Controlling flexural waves in semi-infinite platonic crystals
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 resonators attached to the top surface of the flexural plate and their limiting case of concentrated point masses. We also analyse the special case of resonators attached to opposite faces of the plate. The problem is reduced to a functional equation of the Wiener-Hopf type, whose kernel varies with the type of scatterer considered. A novel approach, which stems from the direct connection between the kernel function of the semi-infinite system and the quasi-periodic Green’s functions for corresponding infinite systems, is used to identify special frequency regimes. We thereby demonstrate dynamically anisotropic wave effects in semi-infinite platonic crystals, with particular attention paid to designing systems to exhibit dynamic neutrality (perfect transmission) and localisation close to the structured interface.
Since the 1980’s, there has been substantial attention devoted to wave interaction with periodic structures leading to the recent surge of interest in designing metamaterials and micro-structured systems that are able to generate effects unattainable with natural media. These are artificially engineered super-lattice materials, designed with periodic arrays of sub-wavelength unit cells; their major concept is that their function is defined through structure. Many of the ideas and techniques originate in electromagnetism and optics but are now filtering into other systems such as the Kirchhoff-Love plate equations for flexural waves. This analogue of photonic crystals, labelled as platonics by McPhedran et al. , features many of the typical anisotropic effects from photonics such as ultra-refraction, negative refraction and Dirac-like cones, see –, amongst others. Recently, structured plates have also been both modelled, and designed, to demonstrate the capability for cloaking applications –.
In this article, we consider a semi-infinite platonic crystal where, by patterning one half of an infinite Kirchhoff-Love plate with a semi-infinite rectangular array of point scatterers, the leading grating acts as an interface between the homogeneous and structured parts of the plate. Haslinger et al.  analysed the case of pinned points, and highlighted effects including dynamic neutrality in the vicinity of Dirac-like points on the dispersion surfaces for the corresponding infinite doubly periodic system, and interfacial localisation, by which waves propagate along the interface. An interesting feature of the discrete Wiener-Hopf method of solution was the direct connection between the kernel function and the doubly quasi-periodic Green’s function, zeros of which correspond to the aforementioned dispersion surfaces.
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 - and - directions, as illustrated in figure 1(a), and the one-dimensional “grating”, with the periodic element confined to the -axis, as illustrated in 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 .
Case 1: point masses, characterised by mass
Case 2: multiple point mass-spring resonators attached to the top surface of the plate, characterised by masses , stiffnesses ;
Case 3: multiple mass-spring resonators attached to both faces of the plate
Case 4: point masses with Winkler foundation (see Biot ), characterised by mass , stiffness .
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 new attributes to the model, most notably an assortment of propagation effects at low frequencies; in contrast, the case of pinned points possesses a complete band gap for low frequency vibrations up to a finite calculable value. The important limiting case of for case 2, (see figure 1b), or equivalently, for case 4 in figure 1(d), retrieves the periodic array of unsprung point masses. The infinite doubly periodic system of point masses has been discussed by Poulton et al , who provided dispersion band diagrams and explicit formulae and illustrations for defect and waveguide modes.
Evans & Porter  considered one-dimensional periodic arrays of sprung point masses (case 4 in figure 1(d) for ), including the limiting case of unsprung point masses. The contributions by Xiao et al.  and Torrent et al.  discussed infinite doubly periodic arrays of point mass-spring resonators, as depicted in figure 1(b); the former for a rectangular array, and the latter for a honeycomb, graphene-like system. The authors provided dispersion relations and diagrams for the platonic crystals, and analysed the tuning of band-gaps and the association of Dirac points with the control of the propagation of flexural waves in thin plates. Examples using finite structures were also illustrated by both , .
In this article, we present the first analysis of semi-infinite arrays for the variety of point scatterers illustrated in figure 1. We demonstrate interfacial localisation, dynamic neutrality and negative 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-infinite line is easily recovered by replacing a quasi-periodic grating Green’s function with the single source Green’s function for the biharmonic operator. A discrete Wiener-Hopf method, incorporating the -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, – 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 the kernel in a general form, and by identifying and studying special frequency regimes, we present the conditions required to predict and observe specific wave effects. This novel approach is used to design structured systems to control the propagation of the flexural waves, without evaluating the explicit Wiener-Hopf solutions, bypassing unnecessary computational challenges. We derive expressions 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 both cases 3 and 4 shown in figures 1(c,d).
