Non-Hermitian acoustic waveguides with periodic electroacoustic feedback Danilo Braghini1Vinicius D. de Lima1Danilo Beli2Matheus I. N. Rosa3 and Jos e R. de F.

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Non-Hermitian acoustic waveguides with

periodic electroacoustic feedback

Danilo Braghini1,*,Vinicius D. de Lima1,Danilo Beli2,Matheus I. N. Rosa3, and Jos´e R. de F.

Arruda1

1School of Mechanical Engineering, University of Campinas, Campinas, S˜ao Paulo 13083-970, Brazil

2Sao Carlos School of Engineering, University of Sao Paulo, Sao Carlos/SP 13563-120, Brazil

3Department of Mechanical Engineering, University of Colorado Boulder, Boulder CO 80309, USA

*Corresponding author: d166353@dac.unicamp.br

October 17, 2022

Abstract

In this work, we investigate non-Hermitian acoustic waveguides designed with period-

ically applied feedback eﬀorts using electrodynamic actuators. One-dimensional spectral

(inﬁnite-dimensional) and ﬁnite element (ﬁnite-dimensional) models for plane acoustic

waves in ducts are used. It is shown that dispersion diagrams of this family of meta-

materials exhibit non-reciprocal imaginary frequency components, manifesting as wave

attenuation or ampliﬁcation along opposite directions for all pass bands. The eﬀects of

diﬀerent feedback laws are investigated. Furthermore, the non-Hermitian skin eﬀect man-

ifesting as topological modes localized at the boundaries of ﬁnite domains is investigated

and successfully predicted by the topology of the reciprocal space. This work extends pre-

vious numerical results obtained for a piezoelectric rod system and contributes to recent

eﬀorts in designing active metamaterials with novel properties associated with the physics

of non-Hermitian systems, which may ﬁnd fruitful technological applications related to

noise control, wave localization, ﬁltering and multiplexing.

Keywords: non-Hermitian systems, skin eﬀect, skin modes, non-reciprocal wave propagation,

metamaterials, metastructures, acoustic ducts

arXiv:2210.05948v2 [physics.app-ph] 14 Oct 2022

1 Introduction

In Physics and Engineering, at the ﬁrst steps of the investigation of a system, it is usual to

assume that no energy is lost when losses are small when compared with the total energy

for the investigated periods. This is the basic condition for classifying systems as Hermitian.

However, the exchange of energy between a system and its surrounding environment in amounts

that cannot be assumed small is ubiquitous. When these eﬀects are considered, the system is

deﬁned as non-Hermitian (NH).

In recent years, the investigations regarding the odd bulk boundary correspondence of NH

systems [1, 2, 3, 4, 5] have led to a deeper understanding of the eﬀects of symmetry, such as the

topological skin modes. Diﬀerently from topological modes previously observed in Hermitian

systems, these modes are extremely sensitive to boundary conditions [6, 7, 8]. Particularly, in

tight-biding models it was shown that the sensitivity is exponential [9]. As shall become clearer

in this work, the appropriate boundary conditions must be chosen to observe skin modes on NH

systems, and yet diﬀerent boundary conditions if the topological aspects are to be analyzed.

Non-Hermiticity has also been used to design metamaterials. Topological mechanics has

been applied for the investigation of the dynamics of such metamaterials [10, 11, 12, 13].

Developments associated with geometrical phases in metamaterial research have allowed the

prediction of novel topological matter exhibiting the NH skin eﬀect (NHSE) on both reciprocal

and non-reciprocal systems, both in one-dimensional and in higher-dimensional systems [14,

15, 16, 17].

One of such topological modes happens in non-reciprocal platforms built upon NH periodic

metamaterials. Non-reciprocity has already been the object of investigations in the context of

metamaterial engineering [18, 19, 20]. For such a system, the topological invariant of the bulk

is deﬁned as the winding number of the corresponding dispersion diagram of Bloch-Bands (BB)

and is related to observable NHSE in the real space [21]. Such properties have been explored

in quantum systems, in classical electric devices [17, 22], and, more recently, in mechanical

platforms [23, 24, 25, 26].

