# Theory of truncation resonances in continuum rod-based phononic crystals
with generally asymmetric unit cells

Hasan B. Al Ba’ba’a  Carson L. Willey  Vincent W. Chen  Abigail T. Juhl
Mostafa Nouh<EMAIL_ADDRESS>Dept. of Mechanical and Aerospace Engineering,
University at Buffalo (SUNY), Buffalo, NY 14260, USA  Department of Mechanical
Engineering, Union College, Schenectady, NY 12308, USA  Air Force Research
Laboratory, Wright-Patterson AFB, OH 45433, USA  UES, Inc., Dayton, OH 45432,
USA

###### Abstract

Phononic crystals exhibit Bragg bandgaps, frequency regions within which wave
propagation is forbidden. In solid continua, bandgaps are the outcome of
destructive interferences resulting from periodically alternating layers.
Under certain conditions, natural frequencies emerge within these bandgaps in
the form of high-amplitude localized vibrations near a structural boundary,
referred to as truncation resonances. In this paper, we investigate the
vibrational spectrum of finite phononic crystals which take the form of a one-
dimensional rod, and explain the factors that contribute to the origination of
truncation resonances. By identifying a unit cell symmetry parameter, we
define a family of finite phononic rods which share the same dispersion
relation, yet distinct truncated forms. A transfer matrix method is utilized
to derive closed-form expressions of the characteristic equations governing
the natural frequencies of the finite system and decipher the truncation
resonances emerging across different boundary conditions. The analysis
establishes concrete connections between the localized vibrations associated
with a truncation resonance, boundary conditions, and the overall
configuration of the truncated chain as dictated by unit cell choice. The
study provides tools to predict, tune, and selectively design truncation
resonances, to meet the demands of various applications that require and
uniquely benefit from such truncation resonances.

###### keywords:

phononic crystals , truncation resonances , natural frequency , mode shapes

##  1 Introduction

Over the past few decades, vibration mitigation research has largely
transitioned from traditional active control techniques and pursuit of
materials with stronger damping properties to the notion of artificially
engineered structures which possess a set of unique wave attenuation features
that are otherwise elusive [1, 2]. Most notable among these are Phononic
Crystals (PnCs), a class of periodic structures which come in continuous
(e.g., rods [3], beams [4], and plates [5]) or discrete (e.g., spring-mass
chains [6] and lattices [7]) forms. PnCs exhibit bandgaps, i.e., frequency
regions of forbidden wave propagation, which arise from destructive
interferences at interfaces with impedance mismatches. These mismatches are
most commonly created by layering two or more materials in a spatially
periodic manner with the smallest self-repeating block being referred to as
the PnC’s unit cell. Owing to their ability to manipulate incident
vibroacoustic excitations, PnCs have been extensively used in a broad range of
applications ranging from energy harvesting [8, 9] and attenuation of airborne
sound [10], to nonreciprocal wave transmission [11, 12] and ultrasonic wave
focusing [13].

In their idealized form, PnCs are typically modeled as unbounded systems with
an infinite number of unit cells. By applying the Bloch theorem, a dispersion
diagram which consists of a set of curves relating the frequency of waves in
the elastic medium to their wavenumber can be obtained [14]. These curves
represent “pass bands”, or regions of permissible wave propagations, while
empty spaces between them indicate bandgap regions where no real wavenumber
solutions exist to satisfy a given frequency, culminating in an exponential
attenuation of incident waves. Intuitively, natural frequencies (or
resonances) of a vibrating PnC would therefore lie within the different pass
bands of the dispersion relation. However, PnCs with a finite number of cells
have been shown to exhibit natural frequencies inside bandgap regions, known
as truncation resonances – named as such since they are a direct outcome of
truncating the infinite medium and introducing boundary conditions [15]. While
truncation resonances had been repeatedly observed in research involving PnCs,
they were never explicitly investigated until 2011 [16]. Mode shapes
corresponding to truncation resonances are uniquely characterized by a profile
of high vibrational amplitudes which are confined to the boundaries and decay
away from it due to the bandgap effect. These profiles represent a set of
“edge” or “surface” modes which have been instrumental in several emerging
applications including phononic topological insulators [17, 18, 19], flow
control via phononic subsurfaces [20, 21], and nano-particle mass sensing
[22].

Given the breadth and diversity of the number of engineering problems where
localized attenuated modes play a central role, there is a strong motivation
to study the dynamics of truncation resonances and comprehend their underlying
features including onset conditions, patterns, distributions, and sensitivity
to varying boundary conditions. Since truncation resonances are a function of
finite PnCs, such goal can only be achieved by employing tools that go beyond
the traditional Bloch-wave unit cell analysis, in order to capture the
structural response of the finite system. Al Ba’ba’a et al. pioneered the use
of continuous fractions [23] to devise a transfer-function-based model which
depicts the frequency response of a finite spring-mass PnC [24], by utilizing
a mathematical scheme that enables closed-form expressions for eigenvalues of
perturbed tridiagonal matrices [25, 26]. The work set a framework to identify
truncation resonance locations by recognizing natural frequencies which avert
dispersion branches (i.e., fall outside the solutions of the analytical
dispersion relation), and consequently, defined existence criteria for
truncation resonances in free-free diatomic PnCs as a function of both mass
and stiffness ratios as well as finite chain symmetry, as dictated by the
truncation location. Additional studies of finite periodic systems shed light
on additional discrepancies between the dispersion behavior of an infinite
chain and the actual response of the final counterpart such as bandgaps which
are undeveloped in terms of attenuation strength or frequency range [27, 28,
29]. Bastawrous and Hussein extended the truncation resonance existence
criteria to general boundary conditions as well as finite PnCs with end masses
[30].

In this paper, we investigate the vibrational spectrum of finite continuum
rod-based PnCs which consist of two alternating layers that repeat spatially
to form a periodic composite-like structure. The primary goal is to provide
the fundamental factors that contribute to the origination of truncation
resonances. Contrary to their discrete (spring-mass) counterparts [24, 30],
closed-form equations of finite PnC rods cannot be obtained by using
tridiagonal k-Toeplitz matrices, and therefore warrant a different approach.
In here, we will adopt the transfer matrix method (TMM) whose use in vibration
characterization dates back to the 1950s, whether for flexural beams [31] or
elastic lattices [32, 33, 34]. Later on, and owing to its simplicity and
efficiency as a predictive tool for finite vibrational structures, TMM has
been widely adopted in the modeling of elastic periodic continua [35, 36, 37].
Using TMM, we will present an exhaustive analysis of the location,
distribution, and associated wave propagation modes of the rod-based PnC’s
truncation resonances subject to all the possible boundary conditions. As
eluded to in previous literature, the location at which an infinite periodic
chain is truncated influences the overall symmetry of the resultant finite
structure, and by extension, alters its truncation resonances, thus posing an
additional variable to an already complex problem [38, 39]. Furthermore, the
final configuration of the truncated PnC often requires the use of a non-
integer number of unit cells (For example, the analysis of a perfectly
symmetric PnC of the form “A-B-A-B-A” requires the use of 2.5 units cell of
the form “A-B”). The crux of the presented model lies in the choice of unit
cell configuration, as depicted in Fig. 1, which utilizes a single symmetry
parameter $\delta$ to define the boundaries of the self-repeating portion.
This streamlines the process of obtaining a family of finite PnC chains which
share the same unit cell, yet distinct truncated forms, by boiling it down to
a single design parameter made at the early stage of choosing the unit cell,
and then repeating the chosen cell an integer number of times to build the
finite structure. As such, the entire spectrum of possible truncated versions
of a given PnC can be generated by sweeping the symmetry parameter between
$-1$ and $1$.

Following a description of the parameters governing the design of a generally
asymmetric unit cell, the TMM is used to derive expressions for the dispersion
diagram as well as the natural frequencies, frequency response functions, and
characteristic equations of the finite PnC for four possible boundary
conditions, namely, free-free, fixed-fixed, fixed-free and free-fixed. By
closely inspecting the first twelve bandgaps, we then explain the analogies
and interplay between truncation resonances emerging across the four
scenarios; making several references to the accompanying mode shapes and
establishing connections between the localized vibrations and the
corresponding boundary condition at that location. Finally, we track the
truncation resonances and the aforementioned modes for each boundary condition
as the unit cell configuration changes, enabling us to formalize the different
patterns of truncation resonances that exist in a rod-based PnC of any
truncated form. The framework presented here provides a pathway to predict and
evaluate truncation resonances prior to fabrication, in addition to an ability
to tune and selectively place their frequencies in a manner which caters to
the different applications that require truncation resonances.

