Controlling flexural waves in semi-infinite platonic crystals
Stewart G. Haslinger, Natasha V. Movchan, Alexander B. Movchan, Ian S. Jones, Richard V. Craster
CControlling flexural waves in semi-infinite platonic crystals
S.G. Haslinger , N.V. Movchan , A.B. Movchan , I.S. Jones & R.V. Craster Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK Mechanical Engineering and Materials Research Centre, Liverpool John Moores University,Liverpool L3 3AF, UK Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Abstract
We address the problem of scattering and transmission of a plane flexural wave through a semi-infinite array of point scatterers/resonators, which take a variety of physically interesting forms.The mathematical model accounts for several classes of point defects, including mass-spring res-onators attached to the top surface of the flexural plate and their limiting case of concentratedpoint masses. We also analyse the special case of resonators attached to opposite faces of theplate. The problem is reduced to a functional equation of the Wiener-Hopf type, whose kernelvaries with the type of scatterer considered. A novel approach, which stems from the direct con-nection between the kernel function of the semi-infinite system and the quasi-periodic Green’sfunctions for corresponding infinite systems, is used to identify special frequency regimes. Wethereby demonstrate dynamically anisotropic wave effects in semi-infinite platonic crystals, withparticular attention paid to designing systems to exhibit dynamic neutrality (perfect transmis-sion) and localisation close to the structured interface.
Since the 1980’s, there has been substantial attention devoted to wave interaction with periodicstructures leading to the recent surge of interest in designing metamaterials and micro-structuredsystems that are able to generate effects unattainable with natural media. These are artificiallyengineered super-lattice materials, designed with periodic arrays of sub-wavelength unit cells; theirmajor concept is that their function is defined through structure. Many of the ideas and tech-niques originate in electromagnetism and optics but are now filtering into other systems such asthe Kirchhoff-Love plate equations for flexural waves. This analogue of photonic crystals, labelledas platonics by McPhedran et al. [1], features many of the typical anisotropic effects from photonicssuch as ultra-refraction, negative refraction and Dirac-like cones, see [2]–[6], amongst others. Re-cently, structured plates have also been both modelled, and designed, to demonstrate the capabilityfor cloaking applications [7]–[10].In this article, we consider a semi-infinite platonic crystal where, by patterning one half of aninfinite Kirchhoff-Love plate with a semi-infinite rectangular array of point scatterers, the leadinggrating acts as an interface between the homogeneous and structured parts of the plate. Haslinger et al. [11] analysed the case of pinned points, and highlighted effects including dynamic neutralityin the vicinity of Dirac-like points on the dispersion surfaces for the corresponding infinite doublyperiodic system, and interfacial localisation, by which waves propagate along the interface. An1 a r X i v : . [ phy s i c s . c l a ss - ph ] S e p nteresting feature of the discrete Wiener-Hopf method of solution was the direct connection betweenthe kernel function and the doubly quasi-periodic Green’s function, zeros of which correspond tothe aforementioned dispersion surfaces. m c (a) yz yd x (cid:1)(cid:1) m dx (b) z N N dx (cid:1)(cid:1) x (d) xyz (c) z (cid:1) (cid:1) (cid:1)(cid:1) x dx (cid:1)(cid:1) xmdxz c mcmcmcmcmc Figure 1: Four cases for semi-infinite arrays of mass-spring resonators with periodicities d x , d y .(a) Case 1: semi-infinite array of point masses. (b) Case 2: multiple mass-spring resonators onthe top surface of the plate characterised by masses m i , and stiffnesses c i . (c) Case 3: Multiplemass-spring resonators attached to both faces of the plate. (d) Case 4: Winkler-type foundation,the masses are embedded within the top surface of the plate.Here, we analyse four alternative physical settings for the point scatterers making up the semi-infinite periodic array, which we classify as one of two possible periodic systems; the two-dimensional“half-plane” with periodicity defined in both the x - and y - directions, as illustrated in figure 1(a),and the one-dimensional “grating”, with the periodic element confined to the x -axis, as illustratedin figures 1(b-d). All of the analysis presented in this article is for the two-dimensional periodicity,and is easily reduced to the special case of a single semi-line of scatterers for x ≥ • Case 1: point masses, characterised by mass m • Case 2: multiple point mass-spring resonators attached to the top surface of the plate, char-acterised by masses m i , stiffnesses c i ; i ∈ Z + Case 3: multiple mass-spring resonators attached to both faces of the plate • Case 4: point masses with Winkler foundation (see Biot [12]), characterised by mass m ,stiffness c .It will be shown that, for certain frequency regimes, some of the cases are equivalent to one another.The replacement of the rigid pins with more physically interesting scatterers brings several newattributes to the model, most notably an assortment of propagation effects at low frequencies; incontrast, the case of pinned points possesses a complete band gap for low frequency vibrationsup to a finite calculable value. The important limiting case of c → ∞ for case 2, N = 1 (seefigure 1b), or equivalently, c → et al [13], who provided dispersion band diagrams and explicit formulae and illustrations for defectand waveguide modes.Evans & Porter [14] considered one-dimensional periodic arrays of sprung point masses (case4 in figure 1(d) for −∞ < x < ∞ ), including the limiting case of unsprung point masses. Thecontributions by Xiao et al. [15] and Torrent et al. [4] discussed infinite doubly periodic arraysof point mass-spring resonators, as depicted in figure 1(b); the former for a rectangular array, andthe latter for a honeycomb, graphene-like system. The authors provided dispersion relations anddiagrams for the platonic crystals, and analysed the tuning of band-gaps and the association ofDirac points with the control of the propagation of flexural waves in thin plates. Examples usingfinite structures were also illustrated by both [15], [4].In this article, we present the first analysis of semi-infinite arrays for the variety of pointscatterers illustrated in figure 1. We demonstrate interfacial localisation, dynamic neutrality andnegative refraction for the two-dimensional platonic crystals. The problem is formulated for the two-dimensional semi-infinite periodic array of scatterers, from which the special case of a semi-infiniteline is easily recovered by replacing a quasi-periodic grating Green’s function with the single sourceGreen’s function for the biharmonic operator. A discrete Wiener-Hopf method, incorporating the z -transform, is employed to derive a series of Wiener-Hopf equations for the various geometries.This discrete method is less common than its continuous counterpart, but it has been used by,amongst others, [16]–[22] for related problems, mainly in the context of the Helmholtz equation.The characteristic feature of each of the resulting functional equations is the kernel which,for all of the cases featured here in figure 1, includes the doubly quasi-periodic Green’s function,meaning that a thorough understanding of the Bloch-Floquet analysis is required. We express thekernel in a general form, and by identifying and studying special frequency regimes, we presentthe conditions required to predict and observe specific wave effects. This novel approach is usedto design structured systems to control the propagation of the flexural waves, without evaluatingthe explicit Wiener-Hopf solutions, bypassing unnecessary computational challenges. We deriveexpressions to connect the geometries being analysed, including a condition for dynamic neutrality(perfect transmission) that occurs at the same frequency for the two-dimensional versions of bothcases 3 and 4 shown in figures 1(c,d).In conjunction with the Wiener-Hopf expressions for each of the cases considered, we also derivedispersion relations, and illustrate dispersion surfaces and band diagrams. Of particular importanceare stop and pass band boundaries, standing wave frequencies (flat bands/low group velocity) andthe neighbourhoods of Dirac-like points, which support dynamic neutrality effects. The concept ofDirac cone dispersion originates in topological insulators and has more recently been transferred3nto photonics (see for example [23]–[26]). It is associated with adjacent bands, for which electronsobey the Schr¨odinger equation, that meet at a single point called the Dirac point.Typically connected with hexagonal and triangular geometries in systems