lines in Fig. 1), reducingthus considerably the computational memory and time.

For closing the computational space along the otherboundaries, avoiding any back reflection from thoseboundaries, usually absorbing boundary conditions areused. Absorbing conditions are used also in all boundariesif non-periodic systems are treated. Among a variety ofabsorbing boundary conditions which have been discussedand utilized in the literature, the most common are the so-called Murs [46, 52] and Liaos [46, 53] boundary condi-tions. Here we will present the absorbing conditions intro-

duced by Zhou et al. [44, 45], which are first order ab-sorbing conditions, giving almost no reflection from theboundaries, even after long computational times. Zhousconditions are obtained by requiring the reflection at theboundaries to be zero for two angles of incidence (q1, q2);they can be written in the form

A@uu@x B @uu

@y I @uu

@t 0 ; 30

where I is the identity 2 2 matrix, uu is the 2 1 matrixux; uyT (T denotes the transpose of a matrix), and A, Bare 2 2 matrices. For the boundary j jmax the matricesA and B can be expressed as

Aq1; q2 h1h1x2 h2x1

Q2 h2h1x2 h2x1

Q1 ; 31

B q1; q2 x2h1x2 h2x1

Q1 x1h1x2 h2x1

Q2 ; 32

with

Q1 clox

21 ctoh21 clo cto x1h1

clo cto x1h1 cloh21 ctox21

" #; 33

Q2 clox

22 ctoh22 clo cto x2h2

clo cto x2h2 cloh22 ctox22

" #; 34

and xi sin qi, hi cos qi (i 1; 2). clo and cto are, re-spectively, the longitudinal and the transverse wave velo-city in the host material of the composite. For the bound-ary j jmin the expressions of A and B are obtained fromEqs. (31) and (32) by replacing qi by qi p (i 1; 2).

The condition (30) is discretized also using central dif-ferences in space and forward differences in time. For theimplementation of (30) in phononic systems the require-ment of complete absorption for q1 0 and q2 p=4gives in most of the cases exceptionally satisfactory re-sults.

For calculating the transmission, the incident wave thatis usually used is a pulse with a Gaussian envelop inspace. The pulse is formed at t 0 at the left side of thecomposite and propagates along the y-direction (seeFig. 1) A longitudinal pulse like that has the form

uy a sin wt y=clo exp bwt y=clo2 ; 35while for a transverse one uy is replaced by ux and clo bycto. The incident pulse is narrow enough in space as topermit the excitation of a wide range of frequencies.

The components of the displacement vector as a func-tion of time are collected at various detection points de-pending on the structure of interest. They are convertedinto the frequency domain using fast Fourier transform.The transmission coefficient (T) is calculated either bynormalizing the (frequency dependent) transmitted fieldintensity (u2x u2y) by the incident field intensity or bynormalizing the transmitted energy flux vector, J(Ji sij duj=dt for real fields), by the incident one.

The pure acoustic waves case (waves in fluid compo-sites) can also be treated with the FDTD using theEqs. (21)(22), but one has to omit the terms containing


the Lame coefficient m. The equations are discretizedthrough the same procedure as for the full elastic case.The boundary conditions coefficients are calculated againthrough the Eqs. (31)(34) where the velocity cto must bereplaced by clo (this replacement is essential in all thecases where the host material is fluid).

3.3 The calculation of the coefficients sxx, syy, sxy

sxxi 1=2; j; k l 2m i 1=2; j uxi 1; j; k uxi; j; k=Dx li 1=2; j uyi 1=2; j 1=2; k uyi 1=2; j 1=2; k

=Dy ;

36sxxi 1=2; j; k l 2m i 1=2; j uxi; j; k uxi 1; j; k=Dx li 1=2; j uyi 1=2; j 1=2; k uyi 1=2; j 1=2; k=Dy ;

37sxyi; j 1=2; k mi; j 1=2 uxi; j 1; k uxi; j; k=Dy mi; j 1=2 uzi 1=2; j 1=2; k uyi 1=2; j 1=2; k=Dx ;

38sxyi; j 1=2; k mi; j 1=2 uxi; j; k uxi; j 1; k=Dz mi; j 1=2 uyi 1=2; j 1=2; k uyi 1=2; j 1=2; k=Dx ;

39sxyi 1; j 1=2; k mi 1; j 1=2 uxi 1; j 1; k uxi 1; j; k=Dy mi 1; j 1=2 uyi 3=2; j 1=2; k uyi 1=2; j 1=2; k=Dx ;

40sxyi; j 1=2; k mi; j 1=2 uxi; j 1; k uxi; j; k=Dy mi; j 1=2 uyi 1=2; j 1=2; k uyi 1=2; j 1=2; k=Dx ;

41syyi 1=2; j 1; k l 2m i 1=2; j 1 uyi 1=2; j 3=2; k uyi 1=2; j 1=2; k=Dy li 1=2; j 1 uxi 1; j 1; k uxi; j 1; k=Dx ; 42

syyi 1=2; j; k l 2m i 1=2; j uyi 1=2; j 1=2; k uyi 1=2; j 1=2; k=Dy li 1=2; j uxi 1; j; k uxi; j; k=Dx : 43

Part II: Theory-numerical results

4. General remarks

This section serves as an introduction to the vibrationalband structures of periodic elastic composites which is themain theme of this review. Most of the composites nowproposed for practical applications have only two constitu-ents, and we will focus our attention on this class of com-posite materials; albeit improved fabrication technology isexpected to lead to the synthesis of more sophisticatedcomposite materials in the future. An important motivationfor a systematic investigation of the vibrational band struc-ture of phononic crystals came from the analogous theore-tical and experimental findings in photonic crystals. EMwaves in photonic crystals and elastic waves in phononiccrystals represent classical analogues of the quantum me-chanical waves of electrons in crystals. Some contrastingproperties of electronic, photonic, and phononic crystalsare listed in Table 1. Of special interest are phononic crys-tals, tailored from two materials which differ in their elas-tic properties, with a complete absolute frequency-gap;this is a frequency region wherein phonons (vibrations)are prohibited, irrespective of the polarization of the pho-non and of the direction of its propagation in space.

From a fundamental point of view, cleverly synthesizedperiodic elastic composites exhibiting a complete band-gapmay offer a systematic route to realize the Anderson loca-lization of sound and vibrations, just as the Anderson lo-calization of light. The term localization has recently ap-peared in the literature on wave propagation in randommedia, but less commonly than in the theory of disorderedcondensed systems. Shortly after the seminal paper of An-derson on electron localization in disordered systems [54],the problem of phonon localization in random systemswas considered [55]. However, a closely related subject oflocalization of elastic (and acoustic) waves in macroscopicsystems, where the role of the disordered material isplayed by a medium with a sharply varying elastic (andacoustic) properties which can strongly scatter the respec-tive vibrations, has drawn attention only more recently[56][61]. The concept of classical localization focusedinitially on elastic waves in disordered solids. John et al.[6264] studied the localization of phonons in a 2 edimensions using a first-principles theory based on thefield theoretical formulation of electron localization byWegner [65]. It was demonstrated that all finite-frequencyphonons in one and two dimensions are localized withlow-frequency localization length diverging as w2 ande1=w

2, respectively, and that a mobility edge, w*, separat-

ing low-frequency extended states from high-frequency lo-calized states exist for n > 2; n refers to the number ofdimensions. This field theory (or nonlinear s-model) waslater recovered and extended by diagrammatic perturbationtechniques [66]. Other important theoretical [67][76] andexperimental [77][86] developments related with the clas-sical wave localization were exhaustively discussed in [3].