##  2 Unit Cell Analysis

###  2.1 Asymmetric unit cell configuration

Consider a one-dimensional PnC rod consisting of an infinite periodic
arrangement of two segments, A and B, which are comprised of different
materials and geometries, as shown in Fig. 1. The rod’s longitudinal
vibrations at a given location $x$ at any time instant $t$ is given by the
displacement function $u(x,t)$, as shown in Fig. 1(b). Each of the two
segments has an elastic modulus $E_{s}$, a density $\rho_{s}$, a cross-
sectional area $A_{s}$, and a length $\ell_{s}$, where $s=a$ for segment A and
$s=b$ for segment B. The self-repeating portion of the PnC rod, henceforth
referred to as the unit cell, is chosen such that the total length of segment
A is equal to $\ell_{a}=\ell_{a+}+\ell_{a-}$. The values of $\ell_{a+}$ and
$\ell_{a-}$ dictate the level of symmetry of the unit cell as depicted in the
lower panel of Fig. 1(a). For any unit cell choice, a symmetry parameter
$\delta$ is defined as

$\delta=\frac{2d}{\ell_{a}},$ (1)

where $d=(\ell_{a+}-\ell_{a-})/2$ is the distance between the center of
segment B in the chosen unit cell and that in a perfectly symmetric unit cell.
As a result, the unit cell studied here is deemed asymmetric except when
$\delta=0$. A choice of $\ell_{a+}>\ell_{a-}$ results in a positive $\delta$
while a choice of $\ell_{a+}<\ell_{a-}$ results in a negative $\delta$, with
$\delta\in[-1,1]$. Using the definition of $d$, the lengths of $\ell_{a+}$ and
$\ell_{a-}$ corresponding to a given symmetry parameter $\delta$ can be
calculated as

$\ell_{a\pm}=\frac{\ell_{a}}{2}(1\pm\delta).$ (2)

Equation (2) confirms the perfect unit cell symmetry condition,
$\ell_{a+}=\ell_{a-}$, which takes place when $\delta=0$. As the absolute
value of $\delta$ increases, the asymmetry of the unit cell increases. At the
limit of $|\delta|=1$, the unit cell takes the form of a single continuous
segment A of length $\ell_{a}$ which is either followed by segment B (when
$\delta=-1$) or follows segment B (when $\delta=+1$). This limiting case of
$|\delta|=1$ is perhaps the most commonly used unit cell configuration in
studies involving bi-layered PnCs (e.g., [40, 41, 42]). However, the choice of
$\delta$ and the effect it has on the shape of a finite PnC that consists of
an integer number of unit cells (also known as the truncation effect) is
critical and at the heart of the present study. In this work, we will detail
how the choice of $\delta$ dictates the presence of truncation resonances
within the emerging bandgaps as well as the number and location of such
resonances.

Figure 1: (a) Schematic diagram of a PnC rod of two alternating layers, A and
B (top), and the different unit cell choices as a function of the symmetry
parameter $\delta$ (bottom). (b) A three-dimensional model of a generic unit
cell indicating the material and geometric properties of the two constitutive
layers, A and B. Longitudinal vibrations in the PnC rod are described by
$u(x,t)$, where $t$ is time and $x$ is the axial direction.

###  2.2 Transfer matrix of a unit cell

Throughout this paper, the transfer matrix method (TMM) is used as the primary
analytical approach as it facilitates both the wave dispersion analysis based
on a single unit cell, as well as the dynamics of a finite PnC rod made of a
given number of cells which will be presented later. The transfer matrix
$\mathbf{T}$ of a unit cell relates the components of the state vector, in
this case the displacement $u$ and force $f$, at the end of one cell $i$ to
the corresponding components at the end of an adjacent cell $i+1$, as follows
[43]

$\begin{Bmatrix}u_{i+1}\\\
f_{i+1}\end{Bmatrix}=\mathbf{T}\begin{Bmatrix}u_{i}\\\ f_{i}\end{Bmatrix}.$
(3)

For the unit cell defined in Sec. 2.1 with a symmetry parameter $\delta$,
$\mathbf{T}$ is found by multiplying the transfer matrices for the three
individual segments which form the unit cell, i.e.,

$\mathbf{T}=\mathbf{T}_{a_{+}}\mathbf{T}_{b}\mathbf{T}_{a_{-}},$ (4)

where

$\mathbf{T}_{s}=\begin{bmatrix}\cos(k_{s}\ell_{s})&\frac{1}{z_{s}\omega}\sin(k_{s}\ell_{s})\\\
-z_{s}\omega\sin(k_{s}\ell_{s})&\cos(k_{s}\ell_{s})\end{bmatrix}.$ (5)

with $s=a_{+}$, $b$, or $a_{-}$. Here, $z_{s}=A_{s}\sqrt{E_{s}\rho_{s}}$ is
the characteristic impedance of each segment, and $k_{s}=\omega/c_{s}$ is the
stand-alone wavenumber of materials A and B as a function of the angular
frequency $\omega$ and the sonic speed of elastic medium
$c_{s}=\sqrt{E_{s}/\rho_{s}}$. Note that $\delta=\pm 1$ renders one of the
transfer matrices $\mathbf{T}_{a_{\pm}}$ an identity matrix, reducing Eq. (4)
to two terms as expected. To streamline the analysis and reduce the number of
parameters, we introduce $\omega_{s}=c_{s}/\ell_{s}$ as well as the two
following non-dimensional quantities

$\alpha=\frac{\frac{1}{\omega_{a}}-\frac{1}{\omega_{b}}}{\frac{1}{\omega_{a}}+\frac{1}{\omega_{b}}},$
(6a) $\beta=\frac{z_{a}-z_{b}}{z_{a}+z_{b}},$ (6b)

where $\alpha\in[-1,1]$ and $\beta\in[-1,1]$ are defined as the frequency and
impedance contrasts, respectively. Using Eq. (6b), the characteristic
impedance of each material can be rewritten as

$z_{s}=z(1\pm\beta),$ (7)

where $z=(z_{a}+z_{b})/2$ is the average characteristic impedance of both
materials. The plus ($+$) version of Eq. (7) denotes material A while the
minus ($-$) version denotes material B. We also define a non-dimensional
frequency $\Omega=\omega/\omega_{0}$, where $\omega_{0}$ is the harmonic mean
of the frequencies $\omega_{a}$ and $\omega_{b}$ and is given by

$\omega_{0}=\frac{2}{\frac{1}{\omega_{a}}+\frac{1}{\omega_{b}}}.$ (8)

The harmonic mean $\omega_{0}$ can be combined with the definition of $\alpha$
to give

$\frac{1}{\omega_{s}}=\frac{1}{\omega_{0}}(1\pm\alpha),$ (9)

where, once again, the plus and minus versions of Eq. (9) correspond to
materials A and B, respectively. Consequently, the matrices in Eq. (4) can be
expanded, resulting in the following unit cell transfer matrix

$\mathbf{T}=\begin{bmatrix}t_{11}&t_{12}\\\ t_{21}&t_{22}\end{bmatrix},$ (10)

where

$\displaystyle t_{11}$
$\displaystyle=\cos(2\Omega\alpha)-\frac{2}{1-\beta^{2}}\sin\Big{(}(1-\alpha)\Omega\Big{)}\Big{[}\sin\Big{(}(1+\alpha)\Omega\Big{)}-\beta\sin\Big{(}\delta(1+\alpha)\Omega\Big{)}\Big{]},$
(11a) $\displaystyle t_{12}$
$\displaystyle=\frac{\sin(2\Omega)+2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)}{z\omega(1-\beta)(1+\beta)^{2}},$
(11b) $\displaystyle t_{21}$
$\displaystyle=\frac{-z\omega}{1-\beta}\left[\sin(2\Omega)-2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)\right],\text{and}$
(11c) $\displaystyle t_{22}$
$\displaystyle=\cos(2\Omega\alpha)-\frac{2}{1-\beta^{2}}\sin\Big{(}(1-\alpha)\Omega\Big{)}\Big{[}\sin\Big{(}(1+\alpha)\Omega\Big{)}+\beta\sin\Big{(}\delta(1+\alpha)\Omega\Big{)}\Big{]}.$
(11d)

