Buckling-induced transmission switching in phononic waveguides^a)

Phononic circuits are attracting increased interest because they tailor the propagation of elastic and acoustic waves, which is advantageous for signal manipulation. For example, phononic circuits are useful for cellular phone duplexers by serving as acoustic isolators and mirrors.^1–3 In medical ultrasound applications and acoustic nondestructive tests, phononic circuits promise to miniaturize the imaging aperture,^4–6 decouple the electro-acoustic transduction,^7,8 and slow the signal for smaller delay lines.^9–11 Moreover, nanostructural phononics operating in the hypersonic (gigahertz to terahertz) frequencies enable thermal management,^12–14 photonic-phononic interactions,^15,16 and quantum information control.^17,18 Phononic structures offer readily achievable nonlinearities allowing for strong optomechanical nonlinearities,^19,20 targeted-energy transfer,^21,22 and passive structural nonreciprocity.^23–25 

Phononic circuits require accurately designed and fabricated waveguides to spatially constrain the acoustic transmission within a specific frequency range referred to as the passband (or the transmission band). In the passband, the temporal frequencies are linked to the spatial frequencies (i.e., the wavenumbers) through the dispersion relationship of the medium, providing additional control over the acoustic transmission.^1,26 This temporal and spatial selectiveness stems from the dynamic characteristics of the unit cells whose periodic repetition forms the waveguide. Therefore, the unit cell design is directly linked to the waveguide characteristics via the Bloch modes of the unit cell. The Bloch modes are the vibrational modes that the unit cell exhibits under Floquet boundary conditions with a wavenumber spanning the irreducible Brillouin zone (IBZ).²⁶ This approach calculates the possible wavenumber-frequency relationship known as the band structure of the phononic crystal (i.e., the unit cell). This band structure matches the transmission in an infinite periodic waveguide of the same repeated unit cell.²⁶ 

Bloch modes predict the transmission of sufficiently long and weakly disordered waveguides,^{5,6,12,16,27,28} although fabricated waveguides are neither infinite nor perfectly periodic. In these cases, the finite-structure modal frequencies lie within (or close to) the Bloch-mode passbands.²⁶ For example, such a waveguide of $N$ cells possesses at most $N$ finite-structure modes for every passband; increasing $N$ makes the $N$ modes more densely packed within the passband, leading to the continuous Bloch-mode band structure as $N→+∞$⁠. The dense packing of modes originates from the structural periodicity whose absence (i.e., aperiodicity) generates frequency-distant modes that cannot approximate the passbands. In addition, the periodicity causes (spatially) extended mode shapes that permit the transmission of a signal between the ends of the structure.²⁶ These features—the approximate passband and the extended mode shapes—are acoustically attractive and enable a finite periodic structure to operate as a waveguide. The Bloch-mode approach is computationally efficient in linear periodic systems because it enables the tailoring of a single unit cell to estimate the behavior of the entire waveguide. On the other hand, it significantly deviates from experimental results when the number of unit cells is limited, when there is aperiodicity (structural asymmetry) in the devices (whether intentional or uncontrolled), and when nonlinearities are profound.

Considering repetitive arrays of drumhead resonators composed of coupled flexible micro-membranes, we have recently shown that thermoelastic buckling of the membranes can switch the acoustic transmission.²⁹ Waveguides made from coupled drumhead resonators were first proposed in 2013 by Hatanaka et al.,³⁰ who showed that these waveguides sustain megahertz-to-gigahertz mechanical vibrations with high quality factors (high Q’s) and optical finesse, features that are valuable in mechanical, electrical, and optical applications.^31–33 For instance, optomechanical interactions favor large surface-area structures (like the drumhead resonators) over beams/cantilevers.^32–34 Another advantage of the drumhead resonators is their manufacturability via conventional micro/nanofabrication,^32,33 while allowing for in situ structural tunability and actuation via piezoelectric,^30,34 electrostatic,³⁵ and thermal control.²⁹ Therefore, drumhead resonators were applied in tunable optical cavities³⁶ and low-loss nonlinear optomechanical coupling.³⁷ Moreover, coupling drumhead resonators in the form of arrays, like the devices studied in this article, served in realizing phononic transistors,³⁰ tunable one-dimensional (1D) phononic waveguides,³⁵ cavity-switchable waveguides,³⁸ and on-chip two-dimensional (2D) topological insulators.³⁹ 

