Wave scattering on lattice structures involving an array of cracks

Abstract

Scattering of waves as a result of a vertical array of equally spaced cracks on a square lattice is studied. The convenience of Floquet periodicity reduces the study to that of scattering of a specific wave-mode from a single crack in a waveguide. The discrete Green’s function, for the waveguide, is used to obtain the semi-analytical solution for the scattering problem in the case of finite cracks whereas the limiting case of semi-infinite cracks is tackled by an application of the Wiener–Hopf technique. Reflectance and transmittance of such an array of cracks, in terms of incident wave parameters, is analysed. Potential applications include construction of tunable atomic-scale interfaces to control energy transmission at different frequencies.

1. Introduction

Multiple scattering [1] has been researched for more than a century and continues to pose interesting questions, while simultaneously finding applications (e.g. [2], etc.). In the context of the mechanics of solids, the presence of defects, such as cracks, grooves, holes etc. [2,3], leads to scattering of elastic waves. One of the simplest cases occurs for scattering of time-harmonic anti-plane shear waves as it often allows an analytical investigation [3], typically involving the two-dimensional Helmholtz equation and the prescription of Dirichlet or Neumann conditions on a certain boundary. The same equation also occurs in special situations dealing with acoustic and electromagnetic waves. Recall that the scattering of H- or E-polarized electromagnetic waves by an infinite array of parallel plates was originally formulated and solved by Carlson and Heins [4–6]. In a mechanical framework too, such problems have been studied (see [7–10], and references therein); for instance, scattering due to an array of cracks.

In recent years, with advancements in technology, the size of structures has been reduced to a few micrometres or nanometres. In a simplified setting, such structures can be modelled using an established discrete framework [11–14]; in fact, some primitive aspects of such models can be traced back to Newton and Hamilton. The discrete models have been extensively used to study brittle fractures [11,15–18], as well as discrete scattering [19–23] in different geometries [24–26]. In this framework, a discrete analogue of the boundary conditions [19,21] in the continuous case, depending on the nature of the defects, needs to be invoked. For example, a crack [19,20] is modelled by assuming broken bonds between two consecutive rows [11,16].

The present article follows the work of Carlson and Heins [4–6], in the arena of discrete models, as wave scattering due to finite as well as semi-infinite cracks is investigated; the latter employs the method of Wiener & Hopf [27]. Figure 1a illustrates a scenario when the finite cracks are ‘long enough’, so that the analysis of semi-infinite cracks is relevant from the viewpoint of transmission analysis; thus, the study of two parallel arrays of crack tips can be replaced by the study of a single array, i.e. semi-infinite cracks. In this paper, the transmission due to finite cracks as well as semi-infinite cracks is analysed where the latter admits elegant closed-form expressions [23], while the relevant expressions for the former rely on a subsidiary problem involving matrix inversion [20]. From the viewpoint of applications, wave transmission across a periodic arrangement of cracks finds potential relevance in radio frequency devices [29–31]. In certain systems [28], such phononic crystals enable tailorability, controllability and high conversion efficiency at large frequencies. Although reported transmission behaviour [28] (particularly, a narrow transmission band, shown in the schematic of figure 1b) is different from that analysed in the present work, the geometric arrangements of the cracks allows a favourable transmission/blocking of high-frequency lattice waves. In the context of thermal conduction in nano-structures [32,33], phonon transmission and reflection have been found to be appropriately controlled using a periodic arrangement of discrete scatterers (air holes, typically). The present study does not investigate any mechanisms enabling the transduction between photons and phonons or the details of phonon transport in monolayers [29–31,34,35].

Figure 1.

(a) An illustration of an infinite array of ‘long’ but finite cracks with interaction between the two arrays of crack tips. (b) A narrow band of frequency transmitted by the system proposed in [28]. (Online version in colour.)

In this article, §1 provides the lattice model. Section 2 formulates the scattering due to a single crack on a lattice ‘waveguide’ and presents the semi-analytical solution for a finite crack; the elementary details of the calculation of a suitable Green’s function are included. The exact solution for the semi-infinite case is given in §3, whereas §4 provides some key results and relevant discussion. Concluding remarks and three appendices appear at the end of article.

2. Square lattice model

Consider an infinite square lattice, with each particle of unit mass and an interaction with its four nearest neighbours through linearly elastic identical, massless bonds with a spring constant 1/b² [19] (figure 2a). Let $\mathbb{Z}$ denote the set of integers; let ${\mathbb{Z}}^2$ denote $\mathbb{Z}\times \mathbb{Z}$. The lattice contains an infinite array of finite-length cracks (of length ${\mathtt{N}}_b$, i.e. the number of broken bonds) with crack faces

$$\{(x,y)\in {\mathbb{Z}}^2|-{\mathtt{N}}_b+n\mathtt{M}\le x\le -1+n\mathtt{M},y=n\mathtt{N}+\mathrm{\aleph }\hspace{0.17em}\text{or}\hspace{0.17em}y=n\mathtt{N}+\mathrm{\aleph }-1,n\in \mathbb{Z}\},$$ 2.1

where ${\mathtt{N}}_b\in \mathbb{Z}$, $\mathtt{M}\in {\mathbb{Z}}^{+}$ and $\mathtt{N}\in {\mathbb{Z}}^{+}$ with (no loss of generality)

$$\mathrm{\aleph }=\frac{\mathtt{N}}{2}\hspace{0.17em}\text{when }\mathtt{N}\text{is even, whereas}\hspace{0.17em}\mathrm{\aleph }=\frac{(\mathtt{N}-1)}{2}\hspace{0.17em}\text{when }\mathtt{N}\text{is odd}.$$ 2.2

Suppose uⁱ describes the incident lattice wave with frequency $\omega$ and a wavenumber $\kappa$ which is incident on the lattice at an angle Θ ∈ ( − π, π]. The total displacement u^t of a particle satisfies the discrete Helmholtz equation [19]

$$\mathrm{\Delta }{u}_{x,y}^t+{\omega }^2{u}_{x,y}^t=0,$$ 2.3

away from the crack faces (2.1), with Δu_x,y = u_x+1,y + u_x−1,y + u_x,y+1 + u_x,y−1 − 4u_x,y. Specifically, it is assumed that (in terms of the incident angle Θ and incident wavenumber $\kappa$)

$$u_{x,y}^i=\text{A}\hspace{0.17em}\exp (-i\kappa x\cos \mathrm{\Theta }-i\kappa y\sin \mathrm{\Theta }-i{b}^{-1}\omega t),\hspace{0.17em}(x,y)\in {\mathbb{Z}}^2,$$ 2.4

where $\text{A}\in \mathbb{C}$. Throughout the article, $\mathbb{C}$ denotes the set of complex numbers; the real part, Re{z, of a complex number $z\in \mathbb{C}$ is denoted by z₁; its imaginary part, Im z, is denoted by z₂ (so that z = z₁ + iz₂); |z| denotes the modulus for $z\in \mathbb{C}$; and $\arg z$ denotes the argument for $z\in \mathbb{C}$. An illustrated in figure 2c, using the reference to the lattice structure shown in figure 2a,b, let $\alpha$ denote the angle of stagger of the crack array (2.1) relative to the x-axis, i.e.

$$\tan \alpha =\frac{\mathtt{N}}{\mathtt{M}},\text{whereas}\hspace{0.17em}\beta :=\frac{\pi }{2}-\alpha +\mathrm{\Theta };$$ 2.5

here, $\beta$ is the angle of incidence relative to the outward normal to the ‘line’ of the edges. Accordingly, the angle of incidence with respect to the upper side of the edge plane is $\alpha -\mathrm{\Theta }$.

Figure 2.

(a) Schematic of a lattice with an infinite array of finite, staggered cracks (with ${\mathtt{N}}_b=4$, $\mathtt{N}=5$, $\mathtt{M}=2$). (b) The shaded portion (within the red lines) corresponds to a lattice waveguide containing a single crack. (c) Typical angles. (Online version in colour.)

Substituting (2.4) in (2.3) (for an intact lattice), a relation for the triplet $\omega$, $\kappa$, Θ, called the dispersion relation (see [21]), is obtained; it is given by ${\omega }^2=4{\sin }^2(\frac{1}{2}\kappa \cos \mathrm{\Theta })+4{\sin }^2(\frac{1}{2}\kappa \sin \mathrm{\Theta }).$ For convenience, a vanishing amount of damping is introduced in the model as in [27]; therefore,

$$\omega ={\omega }_1+i{\omega }_2,{\omega }_2>0.$$ 2.6

Thus, $\kappa$ is also a complex, $\kappa ={\kappa }_1+i{\kappa }_2,$ ${\kappa }_2>0$ (typically, we consider ${\omega }_2\to 0^{+}$, ${\kappa }_2\to 0^{+}$). In this article, the scattered wave displacement u is defined as the difference between the total displacement u^t and the incident wave displacement uⁱ of an arbitrary particle on the lattice

$$u_{x,y}=u_{x,y}^t-u_{x,y}^i,(x,y)\in {\mathbb{Z}}^2.$$ 2.7

Following [19,20], for a particular crack (say, between $y=\mathrm{\aleph }+n\mathtt{N}$ and $y=\mathrm{\aleph }-1+n\mathtt{N}$, when $\mathrm{\aleph }$ is given by (2.2) while n is an arbitrary integer), the force in the vertical bonds connecting the particles at $y=\mathrm{\aleph }+n\mathtt{N}$ and $y=\mathrm{\aleph }-1+n\mathtt{N}$, ahead of the crack, is defined by

$$v_x^t(n):=\frac{-1}{b^2}{\mathtt{v}}_x^t(n),\hspace{0.17em}x\in \mathbb{Z}\setminus \{-1+n\mathtt{M},\dots ,-{\mathtt{N}}_b+n\mathtt{M}\},\text{with}\hspace{0.17em}{\mathtt{v}}_x^t(n):=u_{x,\mathrm{\aleph }+n\mathtt{N}}^t-u_{x,\mathrm{\aleph }-1+n\mathtt{N}}^t.$$ 2.8

The force on the particle at $(x,\mathrm{\aleph }+n\mathtt{N})$, $x\notin \{-1+n\mathtt{M},\dots ,-{\mathtt{N}}_b+n\mathtt{M}\}$, due to the vertical bond with $(x,\mathrm{\aleph }-1+n\mathtt{N})$, is $v_x^t(n)$, while the force at $(x,\mathrm{\aleph }-1+n\mathtt{N})$ due to the same bond is $-v_x^t(n)$. Since the crack is modelled by assuming broken bonds between two consecutive lattice rows, $v_x^t(n)=0,\hspace{0.17em}x\in \{-1+n\mathtt{M},\dots ,-{\mathtt{N}}_b+n\mathtt{M}\}.$ It is also useful to define the difference in the scattered displacements, $u_{x,\mathrm{\aleph }+n\mathtt{N}}$ and $u_{x,\mathrm{\aleph }-1+n\mathtt{N}}$, as

$${\mathtt{v}}_x(n)=u_{x,\mathrm{\aleph }+n\mathtt{N}}-u_{x,\mathrm{\aleph }-1+n\mathtt{N}},x\in \mathbb{Z}.$$ 2.9

In analogy with (2.8), a part of the force $v_x^t(n)$ occurs owing to the scattered displacement of particles at $(x,\mathrm{\aleph }+n\mathtt{N})$, $x\notin \{-1+n\mathtt{M},\dots ,-{\mathtt{N}}_b+n\mathtt{M}\}$; this is given by $v_x(n)=-(1/b^2){\mathtt{v}}_x(n)$. Let the incident crack opening displacement at $(x,\mathrm{\aleph })$ and $(x,\mathrm{\aleph }-1)$ be denoted by

$${\mathtt{v}}_x^i(n)=u_{x,\mathrm{\aleph }+n\mathtt{N}}^i-u_{x,\mathrm{\aleph }-1+n\mathtt{N}}^i.$$ 2.10

Then, $v_x^i(n)=-(1/b^2){\mathtt{v}}_x^i(n)$ can be interpreted as an ‘external force’ on particles at $(x,\mathrm{\aleph }+n\mathtt{N})$, $x\in \{-1+n\mathtt{M},\dots ,-{\mathtt{N}}_b+n\mathtt{M}\}$.

By virtue of (2.3), (2.4) and (2.7), the scattered wave field also satisfies the discrete Helmholtz equation (2.3) (replace u^t by u) away from the array of cracks. The displacement field on the crack face at $y=\mathrm{\aleph }+n\mathtt{N}$ and $y=\mathrm{\aleph }-1+n\mathtt{N}$ satisfies, respectively,

$$u_{x+1,\mathrm{\aleph }+n\mathtt{N}}+u_{x-1,\mathrm{\aleph }+n\mathtt{N}}+u_{x,\mathrm{\aleph }+1+n\mathtt{N}}+({\omega }^2-3)u_{x,\mathrm{\aleph }+n\mathtt{N}}=-{\mathtt{v}}_x^i(n)$$ 2.11

and

$$u_{x+1,\mathrm{\aleph }-1+n\mathtt{N}}+u_{x-1,\mathrm{\aleph }-1+n\mathtt{N}}+u_{x,\mathrm{\aleph }-2+n\mathtt{N}}+({\omega }^2-3)u_{x,\mathrm{\aleph }-1+n\mathtt{N}}={\mathtt{v}}_x^i(n),$$ 2.12

for $x\in \{-1+n\mathtt{M},\dots ,-{\mathtt{N}}_b+n\mathtt{M}\}$. Here, (2.11) and (2.12) can be interpreted as boundary conditions for (2.3). Then, using the definition of the scattered field u, (2.7), along with the definitions of ${\mathtt{v}}_x(n)$ and ${\mathtt{v}}_x^i(n)$ and the boundary conditions (2.11), (2.12), the linear difference equation [36] formally satisfied by the scattered displacement u is

$$\mathrm{\Delta }{u}_{x,y}+{\omega }^2{u}_{x,y}=-\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{l=-{\mathtt{N}}_b}^{-1}({\mathtt{v}}_l(n)+{\mathtt{v}}_l^i(n)){\delta }_{l+n\mathtt{M},x}({\delta }_{\mathrm{\aleph }+n\mathtt{N},y}-{\delta }_{\mathrm{\aleph }-1+n\mathtt{N},y}),$$ 2.13

where ${\{{\mathtt{v}}_l(n)\}}_{l=-{\mathtt{N}}_b,\dots ,-1;n\in \mathbb{Z}}$ are an infinite number of unknowns. Throughout this article, the symbol δ denotes the Kronecker delta so that δ_a,b equals 0 if a ≠ b while it equals 1 if a = b.