###  2.3 Dispersion analysis

Following the derivation of the transfer matrix $\mathbf{T}$, we obtain a
closed-form expression for the PnC’s dispersion relation. We start by
computing the eigenvalues of the transfer matrix via
$|\mathbf{T}-\lambda\mathbf{I}|=0$. Using the fact that $|\mathbf{T}|=1$, a
unique property of transfer matrices, the characteristic equation can be found
as

$\lambda^{2}-\text{tr}(\mathbf{T})\lambda+1=0,$ (12)

where $\text{tr}(\mathbf{T})=t_{11}+t_{22}$ is the trace of matrix
$\mathbf{T}$. The eigenvalues of $\mathbf{T}$ can be found by solving Eq.
(12), which yields the following solution pair

$\lambda_{\pm}=\frac{\text{tr}(\mathbf{T})}{2}\pm\sqrt{\left(\frac{\text{tr}(\mathbf{T})}{2}\right)^{2}-1}.$
(13)

As such, it immediately follows that
$\lambda_{-}+\lambda_{+}=\text{tr}(\mathbf{T})$ from Eq. (13). Rewriting the
eigenvalues as a function of a non-dimensional wavenumber $q$ in the form
$\lambda_{\pm}=e^{\pm\mathbf{i}q}$, where $\mathbf{i}$ is the imaginary unit,
gives

$\text{tr}(\mathbf{T})=2\cos(q),$ (14)

from which the dispersion relation can be extracted as follows

$\cos(2\Omega)-\beta^{2}\cos(2\Omega\alpha)-(1-\beta^{2})\cos(q)=0.$ (15)

It is evident from Eq. (15) that the dispersion relation is not a function of
the symmetry parameter $\delta$, which is a validating sign since the choice
of the unit cell of an infinite PnC should not affect its dispersion diagram.
Such choice, however, plays an instrumental role in the dynamics and
eigenfrequency analysis of the finite PnC, as will be illustrated in Secs. 3
and 4. The dispersion relation can be plotted analytically by calculating the
non-dimensional wavenumber $q$ corresponding to any non-dimensional frequency
$\Omega$ following a simple rearrangement of Eq. (15)

$q=\cos^{-1}\left[\frac{\cos(2\Omega)-\beta^{2}\cos(2\Omega\alpha)}{1-\beta^{2}}\right],$
(16)

which results in generally complex values of the wavenumber
$q=q_{\text{R}}+\mathbf{i}q_{\text{I}}$. A bandgap emerges when
$q_{\text{I}}\neq 0$ (or $|\lambda|\neq 1$) indicating spatial wave
attenuation in the PnC rod. As can be inferred from Eq. (16), a non-zero
imaginary component of the wavenumber is guaranteed when the following
inequality is satisfied

$\left|\frac{\cos(2\Omega)-\beta^{2}\cos(2\Omega\alpha)}{1-\beta^{2}}\right|>1.$
(17)

Figure 2(a) shows the dispersion diagram for a PnC rod made of an ABS polymer
(Material A) and Aluminum (Material B) corresponding to the parameters listed
in Table 1. For these parameters, $\alpha$ and $\beta$ are found to be equal
to $0.5379$ and $-0.8906$, respectively. The figure depicts the dispersion
relation over the frequency range $\Omega\in[0,20]$, revealing the first
twelve bandgaps of the PnC rod, which are shaded and color coded. In Fig.
2(b), the natural logarithm of the magnitude of the transfer matrix’s
eigenvalues is plotted, emphasizing that the eigenvalues are reciprocal of one
another, i.e., $\lambda_{+}\lambda_{-}=1$ or
$\text{ln}(|\lambda_{+}|)+\text{ln}(|\lambda_{-}|)=0$. Frequency ranges
corresponding to $\text{ln}(|\lambda|)\neq 0$ can be used as an alternative
indicator of bandgap regions.

Table 1: Material and geometric properties of segments A and B of the PnC unit cell. | Material | Density | Young’s Modulus | Area | Length
---|---|---|---|---|---
A | ABS | 1040 kg/$\text{m}^{3}$ | 2.4 GPa | 300 $\text{mm}^{2}$ | 40 mm
B | Aluminum | 2700 kg/$\text{m}^{3}$ | 69 GPa | 600 $\text{mm}^{2}$ | 40 mm
Figure 2: (a) Dispersion diagram of the PnC rod unit cell described in Table
1, and (b) the natural logarithm of the magnitude of the transfer matrix
eigenvalues. The first twelve bandgaps are color coded to facilitate the
interpretation of the forthcoming figures.

##  3 Finite Structure Analysis

###  3.1 Transfer matrix of a finite PnC rod

The unit cell transfer matrix $\mathbf{T}$ derived in Eq. (10) can be used to
relate the displacement and forcing at the two ends of a finite PnC rod which
consists of $n$ unit cells as follows

$\begin{Bmatrix}u_{n}\\\
f_{n}\end{Bmatrix}=\mathbf{T}^{n}\begin{Bmatrix}u_{0}\\\ f_{0}\end{Bmatrix},$
(18)

where $u_{0}$ and $f_{0}$ denote the displacement and force at $x=0$, $u_{n}$
and $f_{n}$ denote the displacement and force at $x=n\ell$, and
$\ell=\ell_{a}+\ell_{b}$ represents the length of one unit cell.
Alternatively, the matrix $\mathbf{T}^{n}$ can be found as a function of the
unit cell’s transfer matrix and the wavenumber $q$ using the following
expression [35]

$\mathbf{T}^{n}=\frac{\cos(nq)}{2\cos(q)}\left(\mathbf{T}+\mathbf{T}^{-1}\right)+\frac{\sin(nq)}{2\sin(q)}\left(\mathbf{T}-\mathbf{T}^{-1}\right).$
(19)

Expanding the terms in Equation (19), it can be shown that

$\mathbf{T}^{n}=\begin{bmatrix}\frac{\cos(nq)}{2\cos(q)}(t_{11}+t_{22})+\frac{\sin(nq)}{2\sin(q)}(t_{11}-t_{22})&\frac{\sin(nq)}{\sin(q)}t_{12}\\\
\frac{\sin(nq)}{\sin(q)}t_{21}&\frac{\cos(nq)}{2\cos(q)}(t_{11}+t_{22})-\frac{\sin(nq)}{2\sin(q)}(t_{11}-t_{22})\end{bmatrix}.$
(20)

Using Eq. (14) and the elements of the transfer matrix defined in Eq. (11),
the transfer matrix of a finite PnC with $n$ unit cells can be finally
rewritten as

$\mathbf{T}^{n}=\begin{bmatrix}t_{11n}&t_{12n}\\\
t_{21n}&t_{22n}\end{bmatrix},$ (21)

where

$\displaystyle t_{11n}$
$\displaystyle=\cos(nq)+\frac{2\beta}{1-\beta^{2}}\frac{\sin(nq)}{\sin(q)}\Big{[}\sin\Big{(}(1-\alpha)\Omega\Big{)}\sin\Big{(}\delta(1+\alpha)\Omega\Big{)}\Big{]},$
(22a) $\displaystyle t_{12n}$
$\displaystyle=\frac{1}{z\omega(1-\beta)(1+\beta)^{2}}\frac{\sin(nq)}{\sin(q)}\left(\sin(2\Omega)+2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)\right),$
(22b) $\displaystyle t_{21n}$
$\displaystyle=-\frac{z\omega}{1-\beta}\frac{\sin(nq)}{\sin(q)}\left(\sin(2\Omega)-2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)\right),\text{and}$
(22c) $\displaystyle t_{22n}$
$\displaystyle=\cos(nq)-\frac{2\beta}{1-\beta^{2}}\frac{\sin(nq)}{\sin(q)}\Big{[}\sin\Big{(}(1-\alpha)\Omega\Big{)}\sin\Big{(}\delta(1+\alpha)\Omega\Big{)}\Big{]}.$
(22d)

###  3.2 End-to-end frequency response functions for different boundary
conditions

By parsing the different elements of the matrix $\mathbf{T}^{n}$, a set of
frequency response functions (FRFs) which describe the ratio between the
displacements at both ends of the finite PnC rod (e.g., $u_{n}/u_{0}$ or
$u_{0}/u_{n}$), or the displacement at one end to the force applied at the
other end (e.g., $u_{n}/f_{0}$) can be obtained for commonly encountered
boundary conditions. First, Eq. (18) is expanded into the following two
equations