governed by Maxwell’sequations, and most notably associated with the electronic transport properties of graphene (see,for example, Castro Neto et al. [27]), analogous Dirac and Dirac-like points have recently beendisplayed in phononic and platonic crystals (see for example [28]–[30], [6]). The presence of Diraccones is generally associated with the symmetries of the system through its geometry. When twoperfect cones meet at a point, with linear dispersion, the cones are said to touch at a Dirac point.In the vicinity of a Dirac point, electrons propagate like waves in free space, unimpeded by themicrostructure of the crystal.In platonic crystals, the analogous points generally possess a triple degeneracy, where the twoDirac-like cones are joined by another flat surface passing through what is known as a Dirac-likepoint. This is analogous to the terminology adopted by Mei et al. [28] in photonics and phononics,where the existence of linear dispersions near the point k = of the reciprocal lattice for thesquare array is the result of “accidental” degeneracy of a doubly degenerate mode (the Diracpoint, without the additional mode) and a single mode. Sometimes known as a “perturbed” Diracpoint, the accidental degeneracy does not arise purely from the lattice symmetry, as for a Diracpoint, but from a perturbation of the physical parameters; in this setting, from the fourth orderbiharmonic operator. We identify Dirac-like points to illustrate neutrality, and “Dirac bridges”(Colquitt et al. [31]) to predict unidirectional wave propagation. We also use dispersion surfacesand the accompanying isofrequency contour diagrams to identify frequencies supporting negativerefraction.The paper is arranged as follows: In section 2, we formulate the problem for the two-dimensionalrectangular array, using the discrete Wiener-Hopf technique; the special case of a semi-infinitegrating is also identified. We provide governing equations, and Wiener-Hopf equations for allcases illustrated in figure 1. In section 3, we analyse these equations, highlighting the conditionsrequired for frequency regimes to support reflection, transmission and dynamic neutrality, whichwe illustrate with examples. We also demonstrate Rayleigh-Bloch-like waves for the semi-infiniteline of scatterers. In section 4, we present special examples of waveguide transmission, whereby thestructured system is designed to specifically exhibit negative refraction and interfacial localisationeffects. Concluding remarks are drawn together in section 5. A thin Kirchhoff-Love plate comprises a two-dimensional semi-infinite array of point scatterersdefined by position vectors r (cid:48) np = ( nd x , pd y ), where d x , d y are the spacings in the x - and y -directionsrespectively, and n, p are integers, as illustrated in figure 2(a). It is natural to consider the systemas a semi-infinite array of gratings aligned parallel to the y -axis. By replacing each of these gratingswith a single point scatterer lying on the x -axis, we recover the one-dimensional case of a singlesemi-infinite grating, as illustrated in figure 2(b). The plate is subjected to a forcing in the formof a plane wave, incident at an angle ψ to the x -axis.We assume time-harmonic vibrations of the Kirchhoff-Love plate, and define equations for theamplitude of the total out-of-plane displacement field u ( r ), with r = ( r x , r y ), which can be expressed4 np u inc yd / rxdy x x (cid:1) dy (a) (b) Figure 2: (a) A semi-infinite array of gratings of point scatterers, whose positions are denoted by r (cid:48) np , with horizontal and vertical spacings d x and d y respectively. (b) A semi-infinite line of pointscatterers for a plane wave incident at angle ψ .as the sum of the incident and scattered fields: u ( r ) = u inc ( r ) + u scatt ( r ) . (1)We express the general governing equation for u ( r ) in the form: ∆ u ( r ) − ρhω D u ( r ) = Φ ( ω, m, c ) ∞ (cid:88) n =0 ∞ (cid:88) p = −∞ u ( r (cid:48) np ) δ ( r − r (cid:48) np ) , r (cid:48) np = ( nd x , pd y ) , n ∈ Z + , p ∈ Z . (2)Here, Φ ( ω, m, c ) is a function of radial frequency ω , and the physical parameters of mass m andstiffness c that define the various mass-spring resonator models shown in figure 1. The functionalforms of the various Φ ( ω, m, c ) are provided later in section 2.2. The characteristic physical pa-rameters for the plate are density per unit volume ρ , thickness h and flexural rigidity D (involvingYoung’s modulus E and the Poisson ratio ν ), and we also adopt the use of the spectral parameter β , which has the dimension of a wavenumber: D = Eh − ν ) ; β = ω (cid:114) ρhD ; ω = β Dρh . (3)Note that the Kirchhoff-Love model incorporates the fourth order biharmonic operator, and givesan excellent approximation of the full linear elasticity equations for a sufficiently small value of theratio h/λ , where λ denotes the wavelength of the flexural vibrations of the plate [32]: λ = 2 πβ = 2 π (cid:18) ρhω D (cid:19) − / , hλ (cid:28) / . (4)5 .1 Governing equations and reduction to a functional equation Assuming isotropic scattering, we express the scattered field in the form of a sum of biharmonicGreen’s functions: u scatt ( r ) = Φ ( ω, m, c ) ∞ (cid:88) n =0 ∞ (cid:88) p = −∞ u ( r (cid:48) np ) G ( β | r − r (cid:48) np | ) , (5)where G is the point source Green’s function satisfying the equation: ∆ G ( β, | r − r (cid:48) | ) − β G ( β, | r − r (cid:48) | ) = δ ( r − r (cid:48) ) . (6)Note that for the one-dimensional case of a semi-infinite line of scatterers r (cid:48) n placed on the x -axis, the sum over p in (5) is absent. All derivations for the two-dimensional array given beloware applicable to the special case of the semi-infinite grating, with appropriate adjustments to thesums and Green’s functions. Referring to equations (1)-(2), we may express the total field u ( r ) as u ( r ) = u inc ( r ) + Φ ( ω, m, c ) ∞ (cid:88) n =0 ∞ (cid:88) p = −∞ u ( r (cid:48) np ) G ( β | r − r (cid:48) np | ) . (7)In particular, at r = r (cid:48) st , s ∈ Z + , t ∈ Z , we have the linear algebraic system u ( r (cid:48) st ) = u inc ( r (cid:48) st ) + Φ ( ω, m, c ) ∞ (cid:88) n =0 ∞ (cid:88) p = −∞ u ( r (cid:48) np ) G ( β | r (cid:48) st − r (cid:48) np | ) , s ∈ Z + , t ∈ Z . (8)Recalling that we consider an incident plane wave, we define u inc in the form: u inc = exp { i ( k x r x + k y r y ) } , (9)where k = ( k x , k y ) = ( β cos ψ, β sin ψ ) is the wave vector. Since the scatterers are infinitely periodicin the y -direction, we impose Bloch-Floquet conditions for u scatt in the y -direction. Hence u ( r x , r y + qd y ) = e ik y qd y u ( r ) and u ( r (cid:48) np ) = u ( r (cid:48) n ) e ik y pd y . (10)Thus, recalling the RHS of equation (8), we have ∞ (cid:88) p = −∞ u ( r (cid:48) np ) G ( β | r (cid:48) st − r (cid:48) np | ) = u ( r (cid:48) n ) ∞ (cid:88) p = −∞ e ik y pd y G ( β | r (cid:48) st − r (cid:48) np | )= u ( r (cid:48) n ) e ik y td y ∞ (cid:88) p = −∞ e ik y ( p − t ) d y G (cid:18) β (cid:113) ( s − n ) d x + ( t − p ) d y (cid:19) . (11)Denoting p − t = α , this simplifies to ∞ (cid:88) p = −∞ u ( r (cid:48) np ) G ( β | r (cid:48) st − r (cid:48) np | ) = u ( r (cid:48) n ) e ik y td y ∞ (cid:88) α = −∞ e ik y αd y G (cid:18) β (cid:113) ( s − n ) d x + α d y (cid:19) , (12)6here the sum on the right is precisely the quasi-periodic grating Green’s function; we shall usethe notation G q ( β, | s − n | ; k y , d x , d y ) = ∞ (cid:88) α = −∞ e ik y αd y G (cid:18) β (cid:113) ( s − n ) d x + α d y (cid:19) , (13)when substituting back into equation (8): u ( r (cid:48) st ) = e i k · r (cid:48) st + Φ ( ω, m, c ) ∞ (cid:88) n =0 u ( r (cid:48) n ) e ik y td y G q ( β, | s − n | ; k y , d x , d y ) . (14)Here, we use u ( r (cid:48) st ) = e ik y td y u ( r (cid:48) s ) . Thus, the algebraic system becomes u ( r (cid:48) s ) = e ik x sd x + Φ ( ω, m, c ) ∞ (cid:88) n =0 u ( r (cid:48) n ) G q ( β, | s − n | ; k y , d x , d y ) , (15)where the integer s denotes the x -position of a grating of scatterers parallel to the y -axis andcentred on the x -axis, and the incident field is in the form exp { ik x sd x } . Equivalently, we may write u s = f s + Φ ( ω, m, c ) ∞ (cid:88) n =0 u n G q ( β, | s − n | ; k y , d x , d y ) , (16)where we replace u ( r (cid:48) s ), u ( r (cid:48) n ) with u s , u n for ease of notation, and f s represents the forcing term.The semi-infinite sum indicates that the discrete Wiener-Hopf method is suitable, see Noble[33], where the application to continuum discrete problems [34], such as gratings, is presented asan exercise (4.10, p.173-4) in [33]. After employing the z -transform, we obtain ∞ (cid:88) s = −∞ u s z s = ∞ (cid:88) s = −∞ (cid:16) e ik x d x z (cid:17) s + Φ ( ω, m, c ) ∞ (cid:88) s = −∞ ∞ (cid:88) n =0 z s u n G q ( β, | s − n | ; k y , d x , d y ) . (17)Letting the index s = n + ζ , we derive a functional equation of the Wiener-Hopf type [33]:ˆ U − ( z ) = ˆ F ( z ) + ˆ U + ( z ) (cid:16) Φ ( ω, m, c ) ˆ G ( z ) − (cid:17) , (18)where ˆ F ( z ) = ∞ (cid:88) s = −∞ (cid:16) e ik x d x z (cid:17) s ; ˆ U − ( z ) = − (cid:88) s = −∞ u ( r (cid:48) s ) z s ; ˆ U + ( z ) = ∞ (cid:88) s =0 u ( r (cid:48) s ) z s , (19) z = e iθ and we have introduced the notation ˆ G ( z ) to represent the functionˆ G ( z ) = ∞ (cid:88) ζ = −∞ G q ( β, | ζ | ; k y , d x , d y ) z ζ , (20)7here G q is the quasi-periodic grating Green’s function given by equation (13). We note that for z = exp ( ik x d x ), i.e. θ = k x d x , we recover the doubly quasi-periodic Green’s function:ˆ G ( z ) = ∞ (cid:88) ζ = −∞ ∞ (cid:88) α = −∞ e iαk y d y e iζk x d x G (cid:18) β (cid:113) ζ d x + α d y (cid:19) . (21)We also note that for the case of a semi-infinite line of point scatterers along x ≥