From practical point of view, periodic elastic compo-sites comprised of two (or, more) different materials arebecoming of increasing importance in modern technology.Such composites allow the tailoring of some special prop-


erties, unavailable in their homogeneous constituents. Aperiodic elastic composite can be synthesized such as toexhibit a complete elastic band-gap. Thus a vibrator or asmall (real) crystal introduced into an otherwise periodiccomposite as a defect would be unable to generate soundor vibrations within the band-gap. This implies that suchperiodic elastic composites could be engineered to providea vibrationless environment for high precision mechanicaldevices in a given frequency range. Ferroelectric, piezo-electric, pyroelectric, and piezomagnetic periodic compo-sites have had long-standing applications as medical ultra-sonic and naval transducers, as well as for related tasks inmedical imaging [87][96]. Such composites were initi-ally constructed and used for sonar applications, and arenow being widely used for ultrasonic transducers. Com-bining, for example, a piezoelectric ceramic and passivepolymer to form a periodic composite allows the transdu-cer engineer to design new piezoelectrics which offer sub-stantial advantages over the conventional piezoelectricceramics and polymers. The novelty of the resultant struc-ture lies not in the constituents but in the way they areassembled to produce materials with properties suitable toeach specific application. The effective properties of thecomposites are usually expressed in terms of averagesover the properties of the constituents. However, examin-ing these composites on a scale in which the substructureof the constituent ceramic and polymer are evident leadsone to understand the principles used to design the com-posite. Often these substructures are just miniature ver-sions of transducers familiar to the design engineer on amuch larger scale.

The extensive research on elastic and/or acoustic peri-odic composites actually started when Sigalas et al. [97]reported a narrow but complete band-gap for Au cylindersin Be matrix. The opening up of the spectral gaps in aperfectly periodic, binary elastic composite owes, in gener-al, to five mismatched parameters involved in the problem.These are the ratio of the mass densities, ratio of the long-itudinal velocities, ratio of the transverse velocities, ratioof the longitudinal and transverse velocities, and the fillingfraction. The prospects of achieving such band-gaps foracoustic waves in periodic binary system of liquids and/orgases would, however, much improve because only long-itudinal modes are supported therein. This means that onlythree parameters will effectively be involved. Both elasticand acoustic cases thus contrast their photonic counter-parts where only two dimensionless parameters are in-volved. It is thus conceivable that the phononic crystalsoffer a richer and more complex behavior, and mayrequire relatively more extreme conditions for the obten-tion of complete band-gaps.

At the outset, it is interesting to remark that in all arti-ficial periodic structures the existence of complete gaps isattributed to the joint effect of the Bragg diffraction andthe Mie scattering. The destructive interference due toBragg diffraction accompanied by the Mie resonances dueto strong scattering from individual scatterer is the concep-tual base of a complete gap. The latter becomes effectivewhen the dimension of the scatterer is close to an integermultiple of wavelength [97]. A complete gap is, by defini-tion, the one that persists independent of the direction of

propagation and of the polarization of the wave. However,if the separability of the z and xy modes is legitimate, asis the case with the two-dimensional (2D) phononic crys-tals, the term absolute is preferred to complete, just toavoid the confusion. As a matter of fact, each of thesemodes can be excited independently of the other, at leastin the 2D phononic crystals. Such a separation of thez- and xy modes suggests an application, namely, a po-larization filter. Suppose that the elastic/acoustic waves ofarbitrary polarization are incident at the surface of a peri-odic composite. Then, provided that their frequency lieswithin a band gap for z modes, the z-polarization compo-nent will be totally reflected, and only the xy modes willbe transmitted.

In conclusion, the literature is a live example that, de-spite some of the scattered articles [98][100] remotelyconcerned with the subject, research in phononic crystalsbecame more solid [101][150]. New ideas emerged forpotential device applications, but the driving force was therich fundamental physics governing the elastic wave pro-pagation in diversely designed phononic crystals. The opti-mum choice for the design of such phononic crystalswhich can (and do) exhibit complete large band gaps, irre-spective of the dimensionality of the system, is governedby the topology of the resultant structure. The early ex-perience of the two above-mentioned research groups ledto infer that it is the cermet topology that favors the crea-tion of elastic/acoustic stop bands as compared to the net-work topology that favors the creation of the optical bandgaps in the photonic crystals. These two topologies areillustrated in Fig. 2. W note that in describing the resultsreported in the following sections, we reserve the termelastic (acoustic) waves for the sound/vibrations propagat-ing in inhomogeneous solids (fluids). In addition, it shouldbe pointed out that we will sometimes recall some equa-tions directly from Ref. [4], without specifying further de-tails. In that sense, the following sections heavily rely onRef. [4].

5. Longitudinal vibrations

In this section we aim at describing the numerical resultson the band structure related problem for longitudinal vi-brations, which include both acoustic and elastic waves.


Fig. 2. The cermet topology which is favored by elastic wave con-sists of high index (low velocity) inclusions in a connected low index(high velocity) background. The network topology favored by EMwaves consists of two interpenetrating, connected components.

While the former are the well-known modes supportedonly by the liquids and gases, the propagation of the latteris allowed only within a one-dimensional inhomogeneousmedium; two- and three-dimensional inhomogeneous med-ia do not support purely longitudinal elastic waves.

5.1 One-dimensional systems

Here we start with a simplest example for understandinghow the band-gaps (or stop-bands) can be realized in aone-dimensional periodic system. This could be visualizedas an infinitely long and thin tube of fluid that has beenperturbed by a periodic (of period d) d-function-like var-iations (increases or decreases) in the density along itslength. In this situation, the dispersion relation betweenthe Bloch vector K and the (bulk) frequency w is given by[99]

cos Kd cos kd s2kd sin kd f kd : 44

Equivalence between Eqs. (44) and (2.42) in Ref. [4] isevident: with k w=cl, where cl refers to the longitudinalspeed in the bulk between the d-functions, ands r1d1=r2d is a measure of the strength of d-functions.

Clearly, Eq. (44) can only be satisfied for jf kdj 1.For frequency values kd where this condition holds, thepropagation through the lattice is allowed; for a range ofkd where it is violated, the band-gaps appear. These re-sults are indicated schematically, for s 2, in Fig. 3.Although this proof of the opening up of band-gaps is onedimensional, the scalar nature of the sound field is closelyparalleled by the scalar (spinless) Schrodinger electron

function that observes energy band structure in manythree-dimensional periodic potentials-Kronig-Penney mod-el, for instance, illustrates the similar picture. From theplots of two step functions q1 jf aj, with a kd,as bold solid line, one can see enclosed by rectangles thefirst three propagation windows, each of which begins ata np; n 0, 1, 2. The condition a np impliesd nlk=2 and hence occurs when an integer number ofhalf wavelengths of a normal-mode wave function fitsacross one period interval of length d. In order to probethe acoustic band, as depicted in Fig. 3, Dowling [99] alsoconsidered the steady-state power output of a localizedsource in the tube of fluid with a periodically varying den-sity. The computed radiated energy from such a source ledhim to infer that it would be possible to quench thesesources at the frequencies lying within the band-gaps,while amplifying their outputs in the propagation win-dows. This is analogous to the behavior of one-dimen-sional atom radiating electromagnetically between one-dimensional mirrors.

Next we turn to the propagation of elastic waves in one-dimensional superlattice systems. Longitudinal modes insuch systems are supported in the situation discussed in thebeginning of Sec. 2.5 in Ref. [4]. Here we are concernedwith a binary superlattice system made up of alternatelayers of Al and epoxy [100]. The band structure for long-itudinal elastic waves propagating in this system is de-picted in Fig. 4. Numerical results in this figure are basedon Eq. (2.41), with cti replaced by cli and the propagationvector ~kkk 0. The layering gives rise to the splitting oflongitudinal waves with Bloch vector q 2np=d; dbeing the period. In general, real q leads to the allowedbands and imaginary q to the band-gaps. Inset in Fig. 4shows the reflectivity as a function of frequency. As it isexpected, the reflectivity approaches unity for the frequen-cies lying within the band-gaps. One also notes (see Fig. 4in Ref. [100]) that the width of the allowed and forbiddenbands decreases with increasing period. This results is, how-ever, not unique to the elastic band structure; similar beha-vior has been noted for the electrons in crystal lattices andfor plasmons in metallic or semiconductor superlattices.