$u_{n}=t_{11n}u_{0}+t_{12n}f_{0},$ (23a) $f_{n}=t_{21n}u_{0}+t_{22n}f_{0}.$
(23b)

For a free-free rod, a force $f_{0}$ can be prescribed at $x=0$ with the
displacement measured at $x=n\ell$. This implies that $f_{n}=0$ which results
in $u_{0}=-t_{22n}/t_{21n}f_{0}$ as can be obtained from Eq. (23b).
Substituting back into Eq. (23a) and taking advantage of the fact that the
determinant of $\mathbf{T}_{n}$ is unity, the following FRF can be computed

$\frac{u_{n}}{f_{0}}=-\frac{1}{t_{21n}}.$ (24)

For fixed-free and free-fixed rods, the FRF can be obtained via a base
excitation of the fixed end and a displacement measurement at the free one. In
the fixed-free case, we prescribe a base displacement $u_{0}$ and set
$f_{n}=0$ for the free end at $x=n\ell$. Following which, we solve for the
force $f_{0}=-t_{21n}/t_{22n}u_{0}$ from Eq. (23b) and plug that into Eq.
(23a) to get

$\frac{u_{n}}{u_{0}}=\frac{1}{t_{22n}}.$ (25)

Analogously, the FRF for the free-fixed rod can be obtained by following a
similar procedure to that of the fixed-free rod but switching the base
excitation and measurement locations, thus resulting in

$\frac{u_{0}}{u_{n}}=\frac{1}{t_{11n}}.$ (26)

A slightly different treatment is required to attain the FRF of the fourth,
and final, boundary condition, namely the fixed-fixed rod. Since both ends of
the rod are fixed and a direct measurement of the displacement at either end
is not possible, the FRF will be obtained as a ratio between the displacement
at a point very close to one of the fixed ends, $u_{n-\Delta}$, which is
measured at $x=n\ell-\Delta$, and a base excitation $u_{0}$ at the opposite
end. Here, $\Delta$ is a distance parameter. As a result, the state vector at
$x=n\ell-\Delta$ can be found by a transfer matrix inverse as follows

$\begin{Bmatrix}u_{n-\Delta}\\\
f_{n-\Delta}\end{Bmatrix}=\mathbf{T}_{\Delta}^{-1}\begin{Bmatrix}u_{n}\\\
f_{n}\end{Bmatrix},$ (27)

where

$\mathbf{T}_{\Delta}^{-1}=\begin{bmatrix}\cos\left(\frac{\omega}{c_{a}}\Delta\right)&-\frac{1}{\omega
z_{a}}\sin\left(\frac{\omega}{c_{a}}\Delta\right)\\\ \omega
z_{a}\sin\left(\frac{\omega}{c_{a}}\Delta\right)&\cos\left(\frac{\omega}{c_{a}}\Delta\right)\\\
\end{bmatrix}.$ (28)

Note that consistent with our generalized unit cell definition in Sec. 2.1 and
Fig. 1, both ends of the finite structure are made out of material A for all
unit cell choices (except for the one unit cell design corresponding to
$\delta=-1$ which ends with material B), thus justifying the use of material A
properties in the $\mathbf{T}_{\Delta}^{-1}$ expression. Assuming a very small
$\Delta$, $\mathbf{T}_{\Delta}^{-1}$ can be linearized about $\Delta=0$ as
follows

$\mathbf{T}_{\Delta,0}^{-1}=\begin{bmatrix}1&-\frac{1}{z_{a}c_{a}}\Delta\\\
\frac{\omega^{2}z_{a}}{c_{a}}\Delta&1\\\ \end{bmatrix}.$ (29)

Consequently, the state vector right before the fixed end (i.e., at
$x=n\ell-\Delta$) can alternatively be computed using Eq. (27), albeit with
$\mathbf{T}^{-1}_{\Delta,0}$ instead of $\mathbf{T}^{-1}_{\Delta}$. Setting
$u_{n}=0$ at the fixed end and using Eqs. (23a) and (23b) to derive an
expression for $f_{n}$ as a function of $u_{0}$, we obtain
$f_{n}=-u_{0}/t_{12n}$, and thus the FRF for this boundary condition can be
shown to be

$\frac{u_{n-\Delta}}{u_{0}}=\frac{\Delta(1+\alpha)}{\ell_{a}\omega_{0}z(1+\beta)}\frac{1}{t_{12n}}.$
(30)

Note that setting $\Delta=0$ in Eq. (30) recovers the boundary condition
$u_{n}=0$. Using the same properties listed in Table 1 and $\delta=-0.8$, the
FRFs for all four boundary conditions for a finite PnC rod made out of a
single unit cell ($n=1$; blue curves) and ten unit cells ($n=10$; black
curves) are plotted in Fig. 3. Furthermore, to validate the analytical
expressions obtained earlier, the same FRFs are also reproduced via a finite
element model (FEM) of the finite structures, coded via MATLAB using two-node
rod elements [44]. For the FEM results, the number of finite elements per unit
cell is $n_{e}=250$, and $\Delta=\ell/n_{e}$ is used in the fixed-fixed case.
All the FEM plots are superimposed on the analytical plots for comparison
(white dashed lines), and an excellent agreement between the analytical and
FEM data can be seen.

Two important observations are made regarding Bragg bandgaps exhibited by the
finite PnC rod. The first is that such bandgaps are recognized in FRFs as
regions of significant u-shaped drops in amplitude, which are more pronounced
in structures comprising a larger number of unit cells (e.g., easier to detect
in the $n=10$ FRFs than the $n=1$ FRFs in Fig. 3). The formation mechanism of
such bandgaps in finite PnCs and the underlying reasons behind the extent of
amplitude drop and the number of unit cells are thoroughly detailed in Ref.
[24]. The second observation pertains to the appearance of resonant peaks
inside some bandgaps in the $n=10$ case, often referred to as truncation [15,
24] or bandgap resonances [30]. Interestingly, for both the free-free and
fixed-fixed cases, we observe that these truncation resonances perfectly
coincide with the natural frequencies of the single unit cell which is not the
case for the two other boundary conditions. Even more intriguingly, we note
that while the truncation resonances in the fixed-free and free-fixed cases do
not necessarily align with the natural frequencies of the single unit cell in
their respective cases, each of these truncation resonances perfectly
coincides with a single cell natural frequency from either the free-free or
fixed-fixed cases, as highlighted in the close-up shown in Fig. 3(e). The
locations and unique features of these truncation resonances will be confirmed
and analytically proven in the subsequent sections by closely examining the
characteristic equations of the finite PnCs subject to different boundary
conditions.

Figure 3: Frequency response functions (FRFs) of finite PnC rods comprised of
a single unit cell ($n=1$; solid blue) and 10 unit cells ($n=10$; solid
black), both corresponding to a symmetry parameter of $\delta=-0.8$,
analytically obtained via TMM. FEM results for both cases are superimposed on
all the results (white dashed curves). The FRFs are presented for four
boundary conditions as follows: (a) Free-free, (b) Fixed-fixed, (c) Fixed-
free, and (d) Free-fixed. The rightmost column panel provides illustrative
schematics of the different excitations and boundary conditions. (e) A close-
up of the FRFs shown in (a-d) for the shaded region $\Omega\in[0,4]$, showing
natural frequencies appearing within the first two bandgaps.

##  4 Truncation Resonances

###  4.1 Natural frequencies for different boundary conditions

To find the natural frequencies of the finite PnC rod, we set the force or
displacement to zero for fixed or free ends, respectively, and use either Eq.
(23a) or (23b) to find the characteristic equation. For example,
$f_{0}=f_{n}=0$ in a free-free rod. Substituting these conditions into Eq.
(23b), a non-trivial solution is only possible if $t_{21n}=0$, which
constitutes the characteristic equation for the free-free case. The solutions
of the characteristic equation represent the finite structure’s natural
frequencies. It should be noted that these solutions also represent the roots
of the denominator of the transfer function in Eq. (24), which are commonly
known as the system’s poles in controls and linear systems theory [45]. A
similar process can be adopted for the rest of the boundary conditions to
obtain the remaining characteristic equations, which are summarized in Table 2
for ease of reference.

