Valley edge states with opposite chirality in temperature dependent acoustic media

The valley degree of freedom in phononic crystals and metamaterials holds immense promise for manipulating acoustic and elastic waves. However, the impact of acoustic medium properties on valley edge state frequencies and their robustness to one-way propagation in valley topological phononic crystals remains unexplored. While significant attention has been devoted to scatterer design embedded in honeycomb lattices within acoustic and elastic media to achieve valley edge states and topologically protected nontrivial bandgaps, the influence of variations in acoustic medium properties, such as wave velocity and density affected by environmental temperature, has been overlooked. In this study, we investigate the effect of valley edge states and topological phases exhibited by topological phononic lattices in a temperature-dependent acoustic medium. We observe that a decrease in wave velocity and density, influenced by changing environmental temperature, shifts the topological valley edge states to lower frequencies. Therefore, alongside phononic lattice design, it is crucial to consider the impact of acoustic medium properties on the practical application of acoustic topological insulators. This issue becomes particularly significant when a topological phononic crystal is placed in a wave medium that transitions from incompressible to compressible, where wave velocity and density are no longer constant. Our findings offer a novel perspective on investigating topological insulators in variable acoustic media affected by changing thermodynamic and fluid properties.


Introduction
Topological acoustics has emerged as a rapid and fascinating branch of topological physics followed by groundbreaking discoveries in solid-state physics that revealed the existence of topological states of matter [1][2][3][4][5][6][7][8].Since then, researchers have tried to create acoustic analogue of selected mechanisms and they are majorly classified into three categories: acoustic quantum Hall effect (QHE), quantum spin Hall effect (QSHE), and quantum valley Hall effect (QVHE).The acoustic QHE is designed by breaking the time-reversal symmetry [9].The acoustic QSHE with fourfold Dirac degeneracy is related to high symmetry at the centre point of the first irreducible Brillouin zone (FIBZ), i.e.Γ point.By manipulating the lattice symmetry, the band inversion triggered by topological phase transition with degenerate hybridized modes carrying opposite spins at the topological nontrivial bandgap bounding edges can be observed [10][11][12].The acoustic QVHE with single Dirac cone degeneracy at the K (K ′ ) point of FIBZ can be achieved by breaking the space-inversion symmetry (SIS) of the honeycomb lattice structure [12][13][14].This results in a pair of separated valley edge states with opposite chirality at the bounding edges of the topological bandgap [15,16].Similar to the prior two mechanisms, the topological valley edge states can robustly propagate along the interface of two domains with opposite topological valley phases [17,18].
The spin magnetic moment of the electron and the vortex pseudospin in phononic crystals are two intriguing phenomena that share some similarities.Both involve a form of angular momentum associated with the particles or excitations in the system.However, they arise in distinct physical contexts and have different underlying mechanisms.The spin magnetic moment of the electron is a fundamental property of electrons in quantum mechanics [19].It arises from their intrinsic spin angular momentum, which is a quantum mechanical property that gives rise to a magnetic moment even in the absence of orbital motion.This magnetic moment interacts with external magnetic fields, leading to phenomena such as the Zeeman effect [20] and the Stern-Gerlach experiment [21].
On the other hand, vortex pseudospin in phononic crystals [15] is a concept that emerges in the study of wave propagation in structured media such as phononic crystals.In certain types of phononic crystals with topological properties, wave modes can exhibit a pseudospin degree of freedom analogous to the spin of electrons.This pseudospin can be associated with the chirality or rotational properties of the wave modes, leading to phenomena such as topologically protected edge states and unconventional wave transport [7,22].
The field of acoustic topological QVHE is experiencing rapid advancement due to its adaptability and variety, offering a new avenue for manipulating the propagation of acoustic waves [23][24][25][26][27]. Numerous captivating subjects related to valley properties have been explored, such as topological oneway edge states [2,3,28], acoustic topological insulators [29,30], acoustic topological rainbow trapping [31,32] and topological valley transport [14,33,34].Typically, achieving the valley edge states involve conveniently adjusting the phononic crystal (PnC) scatterers' orientation such that to lower the symmetry from C 3v to C 3 , so called breaking SIS.While number of studies have investigated different honeycomb lattice designs with broken SIS characteristic, to the best of our knowledge, no study has considered the effect of acoustic wave medium properties such as varying acoustic wave speed (influenced by environmental temperature and pressure) and density on topologically protected valley edge state eigenfrequencies and robustness of acoustic wave propagation at the interface.Beside other reported works that began with fascinating topological PnC design with SIS, we considered a relatively simple structure that can support valley edge states in an acoustic medium with varying wave speed and densities.The proposed topological scatterer also supports structural reconfigurability [35] and detailed discussion on this structural attribute is outside the scope of this study.
The motivation for this study arises from a curiosity about how variations in the acoustic properties of wave media influence the valley edge states and the transportation of acoustic waves at the protected interface.In this paper we have considered acoustic wave propagating media whose properties are varying with working environmental temperature and atmospheric pressure.Since experimental data on acoustic variations for methyl nonafluorobutyl ether (MNE) and ethyl nonafluorobutyl ether (ENE) are available [36], we have majorly focused on studying topological valley edge states and topological phases in MNE and ENE media.However, the study implication is not limited to MNE and ENE and can be applied to other acoustic wave medium whose acoustic properties are changing under the influence of external stimuli.This includes the application of acoustic topological insulators in incompressible (constant density) and compressible (varying density) fluid media.MNE and ENE are sustainable biochemical solvents with a wide range of applications in cosmetic, biochemicals, and pharmaceutical products.Recently, Muhammad [37] discovered Fano resonance phenomena and proposed a new application of PnC as a bio-chemical sensor to detect the acoustic variations in MNE and ENE fluids.
In this study, we propose a two-dimensional PnC hexagonal lattice with C 3v symmetric scatterer that is comprised of three rectangular steel rods.The SIS in the proposed structure can be broken by rotating the three-legged rectangular steel rods inside the hexagonal lattice and topological valley edge states with opposite chirality can be realised.The presence of two separate chiral valley pseudospins, each with a distinct nonzero Chern number [14,15], resulted in the band inversion and subsequently triggered a topological phase transition.The three rods are placed at an angle of 120 degrees inside the hexagonal lattice and by changing the orientation of the PnC scatterer, we can increase the bandwidth of the topological bandgap.The PnC scatterer is placed in the MNE and ENE media and the effect of varying acoustic properties i.e. sound speed influenced by changing temperature [36] on the topological nontrivial bandgap and valley edge states is investigated.The effect of MNE and ENE temperature on Dirac degeneracy frequency is discussed.By varying the acoustic properties of MNE and ENE, we have reported robust acoustic wave propagation at the straight and zigzag interfaces without changing the geometric parameters of the scatterer.The study contributes by providing new findings on understanding the impact of acoustic medium properties on valley topological phases and interface state frequencies.It is expected that our findings will open a new avenue for the application of acoustic topological insulators in compressible and incompressible fluid media where external stimuli (temperature and pressure) influence acoustic medium properties.
The remaining paper is organized as follows: section 2 discusses the methodology and valley edge states with vortex features in a single unit cell PnC honeycomb lattice.The valley edge states with opposite chirality influenced by breaking SIS in the supercell lattice structure is explained in section 3. The acoustic wave propagation at the straight and zigzag interfaces are demonstrated in section 4. Finally, the study conclusion and outlook are given in section 5.