0, we would havethe same Wiener-Hopf functional equation as (18), but replace ˆ G with G q (13). Equations (15)-(20) are the fundamental general equations for all four of the cases depicted infigure 1. The general kernel function K ( z ) = Φ ( ω, m, c ) ˆ G ( z ) − pinned semi-infinite platonic crystal analysed in [11] because of thechange in boundary condition for the point scatterers. Whereas the rigid pins impose zero flexuraldisplacement at r (cid:48) np , the nonzero condition for the scatterers considered here introduce additionalterms in (22). The function Φ i ( ω, m, c ), for i = 1 −
4, determines the characteristic features specificto each model, and also those common to the different cases. We now present the expressions forthe various Φ i ( ω, m, c ), which we go on to explain and derive, where necessary, for each case inturn. • Case 1: Point masses: Φ ( ω, m, c ) = mω /D • Case 2: Multiple mass-spring resonators on the top surface of the plate: – N = 1: Φ N =12 ( ω, m, c ) = cmω D ( c − mω ) – N = 2: Φ N =22 ( ω, m i , c i ) = c D c − ( c − m ω )( c − m ω )( c + c − m ω )( c − m ω ) − c • Case 3: Multiple mass-spring resonators on both faces of plate: Φ N =13 ( ω, m i , c i ) = ω D (cid:16) m c c − m ω + m c c − m ω (cid:17) • Case 4: Winkler foundation point masses: Φ ( ω, m, c ) = ( mω − c ) /D The simplest type of point scatterer is the rigid pin, defined as the limiting case of the radiusof a clamped hole tending to zero. There is a large body of literature covering various problemsincorporating this boundary condition. A selection of relevant papers include Movchan et al. [35],Evans & Porter [14], [36], Antonakakis & Craster [37], [5], Haslinger et al. [38]. A logical extensionis to replace the pins with concentrated point masses of mass m , introducing an additional inertial8erm, and hence non-zero displacement at the point scatterer. We include a schematic diagramin figures 1(a) and 2(b). The governing equation for a semi-infinite half-plane of point masses,following equation (2), is ∆ u ( r ) − ρhω D u ( r ) = mω D ∞ (cid:88) n =0 ∞ (cid:88) p = −∞ u ( r (cid:48) np ) δ ( r − r (cid:48) np ) . (23)The total flexural displacement field u j = u ( r (cid:48) j ), of the form (16), is given by u j = f j + mω D ∞ (cid:88) n =0 u n G q ( β, | j − n | ; k y , d x , d y ) , (24)and the corresponding Wiener-Hopf-type functional equation isˆ U − ( z ) = ˆ F ( z ) + ˆ U + ( z ) (cid:16) Φ ( ω, m ) ˆ G ( β ; z ) − (cid:17) , Φ ( ω, m ) = mω D . (25)
We attach mass-spring resonators at each point r (cid:48) np = ( nd x , pd y ) , n ≥
0. Each resonator consistsof N point masses attached to N springs. For the most general case, the finite number of masses m , m , ..., m N are connected by springs of stiffness c , c , ..., c N (see figure 1(b)). For the sake ofsimplicity, we derive the governing equations, and their reduction to functional Wiener-Hopf typeequations, for the cases N = 1 and N = 2, but the procedure for arbitrary N is a simple extension. Single mass-spring resonator, N=1
We assume a semi-infinite rectangular array of simple resonators consisting of point masses at-tached with springs to the plate at points shown in figure 2(a), with the parameters illustrated infigure 1(b). We assume uniform mass m and uniform stiffness c , and negligible effect of gravity. Wederive the equation for Φ N =12 ( ω, m, c ), and the accompanying governing and discrete Wiener-Hopfexpressions, by applying Newton’s 2nd law and Hooke’s law for an arbitrary scatterer placed at r (cid:48) n = ( nd x , y -axis, centred at r (cid:48) n .We denote the flexural displacement of the plate at r (cid:48) n by u ( r (cid:48) n ) = u n . The transverse dis-placement of the mass m is denoted by v ( r (cid:48) n ) = v n , with the forces applied to the plate by thespring, of stiffness c , given by A u ( r (cid:48) n ), and to the connected mass by the spring, as A v ( r (cid:48) n ). Theequation of motion for the sprung mass is written in the form: mω v n = − A v ( r (cid:48) n ) = c ( v n − u n ) . (26)Transverse displacements v n and u n are evaluated with respect to r (cid:48) n , but for the sake of simplicity,we adopt the abbreviated notation with the subscript n . We write v n in terms of u n using (26),and referring to equations (2),(16) and recalling that the flexural rigidity of the plate is D , we have u j = f j + cmω D ( c − mω ) ∞ (cid:88) n =0 u n G q ( β, | j − n | ; k y , d x , d y ) , (27)9ith Φ N =12 ( ω, m, c ) = cmω D ( c − mω ) . (28)Note that in the limit as c → ∞ , equation (27) tends towards the equation for unsprung mass-loadedpoints (24), and v n = u n from (26); the flexural vibrations of the plate and masses are identicalfor infinite stiffness, which is physically consistent with infinitely stiff springs. As m → ∞ , thecoefficient multiplying the sum tends to − c/D , and this may be interpreted physically as the platebeing attached to a rigid foundation with springs of stiffness c .The discrete Wiener-Hopf functional equation is obtained by substituting the expression for Φ N =12 ( ω, m, c ) into equation (18):ˆ U − ( z ) = ˆ F ( z ) + ˆ U + ( z ) (cid:18) cmω D ( c − mω ) ˆ G ( β ; z ) − (cid:19) . (29)Recalling equation (22), the kernel K N =12 ( z ) for case 2, N = 1 is given by K N =12 ( z ) = cmω D ( c − mω ) ˆ G ( β ; z ) − cmω D ( c − mω ) K pins ( z ) − , (30)where we introduce the notation K pins ( z ) as the kernel for the case of rigid pins, see [11].Observing that a similar expression follows for the limit case of point masses (25), this connectionwith the pinned case enables us to employ a similar kernel factorization. First, we rewrite (30) interms of the dimensionless parameters:˜ m = mρhη , ˜ c = cη D , ˜ β = βη, (31)where we introduce a length scale determined by the periodicity of the system η = min ( d x , d y ),which for the sake of simplicity is taken to be d x throughout this article. We also introduce non-dimensional versions of Green’s functions, and their arguments, which possess the dimension of L owing to the factor 1 /β : ˜ˆ G ( ˜ β ; ˜ z ) = β ˆ G , ˜ z = z/η. (32)Hence, ˜ K N =12 ( ˜ β ; ˜ z ) = ˜ m ˜ β (cid:32) − ˜ m ˜ β ˜ c (cid:33) ˜ˆ G ( ˜ β ; ˜ z ) − . (33)We express (33) as ˜ K N =12 ( ˜ β ; ˜ z ) = ˜ K +2 ( ˜ β ; ˜ z ) ˜ K − ( ˜ β ; ˜ z ) , (34)noting that the factorization obviously also applies to the dimensional form of the kernel. Explicitly,we have ˜ K +2 ( ˜ β ; ˜ z ) = ˜ Φ ( ˜ β, ˜ m, ˜ c ) ˜ˆ G + ( ˜ β ; ˜ z ); ˜ K − ( ˜ β ; ˜ z ) = ˜ˆ G − ( ˜ β ; ˜ z ) − Φ ( ˜ β, ˜ m, ˜ c ) ˜ˆ G + ( ˜ β ; ˜ z ) , (35)where the reciprocal of the + function ˜ˆ G + ( ˜ β ; ˜ z ) is a − function.10 ultiple mass-spring resonators, N=2 As for the case N = 1, we derive the equation for Φ N =22 ( ω, m, c ) for an arbitrary scatterer placedat r (cid:48) n = ( nd x , m i is v ( i ) n , i ∈ [1 , N ]. For N = 2, the equations of motion are given by( c + c − m ω ) v (1) n = c u n + c v (2) n ( c − m ω ) v (2) n = c v (1) n . (36)The normalised force acting on the plate at r (cid:48) n , in terms of the out-of-plane displacement, is givenby: A u ( r (cid:48) n ) = c D ( v (1) n − u n ) , (37)where the reciprocal of the plate’s flexural rigidity is the normalisation factor. Thus, referring toequation (27) we may write the total flexural displacement amplitude at u ( r (cid:48) j ) = u j as u j = f j + c D ∞ (cid:88) n =0 ( v (1) n − u n ) G q ( β, | j − n | ; k y , d x , d y ) . (38)We eliminate v (2) n from (36) and derive the expression for v (1) n in terms of u n only: v (1) n (cid:18) c + c − m ω − c c − m ω (cid:19) = c u n . Hence, we rewrite equation (38) in the form, u j = f j + Φ N =22 ( ω, m i , c i ) ∞ (cid:88) n =0 u n G q ( β, | j − n | ; k y , d x , d y ) , (39)such that Φ N =22 ( ω, m i , c i ) = c D c − ( c − m ω )( c − m ω )( c + c − m ω )( c − m ω ) − c . (40)As in previous cases, we employ the z -transform to obtain the discrete Wiener-Hopf equation:ˆ U − ( z ) = ˆ F ( z ) + ˆ U + ( z ) (cid:16) Φ N =22 ( ω, m i , c i ) ˆ G ( β ; z ) − (cid:17) . (41)This Wiener-Hopf equation resembles that of (29) and in the limit as c →
0, we recover preciselythat equation, and similarly (27) from (40).
