# Edge resonance in discrete elastic waveguides and its convergence to isotropic continuous media

G. Carta, M.J. Nieves, M. Brun, Vincent Pagneux

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G. Carta, M.J. Nieves, M. Brun, Vincent Pagneux. Edge resonance in discrete elastic waveguides and its convergence to isotropic continuous media. International Journal of Engineering Science, 2025, 215, pp.104307. 10.1016/j.ijengsci.2025.104307. hal-05383695

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Full length article

# Edge resonance in discrete elastic waveguides and its convergence to isotropic continuous media

G. Carta<sup>a, ID</sup>, M.J. Nieves<sup>b, ID, *</sup>, M. Brun<sup>a, ID</sup>, V. Pagneux<sup>c, ID</sup>

## ARTICLE INFO

### Keywords:

Edge resonance

Micro-structured elastic waveguide

Triangular lattice

Homogenisation

Mode matching

Z-transform

## ABSTRACT

We discuss the problem of edge resonance for Lamb waves in a semi-infinite discrete elastic strip, represented by a triangular lattice. In analogy with the reflection problem in the corresponding continuum, for real frequencies the edge resonance phenomenon for the lattice strip is characterised by localised vibrations at its free edge. By characterising the reflection coefficient in the complex frequency plane, we then verify the existence of a *complex edge resonance frequency* for the lattice, associated with a mode of the homogeneous problem without incident wave. Importantly, when the number of rows in the strip of fixed width is large, we show that the lattice edge resonance frequency converges to the corresponding frequency in the analogous continuum problem for the effective strip. Interestingly, convergence to the *complex edge resonance frequency* is monotonic only with respect to its real part, while its imaginary part exhibits a minimum absolute value, numerically undistinguishable from zero, for a lattice strip with 65 rows in the transverse direction.

## 1. Introduction

Resonance is a fundamental phenomenon in mechanics that occurs when a system is excited at a frequency close to one of its natural frequencies, leading to significantly amplified oscillations. As a consequence, resonance can cause excessive and hazardous vibrations in structures, that could produce damage, malfunctions (Randall, 2011) or possible failure (Slepyan, 2002). While resonance can be detrimental to structural integrity, it offers a useful mechanism for concentrating vibrations or stresses in a confined region (*energy localisation*), for converting wasted vibrational energy into usable power (*energy harvesting*) (Wei & Jing, 2017), and for manipulating and inhibiting dynamic failure processes (Zhang & Wang, 2024).

An important tool in realising these applications is the use of structured mechanical waveguides that open new pathways in controlling mechanical waves. Often, these systems are treated as periodic arrays of point masses connected by elastic links that supply different responses. These models can be used to provide insights into wave propagation and other physical processes within the micro-level of a medium that continuum models are not capable of predicting, e.g. multi-scale dynamic processes in material fracture (Piccolroaz, Gorbushin, Mishuris, & Nieves, 2020; Slepyan, 2002). The discrete models may also be adopted to study wave motion in civil and mechanical engineering systems. The associated solutions in lattice models admit finite physical fields, i.e. stresses in the vicinity of defects or cracks, that appear infinite in analogous continuum models. In addition, a periodic micro-structured medium presents a finite set of dispersion curves and its study can be performed almost completely through analytical methods based on finite algebraic techniques.

<footnote><sup>a</sup> University of Cagliari, Department of Mechanical, Chemical and Materials Engineering, Cagliari, 09123, Italy</footnote>

<footnote><sup>b</sup> Keele University, School of Computer Science and Mathematics, Keele, ST5 5BG, UK</footnote>

<footnote><sup>c</sup> Laboratoire d'Acoustique de l'Université du Maine (LAUM) - CNRS UMR 6613, Le Mans, 72085, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: m.nieves@keele.ac.uk (M.J. Nieves).</footnote>

<footnote>https://doi.org/10.1016/j.ijengsci.2025.104307</footnote>

<footnote>Received 16 March 2025; Received in revised form 1 May 2025; Accepted 7 May 2025</footnote>

<footnote>Available online 20 June 2025</footnote>

<footnote>0020-7225/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</footnote>
The scalar mechanical wave, representing out-of-plane wave propagation in a strip, was firstly examined in Eremeyev and Sharma (2019) and Sharma (2017, 2018). Regarding the vectorial problem, where the motion of the system is described by a vector of in-plane displacements, we mention the recent papers (Carta, Nieves, & Brun, 2023a, 2023b), where the propagation of Lamb waves in lattice strips of various types was analysed in detail. Rayleigh waves for discrete elastic half-spaces have been studied in Nieves, Carta, Pagneux, and Brun (2020) and Slepyan (2002). In connection with the wave scattering in discrete media, the interaction of in-plane waves with inertial defects in structured elastic systems has been recently addressed in Nieves and Sharma (2024). Similar models based on discrete systems have been shown to describe wave transport in carbon nanotubes (Sharma, 2018), high-contrast laminates and sandwich plates (Aydin, Erbas, Kaplunov, & Prikazchikova, 2018; Kaplunov, Prikazchikov, & Prikazchikov, 2017; Kaplunov, Prikazchikova, & Alkinidri, 2021), as well as elastic media with thin, contrasting layers or substrates along their boundaries (Fu, Kaplunov, & Prikazchikov, 2020; Kaplunov, Prikazchikov, & Sultanova, 2019; Sharma & Eremeyev, 2019). Embedding locally resonant properties and periodic materials within the surfaces of elastic systems have enabled the design of seismic metasurfaces (Amanat, Rafiee-Dehkharghani, Bitaraf, & Bansal, 2022; Argatov & Sabina, 2024; Pu, Palermo, Cheng, Shi, & Marzani, 2020). Metamaterial design approaches have recently been exploited to develop a deeper understanding of wave transport, scattering and filtering at interfaces in quantum graphs (Lawrie, Tanner, & Chaplain, 2025; Lawrie, Tanner, & Chronopoulos, 2022) as well as localisation of modes in finite locally resonant discrete systems (Ammari et al., 2024).

In the design of mechanical waveguides, special attention is devoted to edge resonance, since the latter can have important implications in acoustic sensing and structural health monitoring. Edge resonance is often observed in thin structures, such as plates and rods, and it occurs due to the interaction of sound or vibrations with an edge of the waveguide. The first experimental evidence of quasi-resonance, characterised by large vibrations at the boundary, was obtained in 1956 in barium titanate disks by means of optical interference techniques (Shaw, 1956). Successive experiments have confirmed the existence of this phenomenon in rods of different cross-sections (de Billy, 1996) and plates (Cès, Clorennec, Royer, & Prada, 2011; Le Clezio, Valentin Predoi, Castaings, Hosten, & Rousseau, 2003).

Analytical and numerical models for the scattering of waves at the edge of a semi-infinite plate, involving the interaction of Lamb modes propagating along the plate from infinity and interacting with the plate edge, have revealed real frequencies where a localised vibration at the plate edge can be sustained (Auld & Tsao, 1977; Gregory & Gladwell, 1983; Koshiba, Karakida, & Suzuki, 1983; Torvik, 1967). Typically, the real quasi-resonance frequency for the plate edge appears in the mathematical formulation as a result of a complex edge resonance frequency situated in the vicinity of the real frequency axis. At this *complex resonance frequency*, the coefficients for the reflected modes induced by the incident wave have a pole. This effect is linked to the algebraic system that enables the identification of the reflection coefficients in the scattering problem. Further, at such a complex frequency, there is an eigenmode of the semi-infinite elastic strip written completely in terms of reflected modes and that solves the homogeneous problem for the strip without an incident wave, see also Evans, Levinit, and Vassiliev (1994).

In continuous elastic media, it has been proved that there exist exceptional Poisson ratios of the elastic strip, namely $v = 0$ (Roitberg, Vassiliev, & Weidl, 1998) and $v \sim 0.2248$ (Pagneux, 2006; Zernov, Pichugin, & Kaplunov, 2006) (with $v = 0.29$ for plane stress Kaplunov, Pichugin, & Zernov, 2009), where the edge resonance can occur at real frequencies. This is due to the decoupling of the propagating Lamb (symmetric) mode and all the other evanescent modes composing the reflected field, leading to an edge resonance where energy transport is absent (Lawrie, 2023; Pagneux, 2006). This effect has also been investigated for semi-infinite elastic cylinders (Pagneux, 2012). An extensive review on edge resonance in continuous elastic systems is given in Lawrie and Kaplunov (2011). The determination of edge resonance frequencies for the continuous elastic semi-infinite strip is still an active area of research and, more recently, a mode matching approach has led to an exact condition for predicting the vibration frequencies and Poisson ratios that lead to edge resonances (Lawrie, 2023). Further, a new solution method to determining the reflected field in the continuum problem has been proposed in Davey, Assier, and Abrahams (2022), which improves upon convergence issues near the plate corners that is encountered in the classical modal expansion of this field in terms of Lamb waves.

On the other hand, this type of edge resonance in discrete elastic in-plane systems has not been deeply investigated so far. The aim of this paper is to bridge this gap by (i) determining the complex edge resonance frequencies in elastic lattice strips with different numbers of rows, and (ii) understanding the convergence of the lattice strip edge resonance frequency to the analogous quantity in the continuum limit.

The structure of the paper is outlined as follows. In Section 2, we discuss the formulation of the approach based on mode matching, focussing the attention on symmetric Lamb modes in the time-harmonic regime. In Section 3, we obtain the solution by means of the Z-transform and we introduce dynamic homogenisation techniques to derive the continuum limit. In Section 4, we illustrate the main results of the work concerning resonance at the free edge of the strip: first, in Section 4.1, we describe extensively the case when the number of rows is 5; second, in Section 4.2, we present the complex edge resonance frequencies for increasing values of the number of rows and their trends to the continuum limit. In Section 5, we summarise the key findings of the study. Finally, in the Appendices, we include the analytical details of the two methods developed in this work.

## 2. Problem formulation based on mode matching

We consider a two-dimensional triangular lattice, made of point masses *M* linked by elastic springs of stiffness $\Gamma$ and length *L*. The lattice is semi-infinite along the longitudinal $x_1$-direction and finite along the transverse $x_2$-direction, as shown in the two typical examples of Fig. 1. At the three boundaries of the lattice domain, the masses are not subjected to external forces and are free to move, namely we assume stress-free boundary conditions.
<figure><img src="image_3.png" /><figcaption>Fig. 1. 2D lattice, consisting of a triangular array of masses *M* connected by elastic links of stiffness *Γ* and length *L*, that is semi-infinite in the *x*₁-direction and finite in the *x*₂-direction. Free Neumann conditions are imposed at the boundaries. The *m*th elementary cell for the medium is represented as the shaded ‘zig-zag’ region. Two types of lattices are shown, characterised by the number of rows: (a) *N* = 5 and (b) *N* = 7. The convention used for node enumeration in these systems is also provided.</figcaption></figure>

Each lattice particle is identified by two integers (*m*, *n*)<sup>Τ</sup>, with *m*, *n* ∈ N<sub>0</sub>, *m* ≥ 0 and 0 ≤ *n* ≤ *N* − 1, where *N* is the number of rows (see Fig. 1). In this paper, we assume that (i) *N* is odd and (ii) *N* ≥ 5. Here (i) allows us to preserve an axis of symmetry in the medium located along the row *n* = ( *N* − 1)/2, *N* ≥ 5, regardless of the strip width. Assumption (ii) avoids the trivial case *N* = 1, corresponding to a single mass-spring chain, and the case *N* = 3, that possesses unique dynamic features not encountered for *N* ≥ 5 and worth separate attention.

Depending on the value of *N*, the geometry of the termination of the lattice strip differs. For instance, in Fig. 1(a) and (b) we present the geometries of the lattices with *N* = 5 and *N* = 7, respectively, representing the more general cases *N* ≡ 1 (mod 4) and *N* ≡ 3 (mod 4) ('mod' implies modulo arithmetic is used), respectively. In accordance with Fig. 1(a) and (b), the lattice with *N* + 2 rows is constructed by adding one row below and one row above to the lattice with *N* rows and, with respect to this procedure, we alternate the design between the two configurations shown in Fig. 1(a) and (b) for *N* = 5 and *N* = 7 when increasing *N*.

2.1. Recurrence relation, dispersion equation and boundary conditions at the free edge

2.1.1. Time-dependent equations of motion

We denote by **u**<sup>(m,n)</sup>(t) = (u<sub>1</sub><sup>(m,n)</sup>(t), u<sub>2</sub><sup>(m,n)</sup>(t))<sup>T</sup> the in-plane displacement vector of the node at (m,n), that is a function of time t.
The equation of motion for each node composing the strip takes the general form

$$M\ddot{u}^{(m,n)} = f^{(m,n)}, \quad m \ge 0, \quad 0 \le n \le N-1, \qquad (1)$$

where the dot atop of the displacement vector represents the time derivative. Eq. (1) follows from the linear momentum balance for each node shown in Fig. 1(a) and (b). The right-hand side in (1) corresponds to the resultant force experienced by the node (m, n) when taking into account the elastic springs connected to the nearest neighbours. The form of the force $f^{(m,n)}$ varies with respect to the row considered, how the strip terminates at the edge and the number of elastic interactions that each node experiences. The representation of this force for any possible case is given in Appendix A.

### 2.1.2. Steady-state vibrations and normalisation

In the time-harmonic regime, the displacement vector and forcing term of each node in (1) are assumed to be given by

$$u^{(m,n)} = U^{(m,n)}e^{-i\omega t} \quad \text{and} \quad f^{(m,n)} = F^{(m,n)}e^{-i\omega t}, \qquad (2)$$

respectively, where $\omega$ stands for the radian frequency, while $U^{(m,n)}$ and $F^{(m,n)}$ denote the displacement and force amplitude vectors.