Acoustic valley edge states and opposite chirality
The PnC hexagonal unit cell structure is shown in figure 1(a).The unit cell structure consists of three-legged rectangular steel rods (as scatterer) placed 120 degrees apart inside an acoustic media.The acoustic media are comprised of MNE and ENE fluids.The central region enveloped by the rectangular steel rods is a regular triangle.The topological acoustic valley edge states in an air medium with a similar structure were recently studied by Xi et al [35].We utilized this structure to investigate the acoustic valley edge states and topologically protected interface mode propagation in the MNE and ENE media subjected to varying temperatures.The steel scatterer has material properties as follows: mass density ρ = 7850 kg m −3 , Young's modulus E = 210 GPa, and Poisson ratio ν = 0.33.The acoustic properties of MNE and ENE fluids under varying temperatures are given in table 1 [36].The lattice constant of the hexagonal lattice is a = 40 mm with geometric parameters of the rectangular steel rods as follows: width w = 9 mm, height h = 13 mm.The side length of the hexagon is a/2.All three rectangular rods can rotate around their centre with an angle θ to break the SIS of the unit cell structure.The matched mirror symmetry between the hexagonal lattice and scatterer enables the structure to exhibit a single Dirac point at the high symmetry K and K ′ points on the FIBZ.Due to high symmetry in the proposed hexagonal lattice, Dirac degeneracy can be inevitably produced in the FIBZ.Due to inherent property of the proposed lattice structure, by changing the angle θ, the mirror symmetry of C 3v can be broken and transformed to C 3 symmetry.This will result in lifting Dirac cone degeneracy and opening of topologically protected nontrivial bandgap bounded by the distinct valley edge states and topological phases.
The present study is conducted by using finite element code COMSOL Multiphysics v6.2.We used the pressure acoustics physics to model MNE and ENE media.The three-legged steel scatterers are modelled using solid mechanics physics.The Floquet Bloch periodicity condition is applied on all six edges of the hexagonal lattice to make the unit cell structure infinitely periodic in the x and y directions.The COMSOL Multiphysics eigenfrequency study is used to solve the wave dispersion relation and analyse the topological phases with opposite chirality.We used the frequency domain study to demonstrate acoustic wave propagation in topologically protected supercell lattices with straight and zigzag interfaces.
For the proposed lattice structure, when the angle of rotation θ is 30 degrees, the Dirac cone degeneracy is observed, see figure 1(b).By rotating the rectangular steel rods ±30 degrees i.e. θ = 0 • or θ = 60 • , the SIS can be broken as a result Dirac cone degeneracy that was initially observed for θ = 30 • can be lifted and a complete topologically protected nontrivial bandgap with distinct Chern number and topological phases can be observed.Figure 1(c) shows the simulated acoustic energy and intensity fields for two different valley edge states labelled as p − and q + on the first two acoustic bands when θ = 0 • (Type I) and θ = 60 • (Type II).From the acoustic energy and intensity fields for p − and q + modes, one can clearly observe the typical vortex profiles with clockwise and counterclockwise acoustic energy flow, so-called valley pseudospin effect around the honeycomb lattice centre, respectively.In addition, the six vertices of the hexagonal lattice are labelled with p and q alternately, see figure 1(a) red dashed line.The three p and central triangular regions show clockwise acoustic energy flow for p − mode, see blue arrows in figure 1(d).Likewise, the acoustic pressure intensity profile for q + mode shows the counterclockwise acoustic energy flow for the other three hexagonal vertices labelled with q and central triangular region, see black arrows in figure 1(d).
To reveal the topological phase transition of the twodimensional hexagonal lattice, valley Hall transition in the FIBZ is studied by determining the eigenfrequencies and eigenmodes of the valley edge states for different rotation angle θ.Since each rectangular steel rod is separated 120 degrees apart, we cannot rotate the three legs of the scatterer more than 60 degrees to get different topologies.Therefore, the valley edge state eigenfrequencies and eigenmodes are determined in the range of −60 • ⩽ θ ⩽ 60 • for MNE and ENE acoustic media separately.Figure 2 shows the band transition with Dirac degeneracy opening, closing, and reopening of the nontrivial topological bandgap for MNE and ENE fluids at room temperature of 25 • C. When θ = 30 • , Dirac degeneracy is observed at 8.075 kHz (MNE) and 8.372 kHz (ENE) at K point of FIBZ.By varying the angle θ above or below 30 • , Dirac degeneracy is lifted by breaking SIS and a topological nontrivial bandgap emerges.The topological valley phases at the bounding edges of the bandgap showed that band inversion has occurred at θ = 30 • .This can be observed from clockwise (p − mode) and counterclockwise (q + mode) acoustic energy flow/mode polarization in the hexagonal lattice.
Furthermore, the valley edge eigenfrequencies are determined for both MNE and ENE under varying environmental temperatures ranging from 10    When θ = ±60 • , the lower bounding edge of the topological bandgap has clockwise polarization/ acoustic energy flow i.e. p − mode, and the upper bounding edge has counterclockwise polarization/acoustic energy flow i.e. q + mode as shown in figure 3(a).With decreasing rotation angle θ, the topological bandgap becomes narrower until θ = −30 •  where Dirac degeneracy or topological bandgap closing is observed.Such band inversion usually occurs across a nontrivial topological bandgap that is bounded by the trivial and nontrivial bounding edges with opposite chirality.When θ is further increased beyond θ = −30 • , the topological nontrivial bandgap opens again, however, opposite chirality and valley pseudospin behaviour are observed.Even though θ = 0 • and θ = −60 • have identical eigenfrequencies i.e. wave dispersion  curves, they show opposite chirality in their eigenmodes.For −30 • ⩽ θ ⩽ 0 • range, the lower bounding edge of the topologically protected nontrivial bandgap has counterclockwise polarization i.e.q + mode, see figure 3(a).While the upper bounding edge has clockwise polarization i.e. p − mode, which is opposite to valley edge state polarisation behaviour observed in−60 • ⩽ θ ⩽ −30 • range.Identical behaviour with interchangeable valley edge state polarization is observed for the other second half part of the Dirac plot where opening, closing, and reopening of the nontrivial topological bandgap is observed.This showed that before and after θ = ±30 • when Dirac degeneracy is lifted, nontrivial topological bandgap with exchange in valley edge state polarization has occurred.These Dirac plots for MNE and ENE also show that with variation in acoustic properties of wave media, the eigenfrequency of valley edge states has changed while eigenmodes remained the same.An increase in wave velocity and/or acoustic wave medium density shifted the valley edge states and Dirac degeneracy to a lower frequency.
By applying k • p perturbation method [38,39], the Dirac degeneracy and the so-called valley Hall phase transition around K point of FIBZ can be described by θ-dependent continuum Hamiltonian H K (δk) = υ D (δk x σ x + δk y σ y ) + mυ 2 D σ z where υ D is Dirac velocity, δk is the moment derivation of k − k K from the valley centre K, and σ i (i = x, y, z) are Pauli matrices that operate on the vortex pseudospins.The effective mass is m = (ω q+ − ω p− ) /2υ 2 D .The Hamiltonian depends upon the rotation angle θ through the bounding edge frequencies ω q+ and ω p− in the effective mass term.The Berry curvature can be derived from this Hamiltonian using the expression . Further integration of the Berry curvature will lead us to an expression where C K is Chern number and γ is Berry phase.The theoretical valley Chern number for K and K ′ valleys can be calculated using 2 respectively [38].

