## Abstract

We consider the problem of localized flexural waves in thin plates that have periodic structure, consisting of a two-dimensional array of pins or point masses. Changing the properties of the structure at a single point results in a localized mode within the band-gap that is confined to the vicinity of the defect, while changing the properties along an entire line of points results in a waveguide mode. We develop here an analytic theory of these modes and provide semi-analytic expressions for the eigenfrequencies and fields of the point defect states, as well as the dispersion curves of the defect waveguide modes. The theory is based on a derivation of Green's function for the structure, which we present here for the first time. We also consider defects in finite arrays of point masses, and demonstrate the connection between the finite and infinite systems.

## 1. Introduction

The last 20 years have seen remarkable developments in the study and control of waves in periodically structured media. Much of this research has concentrated on electromagnetic waves, in which photonic crystals have been shown to exhibit a number of important applications in both the applied and fundamental sciences; one specific area in which progress has been particularly impressive is the creation of high quality-factor (high-*Q*) resonant cavities (Akahane *et al.* 2003) and waveguides in photonic crystals (Gersen *et al.* 2005). Using an older terminology, these would be referred to as point and line defects in crystals—in the photonic case, these defects can be exploited to trap light for optical switching, and to control the speed at which pulses propagate, leading to dramatically enhanced nonlinear effects (Monat *et al.* 2010).

The analysis of elastic solids with periodic systems of defects (inclusions or voids) brings new challenges that do not appear in the problems of optics and electromagnetism governed by Maxwell's equations. In particular, isotropic elastic media are characterized by two types of waves, dilatational and shear, which are coupled via the boundary conditions on the surfaces of embedded elastic inclusions. This brings new dispersion patterns for elastic Bloch waves, unseen in problems of electromagnetism (Sigalas & Economou 1993). Equally challenging is the study of flexural waves in elastic plates, which are governed by a fourth-order partial differential equation. The analysis of dispersion properties of Bloch waves and simulation of localization around bounded defects were performed by Movchan *et al.* (2007, 2009), McPhedran *et al.* (2009) and Poulton *et al.* (2010) for Kirchhoff plates with periodic arrays of defects, and by Movchan *et al.* (2011) for structured Mindlin plates. Analytical and numerical scalar models of localized waveforms in two-dimensional lattice systems have been developed by Ayzenberg-Stepanenko & Slepyan (2008). The analysis of the dynamic anisotropy in vector problems of elasticity, including special classes of standing waves in lattice systems with rotations, was presented by Colquitt *et al.* (2011); the latter also discussed the effects of focusing and negative refraction in lattice systems with rotations.

The design of point and line defects has generally been accomplished using sophisticated numerical simulations on powerful computers. However, a small number of groups have developed analytic or semi-analytic methods that can deliver more physical insight into why particular structures deliver good performance while others do not (Platts *et al.* 2002, 2003; Botten *et al.* 2005; Sauvan *et al.* 2005; Wilcox *et al.* 2005; Thompson & Linton 2007; Mahmoodian *et al.* 2009*a*,*b*; Busch *et al*. 2011). One difficulty in developing such methods is that ideally one would like to model the influence of changing a localized area in a periodic structure on fields. The fields then cease to be characterized by their behaviour in a period cell, but are spread out over an infinite area in two dimensions, or over an infinite volume in three. This causes computational problems, usually dealt with by computing properties of large but finite systems. However, this is not a completely satisfactory solution, because finite systems can show strong surface wave effects, and internal interference effects of the Fabry–Pérot type not existing in truly infinite systems. One technique to model truly infinite systems has been termed the Fictitious Source Superposition Method by Wilcox *et al.* (2005) and the array scanning method by Thompson & Linton (2007). These methods consist of solving the quasi-periodic array problem for a net of values of the Bloch vector sampling the Brillouin zone (BZ), and then superimposing the solutions to arrive at a defect Green's function with its source located at a single point in the crystal lattice. This Green's function can then be related to the required point defect solution. For a line defect the method is similar, except that a propagation vector component, say *k*_{x}, is chosen, and the required superposition is only over a set of values sampling the component *k*_{y}. The earliest references we know of relating to this method are Zolla *et al.* (1994) and Figotin & Goren (2001).