From the mass-spring system characterised by $M$, $\Gamma$ and $L$, we consider the normalisations

$$U = \tilde{U}L, \quad t = \tilde{t}\sqrt{M/\Gamma}, \quad F = \tilde{F}\Gamma L, \quad \omega = \tilde{\omega}\sqrt{\Gamma/M}, \qquad (3)$$

where the symbol tilde is appended to denote a non-dimensional quantity. In the following, the symbol tilde will be omitted and all the variables will be assumed to be dimensionless.

Considering time-harmonic conditions, defined by (2), and the normalisations (3), the equations of motion for the lattice particles take the form:

$$-\omega^2 U^{(m,n)} = F^{(m,n)}, \quad m \ge 0, \quad 0 \le n \le N-1. \qquad (4)$$

The expressions for the steady-state forces $F^{(m,n)}$, that depend on the displacement amplitudes of the node at (m, n) and its nearest neighbours, are provided in Appendix A.

### 2.1.3. Modal analysis inside the strip

After some algebraic manipulations, the Eqs. (4) in the interior of the strip ($m \ge 1$) may also be written in the form of a linear second-order vector recurrence relation with constant matrix coefficients:

$$(B + \omega^2 I)U_m + JU_{m+1} + J^T U_{m-1} = 0, \quad m \ge 1, \qquad (5)$$

where the column vector

$$U_m = ((U^{(m,0)})^T, \dots, (U^{(m,N-1)})^T)^T \qquad (6)$$

identifies the $2N$ displacements of the nodes along the $m$th elementary 'zig-zag' cell highlighted in the two examples in Fig. 1(a) and (b). This vector provides a suitable framework with which mode matching can be performed at the edge of the strip. The coefficient matrices of these vectors in (5) are of the size $2N \times 2N$, with $B = B^T$ and $I$ being the identity matrix. The general expressions of $B$ and $J$, which is not invertible, are given in Appendix B.1.

In order to separate the dependence on $m$ and $n$, we represent the displacement amplitude vector in the form

$$U^{(m,n)} = \hat{C}^{(n)} z^m, \quad \text{with} \quad \hat{C}^{(n)} = (\hat{C}_1^{(n)}, \hat{C}_2^{(n)})^T, \qquad (7)$$

where $z$ is a complex quantity that can also be written as $z = e^{ik}$. Hence, the homogeneous system (5) can be presented in the form of the vector equation

$$(B + Jz + J^T z^{-1}) C = -\omega^2 C, \qquad (8)$$

with $C = (\hat{C}_1^{(0)}, \hat{C}_2^{(0)}, \hat{C}_1^{(1)}, \hat{C}_2^{(1)}, \dots, \hat{C}_1^{(N-1)}, \hat{C}_2^{(N-1)})^T$. Here, two approaches are possible for identifying the non-trivial solutions to the preceding system.

If $z = e^{ik}$ is given, with $k$ being the real wavenumber, the preceding Hermitian eigenvalue problem may be used to determine $\omega$, with $C$ being the corresponding eigenvector.

The above procedure leads to the dispersion curves $\omega(k)$ in the $(k, \omega)$-plane for the interior of the strip ($m > 0$).

Alternatively, if the frequency $\omega$ is real and given, as is typical in the time-harmonic regime, owing to the fact that $J$ is not invertible, a singular quadratic eigenvalue problem in $z$ must be solved (with a complex eigenvalue $z$). In such a case, we may rephrase the system (8) in the form of the generalised eigenvalue problem by introducing $D = zC$. In doing so, we arrive at the expanded matrix-vector system representing a generalised eigenvalue problem:

$$\mathfrak{A}v = z\mathfrak{B}v \qquad (9)$$

with

$$v = \begin{pmatrix} C \\ D \end{pmatrix}, \quad \mathfrak{A} = \begin{pmatrix} J^T & O \\ O & -I \end{pmatrix}, \quad \mathfrak{B} = - \begin{pmatrix} B + \omega^2 I & J \\ I & O \end{pmatrix}.$$
The solution of the above singular generalised eigenvalue problem provides the complex eigenvalues $z = z(\omega)$ for given $\omega$ and the corresponding eigenvectors, that will be used for the scattering problem at the edge (see Section 2.3). Note that, if we assume a complex radian frequency $\omega + i\epsilon$, with $\epsilon \to 0^+$, we can distinguish right-going from left-going waves, characterised by $|z| < 1$ and $|z| > 1$, respectively.

### 2.1.4. Matrix equations at the strip edge

The total wave field (detailed in Section 2.3) is required to satisfy the edge conditions obtained from (4) when $m = 0$ and $0 \le n \le N - 1$. Indeed, at the edge we have

$$ (S + \omega^2 I)U_0 + JU_1 = 0. \qquad (10) $$

The representation of $S$, with $S = S^T$, is given in Appendix B.1.

We note that (10) may take an alternative form owing to the fact that the total field is a linear combination of eigenmodes satisfying (2) regardless of the value of $m$. Thus we can combine (5) and (10) to give

$$ G U_0 - J^T U_{-1} = 0, \qquad (11) $$

with the form of $G = S - B$ also provided in Appendix B.1. The preceding equation, based on the introduction of ‘ghost’ nodes with $m = -1$, physically represents the absence of elastic forces between the nodes at the edge of the strip (having multi-index $(0, n)$, $0 \le n \le N - 1$) and the ‘ghost’ nodes.

### 2.2. Symmetric conditions

We recall that the edge resonance effect for the classical continuum problem involving the semi-infinite elastic strip is known to occur with only one propagating mode ($S_0$) below the first cut-on frequency above which modes $S_j$, $j \ge 1$, can be supported (Pagneux, 2006). This motivates the use of the symmetric lattice strip studied here, with a total number of rows $N$ being odd. Thus, in going forward, we will focus on symmetric wave motion in the structured strip only and describe how the above formulation may be reduced in complexity.

Since edge resonance is associated with symmetric modes, we consider in-plane displacements that are symmetric with respect to the symmetry axis of the lattice ($n = (N - 1)/2$):

$$ \bullet U_1^{m,N-1-p} = U_1^{m,p}, \quad U_2^{m,N-1-p} = -U_2^{m,p} \quad \text{for } 0 \le p < (N-1)/2, \qquad (12a) $$

$$ \bullet U_2^{m,(N-1)/2} = 0. \qquad (12b) $$

The conditions (12) allow to halve the sizes of the displacement vectors. Introducing the subscript $s$ to denote the matrices obtained by taking the symmetric conditions (12) into account, the recurrence relation (5) becomes

$$ (B_s + \omega^2 I) V_m + J_s V_{m+1} + J_s^T V_{m-1} = 0, \quad m \ge 1, \qquad (13) $$

with $V_m = [(U^{(m,0)})^T, \dots, (U^{(m,(N-3)/2)})^T, U_1^{(m,(N-1)/2)}/\sqrt{2}]^T$. Here, the introduction of the normalisation by $\sqrt{2}$ in the last component enables us to obtain coupling matrices as found in Coutant et al. (2024). The general expressions of coefficient matrices appearing in the equations above are provided in Appendix B.2. We now follow the same lines as before in Section 2.1.3 to obtain the modes of the strip. Taking $V_m = C_s z^m$, where $C_s = [C_1^{(0)}, C_2^{(0)}, \dots, C_1^{((N-3)/2)}, C_2^{((N-3)/2)}, C_1^{((N-1)/2)}]^T$, the reduced form of (8) is then

$$ (B_s + J_s z + J_s^T z^{-1}) C_s = -\omega^2 C_s. \qquad (14) $$

The eigenfrequencies $\omega(k)$ for $z = e^{ik}$, $k$ real, and mode shapes of symmetric modes can then be determined in a similar way to that outlined in Section 2.1.3.

The dispersion diagram, showing how the (real) radian frequency $\omega$ depends on the (real) wavenumber $k$, is reported in Fig. 2. It contains dispersion curves corresponding to both symmetric and antisymmetric modes, indicated by solid and dashed lines, respectively. For what concerns symmetric modes, on which we focus the attention in this work, we note a semi-infinite band-gap for $\omega \gtrsim 2.393$ and no internal stop-bands. For any value of $k$, $N = 5$ positive solutions of $\omega$ are found (as also detailed in Carta et al. (2023b)). In particular, as the edge resonance effect is known to occur in the reflection problem of the continuum strip below the cut-on frequency for $S_j$, $j \ge 1$, modes, for the lattice strip we consider when $\omega < \omega_c$ (see Fig. 2). There, only the $S_0$ mode of the lattice strip propagates.

In the reflection problem, we are interested in the non-trivial solutions of (14) and corresponding values of $z$ for which this is possible when the frequency $\omega < \omega_c$ is specified (see Fig. 2). This leads to the singular generalised eigenvalue problem analogous to (9) from which the modes of the system can be identified.

The total field $V_m$ ($m \ge 0$) can then be constructed (see (17) below) and it should satisfy the edge conditions for the ‘half-strip’. For symmetric waves, the free edge conditions (10) become

$$ (S_s + \omega^2 I) V_0 + J_s V_1 = 0, \qquad (15) $$

where $S_s$ is detailed in Appendix B.2. On the other hand, the simplified edge conditions (11), taking into account the symmetry conditions (12), are expressed by

$$ G_s V_0 - J_s^T V_{-1} = 0. \qquad (16) $$
<figure><img src="image_4.png" /><figcaption>Fig. 2. Dispersion curves $\omega = \omega(k)$ for $N = 5$. The solid (dashed) curves are associated with symmetric (antisymmetric) modes. The grey region indicates the low-frequency regime below the cut-on frequency $\omega_c$ for the first order symmetric Lamb mode, that represents the frequency interval of interest.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Scattering problem when the number of rows $N = 5$. Only half of the strip can be considered if symmetric conditions are employed.</figcaption></figure>

Here, we refer once more to Appendix B.2 for the representations of $G_s = S_s - B_s$. The latter equation will be employed in solving the scattering problem described in the next section, whereas (14) will be used in developing the solution based on the Z-transform in Section 3.

Further, in Appendix B.3 we provide the coefficient matrices mentioned in this Section for the case $N = 5$.

## 2.3. Scattering problem at the edge

Now that we have determined the right- and left-going waves in the strip for a given frequency, we can study the scattering of a left-going wave with unit amplitude from the free edge. When this wave impinges on the edge, it generates (i) a right-going propagating wave with reflection coefficient $R_0$ and (ii) right-going evanescent waves with reflection coefficients $R_i$ (see Fig. 3 for the case $N = 5$, analysed in detail in Section 4.1). Consequently, the solution can be written as

$$V_m = E_{s,0}^{-} e^{-imk_0} + R_0 E_{s,0}^{+} e^{imk_0} + \sum_{i=1}^{N_{E_s}-1} R_i E_{s,i}^{+} e^{imk_i}, \qquad (17)$$

where $E_{s,0}^{-}$, $E_{s,0}^{+}$ and $E_{s,i}^{+}$ are the symmetric eigenvectors corresponding to the wavenumbers $-k_0$, $k_0$ and $k_i$, respectively, and $N_{E_s}$ is the number of symmetric eigenmodes associated with right-going waves for a fixed value of the frequency, namely

$$N_{E_s} = \begin{cases} (3N + 1)/4 & \text{for } N \equiv 1 \pmod 4, \\ (3N - 1)/4 & \text{for } N \equiv 3 \pmod 4. \end{cases} \qquad (18)$$

For instance, we have $N_{E_s} = 4$ for $N = 5$, whereas $N_{E_s} = 5$ for $N = 7$. We recall that we are interested in the low-frequency range, characterised by a single propagating wave, similarly to the continuous case (Pagneux, 2006); then $\pm k_0$ are the wavenumbers associated with the single couple of conjugate eigenvalues $z_0^{\pm} = e^{\pm ik_0}$ of (9).
Substituting the solution (17) into the edge conditions (15), we obtain:

$$ (S_s + \omega^2 I) \left( E_{s,0}^{-} + R_0 E_{s,0}^{+} + \sum_{i=1}^{N_{E_s}-1} R_i E_{s,i}^{+} \right) + J_s \left( E_{s,0}^{-} e^{ik_0} + R_0 E_{s,0}^{+} e^{-ik_0} + \sum_{i=1}^{N_{E_s}-1} R_i E_{s,i}^{+} e^{-ik_i} \right) = 0. \quad (19) $$

The rank of the preceding system is $N_{E_s}$. Then, identifying and solving the reduced system of $N_{E_s}$ equations from (19), we determine the $N_{E_s}$ unknowns that are the reflection coefficients.

When $\omega$ is real, $|R_0| = 1$ due to energy conservation. Further, if the reflection coefficients of evanescent waves take large values in the vicinity of a specific real frequency, we denote the latter as quasi-resonance frequency. Moreover, extending the analysis to the complex $\omega$-plane as in Pagneux (2006), the reflection coefficient $R_0$ possesses a simple pole and a zero at complex conjugated frequencies in the vicinity of the quasi-resonance frequency. The pole, which we denote by $\omega_R$, has $\mathrm{Im}(\omega_R) < 0$ and it is known as the complex edge resonance frequency, which is the eigenfrequency corresponding to the edge mode obtained from the analysis of the homogeneous problem where the incident wave is absent. When $\omega$ is complex and its imaginary part is positive, it is possible to find a value of the frequency for which $R_0 = 0$, which consequently identifies the resonance conditions (Pagneux, 2006). These results will be shown in Section 4.

### 3. General solution based on the Z-transform

Here, we present an alternative method to the mode matching approach implemented in the previous Section in order to identify the reflected field. This alternative method is based on complex variable techniques. The advantages of this approach are as follows.

Firstly, this formulation provides a compact explicit representation of the reflected field as a contour integral evaluated over the unit circle in $\mathbb{C}$ (see Section 3.1). Secondly, explicit expressions for all reflected modes are obtainable through application of the residue theorem at each pole of the integral kernel (see Section 3.2).