Valley edge states in supercell structures
Following the discussion on the valley edge states and topological phases, we categorise the proposed hexagonal lattice into Type I (θ = 0 • ) and Type II (θ = 60 • ) lattices.For Type I hexagonal lattice with θ < 30 • , the theoretical valley Chern number at K and K ′ points are − 1 2 and + 1 2 respectively.While for Type II hexagonal lattice with θ > 30 • , these are opposite as schematically shown in figure 1(c).These distinct features offer an opportunity to obtain valley edge states at the interface of Type I and Type II hexagonal lattices.Prior to revealing the topologically protected interface states at the interface of Type I & II lattices, band structure studies of 12 Type I and 12 Type II hexagonal lattices are carried out to demonstrate valley edge states in the supercell structure.As shown in figure 4, localization of acoustic energy at the interface of Type I and Type II lattices corresponding to wavenumber k n+ (k p− )with a decaying energy field away from the interface is observed.The superscript p and n show positive and negative wavenumbers while the −ve and +ve superscript symbols show the backward and forward acoustic wave propagation.Even though wavenumber k n+ (k p− ) is symmetric with reference to the FIBZ centre, they have opposite polarization of acoustic energy flow.The k n+ band is shown with blue and k p− band is depicted with red dotted lines.Their distinct clockwise and counterclockwise edge mode polarization at the interface with the direction of acoustic energy flow (white arrows) is shown in the figure inset.The acoustic wave energy at wavenumber k n+ features forward clockwise and counterclockwise pseudospins above and below the interface, respectively.While the acoustic wave energy at the k p− wavenumber has an opposite pseudospin direction with backward acoustic energy propagation.For both wavenumbers k n+ (k p− ) the evolution of pseudospin acoustic pressure field from 0 to 2π is shown at the bottom of figure 4.
The band structure of supercell lattices shown in figure 4 is for MNE and ENE at operating temperature of 10 • C. Next, we changed the temperature for both MNE and ENE within the given temperature range, see table 1, and depicted the interface mode band in a rainbow colour spectrum, see figure 5.For convenience, the k n+ (k p− ) wavenumber is shown with the same colours, hence readers are suggested to keep the discussion on figure 4 into consideration when analysing figure 5. Reminiscent of the findings in figure 3, a decrease in interface mode eigenfrequencies is observed with increasing temperature of MNE and ENE media.The localization of acoustic wave energy at the interface of Type I and Type II lattices is shown in the inset of figure 5.