Fig. 3. We plot here as a solid curve, the right-hand side of the dis-persion relation, Eq. (44), denoted by fsa, where a kd and take ad function of strength s 2. The dashed lines represent fsa 1.From the condition jfsaj 1, we can see that sonic band gaps oc-cur when this inequality is violated and sonic passbands occur whenit is obeyed. If we plot the two step functions q1 jfsaj as boldsolid lines then we see enclosed by solid rectangles the first threepassband regions, each of which begins at a np; n 0, 1, 2. Thecondition a np implies d nlk=2 and hence occurs when an inte-ger number of half-wavelengths of a normal-mode wave function fitsacross one period interval of length d. (After Dowling, Ref. [99]).

Fig. 4. Band structure for the longitudinal elastic waves propagatlingin a periodic layered binary system. Computation was performed forAl (d1 0:09 mm)/epoxy (d2 0:02 mm) system. The inset showsthe reflectivity as a function of frequency for the corresponding semi-infinite system. (After Esquivel-Sirvant and Cocoletzi, Ref. [100]).

This can be understood by realizing that large separation ofelastic films reduces the coupling strength between the (sur-face) excitations on each film, which in turn reduces theband-widths a result similar to the reduction in band-widthfor electronic states when the atoms are moved farther apart.Other factor that does influence the band structure is theratio of the sound speeds in the two media of the unit cell.

Finally, we present some band structure results forlongitudinal (acoustic) wave propagation in a system madeup of N 0 dangling side branches (DSB) periodicallygrafted at each of the N equidistant sites on a slender tube[112]. For the sake of simplicity, we embark on the sim-pler system of air and/or water tubes. Of course, in prac-tice, the gas or liquid within these tubes would be con-tained by means of some latex material. The mass densityand speed of sound in rubber are comparable to those ofwater [see, e.g., C.R.C. Handbook of Chemistry and Phy-sics, 66th Edition (CRC Press, Florida, 1985), p. E-43].Hence for a sufficiently thin latex partition, the presenceof this third extra layer should not affect the calculationssignificantly and, in fact, we will neglect it. We have con-sidered the situation both for open and closed tubes. Evi-dently, the relevant parameters involved in the problem areri, vi, di, and ai; as well as the integers N and N

0; withsubscript i 12 for slender tube (DSB). For the reasonsof space, we will discuss only the results for N 0 1. It isworth mentioning that the validity of our results is subjectto the requirement

ai

p di, l. It should also be pointedout that the methodology employed in this particular workis based on the interface response theory of Dobrzynski[Surf. Sci. Rep. 6 (1986) 119].

Figure 5 shows the band structure and the transmissioncoefficient for the open tubes with identical media inside

the DSB and the slender tube. The opening up of the stopbands in the band structure is very well substantiated bythe transmission spectrum. Most important aspect of theseresults is the cutoff frequency Wc below which no propaga-tion at all is allowed. This gives rise to the utter discretiza-tion of the propagation starting right from zero frequency.As regards the regular repetitive pattern of the band struc-ture (and the transmission spectrum), one can easily under-stand this from Eq. (3) in Ref. [112]. For instance, for iden-tical media inside the DSB and the slender tube, Eq. (3)simplifies to W cos12=3 cos kd1 2np; whereW q1d1=v1 is the reduced frequency, p kd1 p,and n is an integer (including zero). This clearly suggeststhe periodic pattern depicted in Fig. 5, where the cutofffrequency Wc is defined by the lowest frequency of thelowest mode (n 0) at the zone center. Other details re-garding the physical conditions subject to the material (riand vi) and geometrical (di and ai) parameters required toachieve the complete stop bands can be seen in Ref.[119]. Note that the situation is dramatically different atd2=d1 0:5, where the aforesaid conclusion may be seento fail as compared to the neighboring cases every pairof bands (counting from the bottom) becomes degenerateat the zone boundary to form a closed loop. What is im-portant to note is that the existence of the lowest gap be-low Wc is more often the rule than the exception: its mag-nitude varies with the variation of both material andgeometrical parameters, though.

The numerical results for a system of closed tubes withidentical material media inside the slender tube and theDSB are illustrated in Fig. 6. There are two major differ-ences when compared to the case of open tubes: the low-est gap extending up to zero has disappeared, and the unitcell of the repetitive pattern now contains four bands (ascompared to two in Fig. 5. The numerical results for asystem of closed tubes with the identical material mediainside the slender tube and the DSB are illustrated inFig. 6. There are two major differences as compared to thecase of open tubes: the lowest gap extending up to zero


Fig. 5. Band structure (left panel) and transmission spectrum (rightpanel) for the system of open tubes. Reduced wave-vector refers tothe dimensionless Bloch vector kd1 and the reduced frequency is de-fined by W wd1=v1. We consider identical fluid (air or water, forexample) both inside the dangling side branches (DSB) and inside theslender tube. The material and geometrical parameters are such thatr2=r1 1, v2=v1 1, d2=d1 1, and a2=a1 1. We call attentionto the periodic pattern of the band structure and the lowest gap belowthe lowest frequency (referred to as the cutoff or threshold frequencyWc in the text) at the zone center. In the right panel N 20 wasconsidered; N 0 1 everywhere. (After Kushwaha et al., Ref. [112]).

Fig. 6. The same as in Fig. 5, but for a system of closed tubes. Note thatthe unit cell of the periodic pattern now contains four bands as comparedto two in Fig. 4 and that the lowest gap pertaining to the system of opentubes no longer exists. (After Kushwaha et al., Ref. [112]).

has disappeared, and the unit cell of the repetitive patternnow contains four bands (as compared to two in Fig. 5).There are full intracell gaps but no intercell gaps. Therepetitive pattern observed in the band structure (and thetransmission spectrum) can be understood through a sim-ple analysis of Eq. (4) in Ref. [112]. Again, the formationof the band at d2=d1 0:5 is unique where all the closedloops, like the one formed by the second and third bandsfrom the bottom, disappear altogether.). There are full in-tracell gaps but no intercell gaps. The repetitive patternobserved in the band structure (and the transmission spec-trum) can be understood through a simple analysis ofEq. (4) in Ref. [112]. Again, the formation of the band atd2=d1 0:5 is unique where all the closed loops, like theone formed by the second and third bands from the bot-tom, disappear altogether.

Figure 7 illustrates the evolution of the transmissionspectrum with increasing number of N for the case of opentubes. We present the results for the case of identical fluidsinside the slender tube and the DSB. For N 1 (the upper-most panel), the transmission coefficient becomes immea-surably small (but never approaches exactly zero) at themidgap frequency corresponding to the larger N. For smal-ler N, vanishingly small transmission represents the lowdensity of states which are referred to as pseudogaps. It isobserved that as N increases the pseudogaps gradually turninto the complete gaps (with transmission equal to zero).However, it is interesting to note that the magnitude of thegap remains the same for N 5. The number of oscilla-tions in the transmission coefficient within the passbandhas been noted to be unfailingly N or N 1.