Thirdly, and alternatively, when evaluating this contour integral solution, particular attention is required only for the integral kernel pole located near the unit circle, as it corresponds to the reflected propagating wave within the considered frequency range. The contributions from all other evanescent modes to the reflected field can then be computed via a modified contour integral solution, which exclusively involves poles situated at a finite distance from the unit circle. This modification proves particularly advantageous when calculating the field for large $N$ at a fixed strip width. In this context, the sum of evanescent modes in the lattice strip approaches the infinite series encountered in the reflected field of the effective continuum. The modified integral formulation thus enables evaluation of this infinite sum, overcoming issues related to the completeness of Lamb modes, essential for analysing series convergence (Besserer & Malishewsky, 2004). Furthermore, this approach offers an alternative to the method described in Davey et al. (2022), ensuring accurate evaluation of the field at strip corners, where the conventional modal expansion for continuum strips typically faces convergence difficulties. Finally, the proposed contour integral solution illustrates an approach to show that the lattice edge resonance frequency converges to the analogous resonance frequency in the effective continuum strip (see Section 3.3). Before moving on, we note that the approach developed here can be generalised to more exotic configurations/waveguides, e.g. Coutant et al. (2024).

We begin by providing the solution to the difference Eqs. (13) and (15) that determine a causal sequence $V_m$, $m \ge 1$, via the application of the Z-transform.

Once more, we will assume that the frequency dependent coefficients of those equations possess a small positive vanishing imaginary part in connection with the causality principle (Slepyan, 2002) or limiting absorption principle (Shaban & Vainberg, 2001), that enables us to derive a physically acceptable solution by removing the possibility of exponentially growing waveforms in the limit $m \to +\infty$.

#### The symmetric reflected field

Let $A_{\mathrm{inc}}^{(n)} z_{\mathrm{inc}}^m$ and $U_r^{(m,n)}$ be the displacement vectors of the node $(m, n)$ generated by the influence of the incident and reflected fields, respectively, in the entire lattice strip. Then, we represent the total field in the standard form:

$$ U_m = \hat{A}_{\mathrm{inc}} z_{\mathrm{inc}}^m + U_m^r, \quad m \ge 0, \quad (20) $$

where the first term on the right-hand side represents the incident wave, composed by two contributions: (i) $z_{\mathrm{inc}} \in \mathbb{C}$, that is a complex number determined from degeneracies of the eigenvalue problem (9) connected with the equations governing the interior of the strip; (ii) $\hat{A}_{\mathrm{inc}} = [(A_{\mathrm{inc}}^{(0)})^T, \dots, (A_{\mathrm{inc}}^{(N-1)})^T]^T$, that is the associated amplitude of the symmetric mode incident on the strip edge. Additionally, in (20) $U_m^r = [(U_r^{(m,0)})^T, \dots, (U_r^{(m,N-1)})^T]^T$, $m \ge 0$, is the field reflected from the edge of the strip. Note that owing to the additional small dissipation in the governing equations (5) and (10), $|z_{\mathrm{inc}}| > 1$ with $||z_{\mathrm{inc}}| - 1| \ll 1$.

When employing the symmetry conditions (12), governing equations (5) and (10) update to (13) and (15), respectively. Further, we replace (20) by

$$ V_m = A_{\mathrm{inc}} z_{\mathrm{inc}}^m + V_m^r, \quad m \ge 0, \quad (21) $$

with $A_{\mathrm{inc}}$, $V_m^r$ and $V_m$ being the restriction of the corresponding vectors for symmetric modes.
### 3.1. Integral representation of the symmetric reflected field

As shown in Appendix C, we can employ the unilateral Z-transform, defined as:

$$V^{\text{r,F}}(z) = \sum_{m \ge 0} V_m^r z^{-m}, \qquad (22)$$

and derive explicitly the reflected field, defined in (21), implementing the inverse transformation in the form of a contour integral:

$$V_m^r = \frac{1}{2\pi i} \oint_C [K(z)]^{-1} \varphi(z) z^{m-1} dz, \qquad (23)$$

where C is the unit circle in the complex plane,

$$K(z) = B_s + \omega^2 I + z J_s + z^{-1} J_s^T \qquad (24)$$

and

$$\varphi(z) = -[Q(z)V_0^r + P(z_{\text{inc}}^{-1})A_{\text{inc}}], \\ Q(z) = S_s - B_s - zJ_s, \\ P(z) = S_s - B_s - zJ_s^T. \qquad (25)$$

We note that the kernel of (23) embeds information on the edge displacement, appearing in (25). This displacement is found by setting $m = 0$ in (23) and forming an algebraic system from which this displacement is determined as

$$V_0^r = (I + T)^{-1} g, \qquad (26)$$

with

$$g = -\frac{1}{2\pi i} \oint_C [K(z)]^{-1} P(z_{\text{inc}}^{-1}) A_{\text{inc}} z^{-1} dz, \qquad (27)$$

$$T = \frac{1}{2\pi i} \oint_C [K(z)]^{-1} Q(z) z^{-1} dz. \qquad (28)$$

Next we discuss the explicit computation of the contour integrals appearing in (23), (27) and (28), leading to a model representation of the symmetric reflected field.

### 3.2. Explicit modal expansion of the solution

The integrals appearing in (23), (27) and (28) have only simple poles in the complex plane when the incoming wave frequency does not correspond to exceptional points on the dispersion diagram, namely where the group velocity is zero or dispersion curves intersect.

As demonstrated in Appendix D, by application of the residue theorem, the contour integrals above can be evaluated and the reflected field can be written in the alternative equivalent form:

$$V_m^r = \sum_{z_p \in D_s} A_r(z_p) \varphi(z_p) z_p^{m-1}, \qquad (29)$$

where, by applying l'Hôpital's rule, we can determine that

$$A_r(z) = \left( \frac{d}{dz} \det(K(z)) \right)^{-1} \operatorname{adj}(K(z)), \qquad (30)$$

where $\operatorname{adj}(K(z))$ represents the adjoint of the matrix $K(z)$. Additionally, in (29) $z_p$ denotes the value of $z$ at the pole and $D_s$ is the set

$$D_s := \{z \in \mathbb{C} : \det(K(z)) = 0, |z| < 1\}. \qquad (31)$$

Further, the terms $g$ and $T$ in (27) and (28) become:

$$g = - \sum_{z_p \in D_s} A_r(z_p) P(z_{\text{inc}}^{-1}) A_{\text{inc}} z_p^{-1}, \quad T = \sum_{z_p \in D_s} A_r(z_p) Q(z_p) z_p^{-1}. \qquad (32)$$

### 3.3. Connection to the continuous problem via dynamic homogenisation

Here, we present a formal argument demonstrating that the complex edge resonance frequency for the lattice strip converges to the corresponding frequency in the analogous continuum problem when the lattice microstructure is dense in comparison with the strip width. This is carried out by considering the zero of the reflection coefficient obtained from the solution (29) based on the Z-transform and noting that the conjugate of this zero is the complex edge resonance frequency. Firstly, we require some properties of the matrix coefficients in (29).
## Properties of the matrix $A_r$

The term $A_r(z)\varphi(z)$ in (29) defines the amplitudes of waveforms generated by the interaction of the incident wave with the edge of the lattice strip, which may be propagating or evanescent. Note that, except for frequencies corresponding to exceptional points where the group velocity is zero or dispersion curves intersect, the matrix $K(z)$ is degenerate at $z = z_p$ and these points provide simple zeros of its determinant. Additionally:

* For $z_p \in D_s$, the rank of the $N \times N$-matrix $K(z_p)$ is $N - 1$, where we recall that $N$ is the number of rows in the strip.

* It follows that if $X(z_p)$ ($Y(z_p^{-1})$) is the corresponding right eigenvector of $K(z_p)$ ($K(z_p^{-1})$), then

$$A_r(z_p) = \alpha_p X(z_p) \otimes Y(z_p^{-1}), \qquad (33)$$

with the constant $\alpha_p \in \mathbb{C}$.

* In particular, if $z_r \in D_s$ is the complex number that defines the reflected propagating wave in the response of the strip, setting the amplitude of the symmetric incident wave $A_{inc} = Y(z_r^{-1})$ where

$$K(z_r^{-1})A_{inc} = K(z_{inc})A_{inc} = 0, \quad \text{with } z_{inc} = z_r^{-1},$$

we have

$$A_r(z_r) = \alpha_r R_0 \otimes A_{inc}, \qquad (34)$$

where $R_0 := X(z_r)$ is the vector amplitude of the reflected wave and $\alpha_r \in \mathbb{C}$.

* At the conjugate of the complex frequency corresponding to the complex edge resonance frequency, we then obtain

$$A_r(z_r)\varphi(z_r) = 0, \qquad (35)$$

which implies together with (34) that

$$A_{inc} \cdot \varphi(z_r) = 0. \qquad (36)$$

## Edge resonance condition for the effective continuum strip

The identity (36) allows one to identify the complex edge resonance frequency for the discrete strip. There exists an analogous integral identity for the effective continuum strip, obtained when the number of rows in a strip of fixed width grows ($N \to \infty$), involving the fields associated with the homogenised problem. To summarise, we start from the discrete lattice quantity (36) and arrive at a continuous integral identity on the free edge. In Appendix E, we use the solution based on the Z-transform to establish a connection between these identities in the discrete and continuous settings.

In contrast to in Carta et al. (2023b) (see Section 2.5.2), where the Taylor expansion was used to connect the dispersive properties for the structured strip and the effective strip, here we apply a dynamic homogenisation procedure based on the so-called method of meso-scale approximations (Nieves & Movchan, 2022b). This technique has been used in Nieves (2017) and Nieves and Movchan (2023) to provide multi-scale PDEs that lead to point-wise descriptions of displacement fields in continuous and discrete media, respectively. For our purposes, we use this method of meso-scale approximations to replace sums obtained from (36) by integral characteristics over the edge of the strip in the effective problem. In doing so, we can show that when $N \to \infty$ whilst fixing the strip width, the condition (36) is equivalent to

$$\int_E \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{inc}}(\mathbf{x}) \, dx_2 = 0. \qquad (37)$$

Here, $\mathbf{x} = (x_1, x_2)^T$ and $E := \{\mathbf{x} : x_1 = 0, 0 < x_2 < H\}$ is the set of coordinates defining the edge of the continuum strip of length $H$. Further $\mathbf{u}_{\text{tot}}(\mathbf{x})$ and $\mathbf{u}_{\text{inc}}(\mathbf{x})$ are the total and incident fields in the corresponding time-harmonic homogenised problem, respectively, that are also symmetric wavefields with respect to the centreline of symmetry in the waveguide. Moreover, $T_n(\nabla_x)$ is the traction operator for the associated effective medium, which applied to the incident field gives:

$$T_n(\nabla_x)\mathbf{u}_{\text{inc}} = \frac{\sqrt{3\Gamma}}{4} \left[ \frac{3}{x_1} \frac{\partial u_{\text{inc}}^{(1)}}{\partial x_1} (x_1, x_2) + \frac{\partial u_{\text{inc}}^{(2)}}{\partial x_2} (x_1, x_2) \right] \left[ \frac{\partial u_{\text{inc}}^{(1)}}{\partial x_2} (x_1, x_2) + \frac{\partial u_{\text{inc}}^{(2)}}{\partial x_1} (x_1, x_2) \right], \qquad (38)$$

where $u_{inc}^{(j)}$ ($j = 1, 2$) are the components of the incident field in the effective medium. This can be reframed in the form:

$$T_n(\nabla_x)\mathbf{u}_{\text{inc}} = \sigma_{\text{hom}}(\mathbf{u}_{\text{inc}})\mathbf{n},$$

where the unit-outward normal $\mathbf{n} = (-1, 0)^T$ and

$$\sigma_{\text{hom}}(\mathbf{u}_{\text{inc}}) = \lambda_{\text{hom}} \nabla \cdot \mathbf{u}_{\text{inc}} I + \mu_{\text{hom}} [\nabla \otimes \mathbf{u}_{\text{inc}} + (\nabla \otimes \mathbf{u}_{\text{inc}})^T]$$

is the Cauchy stress tensor for the effective elastic strip. In this tensor, $\lambda_{\text{hom}} = \mu_{\text{hom}} = \sqrt{3\Gamma}/4$ are the continuum effective Lamé parameters for the elastic strip (see Carta et al. (2023b), Ostoja-Starzewski (2002) and Slepyan (2002)). In Appendix E, we also show that (37) holds in the associated continuum model for a semi-infinite elastic strip.
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Case N = 5: (a) phase of the coefficient $R_0$ of the reflected propagating wave; (b) moduli of the reflection coefficients $R_i$ ($i = 0, ..., 3$). The low-frequency regime $0 \le \omega < \omega_c$ is shown in the horizontal axis (see also the grey region in Fig. 2).</figcaption></figure>

Alternatively, condition (37) may also be derived by considering the equivalent elastodynamic scattering problem for the continuum strip when the frequency is assumed to be the conjugate of the complex edge resonance frequency for the medium.

In Appendix E, we also show that (37) holds in the associated continuum model for a semi-infinite elastic strip at the conjugate of the complex edge resonance frequency for this medium. There we also illustrate how condition (36) may be rewritten as a scalar product of displacements induced by the total field and the forces produced by incident field on the edge nodes of the discrete strip. When the lattice strip is assumed to have sufficiently dense microstructure, we demonstrate that the sums forming this scalar product can be treated as Riemann sums and rewritten as integrals. These integrals are taken over the edge of the effective lattice strip. They involve products of the total field and derivatives of the incident wave for the effective continuum problem. These integrals then combine to form (37).

This homogenisation process shows that the condition (36), where $\omega = \overline{\omega_R}$ in the lattice strip, converges to (37) which holds for the analogous complex frequency in the effective strip. This convergence of the complex edge resonance frequency in the lattice strip to the same physical quantity in the continuum strip and the nature of this convergence is further investigated in the numerical computations below.

## 4. Edge resonance

In this Section, we demonstrate how the solution to the edge scattering problem developed via mode matching can be exploited to numerically determine the quasi-resonance frequency and the complex edge resonance frequency for the lattice strip of size N. Before moving on, we note that the solution based on the Z-transform described in Section 3 has been employed to verify the computations based on mode matching of Section 2 presented in this Section; additional computations based on this solution can be found in Appendix F.