In this work, we study the mechanism of transmission switching in the drumhead-resonator waveguides reported in Ref. 29, a phenomenon that has previously been attributed to buckling-induced aperiodicity. Specifically, we develop a reduced-order model (ROM) that mimics the experiments observed in Ref. 29 (Sec. II). The ROM accounts for out-of-plane translation, rotation, and coupling to accurately predict the first and second passbands of the waveguides as functions of the buckling state. The ROM uses the concept of the von Mises truss⁴⁰ to capture the effect of buckling on drumhead-resonator waveguides, as illustrated in the electro-thermoelastic tunability of individual drumhead resonators in Ref. 41. In turn, the von Mises trusses permit modeling and predictive analysis of the drumhead-resonator waveguides via lumped springs and rigid masses, presenting simpler models that are amenable to analytical studies compared to finite element models (FEMs) (e.g., continuous beams on elastic foundations).

With the von Mises ROM, we calculate the Bloch modes (Sec. III) and compare them to the transmission of (60-cell) finite waveguides in cases of perfect periodicity and (<5%) weak disorder (Secs. IV and V). We investigate the acoustics of the finite waveguides by subjecting their first cell to nonzero initial velocities and monitoring the resulting free responses in the time and frequency domains as functions of the spatial propagation of wave packets in the waveguide. We find that when the weakly disordered finite waveguides are close to their critical buckling state, the transmission through the first passband vanishes. Stronger disorder results in a larger range of temperatures where the first passband does not transmit elastic waves. This contrasts with the corresponding perfectly periodic finite waveguide (i.e., with no disorder), where the first passband transmits elastic waves at all considered temperatures, even at the onsite of critical buckling. As for the second passband, the acoustic transmission persists for all considered disorders and temperatures.

To thoroughly explain the effect of buckling on the transmission, we inspect the dependences of the mode shapes of the considered waveguides on temperature (Sec. VI). The results show that the transmission switching is associated with converting the mode shapes from extended over the entire waveguide to localized at some cells. This localization of mode shapes with disorders conforms to Anderson's localization originally discovered in electromagnetic waves^42–44 and then applied in elastic settings.^27,28,45 Moreover, the ROM allows us to test the validity of certain assumptions (cf. Sec. VII) and reveal the physical mechanisms governing the acoustics of the considered waveguides (e.g., revealing the necessity of a narrowing of a passband in addition to disorder to affect transmission loss with buckling). Finally, we present an evaluation of this buckling-switchable transmission on a FEM of the experimental waveguide studied in Ref. 29 with 5% disorder far from or close to critical buckling (Mm. ). The FEM simulations agree with the predictions of the ROM, thus, conclusively proving that weak disorder leads to loss of transmission in the repetitive array of drumhead resonators due to buckling.

In this work, we study the phononic waveguides shown in Fig. 1(a). This waveguide consists of repetitive cells capable of transmitting flexural acoustic waves.^29,38,46,47 This waveguide was studied in Ref. 29, where the cells are drumhead-like membranes composed of silicon nitride (SiN_x) suspended by an etched silicon oxide (SiO₂) layer on top of a silicon (Si) substrate. The involved materials and fabrication methods induce residual stresses in the waveguide, whose cells buckle as depicted in Fig. 1(b) by atomic force microscopy (AFM) conducted in Ref. 29.

To better convey the buckling and acoustic mechanisms, we want to clarify some terminologies used in Refs. 29 and 41 and this paper. We refer to the point of critical buckling as the thermoelastic state leading to the minimum stiffness of the waveguide membranes, corresponding to the lowest natural frequency attained as a function of the stress state [e.g., at 230 K in Fig. 1(c) and in Fig. 4 in Ref. 41]. The pre-buckling regime corresponds to the thermoelastic state where the compressive effects (stresses/forces/strains/…) in the membrane are lower than the ones required for critical buckling; the post-buckling regime corresponds to the thermoelastic state with compressive effects larger than the ones at the point of critical buckling. Hence, the point of critical buckling can be detected by locating the state where the membrane attains minimum stiffness (i.e., minimum natural frequency).