Here, we develop a new analytic theory for the study of localized elastic vibrations in thin plates. The theory is based on the formulation of the defect Green's function for a time-harmonic Kirchhoff plate with a periodic array of scattering centres, which can take the form of rigid pins or point masses. The governing equation for this system is the biharmonic equation rather than the Helmholtz equation. The study of such periodically structured thin plates, known as platonic crystals, has been taken up in a number of recent papers (Evans & Porter 2007; Movchan *et al.* 2007; McPhedran *et al.* 2009; Meylan & McPhedran 2011; Movchan *et al.* 2011). In one of these (McPhedran *et al.* 2009), a result was given for the frequency of a defect mode in a finite cluster of pinned points, and Evans & Porter (2007) have discussed waves guided between two gratings, but the literature has otherwise been confined to periodic structures without defects. The generalization we make here opens the study of platonic structures to potentially useful designs of the type that have been intensively investigated in the field of photonic crystals, such as high-*Q* cavities, filters based on point defects coupled to waveguides, slow wave designs and waveguides with tight bends.

The structure of the paper is as follows. In §2, we develop the theory for a defect in an array of rigid pins. Our approach is to first construct a quasi-periodic Green's function and then to apply BZ averaging to derive the defect Green's function for the structure. We then continue to find the equations giving the frequency of both point and line defects, where the defects have been created by changing the mass of a single point or a line of points. Throughout this section, we concentrate on the first bandgap for an array of pinned points in a thin elastic plate. In §3, we analyse Bloch waves and defect modes in an elastic plate containing an array of finite masses, and establish a connection with the rigid pins problem of §2. Section 4 presents the discussion of localized vibration modes in an elastic plate containing a finite cluster of masses, accompanied by numerical simulations. Note that the outline of the theory given is valid for any two-dimensional array; however, for brevity, we will usually refer to rectangular arrays, and give numerical results exclusively for the square array.

## 2. Point and line defects in platonic crystals

### (a) The homogeneous array of pins

We consider a Kirchhoff plate containing a doubly periodic rectangular array of pins. The flexural displacement is assumed to be time harmonic with the radian frequency *ω* and amplitude *u*. The rigid pins are considered as the limit of small circular holes , of radius *a* with clamped edges, as . Here, **x**^{(h)} are the positions of rigid pins within the doubly periodic array on the plane. The function *u* satisfies the equation of motion
2.1together with appropriate boundary conditions—namely, that both the value of *u* and its normal derivative vanish at the edge of each small hole (at the location of the pins):
2.2and
2.3where *r*=|**x**−**x**^{(h)}|, and **h** is the bi-index specifying the position **x**^{(h)} in the array. Here *ρ* is the mass density, *ω* is the radian frequency and *D*=*Es*^{3}/(12(1−*ν*^{2})), with *s* being the plate thickness, *E* and *ν* being Young's modulus and Poisson's ratio of the elastic material. The limit, as , is taken in (2.1)–(2.3) for the rigid pins problem. In addition, waves in a periodic lattice must obey the Bloch–Floquet quasi-periodicity condition, which is
2.4where the phase change from one cell in the lattice to the next is specified by the Bloch vector **K**. For the sake of simplicity, we assume a rectangular lattice, in which the lattice points are **x**^{(h)}=(*md*_{x},*nd*_{y}), where both *m* and *n* are integers. The BZ is then given by the set of points {**K**=(*k*_{x},*k*_{y}):|*k*_{x}|≤*π*/*d*_{x},|*k*_{y}|≤*π*/*d*_{y}}.

To derive the dispersion equation for the Bloch waves, we need the quasi-periodic Green's function *G*_{p}(**x**−**x**′;*β*,**K**) for the Kirchhoff plate, which—from Movchan *et al.* (2007)—is given by
2.5with *g* being the fundamental solution (Evans & Porter 2007) for the homogeneous Kirchhoff plate, so that
2.6with the explicit representation of the form
2.7Here
2.8Then the function (2.5) is re-expanded using the Graf addition theorem as in Abramowitz & Stegun (1965), and we note that the contribution of the regular parts of the expansion from pins not located at the origin is to exactly cancel the regular part of the expansion arising from the central pin (McPhedran *et al.* 2009). To describe the behaviour near the origin, it is sufficient to retain only the monopole terms because the limit of pin radius approaches zero; this results in
2.9where *r*=|**x**−**x**^{′}|, and , are the lattice sums for the array, which are formally defined by (Chin *et al.* 1994)
2.10The sum is conditionally convergent when evaluated in direct space, and is rapidly convergent except when *β* is small; appropriate computable representations for these sums are given *inter alia* by McPhedran *et al.* (2009).