### 4.1. Case N = 5

As an illustrative example, we present the results when the number of rows is $N = 5$ (see also Fig. 3).

The dispersion curves associated with symmetric modes for this structure are found in Fig. 2, which have been obtained by specifying $z = e^{ik}$, $k$ real, and determining $\omega$ from the eigenvalue problem (14). On the other hand, for a fixed value of the frequency $\omega$, 8 complex solutions in $k$ (or $z$) are retrieved, associated with 4 left-going and 4 right-going waves. Considering also that the number of linearly independent equations in (19) is equal to 4 for this lattice, we have 4 unknown coefficients in the scattering problem described in Section 2.3, namely $R_0, R_1, R_2$ and $R_3$. The latter are obtained by solving system (19) and their magnitudes are plotted in Fig. 4(b) in the low-frequency range $0 \le \omega/\omega_c < 1$. In this interval of frequencies, we note that $|R_0| = 1$, as expected. However, there is a value of $\omega$ for which the coefficients of evanescent waves become very large; this frequency denotes a quasi-resonance. We also highlight that at the quasi-resonance frequency, the phase of $R_0$ (shown in Fig. 4(a)) undergoes a jump equal to $2\pi$, in accordance with the findings from the analogous continuum problem (Pagneux, 2006).

As stated in Section 2.3, the coefficient $R_0$ of the propagating reflected wave is characterised by a pole (zero) at the complex edge resonance frequency $\omega_R$ (for $\omega = \overline{\omega_R}$). We demonstrate this statement for the case $N = 5$ in Fig. 5, where the modulus of $R_0$ is shown (in logarithmic scale) as a function of the real and imaginary parts of $\omega$, denoted by $\Re(\omega)$ and $\Im(\omega)$, respectively. The value of the complex edge resonance frequency obtained from this computation is $\omega_R \approx 0.45045 - i0.000535$.
<figure><img src="image_7.png" /><figcaption>Fig. 5. Contour plot of the magnitude of the coefficient $R_0$ in the complex plane of the frequency $\omega$, in particular in the neighbourhood of $\overline{\omega_R}$ and $\omega_R$, where $\omega_R \approx 0.45045 - i0.000535$ is the complex edge resonance frequency for $N = 5$. The modulus of $R_0$ is shown in logarithmic scale.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Displacement amplitude field in a lattice with $N = 5$ for $\omega = 0.45 \approx \Re(\omega_R)$. The system includes adaptive absorbing layers in proximity of the right end, not shown in this figure.</figcaption></figure>

### 4.1.1. Numerical verification of the quasi-resonance frequency

In addition to verifying the solution based on mode matching with the solution from the Z-transform, we perform an independent numerical simulation in Comsol Multiphysics. The finite element model is a truss, with point masses at the nodes connected by non-inertial elastic links, that resembles the lattice in Fig. 3. To simulate a semi-infinite discrete waveguide, adaptive absorbing layers are incorporated near the right end of the domain. We calculate the analytical solution (17) imposing an incident wave of unit amplitude, and we apply the resulting displacements to the nodes of the column defined by $m = 10$. Imposing time-harmonic conditions, for $\omega = 0.45$ (that is very close to the real part of the resonance frequency $\omega_R$) we obtain the displacement amplitude field shown in Fig. 6. The relatively large displacements at the left edge, due to significantly large reflection coefficients of the evanescent waves, indicate the effect of quasi-resonance. In addition, we note that the comparison between the analytical and numerical values of the displacement amplitudes at nodes with different values of $m$ shows a perfect agreement.

### 4.2. Results for different $N$ and convergence to the continuum limit

In this Section, we calculate the *complex edge resonance frequency* for increasing values of the number of rows $N$. When $N \gg 1$, we expect that the *complex edge resonance frequency* for the lattice tends to the value determined for the continuum in Pagneux (2006).
<figure><img src="image_9.png" /><figcaption>Fig. 7. (a) Real part of the resonance frequency as function of the number of rows N. The dashed grey line represents the continuum limit, given by (40a). (b) Log-log plot of the difference between the real parts of the resonance frequencies for the discrete and continuous systems versus the number of rows N. The dashed line is given by Eq. (41).</figcaption></figure>

In order to compare the results for the discrete structure with those for the continuous waveguide, we introduce an alternative form of the dimensionless frequency, defined as

$$ \Omega = \frac{\omega(N - 1)}{\sqrt{2}}. \qquad (39) $$

The expression of $\Omega$ in (39) coincides with the definition given in Pagneux (2006), namely $\Omega^{\text{cont}} = \omega^{\text{cont}}h/c_T$, where $\omega^{\text{cont}}$ is the radian frequency in rad/s in the continuum, $h$ is half the thickness of the waveguide and $c_T$ is the shear wave speed. In Pagneux (2006), the real and imaginary parts of the complex edge resonance frequency $\Omega_R^{\text{cont}}$ are evaluated as functions of the Poisson coefficient $\nu$ (see Eqs. (8) and (9) in Pagneux (2006)). In the continuum strip, following (Pagneux, 2006):

$$ \Re(\Omega_R^{\text{cont}}) = 0.652 \nu^2 + 0.898 \nu + 1.9866, \qquad (40a) $$

$$ \Im(\Omega_R^{\text{cont}}) = - \frac{c \nu^4 (\nu - \nu_2)^2}{1 + \left(\frac{\nu - \nu_3}{a}\right)^2 + \left(\frac{\nu - \nu_3}{b}\right)^4}, \qquad (40b) $$

where $\nu_2 = 0.2248$, $\nu_3 = 0.2062$, $a = 0.1696$, $b = 0.2606$ and $c = 1/0.0313$. As the triangular lattice strip converges to an elastic continuum with $\nu = 1/4$ (Carta et al., 2023b; Ostoja-Starzewski, 2002), we then obtain:

$$ \Re(\Omega_R^{\text{cont}}) = 2.25185 \quad \text{and} \quad \Im(\Omega_R^{\text{cont}}) = -7.42423 \times 10^{-5}. $$

For several values of $N$, we compute the resonance frequency $\Omega_R$. The real part of the resonance frequency as a function of $N$ is reported in Fig. 7(a). The black dots represent the values for the lattice strips, while the horizontal dashed grey line indicates the value of the resonance frequency for the continuous waveguide. The trend of $\Omega_R$ is monotonically increasing with $N$, tending to the continuum limit. In the inset, Fig. 7(b), the discrepancy with the continuum is shown in log-log scale; the dashed straight line is represented by the equation

$$ \log(\Re(\Omega_R) - \Re(\Omega_R^{\text{cont}})) = \log(\alpha) + \beta \log(N), \qquad (41) $$

where $\alpha \approx 4.438$ and $\beta \approx -1$, implying that the rate of convergence to the continuum limit is $1/N$.

The variation of the imaginary part of $\Omega_R$ with $N$ is presented in Fig. 8. Interestingly, the graph is not monotonic: for $N = 5$, the imaginary part of $\Omega_R$ has a relatively large magnitude (of the order of $10^{-3}$); increasing $N$, $\Im(\Omega_R)$ increases abruptly and oscillates around the continuum limit; it reaches the maximum value (undistinguishable from zero, i.e. $-1.24 \times 10^{-8}$) for $N = 65$ and then decreases again monotonically, tending to $\Im(\Omega_R^{\text{cont}})$ from above.

To have a clearer point of view on how the resonance frequency for the lattice strips approaches the continuum limit, we show the trajectory of the complex resonance frequency $\Omega_R$ when converging to the continuum limit in the complex plane of Fig. 9. For small $N$, there is a significant difference between lattices with $N \equiv 1 \pmod 4$ and $N \equiv 3 \pmod 4$ rows. When $N$ is relatively large, the values of $\Omega_R$ are densely packed and tend to the continuum limit slowly but with a clear tendency. For computational reasons, we have interrupted the numerical calculations at $N = 501$.
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Imaginary part of the resonance frequency as function of the number of rows *N*. The dashed grey line represents the continuum limit, given by (40b). A magnified view of the outcomes starting from *N* = 7 is included in the inset.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Real and imaginary parts of the complex edge resonance frequency $\Omega_R$ for the discrete waveguides with increasing *N*, starting from *N* = 7 (the value of the complex edge resonance frequency for *N* = 5 is provided in Section 4.1). The continuum limit $\Omega_R^{\text{cont}}$ is indicated by a grey dot.</figcaption></figure>

### 4.2.1. Numerical simulations

Here, we use the above numerical computations for the complex edge resonance frequency to understand the edge resonance response of the lattice strip. The imaginary part of the complex edge resonance frequency is a measure of the decaying rate of resonance: when $\mathfrak{T}(\Omega_R)$ is relatively large, the decay is slow; on the contrary, when $\mathfrak{T}(\Omega_R)$ is close to zero, we observe trapped modes with decoupling between propagating and evanescent waves (Pagneux, 2006).

In order to test the analytical calculations, we carry out separate finite element simulations in Comsol Multiphysics. We consider two lattices: one with *N* = 11 having a relatively large $\mathfrak{T}(\Omega_R)$, and another one with *N* = 63 characterised by $\mathfrak{T}(\Omega_R)$ close to zero (refer to Figs. 8 and 9). In both scenarios, we run a modal analysis to compute eigenfrequencies and eigenmodes.

In Fig. 10(a) we present the model for the discrete strip, having length $2L_0$ and two symmetry axes. In particular, since we are interested in symmetric modes, the nodes on the longitudinal (horizontal) centreline are characterised by symmetry conditions, while in the transverse (vertical) centreline either symmetric or antisymmetric conditions can be applied. The latter are important especially for the large lattice with *N* = 63, in order to consider only one quarter of the model and, hence, reduce the computational cost of its analysis.
<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Eigenvalue analysis for two lattices with $N = 11$ and $N = 63$ rows. (a) Model of the system, where $L_0 = 39$ in the lattice with $N = 11$ while $L_0 = 240$ in the lattice with $N = 63$. (b) Localised eigenmode (shown in the bottom-left quarter only) for $N = 11$, corresponding to $\omega = 0.2674$ ($\Omega = 1.891$). (c) Localised eigenmode (shown in the bottom-left quarter only) for $N = 63$, corresponding to $\omega = 0.0497$ ($\Omega = 2.179$). (d) Horizontal displacements of the nodes positioned in the symmetry longitudinal axes of the lattices, for $0 \le \xi/L_0 \le 1$. The maximum absolute value of each eigenvector has been normalised to 1.</figcaption></figure>

In Fig. 10(b), we show an eigemode for $N = 11$ localised at the two (identical) ends, obtained for an eigenfrequency $\omega = 0.2674$, corresponding to $\Omega = 1.891$ (see formula (39)), which matches $\Re(\Omega_R)$ found for $N = 11$ (see Fig. 7). The eigenmode has been illustrated only for a quarter of the lattice; for the other three quarters, it is easily obtained by exploiting the symmetry conditions. In Fig. 10(c), a similar eigenmode for the larger lattice with $N = 63$ is obtained, corresponding to the eigenfrequency $\omega = 0.0497$; the latter, that can be expressed as $\Omega = 2.179$ according to (39), is very close to $\Re(\Omega_R)$ obtained analytically for $N = 63$ (refer to Fig. 7). We also note that the eigenfrequencies associated with the deformed shapes in parts (b) and (c) do not change imposing symmetric or antisymmetric conditions in the symmetry transverse axis.

In Fig. 10(d), we report the values of the horizontal components of the displacements of the nodes located in the symmetry longitudinal axes of the two discrete structures. For symmetry reasons, only the nodes in the left half of each strip are considered in the graph. It is apparent that for $N = 11$ the oscillations decay slowly from the edges to the interior of the lattice, since the imaginary part of the *complex edge resonance frequency* is relatively large in this case. On the other hand, in the lattice with $N = 63$
rows the displacements decay very rapidly from the edge, confirming that the imaginary part of the complex edge resonance frequency is close to zero, in full agreement with the analytical results.

## 5. Conclusions

We have investigated the scattering of symmetric Lamb modes at a free edge of a discrete triangular semi-infinite elastic strip. Attention has been given to vibration modes localised at the edge of the lattice strip at certain frequencies, referred to as complex edge resonance frequencies, identified as poles of the coefficient of the reflected propagating wave. The solution to this problem has been obtained both via a mode matching method and an approach based on the Z-transform (or discrete Fourier transform).

A dynamic homogenisation procedure applied here has also allowed us to establish a link between the conditions satisfied by the complex edge resonance frequency in the lattice strip and those in the effective strip, obtained when the number of rows of a strip of fixed width becomes large. The numerical calculations performed in this work have also demonstrated a surprisingly interesting feature in the convergence of the complex edge resonance frequency in the discrete strip to the analogous frequency in the homogenised problem for the continuum strip. Namely, while the real part converges monotonically to the real part in the effective semi-infinite continuous strip, the imaginary part of the lattice strip edge resonance frequency exhibits a non-monotonic convergence to the analogous quantity in the effective continuum, going very close to zero at $N = 65$.

The methods presented here can be easily extended to study similar structures, for instance discrete systems made of flexural ligaments or incorporating resonant or gyroscopic mechanical components. The results of this paper could have applications in the non-destructive testing of a variety of structured materials and engineering structures, e.g. trusses, where the identification of frequencies responsible for hazardous or large deformations is crucial in improving their structural integrity. In addition, there are also applications of the work in the creation of novel mechanical energy harvesting devices, where the understanding and manipulation of resonances play a fundamental role in optimising the harvesting performance.

## CRediT authorship contribution statement

**G. Carta:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. **M.J. Nieves:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. **M. Brun:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. **V. Pagneux:** Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

G.C. and M.B.'s research was partially funded within Project AISAC (Analisi e Ingegnerizzazione dei Sistemi Alimentari Complessi) by the Italian Government (Ministero delle Imprese e del Made in Italy), Fondo per la Crescita Sostenibile - Accordi per l'innovazione di cui al D.M. 31 dicembre 2021 e D.D. 18 marzo 2022, grant no. F/310035/05/X56.