In conjunction with the Wiener-Hopf expressions for each of the cases considered, we also derive dispersion relations, and illustrate dispersion surfaces and band diagrams. Of particular importance are stop and pass band boundaries, standing wave frequencies (flat bands/low group velocity) and the neighbourhoods of Dirac-like points, which support dynamic neutrality effects. The concept of Dirac cone dispersion originates in topological insulators and has more recently been transferred into photonics (see for example –). It is associated with adjacent bands, for which electrons obey the Schrödinger equation, that meet at a single point called the Dirac point.
Typically connected with hexagonal and triangular geometries in systems governed by Maxwell’s equations, and most notably associated with the electronic transport properties of graphene (see, for example, Castro Neto et al. ), analogous Dirac and Dirac-like points have recently been displayed in phononic and platonic crystals (see for example –, ). The presence of Dirac cones is generally associated with the symmetries of the system through its geometry. When two perfect 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 the microstructure of the crystal.
In platonic crystals, the analogous points generally possess a triple degeneracy, where the two Dirac-like cones are joined by another flat surface passing through what is known as a Dirac-like point. This is analogous to the terminology adopted by Mei et al.  in photonics and phononics, where the existence of linear dispersions near the point of the reciprocal lattice for the square array is the result of “accidental” degeneracy of a doubly degenerate mode (the Dirac point, without the additional mode) and a single mode. Sometimes known as a “perturbed” Dirac point, the accidental degeneracy does not arise purely from the lattice symmetry, as for a Dirac point, but from a perturbation of the physical parameters; in this setting, from the fourth order biharmonic operator. We identify Dirac-like points to illustrate neutrality, and “Dirac bridges” (Colquitt et al. ) to predict unidirectional wave propagation. We also use dispersion surfaces and the accompanying isofrequency contour diagrams to identify frequencies supporting negative refraction.
The paper is arranged as follows: In section 2, we formulate the problem for the two-dimensional rectangular array, using the discrete Wiener-Hopf technique; the special case of a semi-infinite grating is also identified. We provide governing equations, and Wiener-Hopf equations for all cases illustrated in figure 1. In section 3, we analyse these equations, highlighting the conditions required for frequency regimes to support reflection, transmission and dynamic neutrality, which we illustrate with examples. We also demonstrate Rayleigh-Bloch-like waves for the semi-infinite line of scatterers. In section 4, we present special examples of waveguide transmission, whereby the structured system is designed to specifically exhibit negative refraction and interfacial localisation effects. Concluding remarks are drawn together in section 5.
A thin Kirchhoff-Love plate comprises a two-dimensional semi-infinite array of point scatterers defined by position vectors , where are the spacings in the - and -directions respectively, and are integers, as illustrated in figure 2(a).
It is natural to consider the system as a semi-infinite array of gratings aligned parallel to the -axis. By replacing each of these gratings with a single point scatterer lying on the -axis, we recover the one-dimensional case of a single semi-infinite grating, as illustrated in figure 2(b). The plate is subjected to a forcing in the form of a plane wave, incident at an angle to the -axis.