3. Reduction to lattice waveguide with ‘Floquet boundary’: Green’s function and solution for finite cracks

Since the array of finite-length cracks extends indefinitely, there is a periodicity induced into the system by virtue of the Floquet–Bloch theorem. This conveniently reduces the scattering problem to the study of scattering of the incident wave (2.4) by a single crack in a subset ${\mathcal{S}}_0$ (defined below) with $\mathtt{N}$ rows (see the shaded region of figure 2b). Suppose the region ${\mathcal{S}}_0$ corresponds to the crack n = 0 in (2.1). Henceforth, in the context of the symbols used for crack opening displacement, the (n) notation is dropped; (0) will be omitted in reference to ${\mathcal{S}}_0$. Thus, a shorter notation and Floquet–Bloch periodicity-based reduction allows a simplification from the system of equations (2.13) to the following equation:

$$\mathrm{\Delta }{u}_{x,y}+{\omega }^2{u}_{x,y}=-\sum _{l=-{\mathtt{N}}_b}^{-1}({\mathtt{v}}_l+{\mathtt{v}}_l^i){\delta }_{l,x}({\delta }_{\mathrm{\aleph },y}-{\delta }_{\mathrm{\aleph }-1,y}),(x,y)\in {\mathcal{S}}_0.$$ 3.1

Observe that the set ${\mathcal{S}}_0$ of lattice sites is infinite in the horizontal direction while it is confined in the vertical direction. We employ the natural notation ${\mathbb{Z}}_a^b$ for the set $\{a,a+1,\dots ,b\}$ ($\subset \mathbb{Z}$). Indeed, ${\cup }_{n\in \mathbb{Z}}{\mathcal{S}}_n={\mathbb{Z}}^2\text{ with }{\mathcal{S}}_n={\mathcal{S}}_0+n\mathtt{N}\widehat{\mathbf{\text{j}}}=\{(x,y+n\mathtt{N})\in {\mathbb{Z}}^2:x\in \mathbb{Z},y\in {\mathbb{Z}}_0^{\mathtt{N}-1}\}.$

The incident lattice wave (2.4), i.e.$u_{x,y}^i=\text{A}\exp (-i\kappa (x\cos \mathrm{\Theta }+y\sin \mathrm{\Theta }))$ in ${\mathcal{S}}_n$, i.e. at one set of $\mathtt{N}$ rows, in the lattice is related to another set ${\mathcal{S}}_{n+1}$ via

$$u_{x+\mathtt{M},y+\mathtt{N}}^i=\psi u_{x,y}^i,\text{where}\hspace{0.17em}\psi =\exp (-i\kappa (\mathtt{M}\cos \mathrm{\Theta }+\mathtt{N}\sin \mathrm{\Theta })).$$ 3.2

By the Floquet–Bloch theorem, the scattered wave field must satisfy the identical condition

$$u_{x+\mathtt{M},y+\mathtt{N}}=\psi u_{x,y}.$$ 3.3

From the perspective of the infinite square lattice, the formal definition of ${\mathcal{S}}_0$ is

$$\{(x,y)\in {\mathbb{Z}}^2|y\in {\mathbb{Z}}_0^{\mathtt{N}-1},u_{x+\mathtt{M},y+\mathtt{N}}=\psi u_{x,y}\}.$$ 3.4

Notably, ${\mathcal{S}}_0$ includes the inherent ‘Floquet’ periodic boundary conditions. The periodically repeating cell (as ${\mathcal{S}}_n$s are copies of ${\mathcal{S}}_0$) is the ‘waveguide’ mentioned earlier.

Classically, the wave field in a scattering problem can be written in terms of an appropriate Green’s function (e.g. [3]). It has been shown that using discrete Fourier transforms [20] a discrete Green’s function (following the traditional terminology) can also be used for the lattice wave scattering. In the present case, the discrete Green’s function $\mathcal{G}$ is sought for the lattice waveguide ${\mathcal{S}}_0$ and it satisfies a difference equation given by

$$\mathrm{\Delta }{\mathcal{G}}_{x,y}+{\omega }^2{\mathcal{G}}_{x,y}={\delta }_{x_0,x}{\delta }_{y_0,y},\hspace{0.17em}(x,y)\in {\mathcal{S}}_0,$$ 3.5

where it is assumed that a source is located at $(x_0,y_0)\in {\mathcal{S}}_0$. Because of (2.6), note that the Green’s function ${\mathcal{G}}_{x,y}\sim \exp (-{\kappa }_2|x|)$ as |x| → ∞. The Green’s function, which is the subject of the following, must satisfy the Floquet periodic boundary conditions of the waveguide. Thus, the difference equation (3.5) is subjected to the condition (using (3.2) and (3.3)) ${\mathcal{G}}_{x+\mathtt{M},y+\mathtt{N}}=\psi {\mathcal{G}}_{x,y},(x,y)\in {\mathcal{S}}_0.$ For the particles at the boundary rows, i.e. at y = 0 and $y=\mathtt{N}-1$, this leads to ${\mathcal{G}}_{x+\mathtt{M},\mathtt{N}}=\psi {\mathcal{G}}_{x,0},\hspace{0.17em}x\in \mathbb{Z},$ and ${\mathcal{G}}_{x+\mathtt{M},\mathtt{N}-1}=\psi {\mathcal{G}}_{x,-1},\hspace{0.17em}x\in \mathbb{Z},$ respectively. Using (3.5), the governing equation for a particle at the boundary of the waveguide (that is, y = 0 and $y=\mathtt{N}-1$, respectively) can be written as

$${\mathcal{G}}_{x+1,0}+{\mathcal{G}}_{x-1,0}+{\mathcal{G}}_{x,1}+{\psi }^{-1}{\mathcal{G}}_{x+\mathtt{M},\mathtt{N}-1}+({\omega }^2-4){\mathcal{G}}_{x,0}=0,\hspace{0.17em}x\in \mathbb{Z}$$ 3.6

and

$${\mathcal{G}}_{x+1,\mathtt{N}-1}+{\mathcal{G}}_{x-1,\mathtt{N}-1}+{\mathcal{G}}_{x,\mathtt{N}-2}+\psi {\mathcal{G}}_{x-\mathtt{M},0}+({\omega }^2-4){\mathcal{G}}_{x,\mathtt{N}-1}=0,\hspace{0.17em}x\in \mathbb{Z}.$$ 3.7

Suppose that the discrete Fourier transform of a sequence ${\{u_m\}}_{m\in \mathbb{Z}}$ is denoted by u^F and defined by $u^{\text{F}}(z)={\sum }_{m=-\mathrm{\infty }}^{+\mathrm{\infty }}{u}_m{z}^{-m}$. Using the discrete Fourier transform (see also [20,26]), the transformed Green’s function can be written as (suppressing z dependence for brevity)

$${\mathcal{G}}_y^{\text{F}}={\sum }_{x=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathcal{G}}_{x,y}{z}^{-x}.$$ 3.8

Based on ${\mathcal{G}}_{x,y}$ with |x| → ∞, the region of analyticity of the above Fourier transform [23] is an annulus ${\mathcal{A}}_g$ in the complex plane centred at the origin, which is given by ${\mathcal{A}}_g=\{z\in \mathbb{C}:\exp (-{\kappa }_2)<|z|<\exp \hspace{0.17em}{\kappa }_2\}.$ The application of the discrete Fourier transform (3.8) to (3.5) results in

$${\mathcal{G}}_y^{\text{F}}(\mathcal{H}+2)-({\mathcal{G}}_{y+1}^{\text{F}}+{\mathcal{G}}_{y-1}^{\text{F}})=-z^{-x_0}{\delta }_{y,y_0},\text{where}\hspace{0.17em}\mathcal{H}=2-z-z^{-1}-{\omega }^2.$$ 3.9

Similarly, application of the discrete Fourier transform (3.8) to the boundary conditions (3.6) and (3.7), respectively, yields

$${\mathcal{G}}_0^{\text{F}}(\mathcal{H}+2)-({\mathcal{G}}_1^{\text{F}}+{\psi }^{-1}{z}^{\mathtt{M}}{\mathcal{G}}_{\mathtt{N}-1}^{\text{F}})=0\text{and}{\mathcal{G}}_{\mathtt{N}-1}^{\text{F}}(\mathcal{H}+2)-({\mathcal{G}}_{\mathtt{N}-2}^{\text{F}}+\psi z^{-\mathtt{M}}{\mathcal{G}}_0^{\text{F}})=0.$$ 3.10

Since (3.9) is a non-homogeneous linear difference equation in y with coefficients independent of y, the solution can be written as [36] ${\mathcal{G}}_y^{\text{F}}={\mathcal{G}}_y^{Fh}+{\mathcal{G}}_y^{\text{F}nh},$ where ${\mathcal{G}}_y^{\text{F}h}$ is the solution to the homogeneous equation ${\mathcal{G}}_y^{\text{F}h}(\mathcal{H}+2)-({\mathcal{G}}_{y+1}^{\text{F}h}+{\mathcal{G}}_{y-1}^{\text{F}h})=0,$ with certain boundary conditions (stated below), and ${\mathcal{G}}_y^{\text{F}nh}$ is a particular solution of

$${\mathcal{G}}_y^{\text{F}nh}(\mathcal{H}+2)-({\mathcal{G}}_{y+1}^{\text{F}nh}+{\mathcal{G}}_{y-1}^{\text{F}nh})=-{\delta }_{y,y_0}{z}^{-x_0}.$$ 3.11

Using elementary calculus [37], a (particular) solution of (3.11) is found

$${\mathcal{G}}_y^{\text{F}nh}=G_0(z){\lambda }^{|y-y_0|}(z),z\in \mathcal{A},$$ 3.12

where [11,19,20]

$$\lambda (z):=\frac{\mathcal{r}(z)-\mathcal{h}(z)}{\mathcal{r}(z)+\mathcal{h}(z)},z\in \mathbb{C}\setminus \mathcal{B},\mathcal{h}(z):=\sqrt{\mathcal{H}(z)},\mathcal{r}(z):=\sqrt{\mathcal{H}(z)+4}.$$ 3.13

The square root function, $\sqrt{\cdot }$, has the branch cut from −∞ to 0. $\mathcal{B}$ denotes the union of branch cuts for λ, borne out of the chosen branch for $\mathcal{h}$ and $\mathcal{r}$ such that $|\lambda (z)|\le 1,z\in \mathbb{C}\setminus \mathcal{B}.$ In the above equations, the annulus $\mathcal{A}$ is given by

$$\mathcal{A}={\mathcal{A}}_g\cap {\mathcal{A}}_{\mathcal{L}},$$ 3.14

with ${\mathcal{A}}_{\mathcal{L}}$ being the annular region where $\mathcal{h}$ and $\mathcal{r}$ (and λ too) are analytic (for ${\mathcal{A}}_g$, see the sentence following (3.8)). The coefficient G₀ in (3.12) is determined by substituting the ansatz of ${\mathcal{G}}_y^{\text{F}nh}$ in (3.11), that is, $G_0{\lambda }^{|y-y_0|}(\mathcal{H}+2)-(G_0{\lambda }^{|y-y_0+1|}+G_0{\lambda }^{|y-y_0-1|})=-z^{-x_0}{\delta }_{y-y_0,0},$ for $z\in \mathcal{A}$, which leads to $G_0=-z^{-x_0}/(\mathcal{H}+2-2\lambda )$, so that a particular solution of the linear non-homogeneous difference equation (3.11) can be written as [36]

$${\mathcal{G}}_y^{\text{F}nh}=-\frac{z^{-x_0}{\lambda }^{|y-y_0|}}{\mathcal{H}+2-2\lambda },z\in \mathcal{A}.$$ 3.15