We now consider an extension of section 2.2.2 by attaching mass-spring resonators on oppositefaces of the plate at the same point of the array depicted in figure 2. This system is illustrated forthe case of 2 N mass-spring resonators in figure 1(c). For introducing the model, we analyse the11implest case here; two masses m , m with associated spring stiffnesses c , c , with the index beingodd for resonators attached to the top surface, and even for the bottom surface, as illustrated infigure 1(c). The derivations are similar to the previous sections with the flexural displacement atan arbitrary defect point r (cid:48) n = ( nd x , , n ≥ u n and the transverse displacements of themasses m i , i = 1 , v ( i ) n . The equation of motion of the resonator mass at a single arraypoint is given by m i ω v ( i ) n = c i ( v ( i ) n − u n ) , i = 1 , , (42)where we have used Hooke’s law for the right-hand side. Recalling the general expression for thetotal flexural displacement field of the plate at the point r (cid:48) j = ( jd x , u j = f j + 1 D ∞ (cid:88) n =0 ( A (1) n + A (2) n ) G q ( β, | j − n | ; k y , d x , d y ) , (43)where the forces A ( i ) n are given by A ( i ) n = c i ( v ( i ) n − u n ) , i = 1 , . (44)Similar to case 2, N = 2, we derive the governing equation in the form u j = f j + Φ ( ω, m i , c i ) ∞ (cid:88) n =0 u n G q ( β, | j − n | ; k y , d x , d y ) , (45)where Φ ( ω, m i , c i ) = ω D (cid:18) m c c − m ω + m c c − m ω (cid:19) . (46)Employing the z -transform in the standard way, the accompanying Wiener-Hopf representation isˆ U − ( z ) = ˆ F ( z ) + ˆ U + ( z ) (cid:16) Φ ( ω, m i , c i ) ˆ G ( β ; z ) − (cid:17) . (47) An alternative model for adding mass-spring resonators is shown in figure 1(d), where point massesare embedded within the plate, and additional springs, attached to a fixed foundation, are addedbelow. Referring to equation (26) for case 2, N = 1, we obtain a similar equation, except that herethe displacement at the point of attachment to the fixed foundation v n , is zero. Hence, mω u n = cu n , (48)and the solution for the total flexural amplitude at r (cid:48) j = ( jd x ,
0) is u j = f j + mω − cD ∞ (cid:88) n =0 u n G q ( β, | j − n | ; k y , d x , d y ) . (49)12e note that with this model, taking the limit as c → N = 1, where c → ∞ retrieved the limitingcase of point masses. Similarly, the Wiener-Hopf equation is easily deduced from the generalequation (18): ˆ U − ( z ) = ˆ F ( z ) + ˆ U + ( z ) (cid:18) mω − cD ˆ G ( β ; z ) − (cid:19) , (50)with the kernel function K ( z ) defined by K ( z ) = mω − cD ˆ G ( β ; z ) − , or ˜ K (˜ z ) (cid:18) ˜ m ˜ β − ˜ c ˜ β (cid:19) ˜ˆ G ( ˜ β ; ˜ z ) − . (51) We identify three important frequency regimes using the kernel equation (22): reflection, transmis-sion and dynamic neutrality. Special cases of waveguide transmission including negative refractionand interfacial localisation are illustrated in the subsequent section 4. The five Wiener-Hopf ex-pressions (25), (29), (41), (47) and (50) are characterized by their respective kernels, which all takethe general form K ( z ) = Φ ( ω, m, c ) ˆ G ( β ; z ) − . (52)An analysis of these functions gives us insight into the behaviour of the possible solutions.There are three natural limiting regimes to consider for a kernel function with this structure- when it is either very large or very small, and when Φ ˆ G tends to 0, i.e. K → −
1. Referringto the general Wiener-Hopf equation (18), the first two cases infer that ˆ U + is respectively verysmall or very large; the physical interpretation of ˆ U + is the amplitude of scattering within theplatonic crystal. Thus, small | ˆ U + | indicates reflection (blocking), and large | ˆ U + | indicates enhancedtransmission, which is of particular interest for a single line of scatterers since it manifests in theform of Rayleigh-Bloch-like modes propagating along the grating itself (see, for example, Evans &Porter [14] and Colquitt et al. [39] for related problems). For K → −
1, the general expression (18)tells us that in the limit, ˆ U − + ˆ U + = ˆ F .
Recalling that ˆ F represents the incident field, and ˆ U − + ˆ U + , the total field, we may interpret thisregime as perfect transmission or dynamic neutrality; the wave propagation is unimpeded by themicrostructure of the platonic crystal, a phenomenon often associated with the vicinity of Dirac orDirac-like points [1]. Summarising the three regimes, we have • reflection (blocking) |K| (cid:29) • waveguide transmission |K| (cid:28) • dynamic neutrality (perfect transmission) K → − mω − c , which defines the resonancefrequency ω r of the individual mass-spring resonators: mω r − c = 0 ⇐⇒ ω r = cm . (53)Crucially, however, the kernel functions differ in that this term is in the numerator for the Winklercase, but in the denominator for the mass-spring resonator case in (29),(30). This indicates that,for example, the transmission condition for the Winkler foundation would correspond to the regimeof reflection ( |K| (cid:29)
1) for the mass-spring resonator case and vice versa.