Taking the limits of equation (2.9) as , we note that
2.11and in addition, we have
2.12Comparing the limits at the pinned points, we can see that, provided the condition
2.13is satisfied, Green's function *G*_{p} obeys the limit conditions (2.2) and (2.3) at the pinned points **x**^{(h)}, as well as the quasi-periodicity conditions required for the solution *u*(**x**). We can then identify
2.14with the dispersion equation (2.13) for Bloch waves in an array of rigid pins.

### (b) The point defect Green's function

We now construct the point defect Green's function—that is, Green's function corresponding to the situation when the rigid pin at the origin is released and replaced by a unit force located at the origin. Up to now, we have regarded *G*_{p} as obeying an inhomogeneous differential equation implied by (2.5) and (2.6), the sources at the lattice points arising from the discontinuity in Green's function's second-order derivative. We could equivalently specify *G*_{p} by excluding the lattice points from the domain on which the function is defined and enforcing the appropriate behaviour as : in this way the inhomogeneous differential equation may be replaced with a homogeneous differential equation with inhomogeneous boundary conditions (Morse & Feshbach 1953). The quasi-periodic Green's function then satisfies the homogeneous differential equation
2.15From the expansion of *G*_{p} given in (2.9), together with the quasi-periodicity condition, we have the following limiting behaviour at the lattice points **x**^{(h)}:
2.16We can regard equation (2.16) as an inhomogeneous boundary condition that replaces the driving term in the differential equation. The function *G*_{p} is then determined to within an additive homogeneous part that must vanish at the lattice points: any such homogeneous solution is of course a Bloch mode of the system, occuring within the band at values (*β*,**K**) for which equation (2.13) is satisfied. For defect modes existing within the band, this homogeneous solution must be chosen so that the mode satisfies an outgoing wave condition; however, within the band gap no such solutions exist and uniqueness, to within a scale factor, of the defect mode is ensured by the condition that the mode vanishes at distances far removed from the defect centre.

We now consider an intermediate function , defined by
2.17Here the notation 〈…〉_{BZ} denotes the operation of averaging the vector **K** over the BZ, so that, if we assume a rectangular array with **x**^{(h)}=(*md*_{x},*nd*_{y}) where *m* and *n* are both integers, we have
We note that satisfies the differential equation
2.18together with the boundary conditions
2.19The integration over the unit cell renders all the boundary conditions at lattice points other than the origin homogeneous. Using the expression (2.9) to expand in the central unit cell, we find that
2.20We can recover the inhomogeneous differential equation satisfied by by re-including the origin and noting that
We find then that
2.21The function therefore represents an inhomogeneous source with weight placed at the origin surrounded by a lattice of rigid pins.

By re-scaling, one can deduce the defect Green's function for the array: 2.22

Using the expansion about the origin in (2.9), one immediately finds that, provided the weight factor , the function *G* satisfies the differential equation
2.23together with pinned boundary conditions at the lattice points **x**=**x**^{(h)}, for *h*≠0. The special case where corresponds to a resonance of the structure; this is discussed in the following section.

### (c) Point defects

We consider the equation for a localized state in an array of pins, created by releasing the pin located at **x**^{(h)}=**0**. The localized state thus created obeys the differential equation at all but pinned points:
2.24In addition, we specify appropriate boundary conditions for an array of pins at lattice locations **x**^{(h)}—namely, that both the value of *U* and its derivatives vanish at **x**^{(h)} when **h**≠**0**. We note here that by releasing the pin at the origin, the earlier mentioned differential equation must be satisfied at **x**=**0**.

We now observe that the function is very close to the solution that we want for *U*, because obeys the differential equation (2.24) at all points except for **x**=**0** and in addition obeys the correct limiting conditions at the lattice points. All that is required is to make the weight of the source term vanish. To see this explicitly, we set
2.25where *p*_{0} is some arbitrary constant. Around the origin, the expansion for *U* is
2.26In order to satisfy the correct boundary condition for a ‘removed’ pin, the first term, corresponding to the irregular part, must vanish identically. We therefore have the condition for the defect state:
2.27The expansion for the defect state in the vicinity of the origin is

The equation (2.27) can easily be implemented numerically to determine the frequency of the defect state in a total band gap. The lattice sum has a first-order singularity at what is called in the photonic crystal literature the *light line* or plane wave propagation condition, which occurs when *β*=|**K**|. Consequently, there is a change of sign at the light line, so that the condition (2.27) requires that the contributions from inside the light line and outside the light line balance each other. Numerically, this gives the value *β*=2.5372, which agrees well with the estimate *β*=2.538 for the defect mode in a finite set (17 by 17) of pins from McPhedran *et al.* (2009). We note that the condition for a resonance (2.27) corresponds to a singularity of Green's function for the array (2.22), as is known from the general treatment in Economou (2006).