After substitution of the particular solution (3.15) of (3.9) in the boundary conditions (3.10) (for y = 0 and $y=\mathtt{N}-1$), we obtain the boundary conditions for the homogeneous solution ${\mathcal{G}}_y^{\text{F}h}$,

$${\mathcal{G}}_0^{\text{F}h}(\mathcal{H}+2)-({\mathcal{G}}_1^{\text{F}h}+z^{\mathtt{M}}{\psi }^{-1}{\mathcal{G}}_{\mathtt{N}-1}^{\text{F}h})=\frac{z^{-x_0}({\lambda }^{|y_0|}(\mathcal{H}+2)-{\lambda }^{|1-y_0|}-{\lambda }^{|\mathtt{N}-1-y_0|}{\psi }^{-1}{z}^{\mathtt{M}})}{\mathcal{H}+2-2\lambda }$$ 3.16

and

$${\mathcal{G}}_{\mathtt{N}-1}^{\text{F}h}(\mathcal{H}+2)-({\mathcal{G}}_{\mathtt{N}-2}^{\text{F}h}+\psi z^{-\mathtt{M}}{\mathcal{G}}_0^{\text{F}h})=\frac{z^{-x_0}({\lambda }^{|\mathtt{N}-1-y_0|}(\mathcal{H}+2)-{\lambda }^{|\mathtt{N}-2-y_0|}-\psi z^{-\mathtt{M}}{\lambda }^{|y_0|})}{\mathcal{H}+2-2\lambda },$$ 3.17

for $z\in \mathcal{A}$ (recall that $\mathcal{H}$ is defined by (3.9)); these conditions determine the unknown coefficients in the general solution for the homogeneous part. After elementary calculation, we find

$${\mathcal{G}}_y^{\text{F}}=-z^{-x_0}\frac{{\mathtt{U}}_{\mathtt{N}-|y-y_0|-1}(\vartheta )+{\mathtt{U}}_{|y_0-y|-1}(\vartheta ){(\psi z^{-\mathtt{M}})}^{\text{sign}(y-y_0)}}{2{\mathtt{T}}_{\mathtt{N}}(\vartheta )-(\psi z^{-\mathtt{M}}+{\psi }^{-1}{z}^{\mathtt{M}})},\text{where}\hspace{0.17em}\vartheta =1+\frac{1}{2}\mathcal{H}.$$ 3.18

It is emphasized that the numerator and denominator in (3.18) involve the Chebyshev polynomials, which are significant in the description of the wave propagation characteristics of lattice waveguides [37,38]. In particular, ${\mathtt{U}}_n$ denotes the Chebyshev polynomials of the second kind [39] defined by ${\mathtt{U}}_n(\vartheta ):=\sin ((n+1)\vartheta )/\sin \vartheta ,n\ge 0$, while ${\mathtt{T}}_n$ denotes the Chebyshev polynomials of the first kind [39], defined by ${\mathtt{T}}_n(\vartheta ):=\cos n\vartheta$.

The Green’s function in the lattice is obtained by inverse Fourier transform of (3.18), i.e.

$${\mathcal{G}}_{x,y}=\frac{1}{2\pi i}{\oint }_C{\mathcal{G}}_y^{\text{F}}(z)z^{x-1}\hspace{0.17em}\text{d}z,$$ 3.19

where C is chosen to be a closed contour that lies inside the annulus $\mathcal{A}$ (3.14). Owing to the vanishingly small imaginary part of the frequency and, hence, the wavenumber all the singularities of the integrand in (3.19) are either inside or outside the unit circle; that is, they are away from the contour (see [27]). Up to this point, our exposition completes the derivation of the discrete Green’s function. It seems that the denominator of the transformed Green’s function (3.18) represents the dispersion relation for a square lattice waveguide with Floquet–Bloch periodic boundaries. This is briefly discussed in appendix A.

The description of the scattered displacement field in terms of the Green’s function (3.19) now follows the well-known approach [13] (see also [20,22] for notation relevant to the manipulations presented below). In fact, the present problem is closely related to the scattering due to a finite crack in an infinite lattice and the description of the scattered field u in terms of the Green’s function has been discussed systematically in [20]. For additional clarity, ${\mathcal{G}}_{x,y;x_0,y_0}$ will be used instead of ${\mathcal{G}}_{x,y}$ below. Note that the expression (3.19) is the solution of equation (3.5), which has the source located at (x₀, y₀) (recall (3.5)).

Using (3.1) and (3.5), by inspection, it can be found that the scattered displacement field due to an infinite array of cracks in the lattice is given by

$$u_{x,y}=-\sum _{l=-{\mathtt{N}}_b}^{-1}({\mathtt{v}}_l+{\mathtt{v}}_l^i)({\mathcal{G}}_{x,y;l,\mathrm{\aleph }}-{\mathcal{G}}_{x,y;l,\mathrm{\aleph }-1}),$$ 3.20

for $(x,y)\in {\mathcal{S}}_0$. In fact, (3.20) provides the unique solution to equation (3.1) in terms of the crack opening displacement ${\{{\mathtt{v}}_l\}}_{l\in \in {\mathbb{Z}}_{-{\mathtt{N}}_b}^{-1}}$. The rigorous aspects of the issue of existence and uniqueness of the solution, for the assumed case ${\omega }_2>0$, are analogous to the results provided in [20] and are, therefore, omitted in the present article. Substitution of (3.20) into (2.9) yields a system of equations of the form

$$\begin{array}{rl}\hspace{0.17em}& \sum _{l=-{\mathtt{N}}_b}^{-1}{\mathtt{c}}_{j,l}{\mathtt{v}}_l=b_j,\hspace{0.17em}\hspace{0.17em}j\in {\mathbb{Z}}_{-{\mathtt{N}}_b}^{-1},\end{array}$$ 3.21

$$\begin{array}{rl}\text{where}& {\mathtt{c}}_{j,l}={\delta }_{j,l}-(-{\mathcal{G}}_{j,\mathrm{\aleph };l,\mathrm{\aleph }}+{\mathcal{G}}_{j,\mathrm{\aleph };l,\mathrm{\aleph }-1}+{\mathcal{G}}_{j,\mathrm{\aleph }-1;l,\mathrm{\aleph }}-{\mathcal{G}}_{j,\mathrm{\aleph }-1;l,\mathrm{\aleph }-1}),\end{array}$$ 3.22

$$\begin{array}{rl}\hspace{0.17em}& b_j={\sum }_{l=-{\mathtt{N}}_b}^{-1}({\delta }_{l,j}-{\mathtt{c}}_{j,l}),\hspace{0.17em}\text{for}\hspace{0.17em}j\in {\mathbb{Z}}_{-{\mathtt{N}}_b}^{-1}.\end{array}$$ 3.23

Introducing $\mathbf{\text{v}}={[{\mathtt{v}}_{-1},{\mathtt{v}}_{-2},\dots ,{\mathtt{v}}_{-{\mathtt{N}}_b}]}^{\text{T}}\in {\mathbb{C}}^{{\mathtt{N}}_b}$ and ${\mathbf{\text{v}}}^i={[{\mathtt{v}}_{-1}^i,{\mathtt{v}}_{-2}^i,\dots ,{\mathtt{v}}_{-{\mathtt{N}}_b}^i]}^{\text{T}}\in {\mathbb{C}}^{{\mathtt{N}}_b},$ (3.21) can be expressed as $\mathbf{\text{v}}={\mathbf{\text{F}}}_{{\mathtt{N}}_b}(\mathbf{\text{v}}+{\mathbf{\text{v}}}^i)$, where ${\mathbf{\text{F}}}_{{\mathtt{N}}_b}$ is an ${\mathtt{N}}_b\times {\mathtt{N}}_b$ matrix with ${[{\mathbf{\text{F}}}_{{\mathtt{N}}_b}]}_{j,l}={\delta }_{j,l}-{\mathtt{c}}_{j,l}$. Therefore, $\mathbf{\text{v}}={({\mathbf{\text{I}}}_{{\mathtt{N}}_b}-{\mathbf{\text{F}}}_{{\mathtt{N}}_b})}^{-1}{\mathbf{\text{F}}}_{{\mathtt{N}}_b}{\mathbf{\text{v}}}^i.$ The matrix ${\mathbf{\text{F}}}_{{\mathtt{N}}_b}$ is a matrix of the Toeplitz form [19,20] owing to the peculiar nature of the Green’s function (3.5). Eventually, the complete displacement field on the lattice (with an array of cracks) can be written by substitution of the components of v in (3.20) and extension to the entire lattice by using the Floquet phase factor (3.2).

4. Semi-infinite cracks: Wiener–Hopf method

In the following, the limiting case as ${\mathtt{N}}_b\to \mathrm{\infty }$, i.e. semi-infinite cracks, is analysed via the method of Wiener and Hopf [27]. We depart from the choice (2.2), also without any loss of generality, and consider the choice $\mathrm{\aleph }=0$ in the context of (2.1). The details of the wave propagation problem concerning the periodically repeating cell ${\mathcal{S}}_0$ are provided in appendix A(a); this is now important for the case of semi-infinite cracks since one possible mode of incidence corresponds to that from the cracked portions (with the assumption that the incidence from the infinite array still satisfies the Floquet condition (3.2)).

After taking the Fourier transform along x, the general solution of the discrete Helmholtz equation (2.3) for the scattered wave field in the lattice sites sandwiched between the two edges of ${\mathcal{S}}_0$, i.e. y = 0 and $y=\mathtt{N}-1$, is given by $u_y^{\text{F}}=a{\lambda }^y+b{\lambda }^{-y},y\in {\mathbb{Z}}_0^{\mathtt{N}-1},$ where λ is defined in (3.13). After solving for a and b in terms of $u_0^{\text{F}},$ $u_{\mathtt{N}-1}^{\text{F}}$, it is easy to see that

$$u_y^{\text{F}}=u_0^{\text{F}}\frac{{\lambda }^y-{\lambda }^{2\mathtt{N}-2-y}}{1-{\lambda }^{2\mathtt{N}-2}}+u_{\mathtt{N}-1}^{\text{F}}\frac{{\lambda }^{\mathtt{N}-1-y}-{\lambda }^{\mathtt{N}-1+y}}{1-{\lambda }^{2\mathtt{N}-2}},y\in {\mathbb{Z}}_0^{\mathtt{N}-1}.$$ 4.1

As observed in the previous section, the phase-modulated periodicity (3.2) implies $u_{\mathtt{N}-1}^{\text{F}}=\psi z^{-\mathtt{M}}{u}_{-1}^{\text{F}}.$ We now consider the discrete Fourier transform as a sum of a pair of half-transforms

$$u^{\text{F}}(z)=u_{+}(z)+u_{-}(z),u_{+}(z)=\sum _{m=0}^{+\mathrm{\infty }}{u}_m{z}^{-m},u_{-}(z)=\sum _{m=-\mathrm{\infty }}^{-1}{u}_m{z}^{-m}.$$ 4.2

Applying the discrete Fourier transform (4.2) to the governing equation for a particle at y = 0 and y = −1, respectively, we get (recall that $\mathcal{H}$ is defined by (3.9))

$$\begin{array}{rl}\hspace{0.17em}& -b^2{\mathit{\text{v}}}_{0;-}^i(z)+u_{0;-}(z)-u_{-1;-}(z)=(\mathcal{H}(z)+2)u_0^{\text{F}}(z)-u_1^{\text{F}}(z)-u_{-1}^{\text{F}}(z),\end{array}$$ 4.3

$$\begin{array}{rl}\hspace{0.17em}& b^2{\mathit{\text{v}}}_{0;-}^i(z)+u_{-1;-}(z)-u_{0;-}(z)=(\mathcal{H}(z)+2)u_{-1}^{\text{F}}(z)-u_{-2}^{\text{F}}(z)-u_0^{\text{F}}(z),z\in \mathcal{A},\end{array}$$ 4.4

$$\begin{array}{rl}\hspace{0.17em}& \text{where}\hspace{0.17em}{\mathit{\text{v}}}_{0;-}^i(z)=b^{-2}\text{A}(1-\exp (i{\kappa }_y)){\delta }_{D-}(z{z}_P^{-1}),\hspace{0.17em}\text{and}\hspace{0.17em}{\delta }_{D-}(z):={\sum }_{n=-\mathrm{\infty }}^{-1}{z}^{-n},|z|<1,\end{array}$$ 4.5

$$z_P:=\exp (-i\kappa \cos \mathrm{\Theta })\in \mathbb{C};\hspace{0.17em}\hspace{0.17em}\mathrm{furthermore},\mathrm{implies}\psi {=\mathrm{z}}_{\mathrm{P}}^{\mathtt{M}}{\lambda }_{\mathrm{P}}^{-\mathtt{N}}\mathrm{with}{\lambda }_{\mathrm{P}}=\lambda ({\mathrm{z}}_{\mathrm{P}})=\exp (\mathrm{i}\kappa \sin \mathrm{\Theta }).$$ 4.6

In (4.5)₂, it is clear that δ_D−(z) = z/(1 − z) (using the formula for the geometric series). Using (4.1) and (3.2), the pair of coupled Wiener–Hopf equations (4.4) can be expressed as

$$\left.\begin{array}{rl}\hspace{0.17em}& \mathbf{\text{{B}}}\left[\begin{array}{c}{u}_{0;+}\\ u_{-1;+}\end{array}\right]+\left(\mathbf{\text{{B}}}-\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\right)\left[\begin{array}{c}{u}_{0;-}\\ u_{-1;-}\end{array}\right]=\left[\begin{array}{c}-1\\ 1\end{array}\right]b^2{\mathit{\text{v}}}_{0;-}^i,\\ \hspace{0.17em}& \text{where}\hspace{0.17em}\mathbf{\text{{B}}}=\left[\begin{array}{cc}{\nu }_{\mathtt{N}}& -(1+z^{-\mathtt{M}}\psi {\mu }_{\mathtt{N}})\\ -(1+z^{\mathtt{M}}{\psi }^{-1}{\mu }_{\mathtt{N}})& {\nu }_{\mathtt{N}}\end{array}\right],{\nu }_{\mathtt{N}}=\frac{{\lambda }^{-\mathtt{N}}-{\lambda }^{\mathtt{N}}}{{\lambda }^{1-\mathtt{N}}-{\lambda }^{\mathtt{N}-1}},{\mu }_{\mathtt{N}}=\frac{{\lambda }^{-1}-\lambda }{{\lambda }^{1-\mathtt{N}}-{\lambda }^{\mathtt{N}-1}}.\end{array}\right\}$$ 4.7