Referring to the general equation (52), reflection (blocking) is predicted for frequency regimes whereeither Φ ( ω, m, c ) or ˆ G , or both functions together, blow up. We recall that the kernel function isprecisely ˆ G ( β ; z ) for the case of a semi-infinite array of rigid pins analysed by [11], and that for z = exp { iθ } , with θ = k x d x and k = ( k x , k y ) the Bloch vector for a doubly periodic system, ˆ G ( β ; z )is a doubly quasi-periodic Green’s function. Much has been written about this Green’s function inthe literature; see for example, McPhedran et al. [1], McPhedran et al. [6], Poulton et al. [13]. Avery important property is that its zeros correspond to the dispersion relation for the infinite doublyperiodic system of rigid pins, which possesses a complete band gap for low frequency vibrations upto a finite calculable value. Here we express ˆ G ( β ; z ) (21) in the form:ˆ G ( β ; z ) = ˆ G ( β ; k ) = i β (cid:18) S H ( β ; k ) + 1 + 2 iπ S K ( β ; k ) (cid:19) , (54)where S H , S K are lattice sums defined over the periodic array of point scatterers r (cid:48) np in the followingway: S H ( β ; k ) = (cid:88) r (cid:48) np (cid:54) = { , } H (1)0 ( β | r (cid:48) np | ) e i k · r (cid:48) np , S K ( β ; k ) = (cid:88) r (cid:48) np (cid:54) = { , } K (1)0 ( β | r (cid:48) np | ) e i k · r (cid:48) np . (55)We also note that the lattice sum over the Hankel functions may be written in the form S H ( β ; k ) = − i S Y ( β ; k ) . (56)The lattice sums S H ( β ; k ), S Y ( β ; k ) are only conditionally convergent, and require an appropriatemethod of accelerated convergence for numerical computations. We adopt the same triply inte-grated expressions originally used by Movchan et al. [35], and more recently by [6]. The dispersion14elation for the doubly periodic pinned array is then given byˆ G ( β ; k ) = i β (cid:18) S H ( β ; k ) + 1 + 2 iπ S K ( β ; k ) (cid:19) = 0 , (57)and has real solutions.The direct connection between the kernel function for the semi-infinite array of pins and thedispersion relation for the infinite doubly periodic array enables one to identify frequency regimes forreflection and transmission of the incoming plane waves. Similarly for the point scatterers featuredin this article, which importantly do not impose zero displacement clamping conditions, the zerosand singularities of the kernel function give us information about, respectively, transmission andreflection, but the kernel (52) now depends on more than ˆ G .The singularities of ˆ G still indicate regimes of stop-band behaviour, but there is additionalreflection behaviour determined by Φ ( ω, m, c ) becoming very large. This is evident for the simplemass-spring resonators with N = 1. The kernel is given by (30) where Φ blows up for the frequencycorresponding to the resonance of the individual mass-spring resonators, ω r = c/m . We wouldtherefore expect to see reflection of incident waves for ω close to this resonant frequency ω r , andstop bands in the corresponding dispersion diagrams (arising for zeros of the kernel) for the mass-spring resonators. (a) (b) (c)(d) (e) (f) Figure 3: Band gap behaviour: Real part of total displacement field for a plane wave incidentat ψ = π/ d x = d y = 1 .
0) array of (a) rigid pins for ˜ β = 2 .
35 and(b) point masses with ˜ m = 1 . β = 3 .
02. (c) ψ = 0 for a semi-infinite rectangular ( ξ = √ m = 1 .
0, ˜ c = 1 . β = 0 .
8. Arrows indicate directionof incident plane wave. (d-f) Dispersion surfaces for corresponding doubly periodic arrays of,respectively, parts (a-c).In figures 3, 4 we illustrate reflection (blocking) for the various systems of figures 1(a-d). Wepresent results for two rectangular arrays, the special case of the square array with d x = d y = 1 . ξ = d y /d x = √
2. We demonstrate blocking for the square arraysof both pins and point masses in figures 3(a, b) together with their respective dispersion surfaces,and corresponding stop bands, in parts (d, e). In figure 3(c), we show the reflective behaviour of asemi-infinite rectangular array with ξ = √ ξ = √ N = 1 in part (a) and for the plate with resonators attached to both faces(DSP) in part (b). The corresponding band diagrams are shown in figure 4(c). Here, the Brillouinzone is assumed to be the rectangle Γ XM Y , with Γ = ( k x , k y ) = (0 , , X = ( π, , M = ( π, π/ √ Y = (0 , π/ √ |K| (cid:28)
1, where the flexural waves propagate through the periodic array, whichwe discuss in more detail in the next section. - - β ˜ Γ X MM Y (a) (b) (c)
Figure 4: Reflection: Real part of total displacement field for a plane wave incident at ψ = π/ ξ = √
2) array of (a) mass-spring resonators for ˜ m = 1 .
0, ˜ c = 1 . β = 1 . β = 1 .
10 with m = 12, m = 108 / c = 400, c = 400 / m red = 1 .
0, ˜ c red = 1 . Γ XM Y , with Γ = ( k x , k y ) = (0 , , X = ( π, , M = ( π, π/ √ Y = (0 , π/ √ (cid:18) mω D ˆ G ( β ; z ) − (cid:19) = 0 , (58)and in terms of dimensionless parameters, this is equivalent to˜ m ˜ β ˜ˆ G ( ˜ β ; ˜ z ) − , (59)which has real solutions (see figure 3(e) for the resulting surfaces). Similarly, the equation for thesprung point masses (case 2, N = 1) is given by˜ m ˜ β (cid:32) − ˜ m ˜ β ˜ c (cid:33) ˜ˆ G ( ˜ β ; ˜ z ) − , (60)where the frequency dependence on the resonators is expressed in terms of dimensionless ˜ β ratherthan ω . We show the first two dispersion curves for N = 1 (dashed curves) in figure 4(c), where16e choose dimensionless mass ˜ m = 1 .
0, and dimensionless stiffness ˜ c = 1 . ω r , see equation (53), or equivalently ˜ β r = 1 . m , m and c , c such that the reduced mass is consistent with ˜ m = 1 .
0, ˜ c = 1 . m red = 1 m + 1 m , m red ≤ m , m , (61)and similarly, reduced (effective) stiffness is defined as1 c red = 1 c + 1 c , c red ≤ c , c . (62)The choices for these parameter values will be explained in section 3.2.1 where it is shown that thedouble-sided plate supports dynamic neutrality (perfect transmission). The resulting total bandgap between the first and second surfaces is clearly visible in figure 4(c) and the example of reflectionfor ˜ β = 1 .
10 and ψ = π/ β and ψ illustrated coincide with the total band gaps observed for2 . < ˜ β < π , and 0 < ˜ β < .
94, in figures 3(e, f) respectively.The equations for the kernel and corresponding dispersion relations immediately give us in-formation about the position of the stop bands. We also plot the kernel for ranges of interest todetermine more information about refection and transmission regimes. For instance, for ψ = π/ k x = 0 (corresponding to the edge of the Brillouin zone Y Γ for the corresponding doubly periodic system) and plot the real parts (a) and moduli (b) of thekernel (solid curves) versus ˜ β in figures 5(a, b) for the semi-infinite rectangular ( ξ = √
2) array ofmass-spring resonators with ˜ m = 1 . , ˜ c = 1 . β r = 1 . β ∈ (1 . , .
40) forwhich reflection/blocking is predicted according to |K| (cid:29)
1, which is consistent with the width ofthe stop band in figure 4(c). For sufficiently small |K| , transmission is predicted, which coincideswith solutions to the dispersion relations. A special case of transmission is dynamic neutrality,which occurs for K = −
1. There is evidence in figures 5(a, b) of an extended neutral regime for themass-spring resonators for ˜ β ≈ .
36, when the solid curves in parts (a, b) tend to ∓ a) (b) Figure 5: Transmission and reflection regimes: Plots of (a) real parts and (b) moduli of, respec-tively, the kernel function (solid), Φ ( ω, m, c ) (dashed) and doubly quasi-periodic Green’s function(dotted) for a semi-infinite rectangular ( ξ = √
2) array of mass-spring resonators for N = 1 with˜ m = 1 . , ˜ c = 1 . ψ = π/ k x = 0 (branch Y Γ for corresponding doubly periodic array).
For sufficiently small values of the kernel |K| (cid:28)
1, the semi-infinite crystals admit propagationof incident waves into the periodic array. Owing to the connection with corresponding infinitedoubly periodic systems, this transmission property is predicted for the semi-infinite systems bythe Bloch-Floquet analysis. Therefore, dispersion surfaces (as illustrated in figures 3(d-f)) giveus information for predicting the propagation of waves into the inhomogeneous half of the plate.Besides standard refracted waves, there are various special cases of transmission, including dynamicneutrality (perfect transmission), interfacial localisation and negative refraction.We demonstrate the conditions for dynamic neutrality using the expression for the kernel func-tion, whereas previous examples of the effect given by [11] for the simpler pinned case, relied upondetermining neighbourhoods of Dirac-like points on the accompanying dispersion surfaces. Theresulting field plots yielded approximate neutrality regimes, where the direction of the propagatingwaves remained unchanged, but the plane wave was replaced with a total field consisting of regularregions of constructive and destructive interference of the incident and scattered fields, reminiscentof the reflected fields in figures 3(a,b) and 4(a,b). The analytic prediction for perfect transmis-sion demonstrated here is a powerful method to locate neutrality regimes, the quality of which isquantified using the condition on the scattering coefficients | A k | = 1. The presence of Dirac-like points for the infinite system give us some insight for the correspondingsemi-infinite arrays, but using the dynamic neutrality condition associated with the kernel functionsenables us to predict these effects more accurately. Recalling the mass-spring resonators attached18o both faces of the plate (DSP), we seek a condition on the kernel such that
K → − K = Φ ( ω, m q , c q ) ˆ G ( β ; z ) − , Φ = ω D (cid:18) m c c − m ω + m c c − m ω (cid:19) , (63)which leads to the condition ω = c c ( m + m ) m m ( c + c ) . (64)We adopt the concept of reduced mass and stiffness, as defined by equations (61), (62), obtaining ω = m + m c + c = c red m red ⇐⇒ m red ω − c red = 0 . (65)Substituting the condition for neutrality (65) into equation (45), we obtain U p = f p + ω D ( c red − m red ω ) Ω ∞ (cid:88) n =0 U n G q ( β, | p − n | ; k y , d x , d y ) , where Ω = ( c − m ω )( c − m ω )( c + c )( m + m ) , (66)an equation that is reminiscent of the simple Winkler-type foundation described by figure 1(d)and equations (49), (50) owing to the factor ( c red − m red ω ) in the numerator. Indeed, for specificchoices of m, c and m red , c red , we expect to observe neutrality in both models for the same normalisedfrequency ω that determines zeros of this factor. - - β ˜ Γ X MM Y (a) (b) (c)
Figure 6: Perfect transmission: Real part of the total displacement field for a semi-infinite rectan-gular array of point scatterers with, d x = 1 . , d y = √ ψ = π/
4, ˜ β = 1 . m red = 1 . , ˜ c red = 1 . m = 1 .