We now consider the case where the central pin is released and a finite point mass perturbation *M* (which can be positive or negative) is introduced instead. Assuming that *ω* lies within the stop band, we deduce that the localized state now obeys the differential equation
2.28Green's function *G* for the localized state obeys equation (2.23), with identical conditions at the lattice points **x**^{(h)} as apply for *U*. Comparing these two equations, it is apparent that we can write
2.29We then require that the consistency condition
2.30be satisfied. Rearranging, we obtain
2.31From equations (2.22) and (2.16), we know that
2.32and so we find that the existence condition for a defect state arising from a single point mass is
2.33The point mass perturbation can be removed entirely by taking the limit as . This recovers the equation (2.27).

### (d) Line defects

We now consider the case in which we modify an entire line of pins oriented along the *x*-axis, either replacing them with point masses or removing them entirely. The structure is periodic in the *x*-direction, and so all solutions must obey the Bloch condition
2.34where *d*_{x} is the lattice pitch in the *x*-direction, is the unit basis vector along the *x*-axis and *k*_{x} is the Bloch wavenumber. Following the reasoning given in the previous section, we define a new function as
2.35The notation 〈.〉_{ky} represents an integral over a line in the BZ with respect to *k*_{y}, such that
The function then obeys the boundary conditions at lattice points **x**^{(h)}:
2.36where we have written the bi-index **h**=(*m*,*n*). As in the case for the point defect, the integral over the BZ has the effect of turning inhomogeneous boundary conditions into homogeneous ones, this time however leaving the phased array of sources at lattice points on the *x*-axis. Following identical reasoning to that of the previous section, Green's function for a line defect is
2.37and it satisfies the differential equation
2.38The solution that corresponds to removing an entire row of pins is then
2.39with the resonance condition
2.40required to remove the array of sources, and thus obtain a solution that satisfies the differential equation everywhere. The left-hand side of this relation is a function of *k*_{x}; therefore, we may expect a curve of solutions (*k*_{x},*ω*). This is the dispersion curve for the waveguide created by removing the pins.

The dispersion diagram for the wave guided by a square array of pinned points with a line defect created by releasing an entire row of pins is shown in figure 1. The guided wave *β* value runs over roughly the top third of the band gap range, and exceeds *π* before decreasing towards it as *β* tends to the edge of the BZ. It is notable that the wave guide range of *β* includes the point defect value *β*=2.5372, as this means it will be possible to construct systems coupling point defects to wave guides.

If we now replace the row of released pins with point masses *M*, the solution will satisfy the differential equation
2.41Comparing (2.38) with (2.41), we can write the solution to the line defect problem as
2.42The consistency condition that must be satisfied by a valid solution is then
2.43or, using the formulae (2.37) and (2.16), we deduce
2.44Removing the row of additional masses entirely by taking the limit as recovers the resonance condition (2.40).

## 3. Band structure for Bloch waves in a plate with a periodic array of masses

We consider a Kirchhoff plate containing a doubly periodic rectangular array of additional point masses *m*. The flexural displacement is assumed to be time harmonic with the radian frequency *ω* and the amplitude *u*. The function *u* satisfies the following equation of motion:
3.1As in §2, the solution must have quasi-periodicity specified by the Bloch vector **K**, thus
3.2

By comparing (3.1) and (3.2) with the equation for the quasi-periodic Green's function (2.9), we deduce
3.3with the compatibility condition
3.4By cancelling *u*(**0**) and using the representation (2.9), we derive the dispersion equation relating *β* to **K** as follows:
3.5It is important to note that in deriving this result, we used
The solutions of the dispersion equation (3.5) representing the first three bands are shown in figure 2, for a square lattice.