Although problem (4.7) appears to be in the realm of matrix Wiener–Hopf kernels [27], there exists a structure which leads to its reduction to a scalar equation. Indeed, after the addition of both component equations in (4.7), it is found that

$$u_{-1;+}+u_{-1;-}=u_{-1}^{\text{F}}=\mathcal{V}{u}_0^{\text{F}}=\mathcal{V}(u_{0;+}+u_{0;-}),\hspace{0.17em}\text{where}\hspace{0.17em}\mathcal{V}=-\frac{{\nu }_{\mathtt{N}}-1-{\mu }_{\mathtt{N}}{z}^{\mathtt{M}}{\psi }^{-1}}{{\nu }_{\mathtt{N}}-1-{\mu }_{\mathtt{N}}{z}^{-\mathtt{M}}\psi }.$$ 4.8

On the other hand, taking the difference of both equations (4.7) (at this point recall (2.9)), using

$${\mathtt{v}}^{\text{F}}={\mathtt{v}}_{+}+{\mathtt{v}}_{-}=u_0^{\text{F}}-u_{-1}^{\text{F}}=(1-\mathcal{V})u_0^{\text{F}},$$ 4.9

and simplifying further (following [23]), a scalar Wiener–Hopf equation results in ${\mathtt{v}}_{\pm }$, i.e.

$${\mathtt{v}}_{+}(z)+\mathcal{L}(z){\mathtt{v}}_{-}(z)=-(1-\mathcal{L}(z)){\text{b}}^2{\mathit{\text{v}}}_{0;-}^i(z),z\in \mathcal{A},$$ 4.10

$$\text{where}\hspace{0.17em}\mathcal{L}=\frac{\mathcal{H}(z){\mathtt{U}}_{\mathtt{N}-1}(\vartheta )}{2{\mathtt{T}}_{\mathtt{N}}(\vartheta )-(z^{\mathtt{M}}{\psi }^{-1}+z^{-\mathtt{M}}\psi )}=\frac{\mathcal{N}}{\mathcal{D}}.$$ 4.11

In fact, by virtue of the presence of the Chebyshev polynomials [37] of argument $\vartheta$ (3.18), an equivalent expression for the kernel is $\mathcal{L}={\mathcal{h}}^2\prod _{j=1}^{\mathtt{N}-1}({\mathcal{h}}^2+4{\sin }^2\frac{1}{2}(j\pi /\mathtt{N}))/(\prod _{j=1}^{\mathtt{N}}({\mathcal{h}}^2+4{\sin }^2\frac{1}{2}\frac{(j-\frac{1}{2})\pi }{\mathtt{N}})-(z^{\mathtt{M}}{z}_P^{-\mathtt{M}}{\lambda }_P^{\mathtt{N}}+z^{-\mathtt{M}}{z}_P^{\mathtt{M}}{\lambda }_P^{-\mathtt{N}}))$; recall that $\psi$ is defined by (3.2) and λ_P, z_P by (4.6).

The discrete Wiener–Hopf equation (4.10) has the same form as equation (2.23) in [19] and in fact is almost identical to (3.8) in [23]; hence, employing the same kind of multiplicative factorization of the kernel $\mathcal{L}$ (following §2.4 of [23]), i.e. $\mathcal{L}={\mathcal{L}}_{+}{\mathcal{L}}_{-}$ on $\mathcal{A}$, we find

$${\mathcal{L}}_{\hspace{0.17em}+}^{-1}(z){\mathtt{v}}_{+}(z)+{\mathcal{L}}_{-}(z){\mathtt{v}}_{-}(z)=\mathcal{C}(z),z\in \mathcal{A},$$ 4.12

where $\mathcal{C}(z)=({\mathcal{L}}_{\hspace{0.17em}+}^{-1}(z)-{\mathcal{L}}_{-}(z))\text{A}(1-\exp (i{\kappa }_y)){\delta }_{D-}(z{z}_P^{-1}),z\in \mathcal{A}.$ Further, an additive factorization of $\mathcal{C}$, following [19], is

$$\mathcal{C}={\mathcal{C}}_{+}(z)+{\mathcal{C}}_{-}(z),{\mathcal{C}}_{\pm }(z)=\mp \text{A}(1-\exp (i{\kappa }_y))({\mathcal{L}}_{\hspace{0.17em}+}^{-1}(z_P)-{\mathcal{L}}_{\pm }^{\mp 1}(z)){\delta }_{D-}(z{z}_P^{-1}),z\in \mathcal{A},$$ 4.13

which leads to the exact solution of (4.10) as

$${\mathtt{v}}_{\pm }(z)={\mathcal{C}}_{\pm }(z){\mathcal{L}}_{\pm }^{\pm 1}(z),z\in \mathbb{C},|z|\gtrless \frac{max}{min}\{{\mathit{\text{R}}}_{\pm },{\mathit{\text{R}}}_L^{\pm 1}\}.$$ 4.14

Recall that z_P is given by (4.6).

Using (4.9), the exact solution (4.14) implies (${\mathtt{v}}^{\text{F}}={\mathtt{v}}_{+}+{\mathtt{v}}_{-}$)

$$u_0^{\text{F}}={(1-\mathcal{V})}^{-1}{\mathtt{v}}^{\text{F}},u_{-1}^{\text{F}}=\mathcal{V}{(1-\mathcal{V})}^{-1}{\mathtt{v}}^{\text{F}}\hspace{0.17em}\text{on}\hspace{0.17em}\mathcal{A}.$$ 4.15

Along with (3.3) and (4.1), (4.15) provides the complete solution of the diffraction problem in integral form. By the inverse discrete Fourier transform (analogous to (3.19)), the displacement of the lattice at (x, 0) and (x, − 1) is given by $u_{x,0}=(1/2\pi i){\oint }_{\mathcal{C}}{u}_0^{\text{F}}(z)z^{x-1}\hspace{0.17em}\text{d}z$ and $u_{x,-1}=(1/2\pi i){\oint }_{\mathcal{C}}{u}_{-1}^{\text{F}}(z)z^{x-1}\hspace{0.17em}\text{d}z$, respectively, with $x\in \mathbb{Z}$, where $\mathcal{C}$ is a rectifiable, closed, counterclockwise contour in the annulus $\mathcal{A}$, and the remaining displacements at other edges are given by (3.3). Analogous to u_x,0 and u_x,−1, ${\mathtt{v}}_x=(1/2\pi i){\oint }_{\mathcal{C}}{\mathtt{v}}^{\text{F}}(z)z^{x-1}dz,x\in \mathbb{Z},$ where ${\mathtt{v}}^{\text{F}}={\mathtt{v}}_{+}+{\mathtt{v}}_{-}$. As x → ±∞, an approximation for $\mathtt{v}$ can be obtained by analysing (4.14) with ${\mathtt{v}}_x=(1/2\pi i){\oint }_{\mathcal{C}}{\mathtt{v}}_{\pm }(z)z^{x-1}\hspace{0.17em}\text{d}z,$ $x\in {\mathbb{Z}}^{\pm }.$ Indeed, after deforming the contour of integration $\mathcal{C}$ (expanding it to a circular contour with infinite radius when x → −∞ while contracting it to zero radius when x → +∞) and applying residue calculus, an approximation is given by

$${\mathtt{v}}_x\sim \pm {\sum }_{|z_{\ast }|\lessgtr 1}\hspace{0.17em}\text{Res}\hspace{0.17em}{\mathcal{C}}_{\pm }(z_{\ast }){\mathcal{L}}_{\hspace{0.17em}\pm }^{\pm 1}(z_{\ast })z_{\ast }^{x-1},x\in {\mathbb{Z}}^{\pm },|x|\gg 1,$$ 4.16

where the additive factors ${\mathcal{C}}_{\pm }$ are given by (4.13). Let the unit circle in the complex plane be denoted by $\mathbb{T}$. We observe that in the limit ${\omega }_2\to 0+$ (recall (2.6)), while considering |x| → ∞, the approximation (4.16) of ${\mathtt{v}}_x$ effectively takes into account only the contributions due to z_* approaching $\mathbb{T}$ appropriately and, evidently, representing the outgoing wave modes, as expected [12]. Also, observe that (4.11) allows the relations ${\mathcal{L}}_{+}={\mathcal{N}}_{+}/{\mathcal{D}}_{+}$ and ${\mathcal{L}}_{-}={\mathcal{N}}_{-}/{\mathcal{D}}_{-}$, factorizing the numerator and denominator, i.e. $\mathcal{N}={\mathcal{N}}_{+}{\mathcal{N}}_{-}$ and $\mathcal{D}={\mathcal{D}}_{+}{\mathcal{D}}_{-}$, respectively, on $\mathcal{A}$. Owing to these observations, we consider the following natural definitions:

$${\mathcal{Z}}^{+}=\{z\in \mathbb{T}|{\mathcal{D}}_{+}(z)=0\}{\mathcal{Z}}^{-}=\{z\in \mathbb{T}|{\mathcal{N}}_{-}(z)=0\}$$ 4.17

corresponding to outgoing waves towards the positive and negative x-axis away from the crack tip, respectively. To keep the same notation, the sets of z belonging to

$${\tilde{\mathcal{Z}}}^{-}=\{z\in \mathbb{T}|{\mathcal{D}}_{-}(z)=0\}\text{and}{\tilde{\mathcal{Z}}}^{+}=\{z\in \mathbb{T}|{\mathcal{N}}_{+}(z)=0\},$$ 4.18

can be easily identified as those corresponding to the incident waves; namely, incident from the positive and negative x-axis towards the crack tip. Naturally, the case of incidence we consider here corresponds to those waves that travel from the bulk lattice so that $z_P\in {\tilde{\mathcal{Z}}}^{-}$. Later, we discuss the issue of incidence from the duct, i.e. corresponding to ${\tilde{\mathcal{Z}}}^{+}$.

Returning to (4.16), after substitution of (4.13), we find

$$\left.\begin{array}{rl}\hspace{0.17em}& {\mathtt{v}}_x\sim \text{A}{a}^i\frac{{\mathcal{D}}_{+}(z_P)}{{\mathcal{N}}_{+}(z_P)}{\sum }_{z_{\ast }\in {\mathcal{Z}}^{+}}\frac{{\mathcal{N}}_{+}(z_{\ast })}{{\mathcal{D}}_{+}^{\prime }(z_{\ast })}\frac{z_{\ast }^x}{z_{\ast }-z_P},x\to +\mathrm{\infty },\\ \hspace{0.17em}& {\mathtt{v}}_x\sim \text{A}{a}^i\left(-z_P^x+\frac{{\mathcal{D}}_{+}(z_P)}{{\mathcal{N}}_{+}(z_P)}\sum _{z_{\ast }\in {\mathcal{Z}}^{-}}\frac{{\mathcal{D}}_{-}(z_{\ast })}{{\mathcal{N}}_{-}^{\prime }(z_{\ast })}\frac{z_{\ast }^x}{z_{\ast }-z_P}\right),\hspace{0.17em}x\to -\mathrm{\infty },\end{array}\right\}$$ 4.19

where

$$a^i\hspace{0.17em}\text{is given by}\hspace{0.17em}{a}^i:=1-\exp (i{\kappa }_y).$$ 4.20

Remark 4.1. —

In (4.19) corresponding to x → +∞, $z_P^{-1}$ is included in sum (${\mathcal{D}}_{+}(z_P^{-1})=0,$ ${\mathcal{D}}_{-}(z_P^{-1})\ne 0$ but ${\mathcal{D}}_{-}(z_P)=0$). In (4.19) for x → −∞, z_P does not occur in sum (${\mathcal{D}}_{-}(z_P)=0,{\mathcal{D}}_{+}(z_P)\ne 0$ but ${\mathcal{D}}_{+}(z_P^{-1})=0$). This is testimony to the preceding sentences.

Using (4.19), after simplification, the total field is given by ${\mathtt{v}}_x^t={\mathtt{v}}_x^i+{\mathtt{v}}_x$. Thus, for a wave incident from the bulk lattice, i.e. region (1) of figure 3 in front of the staggered array, the transmitted waves include the contribution from the residue associated with the incident wave, which cancels another equal contribution, while the reflected waves include the contribution from the residue associated with the reflected wave, determined by the incident wavenumber.

Figure 3.

An infinite array of cracks with a horizontal stagger $\mathtt{M}$ (b), together with directions of the incident wave, reflected wave(s) and duct mode(s) in (a) and (c), in relation to the incidence from the bulk lattice (1) and the portion between the cracks (2). The crack tips are shown schematically in (b). The intact lattice is shown as grey dots and the particles located at the crack faces (lacking a nearest-neighbour bond) as white dots. (Online version in colour.)