We assume time-harmonic vibrations of the Kirchhoff-Love plate, and define equations for the amplitude of the total out-of-plane displacement field , with , which can be expressed as the sum of the incident and scattered fields:
We express the general governing equation for in the form:
Here, is a function of radial frequency , and the physical parameters of mass and stiffness that define the various mass-spring resonator models shown in figure 1. The functional forms of the various are provided later in section 2.2. The characteristic physical parameters for the plate are density per unit volume , thickness and flexural rigidity (involving Young’s modulus and the Poisson ratio ), and we also adopt the use of the spectral parameter , which has the dimension of a wavenumber:
Note that the Kirchhoff-Love model incorporates the fourth order biharmonic operator, and gives an excellent approximation of the full linear elasticity equations for a sufficiently small value of the ratio , where denotes the wavelength of the flexural vibrations of the plate :
2.1 Governing equations and reduction to a functional equation
Assuming isotropic scattering, we express the scattered field in the form of a sum of biharmonic Green’s functions:
where is the point source Green’s function satisfying the equation:
Note that for the one-dimensional case of a semi-infinite line of scatterers placed on the -axis, the sum over in (5) is absent. All derivations for the two-dimensional array given below are applicable to the special case of the semi-infinite grating, with appropriate adjustments to the sums and Green’s functions. Referring to equations (1)-(2), we may express the total field as
In particular, at , , we have the linear algebraic system
Recalling that we consider an incident plane wave, we define in the form:
where is the wave vector. Since the scatterers are infinitely periodic in the -direction, we impose Bloch-Floquet conditions for in the -direction. Hence
Thus, recalling the RHS of equation (8), we have
Denoting , this simplifies to
where the sum on the right is precisely the quasi-periodic grating Green’s function; we shall use the notation
when substituting back into equation (8):
Here, we use
Thus, the algebraic system becomes
where the integer denotes the -position of a grating of scatterers parallel to the -axis and centred on the -axis, and the incident field is in the form . Equivalently, we may write
where we replace , with for ease of notation, and represents the forcing term.
The semi-infinite sum indicates that the discrete Wiener-Hopf method is suitable, see Noble , where the application to continuum discrete problems , such as gratings, is presented as an exercise (4.10, p.173-4) in . After employing the -transform, we obtain
Letting the index , we derive a functional equation of the Wiener-Hopf type :
and we have introduced the notation to represent the function
where is the quasi-periodic grating Green’s function given by equation (13). We note that for , i.e. , we recover the doubly quasi-periodic Green’s function:
2.2 Governing equations for various point scatterers
differs from that of the simpler pinned semi-infinite platonic crystal analysed in  because of the change in boundary condition for the point scatterers. Whereas the rigid pins impose zero flexural displacement at , the nonzero condition for the scatterers considered here introduce additional terms in (22). The function , for , determines the characteristic features specific to each model, and also those common to the different cases. We now present the expressions for the various , which we go on to explain and derive, where necessary, for each case in turn.
Case 1: Point masses:
Case 2: Multiple mass-spring resonators on the top surface of the plate:
Case 3: Multiple mass-spring resonators on both faces of plate:
Case 4: Winkler foundation point masses:
Case 1: Point masses
The simplest type of point scatterer is the rigid pin, defined as the limiting case of the radius of a clamped hole tending to zero. There is a large body of literature covering various problems incorporating this boundary condition. A selection of relevant papers include Movchan et al. , Evans & Porter , , Antonakakis & Craster , , Haslinger et al. . A logical extension is to replace the pins with concentrated point masses of mass , introducing an additional inertial term, and hence non-zero displacement at the point scatterer. We include a schematic diagram in figures 1(a) and 2(b). The governing equation for a semi-infinite half-plane of point masses, following equation (2), is
The total flexural displacement field , of the form (16), is given by
and the corresponding Wiener-Hopf-type functional equation is
Case 2: Multiple mass-spring resonators
We attach mass-spring resonators at each point . Each resonator consists of point masses attached to springs. For the most general case, the finite number of masses are connected by springs of stiffness (see figure 1(b)). For the sake of simplicity, we derive the governing equations, and their reduction to functional Wiener-Hopf type equations, for the cases and , but the procedure for arbitrary is a simple extension.
Single mass-spring resonator, N=1
We assume a semi-infinite rectangular array of simple resonators consisting of point masses attached with springs to the plate at points shown in figure 2(a), with the parameters illustrated in figure 1(b). We assume uniform mass and uniform stiffness , and negligible effect of gravity. We derive the equation for , and the accompanying governing and discrete Wiener-Hopf expressions, by applying Newton’s 2nd law and Hooke’s law for an arbitrary scatterer placed at . We use the fact that the quasi-periodic Green’s function (13) in equations (15)-(20) accounts for all scatterers in a grating parallel to the -axis, centred at .