The crux in this work [112] was on the results for thetransmission spectrum depicted in Fig. 8. This refers to asimplest geometry of a long slender tube with airy DSB.We consider the length of the DSB equal to the period ofthe system (i.e., d2 d1) and the cross-sections of theDSB and the slender tube are also taken to be the same(i.e., a2 a1). The upper (lower) panel demonstrates thetransmission spectrum for the system of open (closed)tubes. Apart from the fact that the transmission coefficientdoes not approach exactly zero it remains at immeasur-ably small height above zero the magnitude of the stopbands for N 1 was seen to be surprisingly the same asfor N 1. This is found to be true for the systems ofboth open and closed tubes. Note that the lowest gap forthe open tubes persists below the cutoff frequency. Thediscretization of the transmission spectrum is attributed tothe presence of the DSB (whose number in the case athand is just one) that suppresses the transmission over al-most the whole range of frequencies except for those de-fined by C2 0 (S2 0) for open (closed) tubes. Thatthis so can easily be seen through a careful diagnosis ofthe expression of transmission coefficient T in Eq. (5) forN 1 [112]. This refers to a simplest geometry of a longslender tube with airy DSB. We consider the length of theDSB equal to the period of the system (i.e., d2 d1) andthe cross-sections of the DSB and the slender tube arealso taken to be the same (i.e., a2 a1). The upper (low-er) panel demonstrates the transmission spectrum for thesystem of open (closed) tubes. Apart from the fact that thetransmission coefficient does not approach exactly zero it remains at immeasurably small height above zero themagnitude of the stop bands for N 1 was seen to besurprisingly the same as for N 1. This is found to betrue for the systems of both open and closed tubes. Notethat the lowest gap for the open tubes persists below thecutoff frequency. The discretization of the transmissionspectrum is attributed to the presence of the DSB (whosenumber in the case at hand is just one) that suppresses thetransmission over almost the whole range of frequenciesexcept for those defined by C2 0 (S2 0) for open(closed) tubes. That this so can easily be seen through a


Fig. 7. Evolution of the transmission spectrum as a function of N fora system of open tubes. Note that as N increases the visible slopes inthe minimum of the transmission diminish and ultimately vanish. Theparameters (both material and geometrical) are the same as in Fig. 5.(After Kushwaha et al., Ref. [112]).

Fig. 8. Discrete transmission spectrum for a system of open tubes ofairy dangling side branches grafted on a slender water tube for N 1.The geometrical parameters are the same as in Fig. 5. Existence of thelowest gap is noteworthy. (After Kushwaha et al., Ref. [112]).

careful diagnosis of the expression of transmission coeffi-cient T in Eq. (5) for N 1 [112].

5.2 Two-dimensional systems

In this section we discuss the exciting possibility of creatingcomplete acoustic band-gaps in a two-dimensional systemmade up of liquids. We chose water pipes of circular cross-section immersed in mercury which can be arranged to havetwo macroscopic geometries of interest-square and hexago-nal lattices. Analytical results for the longitudinal wavespropagating in such systems are describable in the frame-work of Eq. (2.67) in Ref. [112], with ~kk and ~gg being thetwo-dimensional vectors. The relevant expressions for thefilling fraction f and the structure factors FG are given by

f pr0=a2 ; for square lattice2p3

p r0=a2 ; for hexagonal lattice

8>: 45

and

FG 2fJ1Gr0=Gr0 ; 46where a and r0 are, respectively, the lattice constant andthe radius of the pipes. J1x is the Bessel function of thefirst kind of order one. Our choice of the specific materi-als (i.e., water and mercury) was motivated by the densitycontrast; although there is no substantial contrast in thespeed of sound in liquids, in general. The longing ques-tion of how to sustain water pipes in mercury and the roleplayed by the latex material which the walls of the pipesare made of will be discussed later in this section. Thestandard eigenvalue problem represented by Eq. (2.70) inRef. [112] was solved to obtain real eigenfrequencies. Agood convergence (of better than 1%) was achieved bylimiting the number of plane waves to 361, in both squareand hexagonal geometries. To be more explicit, the inte-gers nx and ny, in the reciprocal lattice vector ~GG, werepermitted to take the values between 9 and +9 (i.e., 361plane waves). As discussed below, multiple low-frequencyband-gaps were found for both geometries [105]. How-ever, the width of the lowest (and the widest) band-gaps,for the same value of filling fraction, is found to be largerin the hexagonal pattern than in the square pattern.

Figure 9 illustrates the density of states (DOS) for asquare pattern of infinitely long, circular water pipes inmercury that occupy 35% of the total area. The computa-tion of DOS involves 4950 ~kk-points covering the bound-aries as well as the interior of the irreducible part of thefirst Brillouin zone. DOS curve depicts vividly the exist-ence of two complete acoustic band-gaps appearing withinthe first ten bands. The first gap opens up between thefirst (with MM1max) and the second (with XX2min) bands. Si-milarly, the second band-gap exists between fourth ( XX4max)and fifth ( XX5min) bands. It is worth mentioning that onecan choose to plot as many or as few bands as one likes,albeit the computer does calculate all (i.e., 2n 12bands; n 9 in the present case) bands. We have noticedthat if we plot first 50 bands, there appear several narrowbut complete band-gaps at higher frequencies. Again, thisis true for both square and hexagonal lattices.

We summarize the existence of the lowest, which isalways the widest, band-gap for the whole range of fillingfraction and for both geometries in Fig. 10. The curve de-signated as SWM (HWM) refers to the square (hexagonal)lattice with water pipes in Mercury. The reverse (i.e., mer-cury pipes in water host) is the case with the curves labeledas SMW and HMW. One can easily notice that in each casethere is a certain minimum value of the filling fraction, fmin,for a gap to be opened; and likewise there is a certain max-imum, fmax, where the gaps cease to exist. For the squarelattice, with water pipes in mercury host (SWM), fminfmaxis defined as 0.017 (0.77); and for hexagonal lattice,fminfmax is given by 0.008 (0.79). It is noticeable that thegaps cease to exist before the close-packing is attained close packing corresponds to the geometrical pattern whenthe containers of the inclusions (pipes in the present case)start touching each other. The close-packing in the square(hexagonal) lattice is defined by f 0:7854 (0.9069);these values of f correspond to r0 a=2.

Let us now describe briefly the situation when the mer-cury pipes are embedded in the water as a host. The com-putation of the lowest (and the only ones) band-gaps ver-sus filling fraction is demonstrated by the curves labeledas SMW and HMW, respectively, for square and hexago-nal lattices in Fig. 9. In this case the fminfmax is given by 0:35 (0.7854) for the square lattice; the correspondingvalues for the hexagonal lattice are 0:48 (0.822). It isnoteworthy that while SMW refers to the lowest band-gapexisting between the first and the second bands; HMWstands for the gap occurring between the third (withGG3max) and fourth (with XX4min) bands. In this situationthere are no other band-gaps at least as far as 50th band.This comment is valid for both geometries.


Fig. 9. Computed DOS for acoustic waves in a 2D periodic system ofcircular water pipes in mercury. The filling fraction f 0:35. The com-putation involves 4950 ~kk-points covering the boundaries as well as theinterior of the irreducible triangle GG XX MM of the first Brillouin zone. Thematerial parameters are: r 1:025 (13.5) gm/cm3, cl 1531 (1450) m/sec for sea-water (mercury). We call attention to the two wide spectralgaps: The first gap opens up between the first (with MM1 max) and second(with XX2 min) bands; the second gap occurs between fourth (with XX4 max)and fifth (with XX5 min) bands. (After Kushwaha and Halevi, Ref. [105]).