Using (4.19), we can also construct the asymptotic expansion of the scattered wave field. Note that the far field can be determined in terms of the (propagating) normal modes associated with the two different portions (ahead, indicated by subscript +, and behind, by −). The normal modes for a square lattice waveguide with a fixed or free boundary are known (see also [37]). Let

$$z_{\ast }=\exp (-i\xi ),\lambda (z_{\ast })=\exp (i\eta )\text{and}{\kappa }_{z_{\ast }}\hspace{0.17em}\text{refer to the specific normal mode depending on}\hspace{0.17em}{z}_{\ast }.$$ 4.21

Thus (with J_κ as amplitudes of relevant wave modes),

$$\left.\begin{array}{rl}\hspace{0.17em}& u_{x,y}\sim \text{A}\sum _{z_{\ast }\in {\mathcal{Z}}^{+}}{J}_{{\kappa }_{z_{\ast }}}{a}_{+({\kappa }_{z_{\ast }})y}{z}_{\ast }^x,\hspace{0.17em}x\to +\mathrm{\infty },\\ \hspace{0.17em}& u_{x,y}\sim \text{A}\sum _{z_{\ast }\in {\mathcal{Z}}^{-}}{J}_{{\kappa }_{z_{\ast }}}{a}_{-({\kappa }_{z_{\ast }})y}{z}_{\ast }^x-\text{A}{J}_{{\kappa }^i}{a}_{({\kappa }^i)y}{z}_P^x,x\to -\mathrm{\infty },\end{array}\right\}$$ 4.22

respectively, so that the total field is the desired one. The expression (4.22) also yields ${\mathtt{v}}_x$ as x → ±∞. Note that the wave modes ahead of the array are given by (2), (3), while those behind the scattering edges are given by (5). Evidently, this leads to the total displacement field

$$\left.\begin{array}{rl}\hspace{0.17em}& u_{x,y}^t\sim u_{x,y}^i+\text{A}\frac{{\mathcal{D}}_{+}(z_P)}{{\mathcal{N}}_{+}(z_P)}\sum _{z_{\ast }\in {\mathcal{Z}}^{+}}\frac{a^i{a}_{+({\kappa }_{z_{\ast }})y}{z}_{\ast }^x}{a_{+({\kappa }_{z_{\ast }})0}-{\psi }^{-1}{z}_{\ast }^{\mathtt{M}}{a}_{+({\kappa }_{z_{\ast }})\mathtt{N}-1}}\frac{1}{z_{\ast }-z_P}\frac{{\mathcal{N}}_{+}(z_{\ast })}{{\mathcal{D}}_{+}^{\prime }(z_{\ast })}\\ \text{and}& u_{x,y}^t\sim \text{A}\frac{{\mathcal{D}}_{+}(z_P)}{{\mathcal{N}}_{+}(z_P)}\sum _{z_{\ast }\in {\mathcal{Z}}^{-}}\frac{a^i{a}_{-({\kappa }_{z_{\ast }})y}{z}_{\ast }^x}{a_{-({\kappa }_{z_{\ast }})0}-{\psi }^{-1}{z}_{\ast }^{\mathtt{M}}{a}_{-({\kappa }_{z_{\ast }})\mathtt{N}-1}}\frac{1}{z_{\ast }-z_P}\frac{{\mathcal{D}}_{-}(z_{\ast })}{{\mathcal{N}}_{-}^{\prime }(z_{\ast })},\end{array}\right\}$$ 4.23

as x → +∞ and x → −∞, respectively. In (4.19) and (4.23), the substitution of ${\mathcal{L}}_{\mathtt{N}}$ by $\mathcal{N}/\mathcal{D}$ has been used; in this context, ${\mathcal{N}}_{+}/{\mathcal{D}}_{+}^{\prime }={\mathcal{D}}_{-}{\mathcal{N}}_{+}/{\mathcal{D}}^{\prime },$ ${\mathcal{D}}_{-}/{\mathcal{N}}_{-}^{\prime }={\mathcal{D}}_{-}{\mathcal{N}}_{+}/{\mathcal{N}}^{\prime },$ etc. With details relegated to appendix C, the transmittance, i.e. the energy flux transmitted into the cracked portion per unit incident energy flux, is given by

$$\mathcal{T}=\frac{z_P{\mathcal{N}}_{-}(z_P){\mathcal{D}}_{+}(z_P)}{\overline{{\mathcal{D}}_{-}^{\prime }(z_P){\mathcal{N}}_{+}(z_P)}}{\sum }_{z\in {\mathcal{Z}}^{-}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{z_P}{{(z-z_P)}^2}.$$ 4.24

Analogous result holds for the reflectance $\mathcal{R}$ (3). This is a special form of expression for $\mathcal{R}$ and $\mathcal{T}$ as it coincides with that for the bifurcated waveguides [23].

The analysis of $\mathcal{R}$ and $\mathcal{T}$ for finite cracks is provided in appendix B. Here, $\mathcal{R}$ (resp. $\mathcal{T}$) refers to the energy flux reverted back to the same side as that of the incident wave while treating the infinite array of finite-length cracks as an interface. However, when ${\mathtt{N}}_b\gg 1$ it is natural to resolve the issue of energy flux transmission for the waves generated inside the cracks; figure 1a provides a helpful hint. The following deals with this aspect of semi-infinite cracks.

(a). Wave incidence from the ducts

In order to maintain the convenience of Floquet periodicity, it is assumed a priori that a steady state has been reached for an incident wave field that arrives from all of the infinite number of ducts. This assumption is derived from the scenario presented in figure 1a, employed to tackle the issue of finite cracks when ${\mathtt{N}}_b\gg 1$ by posing the Wiener–Hopf problem for two kinds of incidence. The scattering in this case of wave incidence from the ducts occurs because of the intact bonds ahead of the staggered array of defects (figure 3). Analogous to (2.4), it is assumed that the incident wave is

$$u_{x,y}^i:=\text{A}{a}_{({\kappa }^i)\nu }{P}^{\lfloor y/\mathtt{N}\rfloor }\exp (i{\kappa }_{x}x-i\omega t),(x,y)\in \mathbb{Z}\times {\mathbb{Z}}_{\mathtt{N}},{\kappa }_x>0,\hspace{0.17em}\nu =\text{mod}(y,\mathtt{N}),$$ 4.25

where $a_{({\kappa }^i)}$ denotes the wave mode in any of the portions between the scattering edges and P is the phase factor. Note that the phase factor in the duct located between $y=j\mathtt{N}$ and $y=j\mathtt{N}+\mathtt{N}-1$ is P^j. Since |P| = 1, let $P=\exp (i{\kappa }_y^{\mathrm{\#}}\mathtt{N})$ for some ${\kappa }_y^{\mathrm{\#}}\in [-\pi ,\pi ].$ Thus, the incident wave imposes a Floquet–Bloch multiplier $\exp (i{\kappa }_x\mathtt{M}+i{\kappa }_y^{\mathrm{\#}}\mathtt{N})$; recall (3.2).

Remark 4.2. —

Indeed, $u_{x+\mathtt{M},y+\mathtt{N}}^i=\psi u_{x,y}^i,$ $y\in {\mathbb{Z}}_0^{\mathtt{N}-1},x\in \mathbb{Z},$ where $\psi =\exp (i{\kappa }_x\mathtt{M}+i{\kappa }_y^{\mathrm{\#}}\mathtt{N}).$ Given the wave mode κⁱ in the duct portion, the frequency $\omega$ and incident wavenumber ${\kappa }_x$ are related by the duct dispersion relation. Thus, the scattering of a specific duct mode involves two free parameters, ${\kappa }_x$ and ${\kappa }_y^{\mathrm{\#}}$, where the former yields a specific frequency in κⁱ mode (similar to the case of incidence from the bulk lattice where ${\kappa }_y$ and ${\kappa }_x$ can be chosen arbitrarily and $\omega$ is provided by the square lattice dispersion relation). It is useful to note that ${\kappa }_y$ is pre-determined by the bulk incidence while ${\kappa }_x$ depends on the transmitted wave numbers towards the ‘other’ edge.

Taking into account the intact bonds between y = 0 and y = −1 for x ≥ 0, and also $y=0+\mathtt{N}$ and $y=-1+\mathtt{N}$ for $x\ge \mathtt{M}$ and so on, and the broken nature of all other bonds, the equation that must be satisfied by u at y = 0 is found to be, in terms of (4.7),

$$\begin{array}{rl}\hspace{0.17em}& {\text{b}}^2{\mathit{\text{v}}}_{0;+}^i={\nu }_{\mathtt{N}}{u}_{0;+}-(1+z^{-\mathtt{M}}\psi {\mu }_{\mathtt{N}})u_{-1;+}+({\nu }_{\mathtt{N}}-1)u_{0;-}-z^{-\mathtt{M}}\psi {\mu }_{\mathtt{N}}{u}_{-1;-},\end{array}$$ 4.26

$$\begin{array}{rl}\hspace{0.17em}& -{\text{b}}^2{\mathit{\text{v}}}_{0;+}^i={\nu }_{\mathtt{N}}{u}_{-1;+}-(1+z^{\mathtt{M}}{\psi }^{-1}{\mu }_{\mathtt{N}})u_{0;+}+({\nu }_{\mathtt{N}}-1)u_{-1;-}-z^{\mathtt{M}}{\psi }^{-1}{\mu }_{\mathtt{N}}{u}_{0;-},\end{array}$$ 4.27

$$\begin{array}{rl}\text{where}& {\mathit{\text{v}}}_{0;+}^i={\text{b}}^{-2}\text{A}{a}^i{\delta }_{D+}(z{z}_P^{-1}),z_P:=\exp (i{\kappa }_x)\in \mathbb{C},\hspace{0.17em}{a}^i:=a_{({\kappa }^i)0}-{\psi }^{-1}{z}_P^{\mathtt{M}}{a}_{({\kappa }^i)\mathtt{N}-1}.\end{array}$$ 4.28

Indeed, after addition of both equations, it is found that (4.8) holds, and by taking the difference of both equations, using (4.9), and simplifying further, the resulting equation is found to be

$${\mathtt{v}}_{+}(z)+\mathcal{L}(z){\mathtt{v}}_{-}(z)=(1-\mathcal{L}(z)){\text{b}}^2{\mathit{\text{v}}}_{0;+}^i(z),z\in \mathcal{A},$$ 4.29

which is the scalar discrete Wiener–Hopf equation for $\mathtt{v}$, as desired, where $\mathcal{L}$ is given by (4.11). Note that, as $z_P=\exp (i{\kappa }_x)$, for the dissipative case it is found that |z_P| < 1. The above can be compared and contrasted with the case of and incident wave from the bulk lattice, i.e. the intact part of the waveguide, (4.10), and some relations between involved entities the can be found; for instance, the Wiener–Hopf kernel remains the same. Also, (4.29) has same form as (4.10), hence (4.12) holds with

$$\mathcal{C}(z)=-({\mathcal{L}}_{\hspace{0.17em}+}^{-1}(z)-{\mathcal{L}}_{-}(z))\text{A}{a}^i{\delta }_{D+}(z{z}_P^{-1}),z\in \mathcal{A},\hspace{0.17em}{\delta }_{D+}(z):={\sum }_{n=0}^{+\mathrm{\infty }}{z}^{-n},|z|>1.$$ 4.30

An additive factorization, $\mathcal{C}={\mathcal{C}}_{+}(z)+{\mathcal{C}}_{-}(z),$ is constructed with

$${\mathcal{C}}_{\pm }(z)=\pm \text{A}{a}^i({\mathcal{L}}_{-}(z_P)-{\mathcal{L}}_{\pm }^{\mp 1}(z)){\delta }_{D+}(z{z}_P^{-1}),z\in \mathcal{A}.$$ 4.31

In terms of the one-sided transforms, ${\mathtt{v}}^{\text{F}}$ is given by (4.14), while $u_0^{\text{F}}$ and $u_{-1}^{\text{F}}$ are given by (4.15).

We move now to the question of the asymptotic approximation of the solution deep in the portions away from the crack tips. Using (4.31),

$$\left.\begin{array}{rl}\hspace{0.17em}& {\mathtt{v}}_x\sim \text{A}{a}^i(-z_P^x+\frac{{\mathcal{N}}_{-}(z_P)}{{\mathcal{D}}_{-}(z_P)}\sum _{z_{\ast }\in {\mathcal{Z}}^{+}}\frac{{\mathcal{N}}_{+}(z_{\ast })}{{\mathcal{D}}_{+}^{\prime }(z_{\ast })}\frac{z_{\ast }^x}{z_{\ast }-z_P}),x\to +\mathrm{\infty },\\ \text{and}& {\mathtt{v}}_x\sim \text{A}{a}^i\frac{{\mathcal{N}}_{-}(z_P)}{{\mathcal{D}}_{-}(z_P)}\sum _{z_{\ast }\in {\mathcal{Z}}^{-}}\frac{{\mathcal{D}}_{-}(z_{\ast })}{{\mathcal{N}}_{-}^{\prime }(z_{\ast })}\frac{z_{\ast }^x}{z_{\ast }-z_P},\hspace{0.17em}x\to -\mathrm{\infty }.\end{array}\right\}$$ 4.32

It is observed, in the above case corresponding to x → +∞, that z_P does not occur in the sum (here ${\mathcal{N}}_{+}(z_P)=0,{\mathcal{N}}_{-}(z_P)\ne 0$ but ${\mathcal{N}}_{-}(z_P^{-1})=0$) as anticipated. Recall (4.17) and (4.18) for the definitions of ${\mathcal{Z}}^{\pm }$. In the expression for x → −∞, $z_P^{-1}$ is included in the sum (${\mathcal{N}}_{+}(z_P^{-1})\ne 0,$ ${\mathcal{N}}_{-}(z_P^{-1})=0$ but ${\mathcal{N}}_{+}(z_P)=0$).