0, ˜ c = 1 . m = 1 .
0, ˜ c = 1 .
910 for theWinkler-foundation point masses, we assign values to the parameters m = 12, m = 108 / c = 400, c = 400 / m red = 5 . , c red = 100, which ensures that ˜ m red = 1 .
0, ˜ c red = 1 . β = 1 . m red ω − c red = 0 , ω = (cid:112) c red /m red ; β = ρhω /D. (67)We observe virtually perfect transmission (neutrality) for both physical models for ψ = π/ β = 1 . β is also important for the case of mass-spring resonators attached to thetop surface of the plate, demonstrating stop-band behaviour in figure 4(a). The mω − c factor iscommon to both models, but the relationship between the two models is reciprocal; where Winklergives perfect transmission at ˜ β ∗ = 1 . β ≈ .
36, for which the kernel function
K ≈ −
1. For ˜ β = 2 . , ψ = π/ , ξ = √
2, we plot the total displacement field in figure 7(a).We observe an excellent example of dynamic neutrality where the incident plane wave appearsundisturbed by the interaction with the point scatterers, retaining both its direction and amplitudes.The moduli of the scattering coefficients | A k | are plotted for three values of ˜ β in figure 7(b). Forthe perfect transmission frequency ˜ β = 2 . | A k | ≈ β = 1 .
50 (dashed), and ˜ β = 3 . (a) (b) Figure 7: Perfect transmission: (a) Real part of the total displacement field for a semi-infiniterectangular array ( ξ = √
2) of mass-spring resonators for N = 1, ψ = π/ m = 1 . , ˜ c = 1 . β = 2 .
35. (b) Moduli of scattering coefficients | A k | for ˜ β = 2 .
35 (solid), 1 .
50 (dashed), 3 . .2.2 Waveguide transmission for semi-infinite line of scatterers According to Wilcox [40], localised waves travelling along a grating, in the absence of an incidentwave, are called Rayleigh-Bloch waves. Evans & Porter [14] presented conditions for the existenceof Rayleigh-Bloch waves along a one-dimensional periodic array of point masses or Winkler-sprungmasses in a Kirchhoff-Love plate. Identifying Rayleigh-Bloch regimes is also interesting for thesemi-infinite array problems presented here, since evidence of the characteristic localisation will beapparent for normally incident plane waves for corresponding choices of β . Point masses
For case of rigid pins, no Rayleigh-Bloch modes exist for real β > m . This is evident from theapproximate dispersion curves for an infinite grating shown in figure 8(a), obtained for the case |K | (cid:28) mω D G q ( β ; z ) − , and in dimensionless parameters ˜ m ˜ β ˜ G q ( ˜ β ; ˜ k x ) − , (68)where ˜ k x = k x d x . k x Β m (cid:142) (cid:61)
1, 100 - - k β m ˜= = ρ = d x = d y = Γ X MM (a) (b) ∼ ∼ ∼
Figure 8: (a) Dispersion curves for ˜ m = 1 . m = 100 (solid lower) for 0 ≤ ˜ k x ≤ π . The dashed straight line represents both the solutions for the homogeneous plate, and thesingularities of the dispersion relation for the array of masses. The dashed curve represents thedispersion curve for Γ X for the doubly periodic square array with ˜ m = 1, d x = d y = 1 .
0, which isshown in (b), with irreducible Brillouin zone
Γ XM , Γ = ( k x , k y ) = (0 , , X = ( π, , M = ( π, π ).Two curves for ˜ m = 1 . m = 100 are shown in figure 8(a), along with the first two bandsfor the infinite square array of point masses with ˜ m = 1 . d x = d y = 1 . Γ X branch (i.e. with k y = 0) for the two-dimensional system is striking; in both cases, we see linear-like dispersion for ˜ β k x → π . For direct comparison, weinclude precisely the Γ X branch of the first band from figure 8(b) as the dashed curve in figure 8(a).The presence of the acoustic mode at low frequencies contrasts with that of rigid pins. AsPoulton et al. [13] commented, as the dimensionless mass ˜ m → ∞ , this acoustic band becomesflatter and flatter (compare ˜ m = 1 . m = 100), finally collapsing into the axis ˜ β = 0 in thelimit, thereby recovering the case of rigid pins. One other interesting feature of figure 8(b), andfigure 3(e), is that the XM branch of the second band coincides with the dispersion curve for thehomogeneous plate regardless of the value of ˜ m . This indicates that the propagation of the flexuralwaves in the mass-loaded plate is unaffected by the loading in this direction, which is consistentwith the dynamic neutrality regime we observe in the vicinity of the Dirac-like point at M infigure 3(e). This was first pointed out by McPhedran et al. [1] who observed that the second bandis “sandwiched” between two planes of the dispersion surfaces for the homogeneous plate, wherethe lattice sum S Y (56) diverges.The waveguide transmission regime predicted by zeros of the kernel is demonstrated for the semi-infinite line of scatterers in the form of Rayleigh-Bloch-like standing waves. This is illustrated infigures 9(a, b), where we plot the real part of the total displacement field for a plane wave normallyincident on a truncated semi-infinite grating of 1000 point scatterers with ˜ β = 2 .
0, comparing(a) point masses of ˜ m = 1 . (a) (b) (c) Figure 9: A plane wave is normally incident on an array of 1000 point scatterers with spacing d x = 1 .
0. Real part of the total displacement field for ˜ β = 2 . m = 1 . β = 2 . m = 1 . β = 2 . Mass-spring resonators with N = 1The dispersion relation for the semi-infinite line of mass-spring resonators, as for the analogoushalf-plane, is |K| (cid:28)
1: ˜ m ˜ β (cid:32) − ˜ m ˜ β ˜ c (cid:33) ˜ G q ( ˜ β ; ˜ k x ) − . (69)22s expected, the introduction of springs brings new features to the dispersion picture for the waveg-uide transmission regime. The dispersion diagrams for, respectively, a line and doubly periodicsquare array are shown in figures 10(a, b). The crucial difference is the term in the denominator c ˜ = c ˜ = c ˜ = ∞ k ˜ x β ˜ - - β ˜ Γ X MM (a) (b)
Figure 10: (a) Dispersion curves for an infinite line of mass-spring resonators with ˜ m = 1 . c = 1 . c = ∞ . Asymptotes ˜ β = ˜ β ∗ and ˜ β = ˜ k x are dashed straight lines. (b) Band diagram for square array of sprung point masseswith ˜ m = 1 .
0, ˜ c = 1 .
910 with irreducible Brillouin zone a triangle
Γ XM , d x = d y = 1 . m ˜ β / ˜ c = 1. Physically, these solutions ˜ β ∗ coincide withresonances arising for each mass-spring resonator, and result in branching of the dispersion curves,for both the 1d-case in figure 10(a), and the square array with d x = d y = 1 . m = 1 .
0, we consider dimensionless stiffness ˜ c = 1 .
910 and 9.5,labelled in figure 10(a), which correspond to, respectively, stiffnesses c = 100 , c → ∞ , of point masses for the same ˜ m = 1 .