For an ideal membrane with a symmetric cross section, increasing the compressive effects induces no static bending-induced deformation in the pre-buckled regime; once the ideal membrane attains the point of critical buckling, the static deformation drastically increases with further compression in the post-buckling regime.⁴⁸ However, for a non-ideal membrane with an asymmetric cross section, the membrane exhibits some bending deflection in the pre-buckled regime, as demonstrated by Fig. 1(b) deflections of measured at room temperature (i.e., pre-buckled regime). The rate of static deflection starts to increase as a function of increasing compression, as the membrane reaches critical buckling and then transitions to post-buckling. We observe this behavior in the experiments and simulations of Refs. 29 and 41 [also exhibited later in this work by the ROM static simulations of Fig. 2(a)]. Near the point of critical buckling, the membrane's elasticity is strongly sensitive to the thermo-elastic conditions dictated by the residual stresses, temperature, geometry parameters (e.g., thickness), etc. We experimentally observed this strong sensitivity in the asymmetric single resonators of Ref. 41.

In Fig. 1(c), we show the effect of buckling on the elastic transmission of the waveguide of Fig. 1(a).²⁹ The temperature in Fig. 1(c) controls the state of buckling in the waveguide, where lower temperature increases compressions between cells to provoke stronger buckling. At each temperature, the colormap in Fig. 1(c) corresponds to the frequency response measured at the middle cell of the waveguide due to the electrostatic actuation of the gold (Au) pad covering the first cell [cf. Fig. 1(a)]. At high temperatures in Fig. 1(c) (i.e., above ∼230 K), the waveguide exhibits three frequency regimes of effective transmission corresponding to the first three passbands (labeled as I, II, and III). A decrease in temperature from 280 K down to ∼230 K decreases the mean frequency of all the passbands, indicating a softening behavior. During this softening, passband I diminishes its bandwidth until collapsing at ∼230 K, whereas passbands II and III maintain an almost constant bandwidth. Cooling the waveguide to below ∼230 K eliminates the transmission in passband I and increases the mean frequencies of passbands II and III while possessing almost constant bandwidths. The observed temperature-dependent changes in the frequency passbands and the switch in frequency detuning imply that the waveguide at ∼230 K is in a critical buckling state associated with the softest structural configuration (since buckling indicates a minimum linearized stiffness). Accordingly, the waveguide is pre-buckled for temperatures ≳230 K and post-buckled for temperatures ≲230 K, based on Ref. 29. In both buckling regimes, the frequency detunings of passbands II and III are direct consequences of the buckling state of the waveguide. However, the frequency detuning of passband I and its transmission loss in the post-buckled regime necessitates both buckling and disorder in the waveguide.²⁹ 

The requirement of disorder for inducing transmission loss was proven in Ref. 29 via COMSOL simulations, whose results are depicted in Fig. 1(d). The color-shaded regions in Fig. 1(d) illustrate the transmission predicted by Bloch modes as a function of buckling controlled by the imposed temperature change $ΔT$ in COMSOL (cf. Ref. 29 for detailed info). The COMSOL simulated Bloch modes demonstrate the effect of buckling on detuning the frequencies of passbands I and II, yet the Bloch modes cannot predict transmission loss of passband I because Bloch modes assume perfect periodicity with the defined Floquet boundary conditions. Therefore, to capture the transmission loss of passband I depicted by the scatter points of Fig. 1(d), we had to simulate the sinusoidal response of a finite weakly disordered (<5%) waveguide in COMSOL.²⁹ As illustrated by the results of Fig. 1(d), passband I loses transmission around critical buckling, indicating the possibility of transmission loss over a range of temperatures in experiments. We should note that the simulations differ from the experiments in Ref. 29 by showing a certain degree of wave transmission in passband I at the post-buckled regime [i.e., at $ΔT=$ −160 and −140 K in Fig. 1(d)]. This transmission past the critical buckling point was not observed in experiments [cf. Fig. 1(c) in Ref. 29], which might be due to unmodeled effects at low temperatures.