Analogous to (4.23), resulting from (4.32), the total displacement field is written as

$$\left.\begin{array}{rl}\hspace{0.17em}& u_{x,y}^t\sim \text{A}\frac{{\mathcal{N}}_{-}(z_P)}{{\mathcal{D}}_{-}(z_P)}\sum _{z\in {\mathcal{Z}}^{+}}\frac{a^i{a}_{+({\kappa }_z)y}{z}^x}{a_{+({\kappa }_z)0}-{\psi }^{-1}{z}^{\mathtt{M}}{a}_{+({\kappa }_z)\mathtt{N}-1}}\frac{1}{z-z_P}\frac{{\mathcal{N}}_{+}(z)}{{\mathcal{D}}_{+}^{\prime }(z)},\\ \hspace{0.17em}& u_{x,y}^t\sim \text{A}{a}_{({\kappa }^i)y}{z}_P^x+\text{A}\frac{{\mathcal{N}}_{-}(z_P)}{{\mathcal{D}}_{-}(z_P)}{\sum }_{z\in {\mathcal{Z}}^{-}}\frac{a^i{a}_{-({\kappa }_z)y}{z}^x}{a_{-({\kappa }_z)0}-{\psi }^{-1}{z}^{\mathtt{M}}{a}_{-({\kappa }_z)\mathtt{N}-1}}\frac{1}{z-z_P}\frac{{\mathcal{D}}_{-}(z)}{{\mathcal{N}}_{-}^{\prime }(z)},\end{array}\right\}$$ 4.33

as x → +∞ and x → −∞, respectively. Finally, the transmittance, i.e. the energy flux transmitted into the intact portion per unit incident energy flux from the cracked portion, is given by

$$\begin{array}{rl}\mathcal{T}& =\frac{{(v({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}-}^{-1}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{+}}\frac{|a^i{|}^2}{-(\mathcal{N}(z)/\mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta ))}\frac{1}{\overline{z}-{\overline{z}}_P}\frac{\overline{{\mathcal{D}}_{-}(z)}\overline{{\mathcal{N}}_{+}(z)}}{\overline{{\mathcal{D}}^{\prime }(z)}}\frac{1}{z-z_P}\frac{\mathcal{N}(z)}{{\mathcal{D}}_{+}^{\prime }(z){\mathcal{N}}_{-}(z)}v(\xi )\\ \hspace{0.17em}& =\frac{1}{2}i\frac{{\omega }^{-1}|a^i{|}^2}{(v({\xi }_P))|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}-}^{-1}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{+}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{D}}_{+}^{\prime }(z){\mathcal{N}}_{-}(z)}\frac{z_P}{{(z-z_P)}^2},\end{array}$$ 4.34

while the reflectance is given by

$$\begin{array}{rl}\mathcal{R}& =\frac{{(v({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}-}^{-1}(z_P){|}^2}{\sum }_{z\in {\mathcal{Z}}^{-}}\frac{2\mathtt{N}|a^i{|}^2}{\cos \mathtt{N}{\eta }_{\kappa }(\mathcal{H}(z)+4)\mathcal{D}(z)}\frac{\mathcal{D}(z)|v(\xi )|}{\overline{{\mathcal{N}}^{\prime }(z)}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{1}{\overline{z}-{\overline{z}}_P}\frac{1}{z-z_P}\\ \hspace{0.17em}& =\frac{1}{2}i\frac{{\omega }^{-1}|a^i{|}^2}{(v({\xi }_P))|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}-}^{-1}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{-}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{z_P}{{(z-z_P)}^2},\end{array}$$ 4.35

(recall (3)). The coefficient in front of the sum can be simplified as

$$\frac{1}{2}{\omega }^{-1}i\frac{|a^i{|}^2}{(v({\xi }_P))|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}-}^{-1}(z_P){|}^2}=\frac{z_P{\mathcal{N}}_{-}(z_P){\mathcal{D}}_{+}(z_P)}{\overline{{\mathcal{D}}_{-}(z_P){\mathcal{N}}_{+}^{\prime }(z_P)}}.$$ 4.36

5. Numerical results

A numerical scheme on the lines of that stated in the appendix of [23], for bifurcated waveguides, has been used to solve directly the discrete scattering problem involving the array of semi-infinite cracks as well as finite cracks on the lattice. We omit the graphical results in the main paper but remark that the corresponding results have been found to be in excellent agreement with the semi-analytical solution of §2. It is also found that, when the separation between the adjacent cracks, $\mathtt{N},$ is large, the solution near the edge of any crack in the array allows an approximation by that for a single crack on a square lattice, modulo a suitable phase factor. The main results of this paper concern the aspects of transmission of energy. The reflected (resp. transmitted) energy flux per unit incident energy flux, called reflectance (resp. transmittance), is calculated numerically using an analytical expression that has been derived in appendix B.

In figure 4a, the reflectance ($\mathcal{R}$, see (3)₁) and transmittance ($\mathcal{T}$, see (3)₂) have been plotted against the incident angle Θ. It can be verified that the sum of the reflectance and transmittance is one, which is the consequence of the balance of the mechanical energy. It can also be seen that most of the energy is transmitted while a very small amount is reflected for certain choices of Θ. Such information is anticipated to be useful for the planning and engineering of nanostructures where the high-frequency scattering plays a major role. Figure 4a gives the comparison of the semi-analytical and numerical results also; the two approaches show good agreement.

Figure 4.

Reflectance $\mathcal{R}$ and transmittance $\mathcal{T}$ have been plotted against the incident angle Θ in (a) and against the frequency $\omega$ in (b). The semi-analytical results are plotted in lighter shade and bigger dots whereas the numerical results are plotted in black and grey colours. The parameters used for these figures are $\mathtt{M}=0,$ $\mathtt{N}=5$ and ${\mathtt{N}}_b=5$. (Online version in colour.)

In figure 4b, the reflectance and transmittance versus the frequency $\omega$ is shown. When the incident wave frequency is near zero, the discrete solution approaches that of its continuous counterpart [19]. The physical effects of discreteness become visible for much higher frequencies belonging to the passband; see the right-hand side of figure 4b. As part of the analysis of some key features, note that in figure 4b there are numerous peaks or valleys in the transmittance and reflectance. In figure 4a,b, the numerical oscillations depend upon the domain of numerical calculations. The large domain fixes the oscillations resulting in smooth curves. Such oscillations are absent in the semi-analytical results, as can be seen in figure 4a. The limit of the $\omega$ value in figure 4b corresponds to those frequencies that lie in the fundamental zone, i.e. $(\kappa \cos \mathrm{\Theta },\kappa \sin \mathrm{\Theta })\in {[-\pi ,\pi ]}^2$ in (2.4).

The transmittance (calculated numerically, while an analytical expression has been derived in appendix B) is illustrated further in figure 5. The transmittance versus the frequency (in the passband with the limit of $\omega$ value in figure 5a–d corresponding to that which lies in the fundamental zone, such that $(\kappa \cos \mathrm{\Theta },\kappa \sin \mathrm{\Theta })\in {[-\pi ,\pi ]}^2$ in (2.4)) of the incident wave for some values of the relevant parameters, namely the incident angle relative to the normal to the crack tips (denoted by $\beta$ (2.5)), the spacing between two consecutive cracks $\mathtt{N}$, the stagger between cracks $\mathtt{M}$ and the crack length ${\mathtt{N}}_b$, is shown in figure 5. When $\mathtt{M}=0$, which is shown as a black curve in figure 5a, and the cracks are not offset against each other, the incident wave at normal incidence ($\beta =0^{\circ }$) is transmitted perfectly without being scattered for all frequencies in the passband. The complete transmission is also observed in the case of plates with periodically arranged parallel rectangular slots carrying a longitudinal elastic wave at normal incidence, as reported in [40] (figures 3 and 4). The low-frequency region of figure 5a represents the solution (for $\mathtt{M}=0$) that matches the continuous counterpart presented in [40]. It is not surprising to see that the behaviour is extended for much higher frequencies since it is expected from the assumed simplified model. High transmission can be also seen, at normal incidence, when the cracks are staggered (i.e. when $\mathtt{M}\ne 0$). However, in the presence of stagger, there are certain frequencies in the passband at which the incident waves are (almost) completely reflected (as indicated by dip(s) in the transmission curves in all cases in figure 5a–d. Note that the oscillations present near $\omega =0$ in figure 5a–d are numerical figments and disappear with an increase in the numerical domain size, simulating the infinite lattice. It can be observed that the range of the frequencies reflected back decreases, i.e. the band of frequency reflected shrinks in its width, as the inter-crack spacing $\mathtt{N}$ increases; see figure 5b. At oblique incidence ($\beta \ne 0^{\circ }$) there are no frequencies in the passband at which complete reflection or transmission occurs, as illustrated by the black curve for $\beta ={30}^{\circ }$ in figure 5c. The number of dips, i.e. frequencies corresponding to a completely reflected character, increases as the crack length ${\mathtt{N}}_b$ increases, as shown in figure 5d. However, this phenomenon is limited as we have observed that, when ${\mathtt{N}}_b$ increases to larger values, the dips start to disappear (not illustrated here). Although the transmission behaviour with a narrow transmission band, reported by [28], is different from the one reported in the present work (as is clear from figure 5), almost opposite in fact, what is common is that with some appropriate variations in the geometric arrangements of the cracks a favourable transmission/blocking of high-frequency waves can be achieved.

Figure 5.

Transmittance $\mathcal{T}$ (vertical axis) versus frequency ω (horizontal axis) for the parameters shown. (Online version in colour.)

6. Conclusion

The paper deals with the scattering due to a periodic array of staggered cracks in a two-dimensional lattice. Within the setting of finite cracks, the case of semi-infinite cracks is also considered as it is relevant when the length of the cracks is much larger than the spacing in between. The latter can be seen as a discrete analogue of the work by Carlson and Heins [4–6]. In this sense, the present article considers a formulation on a square lattice when there exists a discrete equivalent of an infinite array of parallel finite or semi-infinite rows with the Neumann condition. Because of the Floquet–Bloch theorem, the problem is reduced to that of scattering due to a single crack on a lattice ‘waveguide’ with Floquet boundary conditions on the outer rows. For the purpose of solving the problem for finite cracks, a discrete Green’s function has been derived that satisfies the Floquet periodic boundary conditions of the waveguide. The crack opening displacements and the exact solution of the scattering problem are obtained by the inversion of a Toeplitz matrix, whose entries are found in terms of the Green’s function [20]. The limiting situation comprises semi-infinite cracks, which admit a refined solution obtained by the discrete Wiener–Hopf method. A low-frequency approximation of the solution [19] in integral form recovers the classical continuum solution of Heins and Carlson; however, such details are omitted but can be carried out following earlier work on the continuum limit for a singe crack [19,20,41].

From a physical point of view, the transmission of mechanical energy has also been explored, via the notions of reflectance and transmittance [12,23], with some of the results being illustrated graphically. A peculiar transmission behaviour is observed for a certain range of incident wave and structural parameters wherein a narrow range of frequencies, in a passband of the infinite lattice, have almost complete reflection. Via an appropriate set of variations in the geometric arrangements of the cracks, therefore, we show that it is possible to construct some tunable atomic-scale interfaces. The transmission of energy in such structures will have favourable complete transmission for most of the frequencies with the exception of an interesting small segment of high-frequency waves that can be blocked. The presented results are also useful in the understanding of phonon transport at low temperature in systems involving superlattices. The presence of periodically distributed interfaces in novel lattice structures provides an additional rationale.

Appendix A. ‘Waveguide’ and wave modes

(a) Wave modes in the ‘bulk’

Consider the ‘wave modes’ on the right-hand side of the strip ${\mathcal{S}}_0$, i.e. $y\in {\mathbb{Z}}_0^{\mathtt{N}-1}$. By application of the Floquet–Bloch condition (3.3) for the equation of motion of the ‘upper’ and ‘lower’ boundary rows, i.e. $u_{x,\mathtt{N}}=\psi u_{x-\mathtt{M},0},u_{x,-1}={\psi }^{-1}{u}_{x+\mathtt{M},\mathtt{N}-1}.$ With $u_{x,y}(t)=a_y\exp (-i\xi x-i{\text{b}}^{-1}\omega t),y\in {\mathbb{Z}}_0^{\mathtt{N}-1}$ and $z=\exp (-i\xi )$, it follows that

$$\begin{array}{rl}-{\omega }^2{a}_y& =(1-{\delta }_{y,\mathtt{N}-1})a_{y+1}+{\delta }_{y,\mathtt{N}-1}{a}_0\psi z^{-\mathtt{M}}\\ \hspace{0.17em}& +(1-{\delta }_{y,0})a_{y-1}+{\delta }_{y,0}{a}_{\mathtt{N}-1}{\psi }^{-1}{z}^{\mathtt{M}}+2\cos \xi a_y-4a_y,y\in {\mathbb{Z}}_0^{\mathtt{N}-1}.\end{array}$$ A 1

In particular, $\omega$ is described by the general form ${\omega }^2=4-2\cos \xi -2\cos {\eta }_{\kappa },$ $\kappa \in {\mathbb{Z}}_1^{\mathtt{N}},$ where ${\eta }_{\kappa }$ are determined by the condition $\sin (\mathtt{N}+1){\eta }_{\kappa }-\sin (\mathtt{N}-1){\eta }_{\kappa }-(\psi z^{-\mathtt{M}}+{\psi }^{-1}{z}^{\mathtt{M}})\sin {\eta }_{\kappa }=0,$ i.e.

$${\mathtt{U}}_{\mathtt{N}}(\vartheta )-{\mathtt{U}}_{\mathtt{N}-2}(\vartheta )-(\psi z^{-\mathtt{M}}+{\psi }^{-1}{z}^{\mathtt{M}})=0.$$ A 2