Recently, the investigation of non-Hermitian dispersion relations regarding boundary con-

ditions and the NHSE were extended to distributed-parameter (inﬁnite number of degrees of

freedom) models [27, 14, 25, 28] rather than tight-binding or lumped-parameter (ﬁnite number

of degrees of freedom) models. Mechanical conﬁgurations able to generate arbitrary topologies

have also been previously reported [29, 26] on quantum systems (modulated ring resonator

with a synthetic frequency dimension) and classical mechanical systems (acoustic cavities),

both in the context of lumped-parameter models. In distributed-parameter models, arbitrary

topologies have been investigated by using non-local feedback interactions [24, 25] or by varying

geometrical parameters of the waveguides [28].

Although all topological aspects of the NHSE - in contrast with Hermitian topological modes

- can be studied in simpler single-band periodic systems with one degree of freedom per unit

cell, distributed-parameter models with an inﬁnite number of bands in the reciprocal space

are more realistic in representing practical applications. This work makes contributions in this

direction, using acoustic one-dimensional waveguides with periodically applied electroacoustic

feedback, following a design strategy that emulates both nearest-neighborhood and long-range

non-reciprocal coupling by applying local and non-local feedback interactions, respectively [24,

25, 30, 20].

We explore diﬀerent possible feedback laws as a way to achieve diﬀerent BB topologies.

The stability of the designed metastructures (ﬁnite systems) is investigated as a previous step

in performing experiments on the designed electroacoustic platforms. These systems may ﬁnd

a myriad of applications in engineering, wherever mechanical waves need to be localized and

ﬁltered. For instance, investigations suggest that they can be used as design strategies to

control ﬁlaments and membranes in biological systems [20] and may also be highly eﬀective for

broad-band energy harvesting, when compared to traditional approaches[30].

2 The system under study

LA/2 LB

x0 x2

x1x3

Gfb

LA/2

Figure 1: Acoustic NH metamaterial unit cell.

Figure 1 shows the conﬁguration of the unit cell of the acoustic NH system. It consists of a

1D acoustic duct with circular cross-section of constant diameter d, equipped with an acoustic

pressure sensor (e.g., a microphone) and an electrodynamic actuator (e.g., a loudspeaker).

The pressure is measured at x1and a feedback volume velocity, deﬁned by the operator H

(controller), is applied at x2using, for instance, a loudspeaker. In Fig. 1, x2is located in the

same cell as x1, deﬁning a local feedback actuation. If the feedback is applied to a diﬀerent

cell, the feedback actuation is referred to as non-local. Three methods were used to study the

eﬀects of NH waveguides: the Finite Element Method (FEM), the Spectral Element Method

(SEM), and the Plane Wave Expansion Method (PWE). Details concerning these methods and

their application in this work are provided in the Appendices.

For all the simulations considered, the methods detailed in B were applied to a 1D acoustic

duct ﬁlled with air at ambient conditions (ρ0≈1.225kg/m3,c≈343m/s), Lc= 50cm,d= 4cm,

LA=LB=Lc

3 Results

3.1 Eﬀects of the feedback law on the system spectrum

Fig. 2 shows the complex frequency plane for two diﬀerent boundary conditions: periodic

boundary conditions (PBC) and open (free) boundary conditions (OBC). A system with OBC

is a system composed of a ﬁnite number of cells with closed ends. Thus, it is ﬁnite in length

and will be dubbed a metastructure herein. To impose PBCs on phononic crystals, the Bloch-

Floquet theorem is usually invoked. However, a novel approach is used herein. If the domain

is one-dimensional, one may “wrap around” the system by connecting its ends as a way to

impose the inﬁnite periodic repetition of the system in a cyclic way. In the periodic NH system

treated here, this was achieved by connecting (imposing continuity and equilibrium) the left

boundary of the ﬁrst cell with the right boundary of the last cell (metastructure ends). We use

this wraparound boundary condition and name it periodic boundary condition (PBC).