We denote the flexural displacement of the plate at by . The transverse displacement of the mass is denoted by , with the forces applied to the plate by the spring, of stiffness , given by , and to the connected mass by the spring, as . The equation of motion for the sprung mass is written in the form:
Transverse displacements and are evaluated with respect to , but for the sake of simplicity, we adopt the abbreviated notation with the subscript . We write in terms of using (26), and referring to equations (2),(16) and recalling that the flexural rigidity of the plate is , we have
Note that in the limit as , equation (27) tends towards the equation for unsprung mass-loaded points (24), and from (26); the flexural vibrations of the plate and masses are identical for infinite stiffness, which is physically consistent with infinitely stiff springs. As , the coefficient multiplying the sum tends to , and this may be interpreted physically as the plate being attached to a rigid foundation with springs of stiffness .
The discrete Wiener-Hopf functional equation is obtained by substituting the expression for into equation (18):
Recalling equation (22), the kernel for case 2, is given by
where we introduce the notation as the kernel for the case of rigid pins, see .
Observing that a similar expression follows for the limit case of point masses (25), this connection with the pinned case enables us to employ a similar kernel factorization. First, we rewrite (30) in terms of the dimensionless parameters:
where we introduce a length scale determined by the periodicity of the system , which for the sake of simplicity is taken to be throughout this article. We also introduce non-dimensional versions of Green’s functions, and their arguments, which possess the dimension of owing to the factor :
We express (33) as
noting that the factorization obviously also applies to the dimensional form of the kernel. Explicitly, we have
where the reciprocal of the function is a function.
Multiple mass-spring resonators, N=2
As for the case , we derive the equation for for an arbitrary scatterer placed at , using Newton’s 2nd law and Hooke’s law. In general, the transverse displacement of each mass is , . For , the equations of motion are given by
The normalised force acting on the plate at , in terms of the out-of-plane displacement, is given by:
where the reciprocal of the plate’s flexural rigidity is the normalisation factor. Thus, referring to equation (27) we may write the total flexural displacement amplitude at as
We eliminate from (2.2) and derive the expression for in terms of only:
Hence, we rewrite equation (38) in the form,
As in previous cases, we employ the -transform to obtain the discrete Wiener-Hopf equation:
Case 3: Multiple mass-spring resonators attached to both faces of plate
We now consider an extension of section 2.2.2 by attaching mass-spring resonators on opposite faces of the plate at the same point of the array depicted in figure 2. This system is illustrated for the case of mass-spring resonators in figure 1(c). For introducing the model, we analyse the simplest case here; two masses with associated spring stiffnesses , with the index being odd for resonators attached to the top surface, and even for the bottom surface, as illustrated in figure 1(c). The derivations are similar to the previous sections with the flexural displacement at an arbitrary defect point given by and the transverse displacements of the masses given by . The equation of motion of the resonator mass at a single array point is given by
where we have used Hooke’s law for the right-hand side. Recalling the general expression for the total flexural displacement field of the plate at the point , we write
where the forces are given by
Similar to case 2, , we derive the governing equation in the form
Employing the -transform in the standard way, the accompanying Wiener-Hopf representation is
Case 4: point masses with Winkler-type foundation
An alternative model for adding mass-spring resonators is shown in figure 1(d), where point masses are embedded within the plate, and additional springs, attached to a fixed foundation, are added below. Referring to equation (26) for case 2, , we obtain a similar equation, except that here the displacement at the point of attachment to the fixed foundation , is zero. Hence,
and the solution for the total flexural amplitude at is
We note that with this model, taking the limit as recovers the case of concentrated point masses (case 1), in contrast to the model for case 2, , where retrieved the limiting case of point masses. Similarly, the Wiener-Hopf equation is easily deduced from the general equation (18):
with the kernel function defined by
3 Analysis of kernel functions: reflection, transmission and dynamic neutrality
We identify three important frequency regimes using the kernel equation (22): reflection, transmission and dynamic neutrality. Special cases of waveguide transmission including negative refraction and interfacial localisation are illustrated in the subsequent section 4. The five Wiener-Hopf expressions (25), (29), (41), (47) and (50) are characterized by their respective kernels, which all take the general form
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 tends to 0, i.e. . Referring to the general Wiener-Hopf equation (18), the first two cases infer that is respectively very small or very large; the physical interpretation of is the amplitude of scattering within the platonic crystal. Thus, small indicates reflection (blocking), and large indicates enhanced transmission, which is of particular interest for a single line of scatterers since it manifests in the form of Rayleigh-Bloch-like modes propagating along the grating itself (see, for example, Evans & Porter  and Colquitt et al.  for related problems). For , the general expression (18) tells us that in the limit,
Recalling that represents the incident field, and , the total field, we may interpret this regime as perfect transmission or dynamic neutrality; the wave propagation is unimpeded by the microstructure of the platonic crystal, a phenomenon often associated with the vicinity of Dirac or Dirac-like points . Summarising the three regimes, we have
dynamic neutrality (perfect transmission) .