Table 2: Characteristic equations and FRFs of commonly encountered boundary conditions of the finite PnC rod. Boundaries | Characteristic Equation | Frequency Response Function
---|---|---
Free-Free | $t_{21n}=0$ | $\frac{u_{n}}{f_{0}}=-\frac{1}{t_{21n}}$
Fixed-Fixed | $t_{12n}=0$ | $\frac{u_{n-\Delta}}{u_{0}}=\frac{\Delta(1+\alpha)}{\ell_{a}\omega_{0}z(1+\beta)}\frac{1}{t_{12n}}$
Fixed-Free | $t_{22n}=0$ | $\frac{u_{0}}{u_{n}}=\frac{1}{t_{22n}}$
Free-Fixed | $t_{11n}=0$ | $\frac{u_{0}}{u_{n}}=\frac{1}{t_{11n}}$

####  4.1.1 Free-free and fixed-fixed cases

We begin with the free-free case with the characteristic equation $t_{21n}=0$,
which reads

$\Omega\frac{\sin(nq)}{\sin(q)}\left(\sin(2\Omega)-2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)\right)=0.$
(31)

From Eq. (31), it can be inferred that the solution $\Omega=0$, which
describes a non-oscillatory rigid body mode, satisfies the characteristic
equation, as expected in a free-free (unconstrained) structure. Additionally,
the roots obtained from $\frac{\sin(nq)}{\sin(q)}=0$ provide closed-form
wavenumber solutions of the following form

$q=\frac{r\pi}{n},$ (32)

for all integer values of $r$ except zero and multiples of $n$, which yield
indeterminate expressions. These values can be then substituted into the
dispersion relation in Eq. (15), to get

$\cos(2\Omega)-\beta^{2}\cos(2\Omega\alpha)-(1-\beta^{2})\cos\left(\frac{r\pi}{n}\right)=0.$
(33)

Given the periodicity of the dispersion relation, the range $q\in[0,\pi]$
defining the irreducible Brillouin zone suffices for the analysis. As a
result, the integer $r$ can be limited to $r=1,2,\dots,n-1$. Equation (33) can
then be solved numerically to compute the subset of the PnC’s natural
frequencies which satisfy the dispersion relation, i.e., the natural
frequencies which lie within pass band regions and correspond to the purely
real values of $q$ in Eq. (32). Each value of the integer $r$ will result in
an infinite number of natural frequency solutions, one per dispersion branch,
regardless of the unit cell symmetry parameter $\delta$. The fact that there
exists an infinite number of natural frequencies is a hallmark feature of
finite continuum structures, contrary to finite spring-mass lattices which
comprise a finite number of degrees of freedom and hence a finite number of
resonant vibrational modes [15, 24]. Finally, the presence of resonant peaks
outside of pass bands (and inside bandgaps) in Fig. 3(a) suggests that another
set of natural frequencies must emanate from the roots of the third factor in
Eq. (31), representing the truncation resonances which correspond to the free-
free boundary condition. These are given by the solutions of

$\sin(2\Omega)-2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)=0.$
(34)

Here, we confirm the observation made earlier in Fig. 3(a) that the truncation
resonances of a free-free PnC rod match the natural frequencies of a single
unit cell with identical boundary conditions. This feature is easily provable
by using Eq. (31) and setting the number of cells $n$ equal to one, and is
inline with earlier results reported for mono-coupled systems [46]. It also
serves as both a valuable and efficient tool in applications requiring a
priori placement of truncation resonances, rather than arbitrary trial and
error. It is, however, important to reiterate that the roots of Eq. (34) may
not always lie inside a bandgap and could, in some cases, end up satisfying a
dispersion branch. Whether or not a given solution lie within a bandgap can
always be checked via the inequality provided in Eq. (17). As such, it is
prudent to refer to the solutions of Eq. (34) as potential truncation
resonances. Simply stated, not every solution of Eq. (34) is necessarily a
truncation resonance but any truncation resonance is a solution of Eq. (34).

Figure 4: Analytical (circles) and FEM-obtained (dots) natural frequencies for
finite PnC rod subject to the following boundary conditions: (a) Free-free,
(b) Fixed-fixed, (c) Fixed-free, and (d) Free-fixed. The natural frequencies
are superimposed on the dispersion relation for reference. The rightmost
column shows the number of natural frequencies that appear in each bandgap,
which are color coded to facilitate figure interpretation and correspond to
the legend in Fig. 2.

Next, we move to the fixed-fixed boundary condition where $t_{12n}=0$
represents the characteristic equation of the finite system. From Eq. (22b),
it can be inferred that all the non-zero pass band natural frequencies of the
fixed-fixed PnC rod are identical to those of the free-free one, and can be
similarly obtained using Eq. (33). However, the potential truncation
resonances in this case need to be obtained from the following equation

$\sin(2\Omega)+2\beta\sin\Big{(}(1-\alpha)\Omega\Big{)}\cos\Big{(}\delta(1+\alpha)\Omega\Big{)}-\beta^{2}\sin(2\Omega\alpha)=0.$
(35)

It is important to note that the only distinguishing feature between Eqs. (34)
and (35) is the sign of the second term, meaning that if the sign of $\beta$
is flipped (i.e., impedance values are swapped), the truncation resonances of
a fixed-fixed rod will become those of a free-free one, and vice versa.
Equally important is the fact that the sign of $\delta$ has no effect on the
truncation resonances here since it is a part of the argument of the cosine
function in both equations.

Using the same parameters as those used to generate Fig. 3, all the natural
frequencies of the free-free and fixed-fixed PnC rods up to $\Omega=20$ are
computed analytically using the equations derived here, and projected on the
dispersion diagram, as shown in Figs. 4(a) and 4(b), respectively.
Additionally, and for verification, the natural frequencies are also computed
via the same FEM model described in Sec. 3.2, and superimposed on the figures
for comparison; generally showing excellent agreement with the analytical
values with minor deviations at higher frequencies consistent with FEM
resolution. The right column of Figs. 4(a) and (b) shows a count of the number
of truncation resonances that exist in each bandgap, revealing a constant
total of one truncation resonance per bandgap in both cases. For convenience,
truncation resonance features (e.g., value and count) follow the same color
code used in Fig. 2, allowing each truncation resonance to be easily traced to
its corresponding bandgap.

Finally, we note that while the analytical derivations detailed here are
established for a PnC rod comprising two different materials, the findings are
generalizable for any free-free and fixed-fixed PnC regardless of the number
of material layers. A simple proof can be obtained by inspecting the form of
the matrix $\mathbf{T}^{n}$ and by comparing Eqs. (20) and (21). It can be
seen that $t_{21n}$ is found from the multiplication of the terms
$\frac{\sin(nq)}{\sin(q)}$ and $t_{21}$. Setting $t_{21n}=0$ also mandates
that $t_{21}=0$, suggesting that the natural frequencies of a free-free PnC
with $n=1$ represent a subset of a free-free PnC natural frequencies with any
$n>1$. By replacing $t_{21n}$ with $t_{12n}$, the same can be said about the
fixed-fixed PnC rod. This conclusion remains true regardless of the number of
material layers constituting the unit cell, a fact that was also proven for a
general class of free-free polyatomic PnC lattices [15].

Figure 5: Modes shapes corresponding to truncation resonances revealing edge
modes which attenuate away from the localized vibrations, provided as follows:
(a) First bandgap for fixed-fixed, free-free, and free-fixed boundary
conditions, and (b) second bandgap for fixed-fixed, free-free, fixed-free, and
free-fixed boundary conditions.

####  4.1.2 Fixed-Free and Free-Fixed cases

As indicated in Table 2, the characteristic equations of the free-fixed and
fixed-free finite PnC rods are $t_{11n}=0$ and $t_{22n}=0$, respectively, and
their explicit forms are found by setting Eqs. (22a) and (22d) equal to zero.
By closely examining these two characteristic equations, it can be also
observed that they can be obtained from one another by either changing the
sign of the contrast parameter $\beta$ or the symmetry parameter $\delta$.
Unlike the free-free and fixed-fixed cases, an analytical expression for the
wavenumber $q$ is generally not available for general system parameters. As
such, the natural frequencies can be found by plugging in Eq. (16) into
$t_{11n}=0$ or $t_{22n}=0$, and numerically finding $\Omega$ solutions that
satisfy the resulting equation. This approach yields all the natural
frequencies, whether they represent pass bands or truncation resonances.
Following which, the natural frequencies $\Omega$ can be used to determine the
corresponding wavenumber from Eq. (16). Figure 4(c,d) shows all the natural
frequencies of the fixed-free and fixed-free PnC rods up to $\Omega=20$. By
comparing the results with those obtained earlier for the free-free and fixed-
fixed cases in Fig. 4(a,b), we observe that each truncation resonance that
appears in either the fixed-free or free-fixed cases must already exist in the
free-free or fixed-fixed ones. This feature is graphically emphasized in Fig.
4 via a series of arrows which trace each truncation resonance in the figure’s
last two rows to its counterpart in the first two rows.