Acoustic wave propagation at interface
In this section we will explain the acoustic wave propagation at the interface of Type I and Type II lattices by arranging the supercell structure in such a way to form straight and zigzag interfaces.A circular region (loudspeaker symbol) is used to induce acoustic plane wave and excite the valley edge states at the interface of topologically distinct PnCs for a frequency range covering the topological bandgap.To avoid incident acoustic wave back reflection from domain boundaries that may obscure the results, the entire system is surrounded by a perfectly matched layer.By using the frequency domain study in COMSOL Multiphysics, a wide range of frequencies are swept to visualize the acoustic wave propagation at the interface of Type I and Type II PnC lattices.For a straight interface formed by the Type II PnC lattice at the top and the Type I PnC lattice at the bottom, robust acoustic wave propagation can be observed in figure 6(a).The interface of lattice structures is shown with the red line.We used two probes to measure the acoustic pressure intensity at the interface and a point away from the interface, see microphone symbols in figure 6(a).Inside the bandgap region of the supercell lattice (see figures 4 and 5), robust acoustic wave propagation at the interface while decaying acoustic pressure intensity fields away from the interface are obtained.We calculated the frequency response spectra at different temperatures for MNE, using data given in table 1.The blue dash-dot line shows the decaying acoustic energy for the point probe away from the interface.While the black solid line shows the robust propagation of acoustic energy along the interface of Type I and Type II PnC lattices.The influence of MNE temperature (that alters acoustic wave velocity and density) on the frequency response spectra is shown in figures 6(b)-(i).Like figures 3 and 5, a decrease in valley edge state frequencies with increasing temperature for MNE is observed.This deduced that a decrease in sound speed in MNE media shifts the interface mode frequencies to a lower region.
Next, we arranged the Type I and Type II PnC lattice in a way to mimic the zigzag interface, see figure 7    C), the frequency response spectra are obtained using two-point probes at the interface (solid black line) and a random point away from the interface (dashed blue line).The robustly propagating acoustic wave pressure at the interface (solid black line) and decaying acoustic wave energy inside the bandgap region (dashed blue line) can be observed.The bandgap region is highlighted in green.(i) The comparison of frequency response spectra with changing temperature at the interface probe.C), the frequency response spectra are obtained using two-point probes at the interface (solid black line) and a random point away from the interface (dashed blue line).The robustly propagating acoustic wave pressure at the interface (solid black line) and decaying acoustic wave energy inside the bandgap region (dashed blue line) can be observed.The bandgap region is highlighted in green.(i) The comparison of frequency response spectra with changing temperature at the interface probe.