By studying the Wiener-Hopf equation and its kernel function, we are able to derive conditions for observing wave effects for the different types of point scatterers. For example, it is instructive to compare the representations for point masses attached to a Winkler-type foundation (50),(51) with the equivalent expressions for the mass-spring resonators attached to the top of the plate given by (29),(30). Both expressions incorporate the term , which defines the resonance frequency of the individual mass-spring resonators:
Crucially, however, the kernel functions differ in that this term is in the numerator for the Winkler case, 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 regime of reflection () for the mass-spring resonator case and vice versa.
Referring to the general equation (52), reflection (blocking) is predicted for frequency regimes where either or , or both functions together, blow up. We recall that the kernel function is precisely for the case of a semi-infinite array of rigid pins analysed by , and that for , with and the Bloch vector for a doubly periodic system, is a doubly quasi-periodic Green’s function. Much has been written about this Green’s function in the literature; see for example, McPhedran et al. , McPhedran et al. , Poulton et al. . A very important property is that its zeros correspond to the dispersion relation for the infinite doubly periodic system of rigid pins, which possesses a complete band gap for low frequency vibrations up to a finite calculable value. Here we express (21) in the form:
where , are lattice sums defined over the periodic array of point scatterers in the following way:
We also note that the lattice sum over the Hankel functions may be written in the form
The lattice sums , are only conditionally convergent, and require an appropriate method of accelerated convergence for numerical computations. We adopt the same triply integrated expressions originally used by Movchan et al. , and more recently by . The dispersion relation for the doubly periodic pinned array is then given by
and has real solutions.
The direct connection between the kernel function for the semi-infinite array of pins and the dispersion relation for the infinite doubly periodic array enables one to identify frequency regimes for reflection and transmission of the incoming plane waves. Similarly for the point scatterers featured in this article, which importantly do not impose zero displacement clamping conditions, the zeros and singularities of the kernel function give us information about, respectively, transmission and reflection, but the kernel (52) now depends on more than .
The singularities of still indicate regimes of stop-band behaviour, but there is additional reflection behaviour determined by becoming very large. This is evident for the simple mass-spring resonators with . The kernel is given by (30) where blows up for the frequency corresponding to the resonance of the individual mass-spring resonators, . We would therefore expect to see reflection of incident waves for close to this resonant frequency , and stop bands in the corresponding dispersion diagrams (arising for zeros of the kernel) for the mass-spring resonators.
In figures 3, 4 we illustrate reflection (blocking) for the various systems of figures 1(a-d). We present results for two rectangular arrays, the special case of the square array with , and the rectangle with aspect ratio . We demonstrate blocking for the square arrays of 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 a semi-infinite rectangular array with for Winkler-type sprung masses, and the corresponding dispersion surfaces are illustrated in part (f). In figure 4, we consider the same for mass-spring resonators with 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 Brillouin zone is assumed to be the rectangle , with , . Note that the dispersion surfaces correspond to zeros of the kernel function, so represent the regime , where the flexural waves propagate through the periodic array, which we discuss in more detail in the next section.