An extensive investigation of this specific two-dimen-sional inhomogeneous system of liquids leads us to inferthat low-density, high-velocity inclusions in the high-den-sity, low-velocity host is the optimum geometry for theobtention of multiple wide band-gaps and hence for thelocalization of the acoustic waves in a weakly disorderedsystem. This finding is in entire agreement with our con-clusion drawn in the case of cubic arrays of spherical gas-eous balloons in air (see the following subsection). How-ever, this does disagree with the conclusion [6] that theoptimum case for the appearance of gaps or for the locali-zation of acoustic waves is low-density, low-velocityspheres occupying a volume fraction of the order of 10%of the host material. It should be emphasized, however,that this was the conclusion arrived at on the basis of acalculation carried out for fictitious material parameters bythe authors of Ref. [6].

Now we turn our attention to the latex material con-sidering the optimum geometry where the water pipes areembedded in mercury host. In the latex material (i.e., in-side the walls of the pipes) both transverse and longitudi-nal vibrations are allowed. As such, it is quite likely thatthese shear oscillations produce a finite DOS within theacoustic band-gaps. We expect, however, that this DOSshould be very small if the walls of the pipes are thinenough. We estimated the error involved in neglecting thelongitudinal vibrations in the latex by calculating the cor-rection to s~GG, Eq. (3.60) in Ref. [4], due to a circularshell of thickness D and mass density rw. Provided thatrw > rb> ri and D r0, this correction is given by [4]

sw~GG r1w d~GG; 0 D

r0

E~GG

; 47

where E~GG is a function of the same order of magnitudeas F~GG, Eq. (3.3) in Ref. [4]. A careful look at Eqs.(2.60) and (3.4) in [4] reveals at once that these compres-sive oscillations in latex wall would not alter our resultssignificantly provided that D=r0 rw=ri rw=rb.The left-hand side of this inequality is assumed to bemuch less than one, whereas the right-hand side is indeedmuch greater than one if ri is considerably smaller thanrb, as is the case for water pipes in mercury. A remarkmade on the rubber as the latex material in the previoussection is still valid here.

It is worth noting that a simple two-dimensional inho-mogeneous system of liquids as the one discussed in thissection exhibited the widest band-gaps reported for elastic,acoustic, or optical waves up to that time. To justify thiscomment, we calculate the gap/mid-gap ratios for both thesquare and hexagonal patterns, respectively, for f 0:34and 0.27. The result is a gap/mid-gap ratio for square(hexagonal) lattice of 0.901 (0.984). These were, to ourknowledge, the largest numbers that define the magnitudeof the acoustic band-gaps; irrespective of whether the sys-tems studied in the past were solids, fluids, or gases[100][122]. Recently, the 2D (3D) systems of airy cylin-ders (air bubbles) in water have been shown to exhibiteven larger band gaps, with a gap/midgap ratio of 1.8[113, 114]; these remain to be the widest gaps ever re-ported for photonic and/or phononic crystals to date. With-in these band-gaps the sound and longitudinal vibrationsare forbidden and the total silence prevails.

The simplest way of realizing such acoustic band-gapsin two-dimensional periodic systems is to embed the infi-nitely long thin water pipes in a substrate that forms abottom of mercury tank. If the bottom is smooth, oneshould obtain the complete acoustic band-gaps other-wise (i.e., if the bottom is rough or non-periodic) oneshould be able to observe the localized acoustic modesexisting within these gaps [79].

5.3 Three-dimensional systems

Now we focus on the longitudinal modes propagating inan inhomogeneous three-dimensional system comprised ofliquids and/or gases. The methodology used for computingthe band structure in such systems is the one discussedabove (see Sec. 2.6 in Ref. [4]). Here we discuss our nu-merical results on the band structure of cubic arrays ofspherical balloons in fcc, bcc and sc arrangements. Thefilling fraction f and the structure factor F~GG for suchgeometries are given by

f n 4p3

r0a

348

and

F~GG 1Vc

a

d3r ei~GG ~rr 49

3fGr03sin Gr0 Gr0 cos Gr0 ; 50

where n; r0; and a are, respectively, the number of spheresin the unit cell, radius of the sphere, and the lattice con-


Fig. 10. Normalized gap-widths of the lowest band-gaps as a func-tion of filling fraction f . The curves designated as SWM (HWM)refer to the square (hexagonal) lattice with water pipes in mercury.The reverse (i.e., mercury pipes in water) is the case with the curveslabelled as SMW and HMW. The material parameter are the same asin Fig. 8. The dashed and dashed-dotted vertical lines refer to theclose-packing values of the filling fraction for square and hexagonallattices, respectively. (After Kushwaha and Halevi, Ref. [105]).

stant. We made use of Eq. (2.70) in Ref. [4] to computethe band structure of fcc, bcc, and sc arrays of hydrogenballoons in air [107]. For a realistic situation where allthree media (i.e., gas inside the balloons, the latex walls,and the background gas) were considered, we obtained thereal eigenvalues and good convergence (of better than 2%)by limiting the number of plane waves to 343. Completeband-gaps were found for fcc and bcc geometries; how-ever no gap was obtained for sc lattice. Numerical resultsfor the fcc arrangement of balloons that occupy 35% ofthe total volume are depicted in Fig. 11. The left panel ofFig. 10 shows the band structure in the principal symme-try directions in the Brillouin zone. The middle panel isthe result of an extensive scanning of j~kkj in the irreduciblepart of the first Brillouin zone including the interior ofthis zone and its surface, as well as the principal direc-tions shown in the left panel. The right panel illustratesthe density of states whose computation is based on thescanning of j~kkj depicted in the middle panel. There ap-pears a complete acoustic band-gap between the first andthe second bands, and there are no more gaps at least asfar as the 50th band. Three panels together establish thatthis is, indeed, a complete gap, irrespective of the direc-tion of propagation. Similar results are obtained for thebcc geometry; the sc geometry did not observe a completegap for any value of filling fraction, however.

The dependence of the (lowest) band-gap on the fillingfraction, for fcc and bcc structures is summarized inFig. 12. It is found that the filling fraction must exceed acertain minimum value, fmin, for opening up of a gap. Ifthe pressure inside the balloons is 1.1 atm, then for fcc(curve A) and bcc (curve E) structures fmin 0:12 and0.21, respectively. The corresponding maximum values arefmax 0.63 and 0.54. For any value of filling fraction, thefcc structure gives a wider band-gap than the bcc struc-ture. For both lattices, the band-gaps are widest whenf 0:38; that is when the balloons occupy 38% of the

space. However, the gap/midgap ratio is considerably lar-ger for fcc array about 0.2 (0.1) for fcc (bcc). Thecurves B, C, and D (in Fig. 12) illustrate what happenswhen the pressure inside the balloons is successively in-creased to 1.32, 1.65, and 1.98 atms considering the fccstructure. This leaves the speed of sound unaltered, thoughthe density contrast decreases. If the pressure inside is al-most twice the pressure outside, the density contrast isabout 7, and the band-gap almost disappears (see curve E).It is then apparent that, for high-velocity balloons in alow- velocity host, the band-gaps can exist only if thedensity of the (gas inside the) balloons is considerablysmaller than the density of host.

A word on the latex material which the wall of theballoons is made of is in order. In this substance trans-verse, as well as longitudinal, vibrations are permitted. Itis quite likely that these shear oscillations produce a finitedensity of states (DOS) within the acoustic band-gaps. Weexpect, however, that this DOS should be very small pro-vided that the balloons wall is thin enough. We estimatedthe error involved in neglecting the longitudinal waves inthe latex by calculating the correction to s~GG, Eq. (2.60),due to a spherical wall of thickness w and mass densityrw: If rw rb> ri and w r0, this correction is [4]

sw~GG r1w d~GG; 0 w

r0

H~GG

; 51

where H~GG is a function of the same order of magnitudeas F~GG, Eq. (50). By comparing Eqs. (51) and (2.60) inRef. [4], we see that the compressive oscillations of thelatex wall are not expected to alter significantly our resultsprovided that w=r0 rw=ri rw=rb. The left-hand


Fig. 11. Acoustic band structure for an fcc array of spherical bal-loons containing hydrogen gas in air (left panel of triptych). Middlepart: frequency eigenvalues as a function of j~kk j (the magnitude of theBloch vector) scanned throughout the irreducible part of the Brillouinzone. DOS as a function of reduced frequency is graphed in the rightpanel of the figure. The hatched area refers to the spectral gap forsound or vibrations. The parameter contrast are r(H2=r(air) 0.076and cl(H2=cl(air) 3.706. (After Kushwaha and Halevi, Ref. [107]).