Conclusion and outlook
In this study, we explore acoustic valley edge states and topological phases within temperature-dependent MNE and ENE acoustic media.A phononic crystal periodic scatterer consisting of three-legged rectangular steel rods is arranged in a honeycomb lattice to investigate wave dispersion, Dirac degeneracy lifting, and pseudospin vortex features of valley edge modes along the bounding edges of the topologically nontrivial bandgap.Experimental data obtained from literature indicate that variations in the working temperature of MNE and ENE result in reduced wave speed and density of the acoustic media.By manipulating the temperature of MNE and ENE acoustic media, we analyse the effect of changing acoustic medium wave speed and density on valley edge state eigenfrequencies and eigenmodes.Our findings reveal that a decrease in the speed of sound in MNE and ENE media shifts the valley edge modes, including the Dirac degeneracy point, to a lower frequency region.We also investigate the pseudospin vortex features of valley edge modes with opposite chirality by constructing supercell structures with distinct topologies and conducting wave dispersion studies.Analysis of the supercell dispersion curves unveils forward and backward propagating acoustic bands at the interface of topologically distinct PnC lattices.We discuss the valley pseudospin with opposite vortex chirality for both unit cell structures and supercell lattices, elucidating their effect on changing acoustic medium properties influenced by temperature variation.Furthermore, we combine two types of topologically distinct unit cell structures to emulate straight and zigzag interfaces.Through frequency domain analysis, we examine the frequency response spectra for probe points at the interface and away from it.Our results demonstrate robust acoustic wave propagation at interface mode frequencies for straight and zigzag interfaces.This study represents a unique exploration of valley edge states in temperature-dependent acoustic media, with significant implications for applications in acoustic topological insulators in thermodynamically controlled acoustic media.

Figure 1 .
Figure 1.(a) Schematic of the PnC honeycomb lattice formed by embedding three-legged steel scatterers inside an acoustic media comprising of MNE and ENE fluids.(b) Dispersion plot of PnC lattice at rotation angle of θ = 0 • , 30 • , 60 • with Dirac degeneracy at θ = 30 • .The SIS can be achieved by changing the rotation angle 60 • ⩽ θ ⩽ 0 • .(c) The valley Chern numbers that characterize the distinct valley-dependent behaviours.(d) Valley pseudospin with opposite chirality (clockwise and counterclockwise acoustic energy flow) observed at the bounding edge of the nontrivial topological bandgap when θ = 0 • and θ = 60 • .The white arrows show direction of in-plane acoustic velocity in spatial frame.