To illustrate the underlying dynamics of truncation resonances associated with
these two boundary conditions, we consider the first bandgap in the free-fixed
case. As depicted in Fig. 4(d), two truncation resonances appear in this
bandgap. By tracing the arrows, the first one at $\Omega=0.897$ corresponds to
a truncation resonance from the fixed-fixed case, while the second one at
$\Omega=1.201$ corresponds to a truncation resonance from the free-free case.
This observation can be further explained by considering the eigenvectors
associated with these two truncation resonances, or their mode shapes, in Fig.
5(a). Since these are resonant modes that appear within a bandgap, i.e., a
region of spatial wave attenuation, mode shapes of truncation resonances
typically show vibrations which are localized or confined to one end of the
structure [15]. At $\Omega=0.897$, the mode shapes of the fixed-fixed and
free-fixed systems resemble each other and show localized vibrations near the
rightmost end of the rod, which is fixed in both cases – explaining the
matching truncation resonances. Similarly, at $\Omega=1.201$, the mode shapes
of the free-free and free-fixed systems resemble each other and show localized
vibrations near the leftmost end of the rod, which is free in both cases.
Additional examples of matching truncation resonances which lie in the second
bandgap are provided in Fig. 5(b). One additional observation from Fig. 5(a)
is that the displacement profiles at $\Omega=0.897$ exhibit a slower spatial
decay of vibrations away from the localized edge when compared to their
counterparts at $\Omega=1.201$. This is attributed to the strength of bandgap
attenuation within the first bandgap at each of these two frequencies, as
dictated by the imaginary component of the wavenumber $q_{\text{I}}$, and
shown in Fig. 2(a). The same observation can be made for truncation resonance
mode shapes falling within the second bandgap, shown in Fig. 5(b).

Interestingly, an analytical proof for the observations above can be derived
by considering a semi-infinite PnC, similar to finding surface modes in
elastic lattices [47] or photonic crystals [48]. We propose the following
proof using a combination of the transfer matrix of the finite PnC rod and the
solution of the displacement and internal forces within an elastic medium.
Starting with the latter, the displacement and internal force at an arbitrary
point $x$ along a rod with longitudinal vibrations can be found using a
combination of forward- and backward-propagating waves as follows

$\hat{u}(x)=a_{1}\text{e}^{-\textbf{i}k_{s}x}+a_{2}\text{e}^{\textbf{i}k_{s}x},$
(36a)
$\hat{f}(x)=-\textbf{i}E_{s}A_{s}k_{s}\left(a_{1}\text{e}^{-\textbf{i}k_{s}x}-a_{2}\text{e}^{\textbf{i}k_{s}x}\right).$
(36b)

At $x=0$, we have $u_{0}=\hat{u}(0)$ and $f_{0}=\hat{f}(0)$ and using Eqs.
(36a) and (36b), a compact matrix form of the two equations can be written as

$\begin{Bmatrix}u_{0}\\\ f_{0}\end{Bmatrix}=\underbrace{\begin{bmatrix}1&1\\\
-\textbf{i}E_{s}A_{s}k_{s}&\textbf{i}E_{s}A_{s}k_{s}\end{bmatrix}}_{\mathbf{U}_{s}}\begin{Bmatrix}a_{1}\\\
a_{2}\end{Bmatrix}.$ (37)

Note that zeroing out the displacement (or force) in Eq. (37) implies that
$a_{1}=-a_{2}$ (or $a_{1}=a_{2}$). Now, let us revisit Eq. (18) which relates
the state vectors at $x=0$ and $x=n\ell$. If we consider a semi-infinite
medium, we can relate these two vectors using the eigenvalues of the transfer
matrix as

$\begin{Bmatrix}u_{n}\\\ f_{n}\end{Bmatrix}=\lambda^{n}\begin{Bmatrix}u_{0}\\\
f_{0}\end{Bmatrix}.$ (38)

Using Eqs. (18), (37) and (38), we arrive at an eigenvalue problem of the form

$\underbrace{\mathbf{U}_{s}^{-1}\mathbf{T}^{n}\mathbf{U}_{s}}_{\mathbf{H}_{s}}\begin{Bmatrix}a_{1}\\\
a_{2}\end{Bmatrix}=\lambda^{n}\begin{Bmatrix}a_{1}\\\ a_{2}\end{Bmatrix}.$
(39)

where the matrix $\mathbf{H}_{s}$ is given by

$\mathbf{H}_{s}=\begin{bmatrix}h&\hat{h}\\\
\hat{h}^{\dagger}&h^{\dagger}\end{bmatrix}.$ (40)

where the superscript $\dagger$ denotes the complex conjugate and

$h=\frac{1}{2}(t_{11n}+t_{22n})+\frac{\textbf{i}}{2}\left(\frac{t_{21n}}{E_{s}A_{s}k_{s}}-(E_{s}A_{s}k_{s})t_{12n}\right),$
(41a)
$\hat{h}=\frac{1}{2}(t_{11n}-t_{22n})+\frac{\textbf{i}}{2}\left(\frac{t_{21n}}{E_{s}A_{s}k_{s}}+(E_{s}A_{s}k_{s})t_{12n}\right).$
(41b)

Considering the first row of Eq. (39) in conjunction with $a_{2}=a_{1}$ from
the zero force condition at $x=0$ pertaining to a free end, we obtain

$t_{11n}-\lambda^{n}+\textbf{i}\left(\frac{t_{21n}}{E_{s}A_{s}k_{s}}\right)=0.$
(42)

Equivalently, the second row of Eq. (39) can be used and shall result in the
complex conjugate of Eq. (39). Note that both real and imaginary components
have to be zero to satisfy Eq. (42). Recall that, for a natural frequency
within the bandgap, the eigenvalues of the transfer matrix are both real and
distinct, and only $|\lambda|<1$ shall be considered for a mode decaying away
from $x=0$, hence, $\lambda^{n}\rightarrow 0$ as $n\rightarrow\infty$. As a
consequence, the result of setting the real and imaginary components of Eq.
(42) equal to zero becomes merely the characteristic equations of the free-
fixed ($t_{11n}=0$) and free-free ($t_{21n}=0$) cases, respectively. This
signals that a truncation resonance must satisfy both the characteristic
equations of the free-fixed and free-free cases, which confirms that these
modes are in fact the same for both boundary conditions when
$n\rightarrow\infty$. If instead we consider a fixed-end scenario (i.e,
$u_{0}=0$ and $a_{2}=-a_{1}$), it can be similarly shown that

$t_{22n}-\lambda^{n}+\textbf{i}\left(E_{s}A_{s}k_{s}\right)t_{12n}=0,$ (43)

and hence $t_{22n}=0$ and $t_{12n}=0$ must both be satisfied when
$\lambda^{n}\rightarrow 0$, which serves as a proof for the truncation
resonance matching between the free-fixed and fixed-fixed cases.

###  4.2 Effect of unit cell symmetry on truncation resonances

The performance of the finite PnC rods shown thus far in Figs. 3 through 5 is
for $\delta=-0.8$. However, the choice of unit cell (i.e., symmetry parameter)
can largely influence the final configuration of the finite structure and thus
the location of the truncation resonances, as will be outlined here. Figure 6
illustrates the effect of $\delta$ on the truncation resonance frequencies.
The plots show a series of loci which trace each truncation frequency (color
coded to represent the bandgap to which they belong) as $\delta$ varies from
$-1$ to $1$. This is done for all four boundary conditions discussed in Sec.
4.1. Since a truncation resonance can be located anywhere between the starting
and ending frequencies of the bandgap in which it resides, the shaded space
between the different curves in Fig. 6 actually represents pass band regions.
As can be seen in the figure, the variation of truncation resonance location
as the unit cell choice changes fluctuates up and down in an oscillatory
manner. More fluctuations occur at larger values of $\Omega$ indicating an
increased sensitivity to the symmetry parameter $\delta$ at higher
frequencies.