The most striking feature in figure 2 is the presence of an acoustic band, absent in the corresponding diagram for an array of rigid pins (McPhedran *et al.* 2009). We note that, as the mass ratio *m*/*ρ* increases towards infinity this acoustic band becomes flatter and flatter, finally collapsing into the axis *β*=0 in the limit. Another interesting feature of figure 2 is that the second band for the segment *X*−*M* coincides exactly with the dispersion curve for the unstructured plate, given by
3.6This is true for all values of *m*/*ρ*, and it shows that the propagation of the elastic waves in the mass-loaded plate is totally unaffected by the loading, for this specific direction of propagation. As noted in McPhedran *et al.* (2009), this occurs because the second band is sandwiched between two planes giving dispersion surfaces for the unstructured plate. At each of these planes, the lattice sum diverges, so that the second band cannot cross either plane, and so coincides with them at their intersection. Note that such intersections occur for every higher pair of bands along symmetry lines, so that the higher bands do not have total band gaps; the segment *Γ*−*M* is the only region where there exists a partial band-gap between bands 2 and 3, which touch at the *X* point. Band 4 touches band 3 at the *Γ* point, and so on for higher bands. Thus, the gap between the acoustic band and the second band is the only total gap, for any value of the mass loading. As illustrated in figure 3, the width of the band gap increases with the increase of *m*/*ρ*. The second and third bands in figure 3 are practically unaffected by the change in *m*/*ρ*, whereas the acoustic band moves downward compared with that in figure 2.

The asymptotic analysis of the dispersion equation (3.5) for small *β* and |**K**| leads to the simple relationship
3.7In particular, when the slope of the corresponding dispersion curve reduces to zero. For finite *m*/*ρs*, it lies below the first band for the unstructured plate.

### (e) A point perturbation within the system of masses

For a frequency *ω* from the stop band separating the first and the second dispersion bands in figure 2 there are no propagating Bloch modes. For such a frequency, we consider the exponentially localized Green's function , which satisfies the equation
3.8Physically, this function represents the flexural displacement in the periodically *m*-mass-loaded plate, with the central mass being replaced by a time harmonic point force of unit amplitude. The quantity *β* is related to the radian frequency *ω* by the formula (2.8).

We wish to relate this Green's function to the localized defect mode created by replacing the mass *m* in the central cell by a different value *m*+*M*≥0. This localized mode *U*(**x**;*β*) satisfies the equation
3.9Next, we introduce the quasi-periodic Green's function *G*_{m} for the mass-loaded plate, with the full periodic array of masses (*m*) being present. This obeys the governing equation
3.10As in §2, the notation 〈⋅〉_{BZ} is used for the average, with respect to **K**, over the BZ. Taking the average of equation (3.10) and noting that
we deduce
3.11By comparing equation (3.8) with (3.11) and using the uniqueness of the localized Green's function , we deduce
3.12A similar rescaling relates the defect mode *U* to the localized Green's function
3.13Putting **x**=**0** in (3.13) leads to the compatibility condition
3.14Let be the radian frequency of the localized defect mode corresponding to the perturbation *M* of the mass in the central cell. Then, combining (3.14) and (3.12), we derive
3.15An equivalent version of this equation gives the perturbation mass *M* as a function of a specified value of *β* chosen within the total band gap
3.16

The final step in this argument is to relate Green's function *G*_{m} to the representation (2.9) of the quasi-periodic Green's function *G*_{p} for the homogeneous Kirchhoff plate. By comparing (3.10), with **x**′=**0**, and (2.5) we deduce
3.17Hence, *G*_{m}(**0**,**0**;*β*,**K**) can be represented via *G*_{p}(**0**,**0**;*β*,**K**) as follows:
3.18Furthermore, after the averaging over the BZ, we obtain the relation
3.19This has been written in a form suitable for computations because goes to zero at ‘light lines’ rather than diverging. Using (3.19) and (3.16), we derive the mass perturbation formula
3.20

For the case of *m*/(*ρs*)=5, the mass perturbation *M*/*m* is plotted as a function of *β* in figure 4, which indicates that to increase the frequency of the defect mode the mass of the central cell should be reduced. In particular, in the limit *M*=−*m*, the corresponding frequency represents the defect mode for the extreme case when the point mass has been entirely removed from the central cell. We also note that the small reduction in mass generates a defect mode near the lower edge of the band gap, and the curve *M*/*m* versus *β* is steepest near that band edge. It appears that the lower edge of the stop band is reached for the perturbation mass ratio *M*/*m* close to −0.32, which suggests that a finite mass perturbation is required to create a defect mode even in the close proximity of the boundary of the stop band.