0. To the right of the straight(dashed) line ˜ β = ˜ k x , the dispersion curves for the mass-spring resonators resemble the analogouscurve for the unsprung point masses. However, these curves veer away from the asymptote ˜ β = ˜ k x for comparatively lower values of ˜ β , tending towards the horizontal asymptote ˜ β = ˜ β ∗ , whichseparates the two branches of the dispersion curve for a fixed ˜ c/ ˜ m .The contribution from the grating Green’s function ˜ G q ( ˜ β ; ˜ k x ) also brings singularities associatedwith the “light line” ˜ β = ˜ k x , meaning that two intersecting asymptotes are associated with eachdispersion curve. The notion of “light surfaces” and “light lines” is well known in the modellingof Bloch-Floquet waves. Originating in electromagnetism, light lines identify frequencies for whichlight propagates in the surrounding homogeneous medium (usually air), and are now well used inproblems of acoustics and elasticity for the unstructured parts of the systems in those physicalsettings. The branch to the left of ˜ β = ˜ k x contributes a second set of solutions for small ˜ k x , whichalso appear to tend very slowly towards the asymptote ˜ β = ˜ β ∗ from above, before hitting, and thenfollowing the “light line” ˜ β = ˜ k x . This behaviour for the line of scatterers is consistent with the Γ X branches of the first two bands of the doubly periodic system, illustrated in figure 10(b). Thisdispersive property of the mass-spring resonator systems suggests that the semi-infinite array ofsprung masses supports a neutrality effect for normally incident ( k y = 0) plane waves for ˜ β > ˜ β ∗ .23n figure 11, we consider the evolution of the total displacement fields for a semi-infinite lineas the frequency parameter ˜ β is increased, whilst keeping ψ = 0, ˜ m = 1 .
0, ˜ c = 1 .
910 constant.For ˜ β = 0 .
5, we observe a long wavelength and a slight phase delay near the location of thepoint scatterers; compare the edge and centre of the wavefronts in figure 11(a). This differencebecomes more pronounced in figure 11(b) for ˜ β = 1 .
0. Referring to the relevant dispersion curvein figure 10(a), the sprung masses’ curve is slightly further away from the “light line” for ˜ β = 1 . β = 0 . (a) (b)(c) (d) Figure 11: Real part of total displacement field for a plane wave normally incident on a line arrayof 1000 point sprung masses with ˜ m = 1 . , ˜ c = 1 . d x = 1 . β = 0 .
5, (b)˜ β = 1 .
0, (c) ˜ β = 1 .
20. (d) ˜ β = 1 . β to 1.20 takes us into the stop band clearly identified in the dispersion diagram.This is illustrated by the displacement field for the resonators in figure 11(c), where the pointscatterers appear to block the propagation of the normally incident waves. As ˜ β is increased to1.35, we observe strong localization along the grating for normal incidence, and the correspondingdispersion curve now appears to travel along ˜ β = ˜ k x in figure 10(a); we observe an increase in themoduli of the scattering coefficients, and the phase difference between the centre and edge of thewavefronts is the opposite way round to the case of frequencies below the stop band.For larger values of ˜ β , we see transmission consistent with coincidence of the dispersion curve andthe straight line ˜ β = ˜ k x . This is indicated in figure 11(d) for ˜ β = 1 .
80, where the phase differencehas switched, such that the centre of the wavefront is slightly ahead of the edge; for ˜ β = 2 . β = 2 . ψ = π/ m ˜ β / ˜ c −
1, with the branchessandwiching the resulting asymptote ˜ β = ˜ β ∗ . Thus, the ratio ˜ c/ ˜ m tells us where the stop bandoccurs, but because of the additional factor ˜ m in equation (69), the width of the band can be alteredby varying ˜ m and ˜ c such that their ratio remains constant. This is illustrated in figure 12(a), wherewe plot the dispersion curves (dashed) for ˜ c = 3 .
819 ( c = 200), ˜ m = 2 and ˜ c = 0 .
477 ( c = 25),˜ m = 0 .
25 along with the case ˜ c/ ˜ m = 1 .
910 (solid) from figure 10(a). For increased ˜ m and ˜ c , theband gap is widened, with the opposite result for simultaneous reduction of ˜ m , ˜ c , whilst maintainingthe constant ˜ c/ ˜ m = 1 . c ˜ = m ˜ = c ˜ = m ˜ = k ˜ x β ˜ (a) (b) (c) Figure 12: Stop-band width control: (a) Dispersion curves for a line of point sprung masses for˜ c/ ˜ m = 1 .
910 with three pairs of dimensionless mass and stiffness: ˜ m = 1 . , ˜ c = 1 .
910 (solid curve);˜ m = 0 . , ˜ c = 0 .
477 (dashed blue curve); ˜ m = 2 . , ˜ c = 3 .
819 (dashed purple curve). Asymptotesdenoted by solid straight lines. Real part of total displacement fields for ˜ β = 1 . , ψ = 0 for (b)˜ m = 0 . , ˜ c = 0 . m = 2 . , ˜ c = 3 . c/ ˜ m , we can design systems thatfilter ˜ β values simply by redistributing the masses and stiffnesses of the resonators, as illustrated infigure 12 (b, c). For ˜ c/ ˜ m = 1 .
910 and ˜ β = 1 .
25, the system with ˜ m = 2 .
0, ˜ c = 3 .
819 blocks normallyincident waves in figure 12 (c), but allows them to pass for ˜ m = 0 .
25, ˜ c = 0 .
477 in figure 12 (b).Similar observations about controlling the width of the stop band were made by [15] for the doublyperiodic square array of mass-spring resonators.
In this section, we provide a collection of illustrative examples for designing systems to harnessnotable transmissive effects, including negative refraction and interfacial localisation, for the rect-angular lattice with ξ = √
2. For the sake of computational efficiency, we present the total displace-ment fields for truncated semi-infinite systems, using the algebraic system of equations adopted byFoldy [41]. We rewrite the general governing equation (16) in the truncated form: Φ ( ω, m, c ) N − (cid:88) n =0 u n G q ( β, | s − n | ; k y , d x , d y ) = u s − f ( i ) s s = 1 , , ...., N, (70)25here we have replaced the incident field notation f s with f ( i ) s . Hence, in terms of matrices weobtain the equation u ( Φ G q − I N ) = − f ( i ) . (71)Here u is a vector representing the total displacement field, f ( i ) is the vector representing thecorresponding incident waves and G q is a matrix of quasi-periodic Green’s functions. We solve thealgebraic system of equations (71) to retrieve the displacements u s . The displacement fields arethen illustrated by plotting the real part of u ( r ) = f ( i ) ( r ) + Φ ( ω, m, c ) N (cid:88) n =0 u n G q ( β, | r − ( nd x , | ; k y , d x , d y ) , (72)where each vertical grating labelled by n is centred at ( nd x ,
0) and associated with a correspondingquasi-periodic Green’s function G q . Referring to figure 4(c), the flat segments for XM , M Y for the first two dispersion curves (solid)for the double-sided plate, and the first dispersion curve for the mass-spring resonators with N = 1(dashed), are of interest. These flat sections correspond to isofrequency contours with sharp cornersthat bring saddle points on the dispersion surfaces. It is well known from the photonic crystalliterature, see, for example, Joannopoulos et al. [42], that this anisotropy gives rise to a numberof interesting wave effects. Following the method of Zengerle [43], one uses wave-vector diagrams,consisting of isofrequency contours for both the platonic crystal and the ambient medium (theunstructured part of the plate here), to investigate wave phenomena in planar waveguides. Thistechnique is also outlined by [42] for photonic crystals in their chapter 10, and was employed recentlyby [11] for platonic crystals. The key point is that the predicted direction of propagation for thegroup velocity of refracted waves into the platonic crystal is perpendicular to the isofrequencycontours, and in direction of increasing frequency.The sharp corners joining straight branches of constant β -contours, which are illustrated infigure 13(a), are significant. Small changes in either the angle of incidence, or the frequency, of theincoming waves, thereby switching from one side of the sharp corner to the other on the isofrequencydiagram, predict a strong modification of the direction of the refractive group velocity.We illustrate an example of negative refraction for the double-sided plate (DSP) in figure 13,where we select the value of ˜ β = 1 . β values for the relatively flat,although slightly increasing, branches of the second band in figure 6(c) (solid) i.e. XM and M Y .We choose the oblique angle of incidence ψ = π/
4, which was used to demonstrate the neutralitycondition in figure 6, for a semi-infinite array of mass-spring resonators attached to both faces of aplate (DSP). The mass and stiffness parameters are the same as those considered for that previousexample, ˜ m red = 1 . , ˜ c red = 1 . β produces significantly different propagationresults.We consider an array of 50 gratings positioned parallel to the y -axis, with period d y = √ d x = 1 .