Next, we shift our focus to the combined effect of $\delta$ and boundary
conditions on truncation resonance locations. The first observation that
stands out is the symmetry of the curves in Figs. 6(a,b), representing the
free-free and fixed-fixed rods, about $\delta=0$. This is understandable given
that the boundary condition is the same on both ends of the structure in both
of these cases, and thus a change in the sign of $\delta$ would not result in
a change in the system’s truncation resonances. This has also been
mathematically confirmed in Eqs. (34) and (35) where $\delta$ appears as a
factor inside the cosine function, where the sign of $\delta$ would not affect
the function’s outcome. On the other hand, the truncation resonance curves are
not symmetric about $\delta=0$ for the fixed-free and free-fixed cases, as
shown in Figs. 6(c,d). This is anticipated due to the asymmetric nature of the
rod under these two boundary conditions. As established in Sec. 4.1.2, each
truncation resonance in these two cases must have an identical counterpart in
either the free-free or fixed-fixed rods. For that reason, it can be observed
that parts of the curves shown in Figs. 6(a,b) appear in Figs. 6(c,d). An
interesting graphical depiction of this phenomenon is shown in Figs. 6(e,f)
which show that the combined truncation resonances of the free-free and fixed-
fixed cases are nearly identical to the combined truncation resonances of the
fixed-free and free-fixed ones. To further elucidate the overall patterns
shown in Fig. 6, we break down the truncation resonances of the free-free case
into two groups. The first represents mode shapes with localized vibrations
near $x=0$ and is referred to as group I, while the second represents mode
shapes with localized vibrations near $x=n\ell$ and is referred to as group
II. Similarly, we break down the truncation resonances of the fixed-fixed case
into groups III and IV. With these four groups in mind, the Venn diagram shown
in Fig. 6(g) summarizes the distribution of truncation resonances across the
four boundary conditions. The outer black circle represents all the possible
natural frequencies of the finite PnC rod within a given frequency range,
subject to any of the four boundary conditions. The two ellipses inside the
black circle represent the entire set of truncation resonances in that
frequency range, while the space outside of the two ellipses contains the
remaining natural frequencies representing pass band resonances. The ellipse
comprising groups I and II represents the free-free truncation resonances and
the ellipse comprising groups III and IV represents the fixed-fixed truncation
resonances. The shared area (intersection) between the two ellipses contains
some truncation resonances which co-exist in both cases, as evident from the
crossing of curves in Figs. 6(e,f). Finally, the combination (union) of groups
II and III constitute the truncation resonances for the fixed-free case, while
the combination (union) of I and IV constitute the truncation resonances for
the free-fixed case.

Figure 6: Effect of the symmetry parameter $\delta$ on the truncation
resonances of finite PnC rods subject to the following boundary conditions:
(a) Free-free, (b) Fixed-fixed, (c) Fixed-free, and (d) Free-fixed. The
combined truncation resonances of the free-free and fixed-fixed, and the
combined truncation resonances of the fixed-free and free-fixed are shown in
(e) and (f), respectively. (g) Venn diagram summarizing the distribution of
truncation resonances across the four boundary conditions. The outer black
circle represents all possible natural frequencies of the finite PnC rod
subject to any boundary condition. The two ellipses inside the black circle
represent the entire set of truncation resonances. The space outside of the
two ellipses contains the remaining natural frequencies representing pass band
resonances. Groups I and II represents the free-free truncation resonances and
groups III and IV represents the fixed-fixed truncation resonances. The shared
area (intersection) between the two ellipses contains truncation resonances
which co-exist in both cases. The combination (union) of groups II and III
constitute the truncation resonances for the fixed-free case, while the
combination (union) of I and IV constitute the truncation resonances for the
free-fixed case.

Figure 7: Mode shapes corresponding to the truncation resonances of the finite
PnC rod are classified into four sets as indicated above depending on the edge
where localized vibrations take place and the type of boundary condition.
Following this classification, truncation resonances from Figs. 6(a-d) are re-
plotted here versus the symmetry parameter $\delta$ for the following boundary
conditions: (a) Free-free, (b) Fixed-fixed, (c) Fixed-free, and (d) Free-
fixed.

To further emphasize the truncation resonance features summarized in Fig.
6(g), the eigenvectors (i.e., mode shapes) corresponding to the truncation
resonances are computed using our FEM model previously described in Sec. 3.2.
These modes are then categorized based on the edge of the structure at which
the vibrations are localized. This is accomplished by calculating a normalized
weighted average $\bar{x}$ of the location using the absolute displacements of
the FEM nodes as weights, denoted as $u_{j}$, and their location in the
spatial domain as the data, denoted as $x_{j}$, which is mathematically
expressed as follows

$\bar{x}=\frac{1}{n\ell}\frac{\sum_{j}x_{j}|u_{j}|}{\sum_{j}|u_{j}|}.$ (44)

A normalized average of $\bar{x}<0.50$ or $\bar{x}>0.50$ indicates that higher
displacement amplitudes are more skewed towards the left or right edges,
respectively. The limiting case of $\bar{x}\rightarrow 0$ or
$\bar{x}\rightarrow 1$ indicates that the amplitude is strongly localized at
the left or right edge, respectively. Figure 7 shows the outcome of this
procedure for all four boundary conditions, namely, (a) free-free, (b) fixed-
fixed, (c) fixed-free, and (d) free-fixed for all unit cell choices spanning
the symmetry parameter range $-1\leq\delta\leq 1$. For ease of reference, the
modes in the figure are color coded to reflect the edge at which the localized
vibrations take place (left vs. right) as well as the boundary condition at
that edge (free vs. fixed), and are therefore grouped in four sets as detailed
in the figure’s legend. As seen in Figs. 7(a,b), the truncation resonances in
both the free-free and fixed-fixed cases consist of left- and right-localized
modes, represented by sets I + II and III + IV, respectively. Additionally, it
can be observed that sets II + III constitute the truncation resonances of the
fixed-free case in Fig. 7(c). This makes perfect sense since these sets
represent the matching boundary conditions at the two respective edges.
Specifically, the eigenvectors in set II represent “right-localized modes for
a free end” while the eigenvectors in set III represent “left-localized modes
for a fixed end”. These two sets, put together, create the fixed-free PnC.
Following the same logic, it is observed that sets I + IV constitute the
truncation resonances of the free-fixed case in Fig. 7(d). Lastly, we make an
important observation from Figs. 7(a,b) regarding a truncation resonance
phenomenon which is not immediately evident from Figs. 6(a,b), which is that
the edge at which the localized vibrations happen flips from left to right or
vice versa upon touching a pass band (remember that all these truncation
resonances are by definition inside bandgaps). It should be noted that a
similar behavior has been previously reported in the context of quasi-periodic
lattices [19].

###  4.3 Absence of truncation resonances in symmetric unit cells ($\delta=0$)

A perfectly symmetric unit cell is a special case for all boundary conditions,
where truncation resonances cease to exist. To analytically prove this, we
start with the free-fixed and fixed-free cases whose characteristic equations
are given by $t_{11n}=0$ and $t_{22n}=0$, respectively. For $\delta=0$, both
of these reduce to

$\cos(nq)=0,$ (45)

for which the only possible wavenumber solutions are real and are given by

$q=\frac{(2r-1)\pi}{2n},\ r=1,2,\dots n.$ (46)

All of these real wavenumbers correspond to natural frequencies that fall
within pass bands. These natural frequencies can simply be found by solving
the dispersion relation in Eq. (15) at these wavenumbers, thereby proving that
no truncation resonances shall appear for these two boundary conditions as
long as $\delta=0$.