G.C., M.J.N and M.B are thankful for the financial support through European Union's Horizon 2020 RISE MSCA EffectFact Grant No. 101008140.

M.J.N. gratefully acknowledges the support the International Centre for Mathematical Sciences, Edinburgh, for their funding through the Research-in-Groups scheme from 17/06/2024 to 17/07/2024.

M.J.N. and V.P. would also like to thank the Isaac Newton Institute for Mathematical Sciences for their support and hospitality during the programme 'Mathematical theory and applications of multiple wave scattering' (MWS), where work on this paper was undertaken and supported by EPSRC, UK grant no EP/R014604/1.

## Appendix A. The resultant forces acting on the nodes of the strip

In this Section, we give the form of the forcing term $f^{(m,n)}$ (with $m \ge 0$ and $0 \le n \le N - 1$) in (1). Prior to this, we discuss some preliminary facts associated with the geometry of the strip, that depends on the number of rows $N$.

As stated in Section 2, we assume that $N$ is an odd number and $N \ge 5$. The equations describing the dynamic response on the structure vary with respect to the choice of $N$, since this leads to a change in the elementary cell used to analyse the structure. We consider two scenarios, namely when $N \equiv 1$ (mod 4) and $N \equiv 3$ (mod 4). Fig. 1(a) and (b) represent examples of the two cases, with $N = 5$ and $N = 7$ respectively; in the figures, the elementary cell for each type of strip is highlighted by the shaded 'zig-zag' pattern.

For $N \equiv 1$ (mod 4), the following rule holds:

$$u^{(m,n)}(t) = u\left(m + \frac{1 + (-1)^{n+1}}{4}, \frac{\sqrt{3}n}{2}, t\right). \qquad (A.1)$$
On the other hand, for $N \equiv 3 \pmod 4$, we have:

$$u^{(m,n)}(t) = u\left(m + \frac{1 + (-1)^n}{4} \cdot \frac{\sqrt{3}n}{2}, t\right).$$

In addition, we introduce the unit vectors

$$\mathbf{a}^{(1)} = (1, 0)^{\mathrm{T}}, \quad \mathbf{a}^{(2)} = (1/2, \sqrt{3}/2)^{\mathrm{T}}, \quad \mathbf{a}^{(3)} = (-1/2, \sqrt{3}/2)^{\mathrm{T}}, \qquad \text{(A.2)}$$

that indicate the directions of the lattice links (see Fig. 1).

### A.1. The resultant elastic forces for all nodes in the strip interior

First, we consider the case $N \equiv 1 \pmod 4$. In the transient regime, for $0 < m < \infty$ the forcing term $f^{(m,n)}$ in (1) is expressed as

• for $n = 0$:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(m+1,0)} + \mathbf{u}^{(m-1,0)} - 2\mathbf{u}^{(m,0)}) \mathbf{a}^{(1)} \\ + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(m,1)} - \mathbf{u}^{(m,0)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(m-1,1)} - \mathbf{u}^{(m,0)}) \mathbf{a}^{(3)}], \qquad \text{(A.3a)}$$

• for $0 < n < N - 1$ and $n$ even:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(m+1,n)} + \mathbf{u}^{(m-1,n)} - 2\mathbf{u}^{(m,n)}) \mathbf{a}^{(1)} \\ + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(m,n+1)} + \mathbf{u}^{(m-1,n-1)} - 2\mathbf{u}^{(m,n)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(m-1,n+1)} + \mathbf{u}^{(m,n-1)} - 2\mathbf{u}^{(m,n)}) \mathbf{a}^{(3)}], \qquad \text{(A.3b)}$$

• for $0 < n < N - 1$ and $n$ odd:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(m+1,n)} + \mathbf{u}^{(m-1,n)} - 2\mathbf{u}^{(m,n)}) \mathbf{a}^{(1)} \\ + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(m+1,n+1)} + \mathbf{u}^{(m,n-1)} - 2\mathbf{u}^{(m,n)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(m,n+1)} + \mathbf{u}^{(m+1,n-1)} - 2\mathbf{u}^{(m,n)}) \mathbf{a}^{(3)}], \qquad \text{(A.3c)}$$

• for $n = N - 1$:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(m+1,N-1)} + \mathbf{u}^{(m-1,N-1)} - 2\mathbf{u}^{(m,N-1)}) \mathbf{a}^{(1)} \\ + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(m-1,N-2)} - \mathbf{u}^{(m,N-1)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(m,N-2)} - \mathbf{u}^{(m,N-1)}) \mathbf{a}^{(3)}]. \qquad \text{(A.3d)}$$

In the time-harmonic regime, the term $F^{(m,n)}$ ($m \ge 1, 0 \le n \le N - 1$) appearing in the right-hand side of the normalised governing equations (4) is simply given by (A.3a)-(A.3d) with $u^{(m,n)}$ replaced by $U^{(m,n)}/\Gamma$.

Conversely, when $N \equiv 3 \pmod 4$, for $n = 0$ (A.3a) is replaced with

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(m+1,0)} + \mathbf{u}^{(m-1,0)} - 2\mathbf{u}^{(m,0)}) \mathbf{a}^{(1)} \\ + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(m+1,1)} - \mathbf{u}^{(m,0)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(m,1)} - \mathbf{u}^{(m,0)}) \mathbf{a}^{(3)}],$$

whereas for $n = N - 1$ (A.3d) is substituted by

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(m+1,N-1)} + \mathbf{u}^{(m-1,N-1)} - 2\mathbf{u}^{(m,N-1)}) \mathbf{a}^{(1)} \\ + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(m,N-2)} - \mathbf{u}^{(m,N-1)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(m+1,N-2)} - \mathbf{u}^{(m,N-1)}) \mathbf{a}^{(3)}].$$

Further, (A.3b) and (A.3c) hold when $n$ is odd and even, respectively.

### A.2. The resultant elastic forces for nodes situated at the boundary

When $N \equiv 1 \pmod 4$, the forcing term $f^{(0,n)}$ ($0 \le n \le N - 1$) in (1) has the form:

• for $n = 0$:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(1,0)} - \mathbf{u}^{(0,0)}) \mathbf{a}^{(1)} + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(0,1)} - \mathbf{u}^{(0,0)}) \mathbf{a}^{(2)}], \qquad \text{(A.4a)}$$

• for $0 < n < N - 1$ and $n$ even:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(1,n)} - \mathbf{u}^{(0,n)}) \mathbf{a}^{(1)} + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(0,n+1)} - \mathbf{u}^{(0,n)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(0,n-1)} - \mathbf{u}^{(0,n)}) \mathbf{a}^{(3)}], \qquad \text{(A.4b)}$$

• for $0 < n < N - 1$ and $n$ odd:

$$\Gamma [\mathbf{a}^{(1)} \cdot (\mathbf{u}^{(1,n)} - \mathbf{u}^{(0,n)}) \mathbf{a}^{(1)} + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(1,n+1)} + \mathbf{u}^{(0,n-1)} - 2\mathbf{u}^{(0,n)}) \mathbf{a}^{(2)} \\ + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(0,n+1)} + \mathbf{u}^{(1,n-1)} - 2\mathbf{u}^{(0,n)}) \mathbf{a}^{(3)}], \qquad \text{(A.4c)}$$
$$
\begin{array}{l}
\bullet \text{ for } n = N - 1 : \\
\Gamma [a^{(1)} \cdot (u^{(1,N-1)} - u^{(0,N-1)}) a^{(1)} + a^{(3)} \cdot (u^{(0,N-2)} - u^{(0,N-1)}) a^{(3)}]. \qquad (A.4d)
\end{array}
$$

The time-harmonic equivalents of these forces appearing in (4) are obtained by replacing $\mathbf{u}^{(m,n)}$ by $\mathbf{U}^{(m,n)}/\Gamma$.

When $N \equiv 3 \pmod 4$, (A.4a) is updated to

$$
\begin{equation}
\Gamma [a^{(1)} \cdot (\mathbf{u}^{(1,0)} - \mathbf{u}^{(0,0)}) \mathbf{a}^{(1)} + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(1,1)} - \mathbf{u}^{(0,0)}) \mathbf{a}^{(2)} \nonumber \\
\qquad + \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(0,1)} - \mathbf{u}^{(0,0)}) \mathbf{a}^{(3)}] , \tag{A.5}
\end{equation}
$$

while (A.4d) becomes

$$
\Gamma [a^{(1)} \cdot (\mathbf{u}^{(1,N-1)} - \mathbf{u}^{(0,N-1)}) \mathbf{a}^{(1)} + \mathbf{a}^{(2)} \cdot (\mathbf{u}^{(0,N-2)} - \mathbf{u}^{(0,N-1)}) \mathbf{a}^{(2)} \\
+ \mathbf{a}^{(3)} \cdot (\mathbf{u}^{(1,N-2)} - \mathbf{u}^{(0,N-1)}) \mathbf{a}^{(3)}]. \tag{A.6}
$$

Additionally, (A.4b) and (A.4c) hold for odd and even $n$, respectively.

Alternatively, as mentioned in Section 2.1.4, due to the absence of some links at the edge of the lattice strip where $m = 0$, we can express the boundary conditions as the absence of the corresponding elastic forces. Consequently, when $N \equiv 1 \pmod 4$, the edge conditions that lead to (11) are given as

$$
\begin{array}{l}
\bullet \text{ for } n = 0 : \\
a^{(1)} \cdot (\mathbf{u}^{(-1,0)} - \mathbf{u}^{(0,0)}) a^{(1)} + a^{(3)} \cdot (\mathbf{u}^{(-1,1)} - \mathbf{u}^{(0,0)}) a^{(3)} = \mathbf{0}, \hfill \text{(A.7a)}
\end{array}
$$

$$
\begin{array}{l}
\bullet \text{ for } 0 < n < N - 1 \text{ and } n \text{ even :} \\
\quad a^{(1)} \cdot (\mathbf{u}^{(-1,n)} - \mathbf{u}^{(0,n)}) a^{(1)} + a^{(2)} \cdot (\mathbf{u}^{(-1,n-1)} - \mathbf{u}^{(0,n)}) a^{(2)} \\
\quad \qquad + a^{(3)} \cdot (\mathbf{u}^{(-1,n+1)} - \mathbf{u}^{(0,n)}) a^{(3)} = \mathbf{0},
\end{array}
\qquad \text{(A.7b)}
$$

$$
\begin{array}{l}
\bullet \text{ for } 0 < n < N - 1 \text{ and } n \text{ odd :} \\
\quad a^{(1)} \cdot (\mathbf{u}^{(-1,n)} - \mathbf{u}^{(0,n)}) a^{(1)} = \mathbf{0}, \hfill \text{(A.7c)}
\end{array}
$$

$$
\begin{array}{l}
\bullet \text{ for } n = N - 1 : \\
a^{(1)} \cdot (\mathbf{u}^{(-1,N-1)} - \mathbf{u}^{(0,N-1)}) a^{(1)} + a^{(2)} \cdot (\mathbf{u}^{(-1,N-2)} - \mathbf{u}^{(0,N-1)}) a^{(2)} = \mathbf{0}, \hfill \text{(A.7d)}
\end{array}
$$

where $\mathbf{u}^{(-1,n)}$ ($0 \le n \le N-1$) represent the displacements of absent nodes.

When $N \equiv 3 \pmod 4$, the boundary conditions connected with (11) become

$$
a^{(1)} \cdot (\mathbf{u}^{(-1,p)} - \mathbf{u}^{(0,p)}) a^{(1)} = \mathbf{0}
$$

for $p=0$ and $N-1$, whereas (A.7b) and (A.7c) hold for when $n$ is odd and even, respectively.

**Appendix B. Stiffness matrices**

Here, we specify the forms of the matrices that follow from the governing equations and the elastic interactions within the strip.
To do this, we need the matrices

$$
\begin{align}
\mathbf{A}_1 &= a^{(1)} \otimes a^{(1)} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, & \mathbf{A}_2 &= a^{(2)} \otimes a^{(2)} = \frac{1}{4} \begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3} & 3 \end{pmatrix}, \nonumber \\
\mathbf{A}_3 &= a^{(3)} \otimes a^{(3)} = \frac{1}{4} \begin{pmatrix} 1 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{pmatrix}, \tag{B.1}
\end{align}
$$

as well as the function

$$
\chi(n) = n \bmod 2.
$$

In the formula above, $\chi$ is simply the characteristic function for odd numbers, on which it is equal to unity and zero for all other numbers.

B.1. Stiffness matrices for the entire strip

We start from the strip geometry characterised by a number of rows $N \equiv 1 \pmod 4$. The $2N \times 2N$ coefficient matrices $\mathbf{B} = [\mathbf{B}_{i,j}]_{i,j=1}^N$ and $\mathbf{J} = [\mathbf{J}_{i,j}]_{i,j=1}^N$, used in the recurrence relations (8) and (5) posed in the interior of the strip with $N$ rows, have the $2 \times 2$ block entries:

$$
\mathbf{B}_{i,j} = \begin{cases}
-2\mathbf{A}_1 - (2 - \delta_{1,i} - \delta_{N,j})\mathbf{A}_2 - (2 - \delta_{1,i} - \delta_{N,j})\mathbf{A}_3, & \text{if } i=j, \\
(1 - \chi(i))\mathbf{A}_2 + \chi(i)\mathbf{A}_3, & \text{if } i-j=1, \\
(1 - \chi(i))\mathbf{A}_3 + \chi(i)\mathbf{A}_2, & \text{if } i-j=-1, \\
\mathbf{O}_2, & \text{otherwise,}
\end{cases} \tag{B.2}
$$
and

$$ J^{i,j} = \begin{cases} \mathbf{A}_1, & \text{if } i = j, \\ (1 - \chi(i))\mathbf{A}_3, & \text{if } i - j = 1, \\ (1 - \chi(i))\mathbf{A}_2, & \text{if } i - j = -1, \\ \mathbf{O}_2, & \text{otherwise.} \end{cases} \qquad (B.3) $$

Note that $\mathbf{O}_2$ is the 2 × 2 null matrix.