Fig. 12. Normalized gap-widths versus filling fraction f for fcc andbcc structures (there is no gap for sc structure). For both structuresthe largest gap is obtained for f 0.38, and, for a pressurep 1:10 atm, the approximate gap/midgap ratios are 0.2 and 0.1 forthe fcc (curve A) and bcc (curve E) lattices, respectively. The varia-tion of the gap with pressure is also shown for fcc structure. As pincreases to 1.32 atm (curve B), 1.65 atm (curve C), and 1.98 atm(curve D), the gap gradually disappears. (After Kushwaha and Halevi,Ref. [107]).

side of this inequality has been assumed to be much smal-ler than one, while the right-hand side is indeed muchgreater than one if ri is substantially smaller than rb, justas is the case for hydrogen balloons in air. We recall theremark made in the previous section on rubber as the latexmaterial.

Within the gaps of Figs. 11 and 12, the perfectly peri-odic phononic crystals, stand still and total silencereigns. The situation is of comparable interest to the fullphotonic band-gaps in periodic dielectric composites[151]. These, however, were realized only with certaincomplex structures of the fcc unit cell not with the sim-ple fcc lattice, and much less for bcc and sc lattices. Thecubic arrays of balloons discussed here are probably thesimplest physical systems that exhibit complete band-gaps.

Similar theoretical investigations on the band structuresfor cubic arrays of spherical water balloons surrounded bymercury host exhibit multiple, complete acoustic stopbands for all the three (fcc, bcc, and sc) arrangements[106]. These stop bands are seen to be widest for a vol-ume fraction f 24% and the corresponding gap/midgapratios are about 0.83, 0.77, and 0.62, respectively, for fcc,bcc, and sc lattices. In the reverse situation, where mer-cury balloons are surrounded by water, the gaps obtainedare found to be surprisingly small.

6. Transverse vibrations

As one can see from the analysis presented in Sec. 5 inRef. [4], the propagation of purely transverse modes ispermissible only in one- dimensional periodic systems,superlattices, for examples, and two-dimensional periodicelastic composites; three-dimensional periodic compositesdo not allow the resolution of this polarization. In this sec-tion, we would discuss the transverse modes propagatingin the infinite and semi-infinite elastic superlattices. Themain emphasis would be laid on the creation of elasticband-gaps in two-dimensional periodic, both square andhexagonal patterns, elastic composites. These elastic com-posites are made up of an array of infinitely long, thinrods of an isotropic solid i and embedded in a differentelastic background b, which is also isotropic. The inter-section of the parallel rods with a perpendicular planeform a square or hexagonal lattice. There is a translationalinvariance in the direction z^z parallel to the rods and thesystem has a two-dimensional periodicity in the transverse(x^x y^y) plane. As already mentioned, we will confine tothe transverse modes with ~uu z^zu and ~rr ~uu 0. The jus-tification lies in the fact that this is the only case when thegeneral wave equation for inhomogeneous solids greatlysimplifies.

6.1 One-dimensional systems

The problem of elastic wave propagation has been thesubject of numerous theoretical and experimental studiesduring the past decade (see for example Sec. 1 in Ref.[4]). The extended states propagating in a superlattice ofinfinite extent form the bulk bands which are separated bysmall gaps. The surface phonon-polaritons, which are the

elastic waves localized at and decaying exponentiallyaway from the interfaces, may exist within these gaps. Thespatial location of these surface modes in the w kk planemuch depends on the way an otherwise perfectly periodicsuperlattice system is truncated. For the details of thesemechanisms, the reader is referred to the recent work byDjafari-Rouhani and collaborators [152, 153]. Here we areinterested to give a taste for the simplest geometry wherethe periodicity of the superlattice is truncated with a semi-infinite homogeneous elastic medium. The dispersion rela-tion for such a semi-infinite superlattice system was de-rived by Camley et al. [154] almost two decades ago. Theresults is

F1 tanh a1d1 F2 tanh a2d2 0 52where the symbols have the same meanings as defined inSec. 2.5 in Ref. [4]. In order to examine the effect of per-turbing the periodicity of the superlattice, one has to solveEq. (2.41) in Ref. [4], which describes the bulk bands, andEq. (57), which traces the dispersion of the surface modes,independently. The existence of the surface modes is char-acterized not only by the relative thickness but also by theparameter contrasts (in the densities and speed of sound inthe two layers of a binary superlattice, as well as in thesurface layer and the truncating medium).

Figure 13 illustrates the numerical example for theNbCu superlattice system. One can notice that the dis-persion curves for the periodic system break up into differ-ent bulk bands depending on the value of Kd, where Kdenotes the Bloch vector and d is the period of the super-lattice. The band edges exist at Kd np, n 0, 1, 2, . . .These bulk bands are depicted by hatched areas. The sur-face modes, drawn by dotted curves, exist below the low-est bulk band and within the gaps between the bulk bands.The existence of the surface modes within the gaps be-


Fig. 13. Band structure for transverse elastic modes propagating in aone-dimensional superlattice system. The hatched areas show the bulkbands for an infinite periodic system and dotted curves refer to thesurface modes in a truncated semi-infinite superlattice. The lowestsurface mode merges with the lower edge of the lowest bulk band asthe parallel component of the wave vector ~kkk ! 0. The edges of thebulk bands correspond to the Bloch vector k np=d, where d is theperiod and n 0, 1, 2, . . . The computation is performed for Nb(d1 1000 A)/Cu (d2 500 A) system. (After Camely et al., Ref.[154]).

tween the bulk bands was, in fact, predicted by Auld andcollaborators [155] much before the superlattice era cameinto being.

Authors of Ref. [154] argue that the surface mode ly-ing below the lowest bulk band is similar to the Lovemodes, which by definition are the guided modes of anunsupported plate, in several respects. This surface modeexists only when the outermost layer has the lower ct. Inthe limit of ~kkk ! 1, the velocity of this mode approachesthe sound velocity of the outermost layer. Another charac-teristic of this surface mode is that it has a sinusoidal var-iation through the layers of lower ct and an exponentialdecay through those of higher ct. The surface modes thatlie within the gaps between the bulk bands, on the otherhand, have sinusoidal variations through both layers, andthey do exist even if the outermost layer has a higher ct.For the relevant details on the surface modes in the semi-infinite medium (Reyleigh waves), in the unsupportedmedia with sagittal polarization (Lamb waves) and shear-horizontal polarization (Love waves), and in the supportedmedia with sagittal polarization (Sezawa waves), the read-er is referred to Ref. [156].

6.2 Two-dimensional systems

By two-dimensional systems we mean the elastic (binary)composites which are synthesized to exhibit two-dimen-sional periodicity in the plane perpendicular to infinitelylong elastic rods embedded in a background with differentelastic properties. This leads us to realize two differentgeometries square lattice and hexagonal lattice. In bothof these geometries, we would confine our attention to theelastic rods of circular cross-section; other possibilities onthe shape of the inclusions have been considered in theliterature (see, for example, Refs. [97] and [103]). It isnoteworthy that although the eigenvalue problem for bothhexagonal and square lattices is describable formally bythe same Eq. (3.38) in Ref. [4], the two structures are dis-tinguishable through the values attained by ~KK, ~GG and thestructure factor F~GG. This is true whether or not the sameshape of inclusions is considered for both geometries.