Although the finite waveguide simulations in COMSOL depict the transmission loss and the necessity of disorder, these simulations are computationally expensive and impractical for design or in-depth computational investigations. For instance, determining the transmission of a 60-cell waveguide at each temperature of Fig. 1(d) requires more than 20 h of simulations on COMSOL (cf. Ref. 29 for detailed information). Hence, to perform further parametric investigation of the relationship between disorder, buckling, and elastic transmission in the drumhead waveguides, we propose the ROM depicted in Figs. 1(e) and 1(f). This ROM captures the thermally mediated elastic buckling based on the ROM of a single cell introduced in Ref. 41 exhibiting very good predictive capacity. Here, we extend the ROM of Ref. 41 to account for the coupling between the cells in the waveguide and model the acoustics of the entire phononic waveguide. Accordingly, we allocate to each cell a translational degree-of-freedom (DoF) (as in Ref. 41) and a rotational DoF to capture passbands I and II, respectively.

As shown in Fig. 1(e), each cell of index $i$ in the waveguide consists of a rigid mass $m i$ with a moment of inertia $J i$⁠. Cell $i$ undergoes the motion illustrated in Fig. 1(f) with translational coordinate $u i$ and rotation angle $θ i$⁠. The translation deforms the grounding springs of stiffnesses $k i B$ and $k i S$ representing the restoring forces for bending and stretching, respectively. As in Ref. 41, these translational bending and stretching springs are confined at distances $d B$ and $d S$ [see Fig. 1(f)] while possessing free (undeformed) lengths $L B$ and $L S$⁠, respectively; clearly, a free length larger than the confinement distance (i.e., $L B> d B$ and $L S> d S$⁠) introduces compressive strains (precompression) in the cell. We assume that the remaining springs in the ROM are undeformed at the initial undeformed configuration of the waveguide. For example, the springs with stiffnesses $T i B$⁠, $k i − 1 C$⁠, $k i C$⁠, $T i − 1 C$⁠, and $T i C$ attached to the cell of index $i$ do not apply any forces or torques in Fig. 1(e). The grounding torsional spring of stiffness $T i B$ lumps the bending effects that oppose the rotation $θ i$ due to the grounded boundary of the drumhead. The coupling springs with stiffnesses $k i C$ and $T i C$ account for the force and torque, respectively, applied by cell $i$ to cell (⁠ $i+1$⁠) due to the deformations illustrated in Fig. 1(f). Last, we represent the lattice length separating two successive cells by the length $L$ [see Figs. 1(e) and 1(f)].

To study the effect of buckling on wave transmission, we evaluate the Bloch modes of the cell at each temperature shown in Fig. 2(a). The Bloch modes correspond to the infinite waveguide of Figs. 1(e) and 1(f) made of cells whose parameters are identical to the considered single cell. In this perfectly periodic infinite waveguide, all the cells attain at $T$ the equilibrium state of $u ¯ i Eqm= u ¯ Ref Eqm T$ and $θ i Eqm=0$ rad for all $i∈ 1 , 2 , 3 , … , + ∞$⁠. At every instant $t$⁠, we track the oscillations of the $ith$ cell about its equilibrium state via the perturbation coordinates,

For all $k x/L∈ 0 , π$ rad, the IBZ is defined by the respective pair of eigenfrequencies $ω$ that zero the determinant of the matrix operating on $p i$ in (8). These eigenfrequencies are the Bloch modes' frequencies forming the dispersion curves in Fig. 2(b) at 390, 370, and 350 K for the single cell. The lower (blue) and upper (green) curves in Fig. 2(b) correspond to passbands I and II of the transmission in the perfectly periodic infinite waveguide, respectively. We depict the transmission of this waveguide in Fig. 2(c), where we collect the extrema (maxima and minima) of passbands I and II [as in Fig. 2(b)] for the temperature $T∈ 350 , 400$ K.

Figure 2(c) shows that cooling from 400 to ∼370 K reduces the mean frequencies of both passbands while narrowing the bandwidth of passband I. Cooling below ∼370 to 350 K increases again the mean frequencies of both passbands while widening the bandwidth of passband I. The first cooling phase from 400 to ∼370 K in Fig. 2(c) resembles the cooling phase in Fig. 1(c) between 280 and ∼230 K. However, the second cooling phase between ∼370 and 350 K in Fig. 2(c) diverges fundamentally from the experimental transmission in Fig. 1(c) between ∼230 and ∼80 K, where the transmission in passband I does not reemerge. Therefore, the ROM buckling cannot eliminate the transmission of passband I in a perfectly periodic infinite waveguide. This loss of transmission with buckling necessitates the consideration of disorder (i.e., the break of perfect periodicity) in the waveguide as previously established in Ref. 29.