The eigenvectors a_(κ) are given by

$$a_{(\kappa )y}={\mathtt{C}}_{\kappa ;\mathtt{N}}(\sin (y+1){\eta }_{\kappa }+{\psi }^{-1}{z}^{\mathtt{M}}\sin (\mathtt{N}-y-1){\eta }_{\kappa }),y\in {\mathbb{Z}}_0^{\mathtt{N}-1}$$ A 3

with ${\mathtt{C}}_{\kappa ;\mathtt{N}}^{-2}=\mathtt{N}\left(1-\frac{1}{4}{(\psi z^{-\mathtt{M}}+{\psi }^{-1}{z}^{\mathtt{M}})}^2\right).$

Furthermore, the branches of the dispersion relation are given by ${\omega }_{\kappa }^2=4{\sin }^2\frac{1}{2}\xi +4{\sin }^2\frac{1}{2}{\eta }_{\kappa },\kappa \in {\mathbb{Z}}_0^{\mathtt{N}-1}.$ The group velocity is easily found to be given by (for $\kappa \in {\mathbb{Z}}_0^{\mathtt{N}-1}$)

$$v_{\kappa }(\xi )=\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }{\omega }_{\kappa }={\omega }_{\kappa }^{-1}\left(\sin \xi +\frac{d{\eta }_{\kappa }}{d\xi }\sin {\eta }_{\kappa }\right)={\mathtt{N}}^{-1}{\omega }_{\kappa }^{-1}(\mathtt{N}\sin \xi -\mathtt{M}\hspace{0.17em}\sin {\eta }_{\kappa }).$$ A 4

Using $\sin n{\eta }_{\kappa }/\sin {\eta }_{\kappa }=({\lambda }^{-n}-{\lambda }^n)/({\lambda }^{-1}-\lambda )$, (A 3) can also be written as $a_y=C_{\mathtt{N}}{\mathtt{U}}_{\mathtt{N}-1}{\lambda }^{\mathtt{N}-(y+1)}.$

(b) Wave modes between the cracks

Consider the ‘wave modes’ on the left-hand side of the strip ${\mathcal{S}}_0$, i.e. $y\in {\mathbb{Z}}_0^{\mathtt{N}-1}$. With $u_{x,y}(t)=a_{(\kappa )y}\exp (-i\xi x-i{\text{b}}^{-1}\omega t),$ $y\in {\mathbb{Z}}_0^{\mathtt{N}-1}$ and $z=\exp (-i\xi )$, the wave modes are given by

$$a_{(\kappa )y}=a_{(\kappa )1}\cos (y+\frac{1}{2}){\eta }_{\kappa }/\cos (\frac{1}{2}{\eta }_{\kappa }),y\in {\mathbb{Z}}_0^{\mathtt{N}-1},{\eta }_{\kappa }=(\kappa -1)\pi /\mathtt{N},\kappa \in {\mathbb{Z}}_1^{\mathtt{N}}.$$ A 5

Note that κ = 1 corresponds to ${{\eta }_{\kappa }}_1=0$ for the Neumann case, so that $a_{(1)\nu }=a_{(1)1}=1/\sqrt{\mathtt{N}}$, i.e. along the vertical direction it is a uniform translation of all $\mathtt{N}$ rows.

Appendix B. $\mathcal{R}$ and $\mathcal{T}$ for finite cracks

The reflectance $\mathcal{R}$ (resp. transmittance $\mathcal{T}$) is the ratio of the energy flux in the outgoing wave ahead (resp. behind) of the cracks to the energy flux carried by the incident wave [12]. Since the cracks in the array are staggered with respect to each other, the energy flux is calculated across a boundary shown in figure 6 with thick solid lines (X is taken far away from the crack tips). Along the boundary BC (resp., B’C’), there are $\mathtt{M}$ vertical bonds and $\mathtt{N}$ horizontal bonds that are intersected. Following [12], the incident energy flux across the segment B’C’ (figure 6) is

$$\begin{array}{rl}{\mathtt{W}}^i& =\text{Re}{\sum }_{n=0}^{\mathtt{N}-1}((u_{X+1,n}^i-u_{X,n}^i)\overline{(-i\omega u_{X,n}^i)})\\ \hspace{0.17em}& +\text{Re}{\sum }_{m=1}^{\mathtt{M}}((u_{X+m,\mathtt{N}-1}^i-\psi u_{X+m-\mathtt{M},0}^i)\overline{(-i\omega \psi u_{X+m-\mathtt{M},0}^i)},\end{array}$$ B 1

and the energy flux in the outgoing wave ahead of the cracks, that is, the energy flux in the reflected wave across the boundary B’C’ (as in figure 6), is

$$\begin{array}{rl}{\mathtt{W}}^r& =\text{Re}{\sum }_{n=0}^{\mathtt{N}-1}((u_{X,n}-u_{X+1,n})\overline{(-i\omega u_{X+1,n})})\\ \hspace{0.17em}& +\text{Re}{\sum }_{m=1}^{\mathtt{M}}((\psi u_{X+m-\mathtt{M},0}-u_{X+m,\mathtt{N}-1})\overline{(-i\omega u_{X+m,\mathtt{N}-1})}.\end{array}$$ B 2

Similarly, the energy flux in the outgoing wave behind the cracks, that is, the energy flux carried by the transmitted wave across the boundary BC (figure 6), is

$$\begin{array}{rl}{\mathtt{W}}^t& =\text{Re}{\sum }_{n=0}^{\mathtt{N}-1}((u_{-X,n}^t-u_{-X-1,n}^t)\overline{(-i\omega u_{-X-1,n}^t)}\\ \hspace{0.17em}& +\text{Re}{\sum }_{m=1}^{\mathtt{M}}((u_{-X-1-m,\mathtt{N}-1}^t-\psi u_{-X-1+m-\mathtt{M},0}^t)\overline{(-i\omega \psi u_{-X-1+m-\mathtt{M},0}^t)}.\end{array}$$ B 3

Using (B 1), (B 2) and (B 3), the reflectance and transmittance can be written as

$$\mathcal{R}=\frac{{\mathtt{W}}^r}{{\mathtt{W}}^i}\text{and}\mathcal{T}=\frac{{\mathtt{W}}^t}{{\mathtt{W}}^i},$$ B 4

respectively. For y > y₀, in the numerator of (3.18),

$$\begin{array}{rl}\hspace{0.17em}& {\mathtt{U}}_{\mathtt{N}-|y-y_0|-1}+{\mathtt{U}}_{|y_0-y|-1}{(\psi z^{-\mathtt{M}})}^{\text{sign}(y-y_0)}\\ \hspace{0.17em}& ={({\lambda }^{-1}-\lambda )}^{-1}({\lambda }^{-\mathtt{N}+|y-y_0|}-{\lambda }^{\mathtt{N}-|y-y_0|}+{\lambda }^{-\mathtt{N}\text{sign}(y-y_0)}({\lambda }^{-|y-y_0|}-{\lambda }^{|y-y_0|}))\\ \hspace{0.17em}& ={({\lambda }^{-1}-\lambda )}^{-1}({\lambda }^{-\mathtt{N}}-{\lambda }^{\mathtt{N}}){\lambda }^{-(y-y_0)},\end{array}$$ B 5

while, analogously, for y < y₀, in the numerator of (3.18),

$${\mathtt{U}}_{\mathtt{N}-|y-y_0|-1}+{\mathtt{U}}_{|y_0-y|-1}{(\psi z^{-\mathtt{M}})}^{\text{sign}(y-y_0)}={({\lambda }^{-1}-\lambda )}^{-1}({\lambda }^{-\mathtt{N}}-{\lambda }^{\mathtt{N}}){\lambda }^{-(y-y_0)}.$$ B 6

Hence, for $y\in {\mathbb{Z}}_0^{\mathtt{N}-1}$, ${\mathtt{U}}_{\mathtt{N}-|y-y_0|-1}+{\mathtt{U}}_{|y_0-y|-1}{(\psi z^{-\mathtt{M}})}^{\text{sign}(y-y_0)}=a_y{\lambda }^{-\mathtt{N}+y_0+1}/C_{\mathtt{N}}.$ In the context of expression (3.18) of the Green’s function ${\mathcal{G}}_y^{\text{F}}$, let

$$D=2{\mathtt{T}}_{\mathtt{N}}-(\psi z^{-\mathtt{M}}+{\psi }^{-1}{z}^{\mathtt{M}})=D_{+}{D}_{-},$$ B 7

where D₋ (resp. D₊) contains those zeros of D that lie outside (resp. inside) the unit circle for ${\omega }_2>0$ (2.6). Recall that ${\omega }_2\to 0+$. Thus, for x → −∞, it follows from (3.19) and (3.18), by an application of the complex residue calculus and (B 7), that

$${\mathcal{G}}_{x,y}\sim -\sum _{D_{-}=0}\frac{z^{-x_0-1}}{D^{\prime }(z)}\frac{{\lambda }^{-\mathtt{N}+y_0+1}}{C_{\mathtt{N}}}{a}_y{z}^x=-\sum _{D_{-}=0}\frac{z^{-x_0-1}}{D^{\prime }(z)}\frac{{\lambda }^{-\mathtt{N}}-{\lambda }^{\mathtt{N}}}{{\lambda }^{-1}-\lambda }{\lambda }^{y_0}{z}^x{\lambda }^{-y}.$$ B 8

Similarly, for x → +∞, ${\mathcal{G}}_{x,y}\sim {\sum }_{D_{+}=0}(z^{-x_0-1}/D^{\prime }(z))({\lambda }^{-\mathtt{N}}-{\lambda }^{\mathtt{N}})/({\lambda }^{-1}-\lambda ){\lambda }^{y_0}{z}^x{\lambda }^{-y}.$ Let

$$f(z)={\sum }_{l=-{\mathtt{N}}_b}^{-1}({\mathtt{v}}_l+{\mathtt{v}}_l^i)z^{-l}.$$ B 9

Thus, for x → −∞,

$$\begin{array}{rl}{u}_{x,y}& \sim \sum _{D_{+}=0}\frac{z^{-1}}{D^{\prime }(z)}\frac{{\lambda }^{-\mathtt{N}}-{\lambda }^{\mathtt{N}}}{{\lambda }^{-1}-\lambda }{\lambda }^{\mathrm{\aleph }}{z}^x{\lambda }^{-y}(1-{\lambda }^{-1})\sum _{l=-{\mathtt{N}}_b}^{-1}({\mathtt{v}}_l+{\mathtt{v}}_l^i)z^{-l}\\ \hspace{0.17em}& \sim \sum _{D_{+}=0}{K}_z{z}^x{a}_y,\hspace{0.17em}{K}_z=\frac{(1-{\lambda }^{-1})z^{-1}}{D^{\prime }(z)}\frac{{\lambda }^{-\mathtt{N}+\mathrm{\aleph }+1}}{C_{\mathtt{N}}}f(z).\end{array}$$ B 10

Similarly, for x → +∞, $u_{x,y}\sim -{\sum }_{D_{-}=0}{K}_z{z}^x{a}_y.$ The incident energy flux in a specific mode is given by $-(1/2)|\tilde{K}{|}^2{\omega }^2{\mathcal{V}}^i$, where $-{\mathcal{V}}^i$ is the group velocity inside the ‘waveguide’. Hence, the reflectance and transmittance of the scatterer array are given by

$$\mathcal{R}=\frac{\sum E_{\text{reflected}}^{\tilde{r}}}{E_{\text{incident}}^{\tilde{r}}}=\frac{{\sum }_{D_{+}=0}|K_z{|}^2\mathcal{V}}{-|\tilde{K}{|}^2{\mathcal{V}}^i}\text{and}\mathcal{T}=\frac{\sum E_{\text{transmitted}}^{\tilde{r}}}{E_{\text{incident}}^{\tilde{r}}}=\frac{\sum _{D_{+}=0}|K_z{|}^2\mathcal{V}}{-|\tilde{K}{|}^2{\mathcal{V}}^i},$$ B 11

respectively. In this expression of the reflectance $\mathcal{R}$,

$$\begin{array}{rl}\frac{|K_z{|}^2\mathcal{V}}{-|\tilde{K}{|}^2{\mathcal{V}}^i}& =-\frac{\mathcal{V}}{{\mathcal{V}}^i}{\left|\frac{(1-{\lambda }^{-1})z^{-1}}{D^{\prime }(z)}\right|}^2{\left|\frac{{\lambda }^{-\mathtt{N}+\mathrm{\aleph }+1}}{C_{\mathtt{N}}}\right|}^2|f(z){|}^2\\ \hspace{0.17em}& =\frac{iz}{-2\omega {\mathtt{U}}_{\mathtt{N}-1}}\frac{1}{{\mathcal{V}}^i}\frac{\mathcal{H}}{\overline{D_{-}(z)D_{+}^{\prime }(z)}}(1-\frac{1}{4}{(\psi z^{-\mathtt{M}}+{\psi }^{-1}{z}^{\mathtt{M}})}^2)|f(z){|}^2.\end{array}$$ B 12

Figure 6.

Schematic of a square lattice with the array of staggered finite cracks for the calculation of energy flux in the incident, reflected and transmitted waves across the boundaries B’C’ and BC. (Online Version in colour.)