Considering the edge of the strip, the matrix $S = [S^{i,j}]_{i,j=1}^N$ is needed to account for fewer interactions occurring at this boundary. This $2N \times 2N$ matrix has the 2 × 2 block entries:

$$ S^{i,j} = \begin{cases} -\mathbf{A}_1 - (2 - \delta_{N,i} - \chi(i))\mathbf{A}_2 - (2 - \delta_{1,i} - \chi(i))\mathbf{A}_3, & \text{if } i = j, \\ (1 - \chi(i))\mathbf{A}_2 + \chi(i)\mathbf{A}_3, & \text{if } i - j = 1, \\ (1 - \chi(i))\mathbf{A}_3 + \chi(i)\mathbf{A}_2, & \text{if } i - j = -1, \\ \mathbf{O}_2, & \text{otherwise.} \end{cases} $$

Finally, with $S$ and $B$ in hand, the matrix $G$ in (11) can be calculated as

$$ G = S - B. $$

When $N \equiv 3 \pmod 4$, we simply replace every occurrence of $\chi(i)$ by $1 - \chi(i)$ in the representations for $B$, $J$ and $S$ to retrieve the corresponding forms for coefficient matrices in this case. Moreover, we impose $S^{i,j} = -\mathbf{A}_1 - (1 + \chi(i) - \delta_{1,i} - \delta_{N,i})(\mathbf{A}_2 + \mathbf{A}_3)$ when $i = j$.

## B.2. Stiffness matrices for the symmetric problem

When the symmetry conditions (12) are employed, one needs the $N \times N$ matrices $B_s$ and $J_s$ appearing in (13).

First note that

$$ V_m = N U_m, \qquad (B.4) $$

with $N$ being the rectangular matrix

$$ N = (W_N, O_N), \quad W_N = \text{diag} \left( \sum_{j=1}^{N} \left(1 - \frac{1}{\sqrt{2}}\delta_{N,j}\right)e_j \right), \qquad (B.5) $$

where $e_j$ is the $N \times 1$ unit vector and $O_N$ is the $N \times N$ null matrix. These matrices are written in the forms

$$ B_s = W_N M \hat{B}_s M^T W_N^{-1} \quad \text{and} \quad J_s = M \hat{J}_s M^T W_N^{-1}, \qquad (B.6) $$

respectively, where $W_N$ is given in (B.5) and the rectangular matrix $M$ is

$$ M = (I_N, \mathbf{0}_N), $$

where $\mathbf{0}_N$ is the zero vector of length $N$. The matrices appearing with the hat in (B.6) are $(N + 1) \times (N + 1)$ matrices and they are quoted below. The left and right multiplication by $M$ in the right-hand sides of (B.6) occurs due to the fact that:

(i) employing the symmetry conditions (12), we impose zero vertical components for the displacements of the nodes along the row $n = (N - 1)/2$, allowing for the removal of the $(N + 1)$th columns of the $(N + 1) \times (N + 1)$ matrices;

(ii) following (i), the $(N + 1)$th rows of resulting matrices contain only zeros and consequently they may also be removed to retrieve the $N \times N$ matrices in left-hand sides of (B.6).

Therefore, the pre- and post-multiplication by $M$ in (B.6) enables the aforementioned redundant rows and columns to be removed from the analysis. Finally the application of $W_N^{-1}$ simply scales the last column of the result by a factor of $\sqrt{2}$ (see (B.5)).

When $N \equiv 1 \pmod 4$, the matrix $\hat{B}_s = [\hat{B}_s^{i,j}]_{i,j=1}^{(N+1)/2}$ has the 2 × 2 block entries:

$$ \hat{B}_s^{i,j} = \begin{cases} -2\mathbf{A}_1 - (2 - \delta_{1,i})\mathbf{A}_2 - (2 - \delta_{1,i})\mathbf{A}_3, & \text{if } i = j, \\ (1 - \chi(i))\mathbf{A}_2 + \delta_{(N+1)/2,i}\mathbf{A}_2\mathbf{R} + \chi(i)\mathbf{A}_3, & \text{if } i - j = 1, \\ (1 - \chi(i))\mathbf{A}_3 + \chi(i)\mathbf{A}_2, & \text{if } i - j = -1, \\ \mathbf{O}_2, & \text{otherwise.} \end{cases} \qquad (B.7) $$

In the above definition of the entries, we have used the matrix

$$ R = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \qquad (B.8) $$

Additionally, $\hat{J}_s = [\hat{J}_s^{i,j}]_{i,j=1}^{(N+1)/2}$ has the block entries $\hat{J}_s^{i,j}$ provided by the right-hand side of (B.3).
At the edge of the medium, $S_s = W_N M \hat{S}_s M^T W_N^{-1}$ is also introduced to write the corresponding boundary condition for the strip. Here, $\hat{S}_s = [\hat{S}_s^{i,j}]_{i,j=1}^{(N+1)/2}$ has the 2 × 2 block entries:

$$ \hat{S}_s^{i,j} = \begin{cases} -\mathbf{A}_1 - (2 - \chi(i))\mathbf{A}_2 - (2 - \delta_{1,i} - \chi(i))\mathbf{A}_3, & \text{if } i = j, \\ (1 - \chi(i))\mathbf{A}_2 + \chi(i)\mathbf{A}_3 + \delta_{(N+1)/2,i} \mathbf{A}_2 \mathbf{R}, & \text{if } i - j = 1, \\ (1 - \chi(i))\mathbf{A}_3 + \chi(i)\mathbf{A}_2, & \text{if } i - j = -1, \\ \mathbf{O}_2, & \text{otherwise.} \end{cases} \quad (B.9) $$

Moreover, similar to Appendix B.1, the matrix $G_s$ (defined below (15)) is given by

$$ G_s = S_s - B_s. $$

When $N \equiv 3 \pmod 4$, the same considerations drawn at the end of Appendix B.1 apply.

### B.3. Stiffness matrices when $N = 5$

As an example, when the number of rows in the lattice is equal to 5, the matrices $J_s$ and $B_s$ appearing in (14) and defined in (B.6) are

$$ J_s = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \frac{1}{4} & -\frac{\sqrt{3}}{4} & 1 & 0 & \frac{\sqrt{2}}{4} \\ -\frac{\sqrt{3}}{4} & \frac{3}{4} & 0 & 0 & \frac{\sqrt{6}}{4} \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \quad (B.10) $$

and

$$ B_s = \begin{pmatrix} -\frac{5}{2} & 0 & \frac{1}{4} & \frac{\sqrt{3}}{4} & 0 \\ 0 & -\frac{3}{2} & \frac{\sqrt{3}}{4} & \frac{3}{4} & 0 \\ \frac{1}{4} & \frac{\sqrt{3}}{4} & -3 & 0 & \frac{\sqrt{2}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} & 0 & -3 & -\frac{\sqrt{6}}{4} \\ 0 & 0 & \frac{\sqrt{2}}{4} & -\frac{\sqrt{6}}{4} & -3 \end{pmatrix}, \quad (B.11) $$

respectively.

In addition, the matrix $S_s$ in (15) has the form

$$ S_s = \begin{pmatrix} -\frac{5}{4} & -\frac{\sqrt{3}}{4} & \frac{1}{4} & \frac{\sqrt{3}}{4} & 0 \\ -\frac{\sqrt{3}}{4} & -\frac{3}{4} & \frac{\sqrt{3}}{4} & \frac{3}{4} & 0 \\ \frac{1}{4} & \frac{\sqrt{3}}{4} & -2 & 0 & \frac{\sqrt{2}}{4} \\ \frac{\sqrt{3}}{4} & \frac{3}{4} & 0 & -3 & -\frac{\sqrt{6}}{4} \\ 0 & 0 & \frac{\sqrt{2}}{4} & -\frac{\sqrt{6}}{4} & -\frac{3}{2} \end{pmatrix}. \quad (B.12) $$

Finally, the matrix $G_s$ is

$$ G_s = \begin{pmatrix} \frac{5}{4} & -\frac{\sqrt{3}}{4} & 0 & 0 & 0 \\ -\frac{\sqrt{3}}{4} & \frac{3}{4} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{3}{2} \end{pmatrix}. \quad (B.13) $$

From above it is evident that $J_s$ and $G_s$ are singular matrices, with both having the same rank 4.

## Appendix C. Derivation of the solution to the scattering problem via the Z-transform

Here, we derive the solution to the considered problem in the lattice strip by applying the Z-transform.

First, we note that (13) and (15) can be combined into a single equation using the Kronecker delta:

$$ [(S_s + \omega^2 I)\delta_{m,0} + (1 - \delta_{0,m})(B_s + \omega^2 I)]V_m + J_s V_{m+1} + (1 - \delta_{m,0})J_s^T V_{m-1} = 0, $$
for $m \ge 0$. Inserting (21) leads to the difference equations satisfied by the reflected field $V_m^r$ as

$$ \begin{aligned} & [ (S_s + \omega^2 I) \delta_{m,0} + (1 - \delta_{0,m}) (B_s + \omega^2 I) ] V_m^r + J_s V_{m+1}^r + (1 - \delta_{m,0}) J_s^T V_{m-1}^r \\ & = F_m(z_{\text{inc}}) \delta_{m,0}, \quad m \ge 0, \end{aligned} \quad (C.1) $$

with

$$ F_m(z_{\text{inc}}) = -(S_s - B_s - z_{\text{inc}}^{-1} J_s^T) A_{\text{inc}} z_{\text{inc}}^m = -P(z_{\text{inc}}^{-1}) A_{\text{inc}} z_{\text{inc}}^m, $$

where (13) was used for the incident wave.

Next, we apply the unilateral transform in $m$ given in (22) with respect to the variable $m$, that defines an analytic function outside the unit circle $C$ in the complex plane. To do this, we multiply (C.1) by $z^{-m+1}$ and sum over $m \ge 0$, where $m \in \mathbb{Z}$. Additionally, we need the results:

$$ \sum_{m \ge 0} V_{m+1}^r z^{-m+1} = z^2 (V^{r,F} - V_0^r), \quad (C.2) $$

$$ \sum_{m \ge 0} (1 - \delta_{m,0}) V_{m-1}^r z^{-m+1} = \sum_{m \ge 1} V_{m-1}^r z^{-m+1} = V^{r,F} \quad (C.3) $$

and

$$ \sum_{m \ge 0} (1 - \delta_{m,0}) V_m^r z^{-m+1} = \sum_{m \ge 1} V_m^r z^{-m+1} = z (V^{r,F} - V_0^r), \quad (C.4) $$

which are derived by applying appropriate changes of summation variables and the completion of semi-infinite sums yielding terms involving (22). Further, we apply the standard results

$$ \sum_{m \ge 0} \delta_{m,0} V_m^r z^{-m+1} = z V_0^r \quad \text{and} \quad \sum_{m \ge 0} \delta_{m,0} z_{\text{inc}}^m z^{-m+1} = z, \quad (C.5) $$

where the latter gives the transformation of the load appearing in (C.1):

$$ \sum_{m \ge 0} F_m(z_{\text{inc}}) \delta_{m,0} = -P(z_{\text{inc}}^{-1}) A_{\text{inc}} z. \quad (C.6) $$

Combining (C.2)–(C.5), we obtain that the Z-transform of (C.1) is

$$ z \mathbf{K}(z) \mathbf{V}^{r,F} + z(S_s - B_s - z \mathbf{J}_s) \mathbf{V}_0^r = -P(z_{\text{inc}}^{-1}) \mathbf{A}_{\text{inc}} z $$

or, owing again to (25), we have

$$ \mathbf{K}(z) \mathbf{V}^{r,F} = -P(z_{\text{inc}}^{-1}) \mathbf{A}_{\text{inc}} - \mathbf{Q}(z) \mathbf{V}_0^r. $$

This is a functional equation for the transformed reflected field, with an inhomogeneous term containing the unknown constant vector in the form of the edge displacement. Rearranging for $V^{r,F}$ and inverting the Z-transform then yields (23).

## Appendix D. Evaluation of coefficients in the explicit modal expansion

In this Section we derive (29)–(32).

If the frequency $\omega$ possesses a small positive imaginary part, the singularities of the integral kernel in (23) (corresponding to the simple zeros of the determinant of $K(z)$) undergo a small perturbation in $\mathbb{C}$ from their locations obtained when $\omega$ is purely real. This is demonstrated in Fig. D.1, where the imaginary part of $\omega$ has been chosen to visibly demonstrate the effect of this perturbation.

When the frequency is purely real, those singularities with unit modulus correspond to waves having a constant amplitude that propagate in the strip. There are 2 such singularities in Fig. D.1(a). In addition, there are singularities situated at a finite distance from the unit circle $C$ that define evanescent waves. If one considers an infinite uniform strip, these modes decay exponentially when $m \to \infty$ ($m \to -\infty$) if the associated singularity is inside (outside) $C$. In Fig. D.1(a), there are 3 singularities inside $C$ defining possible evanescent modes, with one situated on the real axis at approximately $z = 0.36$.

The singularities located along the unit-circle $C$ in $\mathbb{C}$ shift to inside or outside this circle when the small imaginary part is introduced (see the illustration given in Fig. D.1(b)). Further, we note that the new locations of these points relative to the unit circle $C$ are determined by the group velocity of the associated waves (Slepyan, 2002).

In the evaluation of the reflected field through (23), we introduce a small positive imaginary part in the frequency within the kernel on the right-hand side. The integral can be fully evaluated by the consideration of residues in connection with the kernel singularities. In particular, the resulting field should be bounded for $m \to \infty$. Hence, we evaluate the contour integral using only the singular points inside $C$, that form the set $D_s$ in (31).