0. The real part of the total displacement field is shown in figure 13(b),26 ����������� ��������� �� �� - - - - - (a) (b) Figure 13: Semi-infinite rectangular array ( ξ = √
2) of mass-spring resonators on both faces ofthe plate with ˜ m red = 1 . , ˜ c red = 1 . ψ = π/
4, ˜ β = 1 . The platonic crystals featured in this paper display several Dirac-like points, some of which areillustrated in figures 3(d-f) for, respectively, square arrays of pins and point masses, and the rectan-gular array of Winkler-type masses. Here in figure 14 we consider the rectangular arrays ( ξ = √ (a) (b) (c) MX DP
Figure 14: Dispersion surfaces for rectangular array of point scatterers with ξ = √ m = 1 . X and M in part (b) indicated by arrows. TheDirac point at ( π/ , β ≈ .
724 is labelled by DP .of (a) pins and (b) point masses. Two Dirac-like points for the latter case are labelled at X and M of the irreducible Brillouin zone in figure 14 (b), and are also indicated by the arrows in the27orresponding band diagram in figure 14(c). The triple degeneracy, where the two Dirac-like conesare joined by another locally flat surface passing through, is clearly evident in figure 14(c).As discussed by Colquitt et al. (2016) for an elastic lattice, Dirac cones are often connected byrelatively narrow flat regions on the dispersion surfaces, which the authors term “Dirac bridges”.Dirac bridges possess resonances where the dispersion surfaces are locally parabolic, and give riseto highly localised unidirectional wave propagation. In this section, we consider one such regimefor the point masses in the vicinity of a Dirac point at ( π/ , β ≈ . DP in figure 14), and the two Dirac-like points highlightedin figures 14(b,c). However, we observe additional steeply increasing sections of the third surface infigure 14(b) for the case of point masses, which replace the flat parabolic profile parallel to k x = 0for the pins for the second surface in figure 14(a).We investigate a semi-infinite rectangular array of 500 gratings of point masses with ˜ m = 1 . ξ = √ β = 4 .
60. The third dispersion surface for the corresponding infinite system is shownin figures 15(a, b) by, respectively, isofrequency contours and the surface itself. With reference tothe ˜ β = 4 .
60 contour of figure 15(a), the parameter setting of ψ = 0 .
11 (with associated Blochparameter k y = β sin ψ = 0 . y -direction) is selected to support a refracted wavedirected parallel to the k y -axis. Recall that since we are considering a finite array of 500 gratings,the information we obtain from the infinite doubly periodic system is only an approximate guidefor the design choices of ψ and k y for the corresponding finite system.The designed system displays the interfacial localisation for the point masses in figure 15(c);the preferred direction for the group velocity of the resultant refracted wave is perpendicular tothe isofrequency contour, and in the direction of increasing frequency, i.e. parallel to the k y -axis(indicated by an arrow in part (a) of figure 15). In contrast, for a slight increase of ψ , and k y accordingly, the point of interest would move to the other side of the corner, parallel to the k y -axis.The predicted direction would then be approximately parallel to the k x -axis, and into the periodicpart of the plate. We note that examples for interfacial localisation are easier to find for pointmasses rather than mass-spring resonators, which possess internal resonances at nearby frequencies˜ β ∗ = (˜ c/ ˜ m ) / . The observation of interfacial localisation, illustrated in figure 15, is linked to the analysis ofthe dispersion surfaces and stationary points of a certain type. Special attention was given to“parabolic” regimes, i.e. locally parabolic dispersion surfaces, which correspond to a unidrectionallocalisation of waveforms. Here, we offer an alternative viewpoint, based on the analysis of thehomogeneous equation ˆ U − = ˆ U + K , (73)where the external forcing term is absent (compare with equation (18) for the general case). In thering of analyticity, the kernel can be written as K = K + K − , and the factorised equation (73) takesthe form: ˆ U − / K − = ˆ U + K + = const (cid:54) = 0 . (74)28 b)(a)(c) (d) Figure 15: Semi-infinite rectangular array of point masses with ˜ m = 1 . ξ = √
2. (a) Isofrequencycontours for the third surface. The contour ˜ β = 4 .
60 is highlighted in bold. (b) Third dispersionsurface. Real part of scattered field for ˜ β = 4 .
60 for (c) ψ = 0 .
11, ˜ k y = k y d x = 0 . K − and K + versus k x d x ∈ [0 , . U + and ˆ U − represent z − transforms of the displacements on the right and the left half-planes,respectively, as defined in (19), and the kernel factors for a general Φ are analogous to those givenfor the mass-spring resonators in equation (35):˜ K + ( ˜ β ; ˜ z ) = ˜ Φ ( ˜ β, ˜ m, ˜ c ) ˜ˆ G + ( ˜ β ; ˜ z ); ˜ K − ( ˜ β ; ˜ z ) = ˜ˆ G − ( ˜ β ; ˜ z ) − Φ ( ˜ β, ˜ m, ˜ c ) ˜ˆ G + ( ˜ β ; ˜ z ) . (75)For the case of point masses illustrated in figures 15(a-c), ˜ Φ = ˜ m ˜ β in (75), and from (74)we seek a solution corresponding to localised interfacial waveforms such that K − vanishes, whilst K + remains finite. In turn, for this set of parameters the quantity ˆ U − also vanishes, whereas anon-trivial solution ˆ U + represents the interfacial waveform within the grating stack, as illustratedin figure 15(c). In figure 15(d), we verify that the parameters ˜ β = 4 . k x ≈ . , k y ≈ . K − and ˜ K + are plotted, on the same figure 15(d), versus k x d x ∈ [0 , .
5] for the vicinity of theestimate for k x d x denoted by the position of the arrow in figure 15(a). For k x d x = 0 .
99, the former29unction does indeed have a local minimum ≈
0, whilst the latter function is finite and nonzero forthe same k x d x . The ability to control flexural wave propagation is important in numerous practical engineeringstructures such as bridges, aircraft wings and buildings, many of whose components may be mod-elled as structured elastic plates. In this article, we have modelled a collection of potential platoniccrystals, where a Kirchhoff-Love plate is structured with a semi-infinite array of point scatterers,including concentrated point masses, mass-spring resonators positioned on either, or both, faces ofthe plate and Winkler-sprung masses. We have considered semi-infinite rectangular arrays, definedby periodicities d x , d y , but the methods are equally applicable for alternative geometries of theplatonic crystal such as triangular or hexagonal lattices.The introduction of resonators, and their mass and spring stiffness parameters, significantlybroadens the frequency range that supports interesting wave effects, compared with the simplifiedpinned plate model [11]. Here, we have shown examples of perfect transmission and negativerefraction for various mass-spring resonator configurations at frequencies that would fall into thezero-frequency stop band imposed by the rigid pins.A discrete Wiener-Hopf method was employed to determine the scattered and total displacementfields for a plane wave incident at a specified angle. The characteristic feature of each of the resultingfunctional equations is the kernel which, for all of the cases featured here, incorporates a doublyquasi-periodic Green’s function: K ( z ) = Φ ( ω, m, c ) ˆ G ( z ) − , (76)and a function Φ ( ω, m, c ) of frequency, mass and stiffness determined by which of the four featuredsystems is being analysed. By identifying and deriving conditions for specific frequency regimes ofthe kernel function, we predict and demonstrate various scattering effects. In this article, we haveillustrated examples of reflection, dynamic neutrality or perfect transmission, interfacial localisationand waveguide transmission. For certain regimes, we have also established a direct connectionbetween alternative scatterers, including a condition for dynamic neutrality that occurs at thesame frequency, shown in figure 6, for a plate with mass-spring resonators attached to both facesof the plate and Winkler-sprung masses.The important observation that the semi-infinite system’s kernel function is directly connectedwith the dispersion relation for the infinite doubly periodic platonic crystals, means that a thoroughunderstanding of the Bloch-Floquet analysis provides great insight. Moreover, an understanding ofthe kernel function is sufficient to design the system for predicting and illustrating wave effects ofinterest, avoiding the necessity for lengthy computations for the evaluation of the explicit Wiener-Hopf solution. In section 4.2.1, we introduce an alternative approach to predicting interfaciallocalisation frequency regimes, based on solving the homogeneous functional equation. This is aninherently interesting problem in itself, and we illustrate its viability with the example of figure 15obtained using wave-vector diagram analysis.The numerous wave effects demonstrated here suggest that these semi-infinite platonic crystalshave potential applications in the control and guiding of flexural waves in structures comprisingthin plates. We have presented an overview of an assortment of practically interesting designs forsemi-infinite platonic metamaterials. Any one of these models could be studied in its own right,with its parameters tuned to improve the resolution of the perfect transmission and interfacial30ocalisation illustrated here. These effects are inherited by finite cluster subsets of the semi-infinitemodel, which could be used as a basis for the design and manufacture of semi-infinite platonicmetamaterials. Acknowledgements
All of the authors thank the EPSRC (UK) for their support through the Programme GrantEP/L024926/1. SGH thanks Dr G. Carta and Dr D. J. Colquitt for valuable discussions about theuse of finite element software packages.
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