As an interesting special case, we consider a system approaching the defect in an array of pinned points created by removing the constraint on the central point. Formally, as *β*>0, the array of pinned points is obtained by letting *m* tend to infinity. To obtain the equation for the defect frequency and the corresponding *β*, we replace *M* by −*m*+*ϵ* and take the limit as . Referring to (3.16) and (3.19), we derive the first-order expansion
and hence
In the limit as , we obtain the formula that defines the frequency of the defect mode for the plate containing an array of pinned points:
3.21

### (f) A line defect within an array of point masses

Consider a waveguide created by altering the masses located on the *x*-axis, so that each mass *m* is replaced by *m*+*M*. The displacement amplitude *U* is assumed to satisfy the one-dimensional quasi-periodicity condition
3.22It also satisfies the differential equation
3.23where the delta function terms represent the inertial contribution from point masses. In turn, the ‘phased line defect’ Green's function for the mass-loaded plate satisfies the equation
3.24The comparison of (3.23) and (3.24) leads to the relation
3.25

In a way similar to (3.12), we obtain the connection between and the quasi-periodic Green's function *G*_{m} for the doubly periodic array of point masses:
3.26where 〈⋅〉_{ky} stands for the average with respect to *k*_{y} within the BZ. Furthermore, similar to (3.15) we derive the dispersion equation for flexural waves localized within the waveguide along the *x*-axis:
3.27One can also take into account the connection (3.19) between the quasi-periodic Green's functions *G*_{p} and *G*_{m}. In the limit, as and we obtain the dispersion equation for waves propagating through a waveguide, along the *x*-axis, surrounded by the arrays of rigid pins. Similar to (3.21), we deduce
3.28which is equivalent to
3.29

## 4. Discussion: localized vibrational modes in a finite cluster

The band diagram in figures 2 and 3 contains a band gap, and the possible location of the frequency of a defect mode is in the total band gap of an infinite periodic system. The presence of the full band gap in figures 2 and 3 suggests that a strong localization of waves will occur in this frequency region. This means that the study of wave properties and defect modes in relatively small finite systems could be expected to yield accurate predictions for the periodic case.

Consider a finite array of point masses *m* arranged at points **O**^{(q)}(*q*_{1},*q*_{2}), where *q*_{1} and *q*_{2} are integers running from −*N* to *N*, omitting the point (0,0) at the origin, and the notation **q**=(*q*_{1},*q*_{2}) is used for a multi-index. A mass *m*+*M* is placed at **O**^{(0)}, at the centre of the cluster. The notations *Π*,*Π*_{0} are used for the cluster excluding the origin, and for the complete cluster with the origin. The equation of motion for the amplitude *W* of the time-harmonic transverse displacement is
4.1The notation *W*^{(h)} is also used for the nodal displacements *W*(**O**^{(h)}). By representing *W* via Green's function *g*(**x**,*β*) from (2.7)
4.2we write the consistency equations for the nodal amplitudes *W*^{(q)} in the form
4.3The equation (4.3) can be programmed in matrix form, where it is required that the determinant of the matrix is zero. A better conditioned form of the equations for this purpose is
4.4where we also note that *g*(0,*β*)=−*i*/(8*β*^{2}).

In the numerical implementation of (4.4), the matrix is of dimension (2*N*+1)^{2}×(2*N*+1)^{2}. The magnitude of the determinant of the matrix has a sharp minimum as soon as *N* is sufficiently large (*N*≥3). Subject to this condition, the mass ratio *M*/*m* for a given *β* is in excellent agreement with the value coming from the treatment for the defect in the infinite array as described in §3. Given the value of *M*/*m* for which the determinant is minimized, the vector of displacement amplitudes *W*^{(q)} at the nodal points of the cluster can be found by solving a system of linear algebraic equations (after putting *W*^{(0)}=1).

We give in figures 5 and 6 plots of the real part of the displacement amplitude versus position within the cluster. Note that the displacement amplitude has only been computed at integer mesh points, but an interpolation procedure has been applied to generate a smooth surface passing through the calculated points, to enable better visualization of the displacement pattern. The imaginary part of the displacement amplitude takes values that are negligible compared with those of the real part.

The first value of *β* chosen (figure 5) lies in the middle of the band gap, which runs between 2.018 and *π*. The mode is accordingly tightly confined at the centre of the cluster. As we move down in *β*, we approach the edge of the stop band, and the mode becomes less localized, and it develops an oscillating part surrounding the central peak. This is particularly evident for figure 6*b*, which has *β* very close to the band edge.

## Acknowledgements

C.G.P. and R.C.M. acknowledge support from the Australian Research Council through its Discovery Grants Scheme, and, in the latter case, from the Liverpool Research Centre in Mathematics and Modelling.

- Received October 6, 2011.
- Accepted December 2, 2011.

- This journal is © 2012 The Royal Society