Table 1 .
Acoustic properties of MNE and ENE with varying temperatures (10 • C-40 • C) at atmospheric pressure of 1 atm.The data are obtained from Pin ¯eiro et al [36].Temperature ( o C) MNE ENE Density (kg m −3 ) Wave speed (m s −1 ) Density (kg m −3

Figure 2 .
Figure 2. Band structure of PnC scatterers in MNE and ENE acoustic media at temperature of 25 • C for θ = 0 • , 30 • , 60 • across the boundary of FIBZ.Dirac degeneracy with band inversion at K point for θ = 30 • can be observed.An opening of a complete topologically protected nontrivial bandgap is achieved for θ = 0 • , 60 • .The acoustic energy flow direction (black and blue arrows) and eigenmode polarization (clockwise p − and counterclockwise q + eigenmodes) are shown in the figure inset.

Figure 3 .
Figure 3. (a) Topological phase diagram exhibiting topological phase transition with changing rotation angle θ ranging from −60 • to 60 • for acoustic wave propagating in MNE and ENE media with changing temperature.The eigenmodes at the left side show the specific vortex feature of the valley edge states at the bounding edge of the nontrivial topological bandgap when θ = ±60 • and θ = 0 • .(b) The zoomed Dirac degeneracy frequency with band inversion for MNE and ENE at different temperatures.(c) The Dirac degeneracy frequency with changing temperature for MNE and ENE acoustic wave media.

Figure 4 .
Figure 4. Band structure of supercell lattice structure consisting of 12 Type I and 12 Type II PnC lattices.The localization of acoustic energy at the interface of Type I and Type II lattices is shown at the top.Such interface mode can be observed for eigenfrequencies positioned at the k n+ and k p− wavenumbers.The specific vortex feature of valley edge states at different values of k n+ and k p− wavenumbers are shown.A forward acoustic wave energy flow at k n+ and backward acoustic wave energy flow at k p− wavenumbers can be observed.The bulk acoustic modes are depicted with black asterisks.
(a).The acous-tic wave is excited in the ENE media and frequency response spectra are calculated.The zigzag interface further demonstrates the robustness of topologically protected valley edge states at the bends and sharp corners.Figures7(b)-(i)shows the frequency response spectra for ENE with varying temperatures in the PnC lattice structure with a zigzag interface.

Figure 5 .
Figure 5. Band structure of PnC supercell lattices in the MNE and ENE acoustic media subject to varying temperatures.The variation in interface mode frequencies at k n+ and k p− wavenumbers with changing temperature can be observed.Any eigenfrequency lying inside the topologically nontrivial bandgap region exhibits interface mode, as shown in the figure top.

Figure 6 .
Figure 6.(a) A straight interface formed by the arrangement of Type I and Type II PnC lattices.The robust acoustic wave propagation at the straight interface can be observed.The excitation source (speaker and probe points (microphone symbol) are shown.(b)-(h) For MNE subject to varying temperatures (10 • C−40• C), the frequency response spectra are obtained using two-point probes at the interface (solid black line) and a random point away from the interface (dashed blue line).The robustly propagating acoustic wave pressure at the interface (solid black line) and decaying acoustic wave energy inside the bandgap region (dashed blue line) can be observed.The bandgap region is highlighted in green.(i) The comparison of frequency response spectra with changing temperature at the interface probe.

Figure 7 .
Figure 7. (a) A zigzag interface formed by arrangement of Type I and Type II PnC lattices.The robust acoustic wave propagation at the zigzag interface can be observed.The excitation source (speaker symbol) and probe points (microphone symbol) are shown.(b)-(h) For ENE subject to varying temperatures (10 • C−40• C), the frequency response spectra are obtained using two-point probes at the interface (solid black line) and a random point away from the interface (dashed blue line).The robustly propagating acoustic wave pressure at the interface (solid black line) and decaying acoustic wave energy inside the bandgap region (dashed blue line) can be observed.The bandgap region is highlighted in green.(i) The comparison of frequency response spectra with changing temperature at the interface probe.
• C-40 • C, see table 1.The temperature variation altered the density and sound speed of