Note that we adopt the relationships in (7) to emulate the experimental transmission in Fig. 1(c) between 280 and ∼230 K. For this reason, we select $Λ i C T$ in (7a) to decrease until the transmission vanishes at the point of minimum frequency, $min 350 → 400 K Λ i G T,$ leading to the shrinkage of passband I between 400 and ∼370 K in Fig. 2(c). In (7b), we assume that the rotation of the cell centerline (of length $L$⁠) deflects an elastic foundation of stiffness density $k i B/L$⁠. In (7c), we impose a temperature-constant bandwidth for passband II like the measurements in Fig. 1(c). The temperature-detuning of the mean frequencies of the passbands in Fig. 2(c) is considered for the identified parameters (i.e., $Λ Ref B$⁠, $β 0$⁠, $β 1$⁠, $β 2$⁠, $γ 0$⁠, and $γ 1$⁠) of the ROM in Ref. 41, which deviate from those in the devices used in Fig. 1(c) (extracted in Ref. 29).

The holes at the center of the cells in Figs. 1(a) and 1(b) are etching holes through which the etchant attacks the underlying layer and suspends the cells. Thus, there is a higher (linear) density of etching holes at the middle of the waveguide [of index $(N+1)/2$] compared to the ends (of indices $1$ and $N$⁠). This higher etching-hole density increases the etching rate, leading to over-etching at the middle of the waveguide compared to its ends.²⁹ We model this over-etching by $s i$ in (9) as a linear distribution of the cell position from the middle of the waveguide. Figures 3(a) and 3(b) show two examples of thickness variation with $σ h=0%$ (i.e., perfectly periodic waveguide) and $σ h$ = 5%, respectively.

We solve (11) via “fsolve” (gradient descent method) in matlab^®. In the numerical solver, the starting guesses for $u ¯ i Eqm$ in (11) correspond to the equilibria of individual cells in (1) [see Fig. 2(a)]. We assign the differences $u ¯ i + 1 Eqm − u ¯ i Eqm$ as starting guesses for $L ¯ θ i Eqm$ for $i∈ 1 , 2 , … , N − 1$ and $L ¯ θ N − 1 Eqm$ as the guess for $L ¯ θ N Eqm$⁠. Figures 3(c) and 3(d) display the computed equilibria of (11) at different temperatures in the perfectly periodic and weakly disordered waveguides of Figs. 3(a) and 3(b), respectively. In Figs. 3(c) and 3(d), each segment (thick dashed line) represents a cell in the ROM of Figs. 1(e) and 1(f) translated and rotated according to the equilibrium of (11).

In Fig. 3(c), the cells at equilibrium undergo the same translational deflections without rotation, which results from the perfect periodicity of the waveguide of Fig. 3(a). For instance, the weakly disordered waveguide of Fig. 3(b) attains equilibrium with different cell translations and rotations, as shown in Fig. 3(d). For both waveguides of Figs. 3(c) and 3(d), the cooling increases the cells' baseline translational deflections, going from negative to positive values between 400 and 350 K [like the individual cell in Fig. 2(a)]. Moreover, each 10 K of cooling induces larger deflections at lower temperatures in Figs. 3(c) and 3(d), which mirrors the buckling susceptibility observed in Fig. 2(a).

Figures 4(a) and 4(b) show the translational and rotational time-responses, respectively, for wavepackets propagating through the waveguide of Fig. 3(a) at a temperature of 390 K. We observe that the initial velocity imposed at cell 1 by (16) provokes a wavepacket that propagates to the last cell (with index 60) (cf. also Mm. ). The propagating front of this wave consists mainly of two distinct wavepackets with different wave speeds, namely, a “fast” wavepacket reaching cell 60 within ∼17  $μ$s [cf. $h ¯ 60$ in Fig. 4(b)] and a “slow” wavepacket reaching that boundary cell within ∼38  $μ$s [cf. $v ¯ 60$ in Fig. 4(a)]. Figure 4(a) shows the fast wavepacket characterized by low translational amplitudes compared to the slow wavepacket. Figure 4(b) shows both wavepackets possessing similar rotational amplitudes.