While no truncation resonances exist for the free-free and fixed-fixed
boundary conditions when $\delta=0$, proving so is not as straightforward. In
these two cases, the portion of the characteristic equations responsible for
truncation resonances, i.e., Eqs. (34) and (35) reduce to

$\Big{(}\cos(\Omega)\mp\beta\cos(\Omega\alpha)\Big{)}\Big{(}\sin(\Omega)\pm\beta\sin(\Omega\alpha)\Big{)}=0,$
(47)

for free-free $(-,+)$ and fixed-fixed $(+,-)$ boundary conditions. Both of the
bracketed expressions in Eq. (47) are a multiplication of two terms, which are
in fact related to the equations that govern the bandgap limits. To elaborate,
let’s substitute $q=0$ and $q=\pi$, the two values which mark the beginning
and end of all dispersion branches, in the dispersion relation given by Eq.
(15), which results in the following

$\cos(2\Omega)-\beta^{2}\cos(2\Omega\alpha)\pm(1-\beta^{2})=0.$ (48)

Equation (48) provides the limits for odd $(+)$ and even $(-)$ numbered
bandgaps, and can be further simplified to

$\Big{(}\cos(\Omega)-\beta\cos(\Omega\alpha)\Big{)}\Big{(}\cos(\Omega)+\beta\cos(\Omega\alpha)\Big{)}=0.$
(49a)
$\Big{(}\sin(\Omega)-\beta\sin(\Omega\alpha)\Big{)}\Big{(}\sin(\Omega)+\beta\sin(\Omega\alpha)\Big{)}=0,$
(49b)

One can now see the connection between the equations defining bandgap limits
and the characteristic equations of the free-free and fixed-fixed cases when
$\delta=0$. Specifically, the two equations in Eq. (47) can be simply obtained
by multiplying one term in Eq. (49a) by another in Eq. (49b). This effectively
proves that all resonances corresponding to these two boundary conditions when
$\delta=0$ are indeed within pass bands. Similar results have been reported by
Mead for mono-coupled systems, where some natural frequencies for asymmetric
elements arise within attenuation zones and coincide with bounding frequencies
of bandgaps for symmetric elements [46]. The same phenomenon has also been
observed in some configurations of diatomic lattices with free boundaries
[24].

Figure 8: (a) Relationship between the truncation resonances patterns and the
contours of Eqs. (34) and (35). Color-coded solid lines between the contours
represent points that render the contour’s magnitude equal to zero, i.e.,
roots of Eqs. (34) and (35). (b) The combined patterns of the free-free and
fixed-fixed truncation resonances, given by the lines described in Eqs. (50)
and (51). Figure 9: Effect of system parameters, namely the frequency contrast
$\alpha$ and the impedance contrast $\beta$, on the overall patterns of the
truncation resonances across all the possible unit cell configurations, as
dictated by the full $\delta$ range.

###  4.4 Patterns of truncation resonances

The general pattern of the truncation resonance curves is best explained using
Eqs. (34) and (35). For given values of $\beta$ and $\alpha$, a contour plot
can be obtained by sweeping $\Omega$ and $\delta$, as can be seen in Fig. 8(a)
for the free-free and fixed-fixed cases. Whenever a contour has a magnitude of
zero, it represents a solution of the equation (i.e., a truncation resonance),
which are indicated by the color-coded solid curves in the figure. The proof
presented earlier confirming the absence of truncation resonance for free-free
and fixed-fixed PnCs which are formed of symmetric unit cells (i.e., when
$\delta=0$) can be used to explain the general patterns of the truncation
resonances as a function of $\delta$. We start by pointing out that the
magnitude of the cosine function $\cos\big{(}\delta(1+\alpha)\Omega\big{)}$ in
Eqs. (34) and (35) becomes unity when $\delta=0$ and results in the simplified
characteristic equations in Eq. (47). Second, given its periodicity, the
cosine function may assume the values of $\pm 1$ at different combinations of
$\Omega$ and $\delta$. As such, one can derive an expression that relates the
frequency $\Omega$ to the symmetry parameter $\delta$ that satisfies such
condition, which results in

$\hat{\Omega}_{p}=\frac{\pm p\pi}{(1+\alpha)}\frac{1}{\delta},$ (50)

where $p=0,1,2,\dots,\infty$. These lines intersect with the truncation
resonance curves when they touch the bandgap limits, as observed in Fig. 8(b).
In addition, it is possible that the magnitude of
$\cos(\delta(1+\alpha)\Omega)$ becomes zero at some combinations of $\Omega$
and $\delta$, resulting in

$\bar{\Omega}_{p}=\frac{\pm(2p+1)\pi}{2(1+\alpha)}\frac{1}{\delta}.$ (51)

In such a case, the curves for free-free and fixed-fixed truncation resonances
intersect since $t_{12n}=0$ and $t_{21n}=0$ have identical roots, and the
intersection point satisfies Eq. (51). This also aligns with the shared area
between the two ellipses representing the free-free and fixed-fixed truncation
resonances in the Venn diagram of Fig. 6(g).

Finally, for completeness, we study the effect of the design parameters on the
truncation resonance curves. As depicted in Fig. 8, such curves fluctuate in
an oscillatory manner as the value of $\delta$ changes, and become more
sensitive to changes in $\delta$ at higher frequencies. Figure 9 shows the
combined truncation resonances for the free-free and fixed-fixed cases for
different values of $\alpha$ (columns) and $\beta$ (rows), from which the
following can be established:

  * 1.

Equations (50) predicts when the solutions of the free-free and fixed-fixed
truncation resonances touch the bandgap limits, while Eq. (51) predict when
the solutions of the free-free and fixed-fixed truncation resonances intersect
with one another, as established earlier.

  * 2.

The frequency of the oscillatory behavior of the truncation resonance curves
depends on $\alpha$. It is seen that a move towards increasingly negative
values of $\alpha$ decreases the frequency by which truncation resonances
fluctuate in response to a change in $\delta$ over the same $\Omega$ range.
This behavior can be well explained by inspecting the lines generated by Eq.
(51), which captures all the intersections between truncation resonance lines,
as shown in Fig. 9. The first solution of this equation becomes larger with a
smaller denominator due to the $(1+\alpha)$ term. As a result, integer
multiples of the first solution grow further apart, thus forcing truncation
resonance curves to oscillate a fewer number of times over the $\delta$ range.

  * 3.

The impedance contrast $\beta$ plays no role in the periodic fluctuations of
the truncation resonance curves as $\delta$ changes. This can be inferred from
Eqs. (50) and (51) since they are not functions of $\beta$. However, the
amplitude of the fluctuation is largely a function $\beta$ and is bounded by
the width of the underlying bandgap.

  * 4.

The sign of $\beta$ does not affect the behavior of the truncation resonances
curves when we collectively consider the free-free and fixed-fixed cases,
owing to the fact that flipping the sign of $\beta$ only interchanges the
curves of the free-free and fixed-fixed cases, as evident in Eqs. (34) and
(35).

##  5 Conclusions

In this paper, we presented a comprehensive analytical investigation of
truncation resonances emerging within the bandgaps of continuum rod-based
finite PnCs. By implementing an approach based on the transfer matrix method
(TMM), expressions for both the unit cell dispersion relation of the infinite
PnC rod and the frequency response functions of a finite one with defined
boundaries were obtained. Different truncated chains were obtained by stacking
a number of unit cells, each consisting of two different material/geometric
segments and a level of symmetry characterized by the parameter $\delta$. It
was proven that the truncation resonances of a free-free and fixed-fixed PnC
rod always coincide with the natural frequencies of a single unit cell with
the same boundary conditions, providing an efficient tool to tune these
truncation resonances. Furthermore, it was shown that the truncation
resonances exhibited by the two other cases (fixed-free and free-fixed) must
co-exist in the two former cases (free-free and fixed-fixed). This feature was
further verified and analytically proven using a semi-infinite PnC
configuration. With the exception of the perfectly symmetric unit cell choice
(i.e., $\delta=0$), it was also shown that truncation resonances can
potentially occur in all the commonly encountered boundary conditions and are
functions of the symmetry parameter $\delta$ as well as the unit cell
properties, defined here using the frequency and impedance contrasts, $\alpha$
and $\beta$. In the free-free and fixed-fixed cases, a maximum of one
truncation resonance may occur inside each bandgap, while in the fixed-free
and free-fixed cases, up to two truncation resonances may emerge inside each
bandgap. The underlying connection between the localized mode associated with
a truncation resonance and the corresponding boundary condition was explained,
enabling us to interpret the different patterns of truncation resonances for a
family of finite PnCs obtained using different asymmetric realizations of a
single unit cell. The ability to understand the origins of truncation
resonances beyond lumped-parameter systems, and to accurately predict these
resonances in continuum rod-based PnCs provides an intriguing road map for a
broad range of applications that demonstrably benefit from the presence of
truncation resonances.

## Funding

M. Nouh acknowledges support of this work by the US National Science
Foundation through research award no. 1847254 (CAREER) and 1904254. A. T. Juhl
acknowledges support of this work by the Air Force Office of Scientific
Research via award no. 20RXCOR058.

## Data availability

All the data used and/or analysed during the current study is included in the
manuscript.

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