In both cases - OBC and PBC -, FEM was used to obtain the dynamic stiﬀness matrix of

the closed-loop feedback metastructure, which leads to a ﬁnite-dimensional eigenproblem. The

correspondence between the eigenspectrum and the complex plane of the dispersion relation

computed via SEM is direct, by the change of variables sto f=−js

2π, where sis any eigenvalue,

fis the complex frequency and jdenotes the imaginary unit. Figure 3.1 shows that, with

PBC, the eigenfrequencies are on the dispersion curve. This is due to the fact that the FEM

mesh is a discretization (lumped model) of the system, and, thus, the eigenspectrum found is

actually a discretization of the corresponding continuum spectrum of the metamaterial (inﬁnite

system [31]) depicted in solid lines. In Fig. 3.1, with OBC, the eigenmodes found by FEM are

placed on the real and imaginary axes and represent a discretization of the metastructure

eigenspectrum. The diﬀerence between results on Fig. 3.1(a) and (b) expose the unique bulk

boundary correspondence of NH systems, where eigenmodes are extremely sensitive to the

boundary conditions.

These results are in agreement with theorem 1 and Eq.(5) of reference [7], which state that

the eigenspectrum of the system under OBC is contained on the set formed by the union of

the eigenspectrum of the metamaterial (which forms closed paths on the complex plane) with

the subset of the plane divided by the paths that have a non-zero winding number (i.e., inside

the closed paths). Also, since the matrices are real, the spectrum is symmetric relative to the

imaginary frequency axis. Thus, we will herein consider only positive real frequencies, which

have physical meaning from the wave propagation viewpoint.

(a) (b)

Figure 2: Complex frequency plane of the system with the feedback deﬁned in Eq. (33) for local

(a= 0) and integral action (γD=γP= 0), with gain γI= 0.015. Dispersion relations (solid

curves) and eigenmodes of the structure (circles) are compared under (a) PBC and (b) OBC.

In the sequence, the eigenspectrum of metastructures with each individual component of

a typical proportional-integral-derivative (PID) controller used as the feedback interaction in

each unit cell is depicted over a wide range of frequencies (0 −30kHz) using 21 ﬁnite elements

per unit cell and applying PBC as a way to approximate the dispersion relation.

In Fig. 3 it can be seen that a non-trivial topology was achieved with proportional feedback,

as indicated by the closed paths on the complex plane. Moreover, the imaginary part of the

frequency tends to increase up to an optimal frequency, and decrease for higher frequencies.

It should be observed that, with lower values of feedback gain, this proportional feedback

law provides a non-Hermitian trivial topology, as depicted in Fig. 4, which is diﬀerent from the

Hermitian one (purely real frequency). Even though the real part of the dispersion relation

shows no diﬀerence from the Hermitian counterpart of the acoustic system (passive and without

damping), the imaginary parts exhibit a small but non-zero value, with ranges of wavenumbers

for which the wave response is attenuated (positive imaginary frequency), as well as ranges

for which it is ampliﬁed (negative imaginary frequency). Moreover, the diagram is reciprocal,

which indicates the absence of the NHSE.

(a) (b)

Figure 3: (a) Complex frequency plane of the system under PBC with local proportional

feedback -γP= 1e−5. (b) zooming in the low frequencies.

(a) (b)

Figure 4: Dispersion relation (a) real frequency against wavenumber (b) imaginary frequency

against wavenumber.

As can be seen in Fig. 5, with the same number of ﬁnite elements, the derivative feedback

concentrates the imaginary part of the frequency, related to non-reciprocal attenuation or am-

pliﬁcation, on loops that get larger with increasing frequency. On the other hand, integral

feedback causes loops that get smaller as frequencies gets higher. Thus, we can conclude that

the derivative feedback eﬀect on the BB dominates at high frequencies, whereas the integral

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Non-HermitianacousticwaveguideswithperiodicelectroacousticfeedbackDaniloBraghini1,*,ViniciusD.deLima1,DaniloBeli2,MatheusI.N.Rosa3,andJoseR.deF.Arruda11SchoolofMechanicalEngineering,UniversityofCampinas,Campinas,S~aoPaulo13083-970,Brazil2SaoCarlosSchoolofEngineering,UniversityofSaoPaulo,SaoCarlos/S...

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Non-Hermitian acoustic waveguides with periodic electroacoustic feedback Danilo Braghini1Vinicius D. de Lima1Danilo Beli2Matheus I. N. Rosa3 and Jos e R. de F.

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