Appendix C. $\mathcal{R}$ and $\mathcal{T}$ for semi-finite cracks: bulk incidence

For the purpose of the manipulations presented below, consider $\vartheta (z,\omega )$, i.e. as a function of z and $\omega$; the same consideration applies to other relevant functions. Thus, the following relations are obtained concerning the group velocity of wave modes, on either side of the scatterer,

$$\frac{v}{{\mathcal{N}}^{\prime }(z)}=\frac{iz}{\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{N}(z,\omega )}{|}_{z=\exp (-i\xi )}\text{and}\frac{v}{{\mathcal{D}}^{\prime }(z)}=\frac{iz}{\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{D}(z,\omega )}{|}_{z=\exp (-i\xi )}.$$ C 1

Note that, by reference to (4.11), we have $\mathcal{N}(z)=\mathcal{H}(z){\mathtt{U}}_{\mathtt{N}-1}(\vartheta ),$ $\mathcal{D}(z)=2{\mathtt{T}}_{\mathtt{N}}(\vartheta )-(z^{\mathtt{M}}{z}_P^{-\mathtt{M}}{\lambda }_P^{\mathtt{N}}+z^{-\mathtt{M}}{z}_P^{\mathtt{M}}{\lambda }_P^{-\mathtt{N}}),$ where $\vartheta =\frac{1}{2}\mathcal{Q}(z),$ $\mathcal{Q}=\mathcal{H}+2.$ Invoking (A 1)₁ (while using ${\mathtt{U}}_n^{\prime }=(n+1){\mathtt{T}}_{n+1}/({\vartheta }^2-1)$ when ${\mathtt{U}}_n$ is zero) with z such that $\mathcal{N}(z)=0$ (let $\mathcal{R}(z):=\mathcal{H}(z)+4$, recall (3.9)),

$$\begin{array}{rl}\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{N}(z,\omega )& =\mathcal{H}(z)\mathtt{N}\frac{2\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{Q}(z)\right)}{\mathcal{Q}{(z)}^2-4}{\mathtt{T}}_{\mathtt{N}}(\vartheta )\hspace{0.17em}\text{or}\hspace{0.17em}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{H}(z)\right){\mathtt{U}}_{\mathtt{N}-1}(\vartheta )\\ \hspace{0.17em}& =-4\omega \frac{\mathcal{H}(z)\mathtt{N}{\mathtt{T}}_{\mathtt{N}}(\vartheta )}{\mathcal{Q}{(z)}^2-4}\text{ or }-2\omega {\mathtt{U}}_{\mathtt{N}-1}(\vartheta )=-4\omega \mathtt{N}\frac{{\mathtt{T}}_{\mathtt{N}}(\vartheta )}{\mathcal{R}(z)}\hspace{0.17em}\text{or}\hspace{0.17em}-2\omega {\mathtt{U}}_{\mathtt{N}-1}(\vartheta ).\end{array}$$ C 2

Similarly, invoking (C 1)₂, while using ${\mathtt{T}}_n^{\prime }=n{\mathtt{U}}_{n-1}$ with z such that $\mathcal{D}(z)=0$, $\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{D}(z,\omega )=\mathtt{N}(\frac{\mathrm{\partial }}{\mathrm{\partial }\omega }\mathcal{Q}){\mathtt{U}}_{\mathtt{N}-1}(\vartheta )=-2\omega \mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta ).$ Hence, behind and ahead of the scatterer, respectively, $v/{\mathcal{N}}^{\prime }(z)=iz\mathcal{R}/(-4\omega \mathtt{N}{\mathtt{T}}_{\mathtt{N}}(\vartheta ))$ or $iz/(-2\omega {\mathtt{U}}_{\mathtt{N}-1}(\vartheta )),$ $v/{\mathcal{D}}^{\prime }(z)=iz/(-2\omega \mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta )).$ Using the normal modes for a lattice strip of width $\mathtt{N}$, which is free on the upper and lower boundary (using $\sin \mathtt{N}{\eta }_{\kappa }=0$, which implies $\cos (\mathtt{N}-\frac{1}{2}){\eta }_{\kappa }=\cos \mathtt{N}{\eta }_{\kappa }\cos \frac{1}{2}{\eta }_{\kappa }$), $|a_{-({\kappa }_z)0}-{\psi }^{-1}{z}^{\mathtt{M}},a_{-({\kappa }_z)\mathtt{N}-1}{|}^2=\frac{1}{2\mathtt{N}}\cos \mathtt{N}{\eta }_{\kappa }\mathcal{R}(z)\mathcal{D}(z).$ But ${\mathtt{U}}_{\mathtt{N}-1}=0$ implies that ${\mathtt{U}}_{\mathtt{N}-2}=-{\mathtt{U}}_{\mathtt{N}},$ i.e. $2{\mathtt{T}}_{\mathtt{N}}=2{\mathtt{U}}_{\mathtt{N}}.$ More directly, $\vartheta =\cos \frac{j\pi }{\mathtt{N}},$ so that $2{\cos }^2\frac{1}{2}{\eta }_{\kappa }=1+\cos {\eta }_{\kappa }=1+\vartheta =2{\cos }^2\frac{j\pi }{2\mathtt{N}},{\eta }_{\kappa }=\pm \frac{j\pi }{\mathtt{N}}$. Also $(\vartheta -1){\mathtt{U}}_{\mathtt{N}}=\frac{1}{2}({\mathtt{U}}_{\mathtt{N}+1}+{\mathtt{U}}_{\mathtt{N}-1}-2{\mathtt{U}}_{\mathtt{N}})$. Note that ${\mathtt{U}}_{\mathtt{N}}^2-{\mathtt{U}}_{\mathtt{N}-1}{\mathtt{U}}_{\mathtt{N}+1}=1$, which implies ${\mathtt{U}}_{\mathtt{N}}^2=1$ when ${\mathtt{U}}_{\mathtt{N}-1}=0,$ so that ${\mathtt{T}}_{\mathtt{N}}={\mathtt{U}}_{\mathtt{N}}=\pm 1.$ The transmittance is

$$\begin{array}{rl}\mathcal{T}& =\frac{(\mathtt{N}v{({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2}{\sum }_{z\in {\mathcal{Z}}^{-}}\frac{|-2i\sin \frac{1}{2}{\eta }_{\kappa }({\vartheta }_{\mathtt{N}}(z_P)){|}^2}{\frac{1}{2\mathtt{N}}\cos \mathtt{N}{\eta }_{\kappa }\mathcal{R}(z)\mathcal{D}(z)}\frac{\mathcal{D}(z)}{\overline{{\mathcal{N}}^{\prime }(z)}}(v(\xi ))\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{1}{\overline{z}-{\overline{z}}_P}\frac{1}{z-z_P}\\ \hspace{0.17em}& =\frac{(\mathtt{N}v{({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{-}}\frac{4{\sin }^2\frac{1}{2}{\eta }_{\kappa }({\vartheta }_{\mathtt{N}}(z_P))}{\frac{1}{2\mathtt{N}}\cos \mathtt{N}{\eta }_{\kappa }\mathcal{R}(z)\mathcal{D}(z)}\frac{-i\overline{z}\mathcal{D}(z)}{\overline{\mathtt{N}\frac{2(-2\omega )}{\mathcal{R}}{\mathtt{T}}_{\mathtt{N}}(\vartheta )}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{1}{\overline{z}-{\overline{z}}_P}\frac{1}{z-z_P}\\ \hspace{0.17em}& =-2i\frac{{(\mathtt{N}v({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{-}}\frac{\mathcal{H}(z_P)}{{\mathtt{T}}_{\mathtt{N}}(\vartheta )}\frac{1}{\overline{4\omega {\mathtt{T}}_{\mathtt{N}}(\vartheta )}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{\overline{z}}{\overline{z}-{\overline{z}}_P}\frac{1}{z-z_P}\\ \hspace{0.17em}& =\frac{1}{2}i\frac{{\omega }^{-1}\mathcal{H}(z_P)}{(v({\xi }_P))|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2\mathtt{N}}\sum _{z\in {\mathcal{Z}}^{-}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{N}}_{-}^{\prime }(z){\mathcal{D}}_{+}(z)}\frac{z_P}{{(z-z_P)}^2}.\end{array}$$ C 3

Using (A 2), (A 3) (using $2{\mathtt{T}}_{\mathtt{N}}(\vartheta )=({\psi }^{-1}{z}^{\mathtt{M}}+\psi z^{-\mathtt{M}})=(z^{\mathtt{M}}{z}_P^{-\mathtt{M}}{\lambda }_P^{\mathtt{N}}+z^{-\mathtt{M}}{z}_P^{\mathtt{M}}{\lambda }_P^{-\mathtt{N}})$), $|a_{+({\kappa }_z)0}-{\psi }^{-1}{z}^{\mathtt{M}}{a}_{+({\kappa }_z)\mathtt{N}-1}{|}^2=-\frac{\mathcal{N}(z)}{\mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta )}.$ In (4.19), ${\mathcal{D}}^{\prime }(z)=2{\vartheta }^{\prime }(z){\mathtt{T}}_{\mathtt{N}}^{\prime }(\vartheta )={\mathcal{H}}^{\prime }(z)\mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta ),$ ${\mathcal{N}}^{\prime }(z)=\mathcal{H}(z){\vartheta }^{\prime }(z){\mathtt{U}}_{\mathtt{N}-1}^{\prime }(\vartheta )=\mathcal{H}(z){\vartheta }^{\prime }(z){\mathtt{U}}_{\mathtt{N}-1}^{\prime }(\vartheta )=\mathcal{H}(z)\mathtt{N}\frac{2{\mathcal{H}}^{\prime }}{\mathcal{H}\mathcal{R}}{\mathtt{T}}_{\mathtt{N}}(\vartheta )=\mathtt{N}\frac{2{\mathcal{H}}^{\prime }}{\mathcal{R}}{\mathtt{T}}_{\mathtt{N}}(\vartheta )$ or ${\mathcal{H}}^{\prime }(z){\mathtt{U}}_{\mathtt{N}-1}(\vartheta ),$ where $\vartheta =\cos \theta ,\theta \in [0,\pi ]$. The reflectance is

$$\begin{array}{rl}\mathcal{R}& =\frac{{(\mathtt{N}v({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{+}}\frac{|-2i\sin \frac{1}{2}{\eta }_{\kappa }({\vartheta }_{\mathtt{N}}(z_P)){|}^2}{-(\mathcal{N}(z)/\mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta ))}\frac{1}{\overline{z}-{\overline{z}}_P}\frac{\overline{{\mathcal{D}}_{-}(z)}\overline{{\mathcal{N}}_{+}(z)}}{\overline{{\mathcal{D}}^{\prime }(z)}}\frac{1}{z-z_P}\frac{\mathcal{N}(z)}{{\mathcal{D}}_{+}^{\prime }(z){\mathcal{N}}_{-}(z)}|v(\xi )|\\ \hspace{0.17em}& =-\frac{{(\mathtt{N}v({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{+}}\frac{\mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta )|-2i\sin \frac{1}{2}{\eta }_{\kappa }({\vartheta }_{\mathtt{N}}(z_P)){|}^2}{(\overline{z}-{\overline{z}}_P)(z-z_P)}\frac{\overline{{\mathcal{D}}_{-}(z)}\overline{{\mathcal{N}}_{+}(z)}}{{\mathcal{D}}_{+}^{\prime }(z){\mathcal{N}}_{-}(z)}\frac{-i\overline{z}}{-2\omega \mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta )}\\ \hspace{0.17em}& =-\frac{{(\mathtt{N}v({\xi }_P))}^{-1}}{|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2}\sum _{z\in {\mathcal{Z}}^{+}}\mathcal{H}(z_P)\frac{1}{\overline{z}-{\overline{z}}_P}\frac{1}{z-z_P}\frac{\overline{{\mathcal{D}}_{-}(z)}\overline{{\mathcal{N}}_{+}(z)}}{{\mathcal{D}}_{+}^{\prime }(z){\mathcal{N}}_{-}(z)}\frac{-i\overline{z}}{-2\omega }\\ \hspace{0.17em}& =\frac{1}{2}i\frac{{\omega }^{-1}\mathcal{H}(z_P)}{(v({\xi }_P))|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2\mathtt{N}}\sum _{z\in {\mathcal{Z}}^{+}}\frac{\overline{{\mathcal{D}}_{-}(z){\mathcal{N}}_{+}(z)}}{{\mathcal{D}}_{+}^{\prime }(z){\mathcal{N}}_{-}(z)}\frac{z_P}{{(z-z_P)}^2}.\end{array}$$ C 4

For the incident wave from the bulk lattice, since ${\mathcal{D}}_{-}(z_P)=0$, $v({\xi }_P)$ is given by $v({\xi }_P)=i{z}_P{\mathcal{D}}_{+}(z_P){\mathcal{D}}_{-}^{\prime }(z_P)/(-2\omega \mathtt{N}{\mathtt{U}}_{\mathtt{N}-1}(\vartheta (z_P))).$ The coefficient in front of the sum is simplified as

$$\frac{1}{2}{\omega }^{-1}i\frac{\mathcal{H}(z_P)}{(v({\xi }_P))|{{\mathcal{L}}_{\mathtt{N}}}_{\hspace{0.17em}+}(z_P){|}^2\mathtt{N}}=\frac{z_P{\mathcal{N}}_{-}(z_P){\mathcal{D}}_{+}(z_P)}{\overline{{\mathcal{D}}_{-}^{\prime }(z_P){\mathcal{N}}_{+}(z_P)}}.$$ C 5

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Authors' contributions

Both authors contributed equally to the writing of the manuscript, the problem formulation as well as the discussion/interpretation of the results. The manuscript is partly based on some results found in the PhD thesis of G.M. [42]. The semi-analytical approach to the finite crack problem and its direct numerical solution is due to G.M. The application of the Wiener–Hopf method to the semi-infinite crack problem is due to B.L.S.

Competing interests

We declare we have no competing interest.

Funding

G.M. acknowledges MHRD (India) and IITK for providing financial assistance in the form of a Senior Research Fellowship. B.L.S. acknowledges the partial support of SERBMATRICS grant no. MTR/2017/000013.