Thus, the residue theorem applied to (23) implies that

$$ V_m^r = \sum_{z_p \in D_s} \text{Res}[(\mathbf{K}(z))^{-1} \varphi(z) z^{m-1}, z_p] \quad (D.1) $$
<figure><img src="image_13.png" /><figcaption>Fig. D.1. The roots of det(K(z)) for N = 5 and (a) ω = 0.6, (b) ω = 0.6 + 0.02i. The roots in (a) are indicated by crosses, whereas the roots in (b) are red (blue) dots on the exterior (interior) of the unit circle C. Those circled contribute to the set D, defined in (31), representing symmetric modes in the strip.</figcaption></figure>

where, owing to the fact that z_p are simple poles, the residues at these points are

$$
\mathrm{Res}([\mathbf{K}(z)]^{-1} \varphi(z) z^{m-1}, z_p) = \lim_{z \to z_p} (z - z_p) [\mathbf{K}(z)]^{-1} \varphi(z) z^{m-1}.
$$

Then, using the standard representation for the matrix inverse and the definition of the derivative as a limit, we have

$$
\begin{align}
\mathrm{Res}([\mathbf{K}(z)]^{-1} \varphi(z) z^{m-1}, z_p) &= \lim_{z \to z_p} \frac{(z - z_p)}{\det(\mathbf{K}(z))} \mathrm{adj}(\mathbf{K}(z)) \varphi(z) z^{m-1} \nonumber \\
&= \mathbf{A}_r(z_p) \varphi(z_p) z_p^{m-1}, \tag{D.2}
\end{align}
$$

where $\mathbf{A}_r(z)$ is found in (30). Note that, since $\mathbf{K}$ is degenerate at $z_p$, l'Hôpital's rule has been applied in (30) to compute the limit.
A similar procedure can be applied to evaluate (27) and (28) to obtain (32).

**Appendix E. Dynamic homogenisation of the lattice edge resonance condition**

In this Appendix, we detail how (36) leads to an analogous condition for edge resonance in the continuum strip (37). Firstly, in Appendix E.1 we show that (37) follows from the governing equations for the homogenised problem in the semi-infinite continuum strip. Then, in Appendix E.2 we rewrite (36) in a form amendable to the dynamic homogenisation scheme, that will be applied in Appendix E.3. In particular, in Appendix E.3 we use the method of meso-scale approximations, a technique used to describe the response of solids with inclusions and discrete media (Nieves, 2017; Nieves & Movchan, 2022a, 2023). Here, this method is adopted to formally establish the connection between the identities (36) for the lattice strip and (37) for the effective continuum strip.

E.1. Derivation of (37) from the effective problem for the continuum strip

Scattering problem for the semi-infinite effective strip

Before proving (37) we formulate the scattering problem for the effective semi-infinite elastic strip, which occupies the set
Ω := {x = (x₁, x₂)ᵀ : x₁>0, 0<x₂< H}. We define the total field uₜt as the solution of

$$
\mathcal{L}(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x}) + \rho_{\text{hom}} \omega^2 \mathbf{u}_{\text{tot}}(\mathbf{x}) = \mathbf{0}, \quad \mathbf{x} \in \Omega, \tag{E.1}
$$

$$
T_n(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x}) = \mathbf{0}, \quad \mathbf{x} \in E. \tag{E.2}
$$

Here,

$$
\mathcal{L}(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x}) := \mu_{\text{hom}} \Delta \mathbf{u}_{\text{tot}}(\mathbf{x}) + (\lambda_{\text{hom}} + \mu_{\text{hom}}) \nabla(\nabla \cdot \mathbf{u}_{\text{tot}}(\mathbf{x}))
$$

is the Lamé operator and E is the set describing the edge of the strip at $x_1 = 0$, as defined in Section 3.3.

For edge resonance in the strip, we concern ourselves with symmetric propagating wave fields in the strip. Thus, we assume

$$
\mathbf{u}_{\text{tot}}(x_1, x_2) = R \mathbf{u}_{\text{tot}}(x_1, H - x_2), \qquad (\text{E.3})
$$

with **R** defined in (B.8).
In addition, we define the reflected field $\mathbf{u}_{\text{ref}}$ in the usual way:

$$ \mathbf{u}_{\text{ref}}(\mathbf{x}) = \mathbf{u}_{\text{tot}} - \mathbf{u}_{\text{inc}}(\mathbf{x}), \qquad (\text{E.4}) $$

where the incident field $\mathbf{u}_{\text{inc}}(\mathbf{x}) = \mathbf{a}_{\text{inc}}(x_2)e^{-ik_i x_1}$ and also obeys the symmetry property (E.3). In this vector function, $\mathbf{a}_{\text{inc}}$ is the vector amplitude of the incident wave and $k_i$ is the corresponding wavenumber (see for instance Pagneux, 2006 for further details of their computation). The amplitude is responsible for ensuring that $\mathbf{u}_{\text{inc}}(\mathbf{x})$ satisfies the boundary conditions on the upper and lower faces of the strip.

Note that $\mathbf{u}_{\text{ref}}(\mathbf{x})$ is a solution of (E.1), whereas

$$ T_n(\nabla_x)\mathbf{u}_{\text{ref}}(\mathbf{x}) = -T_n(\nabla_x)\mathbf{u}_{\text{inc}}(\mathbf{x}), \quad \mathbf{x} \in E. \qquad (\text{E.5}) $$

Further, $\mathbf{u}_{\text{ref}}(\mathbf{x})$ is also subject to the radiation condition for $x_1 \to \infty$.

### Proof of (37)

Next we show that (37) holds at the conjugate of the edge resonance frequency for the scattering problem in the effective continuum strip. At this frequency $\text{Im}(\omega) > 0$, and this leads to the wavenumber $k_i$ defining the incident wave possessing a small positive imaginary part, giving an incident field and total field that exponentially grows for $x_1 \to \infty$. The selected frequency also ensures $\mathbf{u}_{\text{ref}}(\mathbf{x})$ is a linear combination of right-going evanescent waves only, exponentially decaying as $x_1 \to \infty$ and associated with finite decay rate constants, whose magnitudes exceed $\text{Im}(k_i)$.

According to (E.1)–(E.5), the integral in the left-hand side of (37) is equivalent to

$$ - \int_E \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) \, dx_2. \qquad (\text{E.6}) $$

Next, we use the Betti formula applied to $\mathbf{u}_{\text{tot}}(\mathbf{x})$ and $\mathbf{u}_{\text{ref}}(\mathbf{x})$ in $\Omega_R := \{\mathbf{x} = (x_1, x_2)^T : 0 < x_1 < R, 0 < x_2 < H\}$. This helps to convert (E.6) to integrals over $\Omega_R$ and $E_R := \{\mathbf{x} = (x_1, x_2)^T : x_1 = R, 0 < x_2 < H\}$ as follows:

$$ \begin{aligned} & - \int_E \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) \, dx_2 \\ &= \int_{\Omega_R} \{\mathbf{u}_{\text{ref}}(\mathbf{x}) \cdot \mathcal{L}(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x}) - \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot \mathcal{L}(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x})\} d\mathbf{x} \\ &\quad + \int_{E_R} \{\mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) - \mathbf{u}_{\text{ref}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x})\} d\mathbf{x}_2. \end{aligned} \qquad (\text{E.7}) $$

Owing to the definition of $\mathbf{u}_{\text{tot}}(\mathbf{x})$ and $\mathbf{u}_{\text{ref}}(\mathbf{x})$, the integral over $\Omega_R$ vanishes and we are left with the identity:

$$ \begin{aligned} & \int_E \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) \, dx_2 \\ &= \int_{E_R} \{ \mathbf{u}_{\text{ref}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x}) - \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) \} \, dx_2. \end{aligned} \qquad (\text{E.8}) $$

Finally, $\omega$ is chosen to be the frequency where the coefficient of the reflected field vanishes (for $\text{Im}(\omega) > 0$), whose conjugate determines the complex edge resonance frequency of the strip. At this frequency, the exponential decay of the reflected field dominates in behaviour of the preceding integrand as $x_1 \to \infty$. Hence, taking the limit as $R \to \infty$ we arrive at

$$ \begin{aligned} & \int_E \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) \, dx_2 \\ &= \lim_{R \to \infty} \int_{E_R} \{ \mathbf{u}_{\text{ref}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{tot}}(\mathbf{x}) - \mathbf{u}_{\text{tot}}(\mathbf{x}) \cdot T_n(\nabla_x) \mathbf{u}_{\text{ref}}(\mathbf{x}) \} \, dx_2 = 0, \end{aligned} \qquad (\text{E.9}) $$

which upon using (E.5) proves (37).

## E.2. Equivalent form of the edge resonance condition

The condition (36) is given by

$$ \begin{aligned} 0 &= \mathbf{A}_{\text{inc}} \cdot \varphi(z_r) \\ &= \mathbf{A}_{\text{inc}} \cdot (\mathbf{S}_s + \omega^2 \mathbf{I} - (\mathbf{B}_s + \omega^2 \mathbf{I}) - z_r^{-1} \mathbf{J}_s^T) \mathbf{A}_{\text{inc}} \\ &\quad + \mathbf{A}_{\text{inc}} \cdot (\mathbf{S}_s + \omega^2 \mathbf{I} - (\mathbf{B}_s + \omega^2 \mathbf{I}) - z_r \mathbf{J}_s) \mathbf{V}_0^r. \end{aligned} \qquad (\text{E.10}) $$

Here, (25) has been used in writing the equality.

Next, noting that the incident field

$$ \mathbf{V}_m^{\text{inc}} = \mathbf{A}_{\text{inc}} \mathbf{z}_{\text{inc}}^m, \quad m \in \mathbb{Z} \qquad (\text{E.11}) $$

in (21), which also satisfies (13) for any $m$, allows us to obtain:

$$ 0 = \mathbf{A}_{\text{inc}} \cdot (\mathbf{S}_s + \omega^2 \mathbf{I} + z_{\text{inc}} \mathbf{J}_s) \mathbf{A}_{\text{inc}} + \mathbf{A}_{\text{inc}} \cdot (\mathbf{S}_s + \omega^2 \mathbf{I} + z_r^{-1} \mathbf{J}_s^T) \mathbf{V}_0^r, \qquad (\text{E.12}) $$
where $z_{inc} = z_r^{-1}$. Taking into account that $S_s$ is symmetric, we get:

$$0 = \mathbf{A}_{\text{inc}} \cdot (\mathbf{S}_s + \omega^2 \mathbf{I} + z_{\text{inc}} \mathbf{J}_s) \mathbf{A}_{\text{inc}} + \mathbf{V}_0^r \cdot (\mathbf{S}_s + \omega^2 \mathbf{I} + z_{\text{inc}} \mathbf{J}_s) \mathbf{A}_{\text{inc}}, \quad (\text{E.13})$$

which is

$$0 = \mathbf{V}_0 \cdot ((\mathbf{S}_s + \omega^2 \mathbf{I})\mathbf{V}_0^{\text{inc}} + \mathbf{J}_s \mathbf{V}_1^{\text{inc}}) \quad (\text{E.14})$$

owing to (21) and (E.11). Here, we mention that the term in the brackets corresponds to the stresses produced at the edge of the lattice strip by the incident field. Another form of this condition can therefore be achieved knowing that the incident wave amplitude satisfies (13). Thus, we obtain the condition:

$$0 = \mathbf{V}_0 \cdot ((\mathbf{S}_s - \mathbf{B}_s)\mathbf{V}_0^{\text{inc}} - \mathbf{J}_s^T \mathbf{V}_1^{\text{inc}}). \quad (\text{E.15})$$

Finally, for the dynamic homogenisation to be applied, we require (E.15) in the dimensional form. This relies on the normalised parameters in (13) and (15) being replaced through (3) by their dimensional equivalents. This procedure yields analogous equations where $\omega^2$ is replaced by $M\omega^2$ and matrices $\mathbf{B}_s$, $\mathbf{S}_s$ and $\mathbf{J}_s$ are scaled by $\Gamma$. In this form, one can find the reflected field in terms of dimensional variables via the approach outlined in Section 3 with obvious modifications. In addition, we also arrive at the equivalent of (36) in terms of variables that carry dimensions. Further, (E.15) is derived in a similar manner as above.

In going forward, we let $\mathbf{V}_m$ and $\mathbf{V}_m^{\text{inc}}$ stand for the dimensional total and incident fields. The condition (E.15) still holds for these fields but the procedure in returning to dimensional variables simply leads to the rescaling of this condition by $\Gamma$, which is omitted for the time being.