6.2.1 Square lattices

Since we consider an array of cylinders of circular cross-section, the structure factor F~GG is specified by Eq. (46).The secular equation used to compute the band structureis Eq. (2.40) in Ref. [4] which corresponds to the standardeigenvalue problem. The integers nx and ny were permittedto take the values between 10 and 10 (441 planewaves). This resulted in a very good convergence. We per-formed the computation for specific materials of Ni(Al)alloy cylinders in Al(Ni) alloy background. Numerical re-sults for a filling fraction f 0:35 are shown in Fig. 14.The figure is comprised of three parts. In the first part,we have plotted the band structure in the three principalsymmetry directions, letting ~kk scan only the periphery ofthe irreducible triangle of the first Brillouin zone. Thereappears a band-gap opened up between the first twobands. For this value of f , there is another very narrowband-gap lying between the fourth and the fifth bands,

with DW 0:02: (There are no higher gaps, at least as faras 50th band.) The middle part of this figure illustrates aninteresting way to present the band structure, namely, herewe plot the eigenvalues Wn as a function of j~kkj, i.e., thedistance of a point in the irreducible triangle of the Bril-louin zone from the origin. In doing so we have scannednot only the periphery but also the interior of the irreduci-ble triangle GG XX MM of the Brillouin zone (see the inset ofFig. 14). This part of the computation embodies 1326 val-ues of the uniformly distributed grid of ~kk-points through-out the irreducible part of the Brillouin zone. Making useof the same number of ~kk-points, we have computed thedensity of states (DOS), plotted in the third part of thefigure. The magnitudes of the elastic band-gaps coincidein all the three parts of the figure, which leads us to inferthat the existing band-gaps extend throughout the Bril-louin zone. This in turn establishes the fact that the wavepropagation in the transverse plane is forbidden for thevibrations parallel to the cylinders. The value of the nor-malized lowest gap in this case is DW 0:12.

Next we examine the magnitude of the lowest band-gap as a function of the parameter contrasts (i.e., DC44and Dr). The geometry is the same as for Fig. 14, i.e., Nialloy cylinders in Al alloy matrix. The numerical results,for f 0:35, are depicted in Fig. 15. This three-dimen-sional plot contains a wealth of information about the ex-istence of elastic band-gaps and the choice of materials tocreate such gaps. For instance, the arrow (on the right-hand side of this surface) indicates our explicit choice ofthe materials generating a gap given by DW 0:12 (seeFig. 14). In other words, the plot in Fig. 15 provides aguide to the feasibility of designing the phononic crystalsthat can possess the elastic band-gaps by an appropriatechoice of the materials for a binary composite. In particu-


Fig. 14. Elastic band structure and density of states for Ni cylindersin an Al matrix square lattice. The figure is comprised of three parts.In the first part, we plot the band structure in three principal symmetrydirections letting ~kk scan the periphery of the triangle GG XX MM. The middlepart of this figure illustrates the eigenvalue Wk as a function of j~kkj; i.e.,the distance of a point in the irreducible part of the Brillouin zone fromthe GG point. The third part depicts the DOS. The material parameters arer 8:9362:697 gm/cm3, C44 rc2t 7:542:79 1011 dyn/cm2for Ni(Al), and f 0:35. Attention is drawn to the vibrational band-gap between the first two bands extending throughout the first Bril-louin zone. (After Kushwaha et al., Ref. [101]).

lar, let us remark that according to Fig. 11 the opening upof large band-gaps require that the contrast Dr and DC44both be large. Here we have explored only the case whereri > rb and C44i > C44b other possibilities are worthattempting, however.

Now we turn to the situation where Al alloy cylindersare embedded in a Ni alloy background. The numericalresults are illustrated by the specific example f 0:75 inFig. 16. We find one elastic band-gap existing between thefirst two bands. The existence and magnitude of this gapis well established by the band structure (middle part ofthis figure) and by the density of states (the third part ofthe figure). It is thus concluded that this elastic band-gapextends throughout the Brillouin zone. The rest of the dis-cussion related to Fig. 14 is still valid. It is worth mention-ing that there are no other band-gaps opening up abovethe second band, or up to the 50th band, at least.

Finally, we scrutinize the width of the lower gap as afunction of filling fraction for both cases. The numericalresults are shown in Fig. 17. The curve marked case A(case B) represents the situation with Ni(Al) alloy cylin-

ders in the Al(Ni) alloy matrix. It is found that the widestband-gap in case A corresponds to the filling fractionf 0:33. Similarly, the widest band-gap in case B opensup at a filling fraction corresponding to the close-packing(f 0:7854). It is noteworthy that, in case A, an absoluteband-gap exists over a large range of filling fraction de-fined by 0.10 f 0:69. In case B, on the other hand,there is no elastic band-gap for f 0:52.

6.2.2 Hexagonal lattices

This section is devoted to discuss the possibility of achievingelastic band-gaps in the two-dimensional periodic elasticcomposites where the periodic array of parallel metallic rodsof circular cross-section forms a hexagonal lattice in the per-pendicular plane. The problem of classical band structure,both photonic and phononic, in the hexagonal pattern hasreceived relatively less attention for the time being. Thisseems to be true in spite of the fact that the hexagonal patternhas exhibited wider band-gaps as compared to its square pat-tern counterpart, for the same value of the filling fraction [97,102, 105, 113]. Since the array of cylinders forms a hexago-nal lattice (of lattice constant a), the reciprocal lattice vec-tor ~GG is given by ~GG 2p=a nxx^x nx 2ny y^y=

3

p ;we allowed the integers nx and ny to take the values in therange defined by 10 nx; ny 10: This implies 441plane waves considered in the computation; with an esti-mated error of less than 1% and hence a very good con-vergence.

Figure 18 illustrates the band structure and the densityof states (DOS) for Ni alloy cylinders in an Al alloy back-ground. The plots are rendered in terms of dimensionlessfrequency W wa=2p rr= CC441=2 versus the dimension-less Bloch vector ~kk a~KK=2p; just as in the plots in thepreceding section. The figure is comprised of three parts.In the first part we plot the lowest ten bands in the threeprincipal symmetry directions, letting ~kk scan only the per-iphery of irreducible triangle of the first Brillouin zone(see inset of Fig. 18). We obtain a wide elastic band-gapopened up between the first two bands. Note that for thisvalue of filling fraction f 0:20 there are no highergaps, at least so far as 50th band. The second part of thisfigure shows a novel way to plot the band structure. Herewe plot Wn as a function of j~kkj. In this (middle) part ofthe figure we have scanned not only the periphery but also


Fig. 15. Normalized magnitude of the lowest band-gap as a functionof the contrasts in the elastic constant and in the density. We empha-size that this three-dimensional plot provides a guide to the feasibilityof designing vibrational band-gaps by an appropriate choice of thematerials for a binary composite in the square geometry. Heref 0:35, just as in Fig. 14. The contrast parameters have the sameunits as defined in Fig. 14. (After Kushwaha et al., Ref. [101]).

Fig. 16. The same as in Fig. 14, but for Al cylinders in Ni host ma-trix. The filling fraction is f 0:75. (After Kushwaha et al., Ref.[101]).

Fig. 17. The width of the lowest band-gap as a function of fillingfraction. Case A (case B) refers to the Ni(Al) cylinders in Al(Ni)background. (After Kushwaha et al., Ref. [101]).

the interior (for 1326 ~kk-points) of the irreducible triangleGG JJ XX of the Brillouin zone. In the third part of the figurewe plot DOS, using the same number of ~kk-points as in themiddle part of the figure. We draw attention to the magni-tude of the gap which coincides in all the three parts ofthe figure. One can thus conclude that the existing gapextends throughout the Brillouin zone, and hence estab-lishes the fact that within this gap, wave propagation inthe transverse plane is forbidden for vibrations parallel tothe cylinders. The value of the normalized gap-width inthis case is DW 0:18.