For comparison, in Fig. 5, we depict the corresponding wave transmission in the (⁠ $σ h=$ 5%) weakly disordered waveguide at 390 K, forced by the same excitation in (16). A drastically different acoustics are observed for the disordered system. We observe that only the early (fast) wavepacket propagates to cell 60 in the presence of disorder. Moreover, as Fig. 5(c) shows, the slow wavepacket (which carries the major portion of the available energy) becomes spatially localized in the first 30 cells, a result that indicates that the weakly disordered waveguide at 390 K cannot transmit the slow wavepacket corresponding to passband I of Figs. 2(b) and 2(c). This transmission loss for wavepackets in passband I is in full agreement with the experimental findings reported in Ref. 29 for a similar waveguide [and summarized in Figs. 1(a)–1(c)]. Therefore, the ROM developed in this work captures the experimental transmission loss mediated by buckling and provides conclusive proof regarding the important role that structural disorder plays for the transmission loss in buckled waveguides at certain temperature ranges.

This transmission loss mechanism is also observed by the FEM of the waveguide studied in Ref. 29, further illustrated in Mm. , presenting the time-series deformations of the centerlines of two waveguides based on the COMSOL simulations of Ref. 29. In particular, we consider two identical 20-cell weakly disordered waveguides at −20 K (i.e., far from critical buckling) and −120 K (close to critical buckling), respectively. In Mm. , we assess the capacity of waveguides to transmit propagating wavepackets with frequency contents inside passband I of their corresponding infinite waveguide (i.e., the passbands based on the Bloch modes of the constitutive unit cell). These FEM simulations show that an elastic wavepacket can propagate only in the weakly disordered waveguide far from critical buckling (cf. Ref. 29 for the FEM geometry and methods). Hence, with the ROM developed in this work, we confirm that buckling-induced transmission loss for waves in passband I is associated with weak disorder and thermoelastic effects, confirming the experimental and FEM results reported in Ref. 29.

Figures 6(a) and 6(b) illustrate that all modal frequencies of the perfectly periodic finite waveguide are inside the passbands (there exist 60 modes in each passband), whereas Figs. 6(c) and 6(d) show certain modes of the (⁠ $σ h=$ 5%) weakly disordered finite waveguide are lying outside these passbands, i.e., in stopbands. Hence, the Bloch modes of the infinite waveguide constitute a perfect estimator of wave transmission only in the perfectly periodic finite waveguide.

In Figs. 6(a) and 6(b), the perfectly periodic finite waveguide corresponds to strong translational and rotational responses at cell 60, respectively, only when their frequency contents are inside the passbands. Outside of the passbands, however, the frequency responses at cell 60 are minimal compared to the response of the excited cell 1, indicating a lack of transmission throughout the waveguide (i.e., stop band). In Figs. 6(c) and 6(d), cell 60 of the (⁠ $σ h=$ 5%) weakly disordered finite waveguide admits weak responses compared to the response of cell 1 in the Bloch modes defining passband I. These frequency responses verify that the transmission loss shown in Fig. 5 corresponds to the transmission loss of wavepackets inside passband I. Notably, concerning wavepackets initiated in passband II [cf. Fig. 6(d)], the rotational response of cell 60 compares in magnitude to cell 1 due to the persistence of the transmission in this passband, as previously illustrated in Fig. 5.

Similar performance to the results of Figs. 4–6 is observed for different temperatures between 400 and 350 K, as shown in Fig. 7. In particular, the perfectly periodic finite waveguide of Fig. 3(a) does not lose transmission for the considered temperatures; the weakly disordered finite waveguide of Fig. 3(c) cannot transmit waves in passband I between 390 and 352 K.