Next, in constructing the edge conditions for the incident field $\mathbf{V}_m^{\text{inc}}$ for $m = 0, -1$ as outlined in Appendix A.2 (taking into account the symmetry conditions along the centre line of the strip), we see that (E.15) can be expressed as the sum of four contribution as follows

$$0 = \sigma_a + \sigma_b + \sigma_c + \sigma_d, \quad (\text{E.16})$$

where

$$\sigma_a = \mathbf{V}^{(0,0)} \cdot \left[ \mathbf{a}^{(1)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,0)} - \mathbf{V}_{\text{inc}}^{(0,0)} \right) \mathbf{a}^{(1)} + \mathbf{a}^{(3)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,1)} - \mathbf{V}_{\text{inc}}^{(0,0)} \right) \mathbf{a}^{(3)} \right], \quad (\text{E.17})$$

$$\sigma_b = \sum_{\substack{n \text{ even,} \\ 0 < n < (N-1)/2}} \mathbf{V}^{(0,n)} \cdot \left[ \begin{aligned} & \mathbf{a}^{(1)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,n)} - \mathbf{V}_{\text{inc}}^{(0,n)} \right) \mathbf{a}^{(1)} \\ & + \mathbf{a}^{(2)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,n-1)} - \mathbf{V}_{\text{inc}}^{(0,n)} \right) \mathbf{a}^{(2)} + \mathbf{a}^{(3)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,n+1)} - \mathbf{V}_{\text{inc}}^{(0,n)} \right) \mathbf{a}^{(3)} \end{aligned} \right], \quad (\text{E.18})$$

$$\sigma_c = \sum_{\substack{n \text{ odd,} \\ 0 < n < (N-1)/2}} \mathbf{V}^{(0,n)} \cdot \left[ \mathbf{a}^{(1)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,n)} - \mathbf{V}_{\text{inc}}^{(0,n)} \right) \mathbf{a}^{(1)} \right] \quad (\text{E.19})$$

and

$$\sigma_d = \mathbf{V}^{(0,(N-1)/2)} \cdot \left[ \begin{aligned} & \mathbf{a}^{(1)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,(N-1)/2)} - \mathbf{V}_{\text{inc}}^{(0,(N-1)/2)} \right) \mathbf{a}^{(1)} \\ & + \mathbf{a}^{(2)} \cdot \left( \mathbf{V}_{\text{inc}}^{(-1,(N-3)/2)} - \mathbf{V}_{\text{inc}}^{(0,(N-1)/2)} \right) \mathbf{a}^{(2)} \\ & + \mathbf{a}^{(3)} \cdot \left( \mathbf{R} \mathbf{V}_{\text{inc}}^{(-1,(N-3)/2)} - \mathbf{V}_{\text{inc}}^{(0,(N-1)/2)} \right) \mathbf{a}^{(3)} \end{aligned} \right], \quad (\text{E.20})$$

where again we note $\mathbf{V}_m = ((\mathbf{V}^{(m,0)})^T, \dots, (\mathbf{V}^{(m,(N-3)/2)})^T, \mathbf{V}_1^{(m,(N-1)/2)})^T$ and $\mathbf{V}_m^{\text{inc}} = ((\mathbf{V}^{(m,0)})^T, \dots, (\mathbf{V}^{(m,(N-3)/2)})^T, (\mathbf{V}_1^{(m,(N-1)/2)})^T$ denote the dimensional fields. Here, in the last condition we note $\mathbf{R}$ in (B.8) has once more been employed as a result of the symmetry conditions (see Section 2.2).

### E.3. Formal dynamic homogenisation scheme

Now equipped with the required tools developed in the previous two Sections, we apply the method of meso-scale approximations to replace the sums in (E.16) by integral characteristics involving fields of the effective problem. In doing so, the condition (E.16) developed for the lattice strip leads to the condition that is connected with the appearance of the edge resonance in the analogous continuum problem.

#### Lattice parameters and homogenisation within strip interior

We assume that, independently of the total number of rows $N$, the strip has a fixed width $H = \sqrt{3d}/2$, where $d > 0$. In this strip, the link length is $L = d/N$. We also assume that the masses at the nodes depend on $N$ as $M = O(N^{-2})$. As discussed in Carta et al. (2023b) and Slepyan (2002), as $N \to \infty$, we obtain an elastic continuum characterised by density $\rho_{\text{hom}} = 2MN^2/(\sqrt{3d^2})$ and Lamé parameters $\lambda_{\text{hom}} = \mu_{\text{hom}} = \sqrt{3}\Gamma/4$.
### Assumptions required for the homogenisation scheme

In (E.17)–(E.20), we have that

$$ \mathbf{V}_{\text{inc}}^{(m,n)} = \mathbf{V}_{\text{inc}}\left(\frac{L}{2}\left[2m + \frac{1 + (-1)^{n+1}}{2}\right], \frac{\sqrt{3}}{2}Ln\right), \quad m = -1, 0, \quad 0 \le n \le N-1 \qquad (E.21) $$

and

$$ \mathbf{V}^{(m,n)} = \mathbf{V}\left(\frac{L}{2}\left[2m + \frac{1 + (-1)^{n+1}}{2}\right], \frac{\sqrt{3}}{2}Ln\right), \quad m = 0, \quad 0 \le n \le N-1. \qquad (E.22) $$

Further, we also assume for $x_2 < H/2$:

$$ \lim_{N \to \infty} \mathbf{V}^{(0,n)} = \mathbf{u}_{\text{tot}}(0, x_2), \quad \lim_{N \to \infty} \mathbf{V}_{\text{inc}}^{(0,n)} = \mathbf{u}_{\text{inc}}(0, x_2), \qquad (E.23) $$

and for large N

$$ \mathbf{V}_{\text{inc}}^{(-1,n)} \sim \mathbf{u}_{\text{inc}}\left(-L\left\{1 + \left[\frac{(-1)^n - 1}{4}\right], x_2\right\}, \right) \qquad (E.24) $$

for $0 \le n \le (N-1)/2$, with $x_2 = \sqrt{3}Ln/2$. Similarly, for large N we assume

$$ \mathbf{V}_{\text{inc}}^{(-1,n \mp 1)} \sim \mathbf{u}_{\text{inc}}(-L/2, x_2 \mp \sqrt{3}L/2), \qquad (E.25) $$

for $n$ even with $0 \le n \le (N-1)/2$, where again $x_2 = \sqrt{3}Ln/2$. The quantities $\mathbf{u}_{\text{inc}}$ and $\mathbf{u}_{\text{tot}}$ in the above right-hand sides (E.23)–(E.25) represent the incident and total fields, respectively, for the effective continuum.

### Dynamic homogenisation of (E.16)

The adopted homogenisation procedure is similar to that developed in Nieves and Movchan (2022b), which enables sums involving quantities linked to the discrete medium to be approximated by integrals in the case when the microstructure is dense in comparison to the overall size of the discrete system.

We begin by inserting (E.21) and (E.22) into (E.17)–(E.20). Following this, we assume $N$ is large and employ (E.23)–(E.25). The Taylor expansion is then utilised to determine the leading order behaviour of those terms as $L \to 0$ ($N \to \infty$).

For example, if $N$ is large, in (E.18) we use (E.23)–(E.25) and employ the Taylor expansion about $(x_1, x_2) = (0, \sqrt{3}Lm/2)$ to the finite differences involving the incident field within the $m$th term of the sum. In doing so, we get:

$$ \lim_{N \to \infty} \sigma_b = - \sum_{\substack{n \text{ even} \\ 0 < n < (N-1)/2}} L \mathbf{u}_{\text{tot}}(0, x_2) \cdot \beta(0, x_2) \Big|_{x_2 = \frac{\sqrt{3}}{2}Ln} + O(L) \qquad (E.26) $$

obtaining the aforementioned expansion, where

$$ \beta(x_1, x_2) = \left[ \begin{aligned} & \frac{5}{4} \frac{\partial u_{\text{inc}}^{(1)}}{\partial x_1}(x_1, x_2) + \frac{3}{4} \frac{\partial u_{\text{inc}}^{(2)}}{\partial x_2}(x_1, x_2) \\ & - \frac{3}{4} \frac{\partial u_{\text{inc}}^{(1)}}{\partial x_2}(x_1, x_2) + \frac{3}{4} \frac{\partial u_{\text{inc}}^{(2)}}{\partial x_1}(x_1, x_2) \end{aligned} \right]. $$

Note, as $N$ is large with $L = O(N^{-1})$, the sum in (E.26) can be treated as a Riemann sum and consequently approximated by integral over the edge $\mathcal{E} := \{\mathbf{x} : x_1 = 0, 0<x_2<H/2\}$ of the continuum strip. This gives for $L \to 0$ that

$$ \sigma_b = - \int_{\mathcal{E}} \mathbf{u}_{\text{tot}}(0, x_2) \cdot \beta(0, x_2) dx_2. \qquad (E.27) $$

A similar approach using the Taylor expansion about $(x_1, x_2) = (0, \sqrt{3}Lm/2)$ in (E.19) gives

$$ \sigma_c = - \sum_{\substack{n \text{ odd} \\ 0 < n < (N-1)/2}} L \mathbf{u}_{\text{tot}}(0, x_2) \cdot \gamma(0, x_2) \Big|_{x_2 = \frac{\sqrt{3}}{2}Ln} + O(L), \qquad (E.28) $$

where

$$ \gamma(x_1, x_2) = \left[ \begin{array}{c} \frac{\partial u_{\text{inc}}^{(1)}}{\partial x_1}(x_1, x_2) \\ 0 \end{array} \right], $$

and upon rewriting the leading order term as an integral we obtain

$$ \sigma_c = - \int_{\mathcal{E}} \mathbf{u}_{\text{tot}}(0, x_2) \cdot \gamma(0, x_2) dx_2 $$

as $L \to 0$.

Finally, following the application of the Taylor expansion about $(x_1, x_2) = (0, 0)$ and $(0, \sqrt{3}(N-1)L/2)$ in $\sigma_a$ and $\sigma_d$, we arrive at $\sigma_a = O(L)$ and

$$ \sigma_d = \mathbf{u}_{\text{tot}}(0, H/2) \cdot \frac{\sqrt{3}}{2} \left( \frac{1}{-\sqrt{3}} \right) u_{\text{inc}}^{(2)}(0, H/2) + O(L), \qquad (E.29) $$
<figure><img src="image_14.png" /><figcaption>Fig. F.1. Behaviour of the reciprocal of the edge displacement magnitude defined by (F.1) as a function of frequency $\omega$. This computation is based on (26) and (32).</figcaption></figure>

respectively. Note that the leading order term in the preceding right-hand side is then zero as a consequence of the symmetry of the wave fields considered. This shows that $\sigma_d = O(L)$. Consequently the terms $\sigma_a$ and $\sigma_d$ vanish in the limit as $N \to \infty$ in (E.16). This result, together with the combination of (E.27) and (E.28) and their appropriate scaling by $\Gamma/\sqrt{3}$, provides (38) with the integration domain being $\mathbb{C}$. The main result of this Section and Section 3.3 is then given by the extension of this integral over $\mathbb{C}$ to one over $E$ through a symmetry argument (e.g. see (E.3)). This then proves that (37) is achieved as the limit of (36) as $N \to \infty$ in the lattice strip as the width of the strip $H$ is fixed.

## Appendix F. Numerical computations based on the Z-transform

In this Section, we present additional computations that utilise the solution derived in Section 3.

### F.1. Frequency dependency of the displacement at the edge of the strip

The solution based on the Z-transform can be exploited to determine the quasi-resonance frequency through the study of the edge displacement $V_0^r$ given by (26) and (32). When the system is close to the edge resonance frequency, we expect this quantity to be large. To verify this, we study the full reflected field at the edge $U_0^r = [U_r^{(0,i)}]_{i=0}^{N-1}$, with $U_r^{(0,i)} = (U_{r,1}^{(0,i)}, U_{r,2}^{(0,i)})^T$ being the displacement generated by the reflected field at node (0, i) at the edge. This is obtained from $V_0^r$ through (21) and by simple application of the symmetry conditions for the response of the strip considered here (see Section 2.2).

We define

$$ \|U_0^r\| := \left( \sum_{i=0}^{N-1} \left( |U_{r,1}^{(0,i)}|^2 + |U_{r,2}^{(0,i)}|^2 \right) \right)^{1/2}, \qquad (F.1) $$

where the internal norm applied to individual components of the quantity $U_0^r$ yields the modulus of their complex value.

The reciprocal of $\|U_0^r\|$ has been plotted in Fig. F.1 as a function of the incoming wave frequency $\omega$. For this computation, the incident wave has been chosen with the vector amplitude defined through the norm in (F.1) equal to 0.05. The incoming wave has a wavenumber determined from the dispersion curves in Fig. 2. In this computation, the imaginary part of the complex frequency has been chosen as $10^{-8}$.

In Fig. F.1, we observe that $\|U_0^r\|^{-1}$ is a smooth function of $\omega$ below the cut-on frequency $\omega_c$ for the first order Lamb mode for the lattice strip. This quantity can become extremely small at a single frequency, that is the quasi-resonance frequency, where $U_0^r$ consequently has a large modulus relative to the lattice spacing. This indicates that the edge of the lattice strip is experiencing a response close to resonance.

As expected, the value of the quasi-resonance frequency obtained with the approach based on the Z-transform coincides with the value determined with the mode matching method (compare Figs. F.1 and 4).

### F.2. Comparison of the solution based on the Z-transform with finite element computations

The analytical formulae describing the reflected field in (29)–(32) can be readily checked against benchmark finite element computations.
<figure><img src="image_15.png" /><figcaption>Fig. F.2. Validation of (29). (a) Comsol computation for the reflected wave in an elastic lattice strip with 5 rows generated by an incoming wave with complex frequency $\omega = 0.6 + 10^{-7}i$. An adaptive absorbing layer, not shown here, has been implemented at the right end of the strip to minimise the reflections from the right-hand boundary of the computational domain. (b) Plot of the solution based on (29) near the edge of the lattice strip.</figcaption></figure>

As an example, Fig. F.2(a) provides the displacements of the 5-row strip associated with the reflected field. This computation has been carried out using Comsol Multiphysics. The numerical simulation has been designed in accordance with Section 4.1, where the load at the left-most edge (e.g. $m = 0$) of the strip induced by incident field has been implemented whereas the nodes along the upper and lower boundaries of the interior of the strip ($m \ge 1$) are free. The incident wave has complex frequency $\omega = 0.6 + 10^{-7}i$, amplitude norm $\|\dot{\lambda}_{inc}\| = 0.28$ (see (20) and (F.1)) and $z_{inc} = 0.771 + 0.636i$. Note that in this example the chosen frequency is relatively far from the quasi-resonance frequency (i.e., $\omega \approx 0.4505$). The length of the strip is 100 unit lattice spacings and an adaptive absorbing layer is introduced at the right end of the strip, having width equal to 10 unit lattice spacings.

The illustration in Fig. F.2(a) shows a wave clearly propagating away from the edge of the strip. Fig. F.2(b) shows the displacements for the strip determined from (29)–(32). The excellent agreement between the finite element simulations and the analytical results is evident.

Further, we stress that the set-up of a semi-infinite numerical model in a finite element software requires the fine tuning of an adaptive absorbing layer. This can be time consuming and can yield small perturbations at isolated frequencies where special responses of the strip can be observed, such as the edge resonance frequency and exceptional points in the dispersion diagram where the group velocity is zero or dispersion curves intersect. On the other hand, the analytical methods adopted here provide a fast and highly accurate alternative to the implementation of the numerical solution.

## Data availability

No data was used for the research described in the article.

## References

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