Next we examine the situation when the filling fractionis increased. It is found that as f increases the magnitudeof the lower gap attains a maximum (at f 0:29) fol-lowed by a decrease for still higher values of f . Now, pre-cisely at f 0:29 another gap opens up between third andfourth bands. This upper gap also attains a maximum atf 0.48. An example for the coexistence of the two gapsis illustrated by plotting the dependence of the gap-widthson the filling fraction in Fig. 19. The solid curves markedas upper gap and lower gap correspond to the present(hexagonal) case while the dashed curve refers to the low-

er gap for the square lattice (see the curve designated ascase A in Fig. 17). The latter (dashed curve) is plottedhere just for the sake of comparison. As one can see, forany value of f , the hexagonal pattern exhibits a wider elas-tic band-gap than the square pattern. Also, the range of ffor the existence of the gaps is larger for the hexagonalcase than for the square lattice. The specific range of thelower (upper) gap for the hexagonal lattice is defined by0:04 f 0.73 (0.29 f 0.71).

From the definition of the normalized frequency W it isclear that all the gaps, both in the square and hexagonal

lattices, are proportional to CC44=rr1=2 and are inverselyproportional to the lattice constant a. For a given f thegap-width also depends on the contrast parameters Dr andDC44.

We examine the magnitude of the lowest band-gap as afunction of the elastic constant and density contrasts. Thegeometrical configuration is the same as before, i.e., infi-nitely long cylinders in a host material. The numerical re-sults, for f 0:20; are depicted by a three- dimensionalplot in Fig. 20. This plot provides us with useful informa-tion about the existence of elastic band-gaps and thechoice of materials to tailor such gaps. For instance, ourspecific choice of the NiAl composite gives rise to a gapgiven by DW 0:18. This choice of alloys is representedby the dot next to the 1. Similarly, the numbers 25correspond to other pairs of alloys (see the figure caption).In other words, the plot in Fig. 20 helps to engineer adesired elastic band-gap by an appropriate choice of thematerials for a binary composite.

It should be pointed out that, unlike the square lattice,we find no gaps for the inverted geometry, namely, Alalloy cylinders in a Ni alloy background. We note that inthis case we have a higher-velocity inclusions (the Al) sur-rounded by a lower-velocity materials (the Ni). It has al-ready been commented that such a situation is less favor-able for the creation of gaps than when a lower-velocityinclusions are surrounded by a higher-velocity material[115]. The larger range of filling fraction for opening upthe wider (lowest) band-gap in the hexagonal lattice, than


Fig. 18. The same as in Fig. 14, but for hexagonal lattice. The irredu-cible part of the first Brillouin zone (see the inset) is GG JJ XX. The fillingfraction is f 0:20. We call attention to a wide band-gap existingbetween the first two bands and extending throughout the Brillouinzone. (After Kushwaha et al., Ref. [102]).

Fig. 19. Normalized widths of the lower and upper gaps (solidcurves) as a function of filling fraction. The dashed curve refers tothe lower gap in the square lattice for the same elastic composite(i.e., Ni cylinders in Al matrix). (After Kushwaha et al., Ref. [102]).

Fig. 20. Normalized magnitude of the lowest band-gap as a functionof the contrasts in the density and in the elastic constant. The dotsmark the gaps, assuming f 0:20, for the following pairs of alloys:(a) NiAl, (2) CuAl, (3) CuSn, (4) PtAu, and (5) FeAl. [Thefirst (second) alloy of the pair corresponds to the cylinder (host).]The dot 1 describes the gap in Fig. 18. We stress that this three-dimensional plot provides a guide to the feasibility of engineeringvibrational band-gaps by an appropriate choice of the materials for abinary composite in the hexagonal geometry. (After Kushwaha et al.,Ref. [102]).

that in the square lattice, is attributed to the fact that theconstant energy surfaces of a hexagonal lattice are closerto the circular shape than those of a square lattice.

It is important to note that purely transverse (or, z^z-po-larized) modes propagating in the two-dimensional peri-odic systems are independent of the mixed (or plane-polar-ized) modes (see Sec. 2.4 in Ref. [4]), have legitimacy intheir own right, and can be excited separately from themixed modes [141][144]. As such, a composite of peri-odic, long cylinders is the simplest system that can giverise to the stop bands in which oscillations of a certainpolarization are forbidden.

The existence of these absolute gaps in two-dimen-sional composites is expected to guarantee, with gradualdisordering, the Anderson localization of transverse vibra-tions. The transition between the localized and the ex-tended states in a two-dimensional disordered system is aquestion of considerable current interest [74][76]. It hasbeen conjectured that there could exist the quasi-mobilityedge(s) in two-dimensional disordered systems [85], whichseparate the strongly localized states from the weakly loca-lized states (or power-law localized states). This has ques-tioned the scaling theory of localization [157] which theo-rizes that no extended states should occur for any amountof disorder in one- and two-dimensional systems. Theore-tically, such systems cannot be treated as regular systemswith a small perturbation. Thus their characterization, par-ticularly in respect of their transport properties, poses asubstantial problem. The simpler systems as discussed inthis section are believed to address the complex issue ofquasi-mobility edge(s) in the classical (elastic) wave loca-lization unambiguously the root-cause of the expectedunambiguity lies, of course, in the separability of thetransverse and the mixed elastic waves.

7. Mixed and coupled vibrations

Before discussing the specific numerical examples on theband structure related problems that belong to the title ofthis section, we should make crystal clear the terms likelongitudinal modes, transverse modes, mixed modes, andcoupled (longitudinal-transverse) modes, which have oftenbeen used in this review. We need to do so at this stage inorder to avoid any confusion associated particularly withthe terms mixed modes and coupled modes. Note that de-fining these terms has much to do with the (periodic) di-mensionality of the system concerned. Remember, we areconcerned with these terminologies in the context of elas-tic and/or acoustic waves in liquids, gases, and solids inthe EM case the same terms are defined in a differentframe.

As we mentioned earlier, the longitudinal waves aresupported only in liquids and gases. One-dimensional peri-odic elastic composites, superlattices, for example, can,however, allow either pure longitudinal or pure transverseelastic waves, provided that the displacement vector ~uu de-pends only on the spatial coordinate along the direction ofthe periodicity. In the case that the displacement vector ~uudepends on the spatial coordinates in the sagittal plane(i.e., plane through ~kkk and surface normal), one-dimen-

sional systems can support either pure transverse elasticwaves or the mixed modes, which are neither longitudinalnor transverse. In the two-dimensional periodic elasticcomposites (see Sec. 2.4 in Ref. [4]), we have seen thatonly pure transverse or mixed modes are allowed. The for-mer are characterized by the displacement vector ~uu alongthe cylinders (z^z-axis) and perpendicular to the plane ofpropagation (x^x y^y); while the latter are characterized byboth ~uu and ~kk in the x^x y^y plane. In the literature the com-plete elastic band-gaps are assigned to the frequency win-dow wherein overlapping of both pure transverse andmixed modes together prohibit the wave propagation in allpossible directions. In the three-dimensional periodic elas-tic composites solid inclusions in a solid background no resolution of any polarization is possible and hence thepropagating modes always remain coupled; only thesemodes will be referred to as coupled (longitudinal trans-verse) modes.

This nomenclature of vibrational modes does not pre-clude the coupling of the longitudinal and transversemodes in the two-dimensional pe

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