In Figs. 7 and 8, the contour plots depict the $E ̃ i , m$ normalized by $max E ̃ j , p$ over all $j∈$ {2, 3,…, $N$} and all $p∈${1, 2,…, $n FFT$} of the respective response. For better visualization, we coerce the contour values to 0 and 1 if they fall below the minimum threshold $E Ths≤$ 2 × 10⁻³ and above the saturation limit $E Sat≥$ 0.5, respectively. In essence, in Figs. 7 and 8, we assess the binary behavior of the considered waveguide by checking whether the energy can transmit to cell $i$ at frequency $ω ̃ m$ under the studied temperature and disorder conditions.

The plots of Figs. 7(a)–7(e) display the wave transmission in the three finite waveguides at 400, 390, 370, 353, and 350 K, respectively. For all these temperatures, the perfectly periodic finite waveguide (⁠ $σ h$ = 0%) transmits energy to the last cell 60, with frequency contents in both passbands I and II—cf. Figs. 7(a)(i)–7(e)(i). This transmission for all temperatures is unique for the perfectly periodic waveguide and does not occur in the weakly disordered finite waveguides considered in Figs. 7(a)(ii)–7(e)(ii) and Figs. 7(a)(iii)–7(e)(iii). For example, we observe transmission loss of waves with frequency content in passband I for the waveguide with $σ h$ = 2.5% at 370 K in Fig. 7(c)(ii) and for the waveguide with $σ h$ = 5% at 390, 370, and 353 K in Figs. 7(b)(iii)–7(d)(iii), respectively. Hence, larger disorders correspond to a more extended range of temperatures with transmission loss in passband I.

Moreover, we notice that the frequency width of passband I is smaller in the transmitting scenarios of the weakly disordered waveguides compared to the perfectly periodic waveguide. This width-narrowing of passband I accompanies a similar narrowing of passband II in the weakly disordered waveguides of Figs. 7(a)(ii)–7(e)(ii) and Figs. 7(a)(iii)–7(e)(iii). However, contrary to passband I, the weakly disordered waveguides keep transmitting energy to the last cell 60 in passband II at all temperatures considered, as depicted in Fig. 7. Therefore, we conclude that passband II is less susceptible to structural disorder than passband I, which fully agrees with what was experimentally witnessed in Ref. 29. Hence, the ROM developed herein accurately captures these acoustic aspects of the phononic lattice under investigation.

To further clarify the dependence of wave transmission on temperature, in Figs. 8(a)–8(c), we study the frequency content of transmitted waves reaching cell 45 in the three finite waveguides with disorder $σ h∈$ {0%, 2.5%, 5%}, respectively. On top of the finite waveguides results, we overlay the corresponding passbands of the infinite waveguides [cf. Fig. 2(c)]. The results in Fig. 8(a) prove that, at all the considered temperatures, the wave transmission in the perfectly periodic finite waveguide has frequency contents solely inside the passbands. Thus, the passbands of the infinite waveguide perfectly estimate the wave transmission in the perfectly periodic finite waveguide at all temperatures, as concluded from Figs. 6(a) and 6(b) at 390 K.

However, this perfect estimation does not hold for the wave transmission in the weakly disordered finite waveguides considered in Figs. 8(b) and 8(c), especially around the critical-buckling temperature (i.e., 370 K). Although, as discussed previously, there is a band-narrowing of both passbands I and II, wave transmission loss is only realized for passband I [cf. Figs. 8(b) and 8(c)]. This behavior confirms that passband I is more susceptible to disorders than passband II, confirming the analogous conclusion illustrated in Fig. 1(c) of the experimental work in Ref. 29. Last, by comparing Fig. 8(b) to Fig. 8(c), we deduce that stronger disorders result in more severe wave transmission losses with thermal-mediated buckling.

To explain the causes of transmission loss due to disorder, we plot in Figs. 9 and 10 the spatial distributions (modeshapes) of two modes of the finite waveguides, specifically modes 30 and 90, at 400, 390, 370, 353, and 350 K. Thick dashed lines represent the normalized translational and rotational deformations of individual cells calculated by the eigenvector $Φ$ of (18) at the considered temperature and waveguide. For better visualization, we join the cells with cubic splines depicted as thin lines to imitate the continuous deformation of the waveguide's centerline. Mode 30 considered in Fig. 9 is inside passband I of the infinite perfectly periodic waveguide (which contains also the first 60 modes of the perfectly periodic finite waveguide with no disorder), whereas mode 90 in Fig. 10 is located inside passband II (which also contains modes 61–120 of the corresponding perfectly periodic finite waveguide with no disorder).

Figures 9(a)(i)–9(e)(i) show that mode 30 of the perfectly periodic finite waveguide deforms with comparable amplitudes over the entire spatial extent of the system, i.e., from cell 1 (the excited cell) up to cell 60, at all the considered temperatures. Such modes with spatially extended amplitude distributions over the entire waveguide are called extended modes that are necessary for wave propagation in the finite waveguides. For instance, this type of extended modes enables energy initially applied to cell 1 to appreciably deform the remaining cells resulting in detectable mechanical energy propagation through the entire spatial length of the waveguide. The extended modes in Figs. 9(a)(i)–9(e)(i) verify the persistence of wave transmission via passband I of the perfectly periodic finite waveguide as depicted in Figs. 7(a)(i)–7(e)(i) and Fig. 8(a)  for all temperatures, even very close to the critical-buckling temperature.

Conversely, in the weakly disordered finite waveguides, not all modes are extended for all temperatures. For example, for disorder level $σ h$ = 5%, mode 30 is an extended mode only at 400 and 350 K, respectively—cf. Figs. 9(a)(ii) and 9(e)(ii). This extended shape directly affects wave transmission through the waveguide: A nonzero initial deformation of cell 1 due to the applied excitation deforms the cells throughout the waveguide as shown by the modeshapes of Figs. 9(a)(ii) and 9(e)(ii), allowing energy to propagate throughout the waveguide [cf. Figs. 7(a)(iii), 7(e)(iii), and 8(c)]. At the other considered temperatures, the disorder shrinks passband I, and, in turn, mode 30 shifts to a stop band and becomes spatially localized. Nevertheless, cell 1 exhibits minimal vibrations in Figs. 9(b)(ii)–9(d)(ii) at 390, 370, and 353 K, respectively, justifying the inability to transmit energy by mode 30 in Figs. 7(b)(iii)–7(d)(iii) and 8(c). The modeshapes in Figs. 9(b)(ii)–9(d)(ii) are “localized modes” where large deformations are confined only locally without extending over the entire waveguide (as in extended modes).

Localized modes characterize aperiodic/disordered structures because extended modes necessitate the periodicity between the constitutive cells in the waveguides. In other words, cells of similar geometry and material configurations form a periodic structure with cells of similar (isolated) modal frequencies, which we refer to as modal periodicity. This modal periodicity is necessary to form the extended modes that enable transmission throughout the structure. In this work, we observe this modal periodicity breaking in the weakly disordered waveguides under the effect of buckling because the cells in the waveguides develop different (isolated) modal frequencies in passband I due to buckling-induced changes in the grounding and coupling stiffnesses (cf. Ref. 41) The buckling-induced differences in modal frequencies lead to localized modes, as in Figs. 9(b)(ii)–9(d)(ii), inhibiting energy transmission throughout the waveguides.

As a general conclusion, buckling and thermoelastic effects lead to shifts of modeshapes in the frequency domain due to disorder, which, in turn, “transforms” certain modes from extended to localized. Therefore, thermal effects and buckling magnify the “modal” disorder in the disordered waveguides, leading to energy localization and confinement, similar to Anderson localization.⁴² Due to this effect, many modes between 1 and 60 (inside passband I of the perfectly periodic finite waveguide) become localized in the weakly disordered waveguides, as seen at 390 K in Mm. . Indeed, at 390 K, all the leading 60 modes are localized in the waveguide with $σ h$ = 5%, prohibiting passband I transmission [cf. Figs. 7(b)(iii) and 8(c)]. In addition, Mm. demonstrates the existence of some extended modes between modes 1 and 60 in the waveguide with $σ h$ = 2.5% at 390 K, which explains the observed passband I transmission at 390 K of Figs. 7(b)(ii) and 8(b). Last, in Mm. , all the leading 60 modes of the perfectly periodic waveguide are extended modes, resulting in broader passband I at 390 K than the waveguide with $σ h$ = 2.5%—compare Fig. 7(b)(i) to Fig. 7(b)(ii). Note that all 60 leading modes of the perfectly periodic waveguide are extended modes even at the critical-buckling temperature of 370 K, as shown in Mm. .