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# Asymmetric transmission and coherent perfect absorption in a periodic array of thermoacoustic cells

Come Olivier, Anis Maddi, Gaëlle Poignand, Guillaume Penelet

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## ► To cite this version:

Come Olivier, Anis Maddi, Gaëlle Poignand, Guillaume Penelet. Asymmetric transmission and coherent perfect absorption in a periodic array of thermoacoustic cells. Journal of Applied Physics, 2022, 131 (24), pp.064012. 10.1063/5.0089989. hal-04395161

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Distributed under a Creative Commons Attribution 4.0 International License
Asymmetric Transmission and Coherent Perfect Absorption in a periodic
array of thermoacoustic cells

Come Olivier,ᵃ) Anis Maddi, Gaelle Poignand, and Guillaume Penelet

Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School

(IA-GS), CNRS, Le Mans Université, France

This paper describes some exotic scattering properties of a one-dimensional network of thermoacoustic cells, and characterizes them experimentally. The considered two-port consists of a waveguide containing a periodic arrangement of porous materials subjected to temperature gradients and separated by empty sections. The interaction of an acoustic wave with the temperature gradients leads to an inherently nonreciprocal phenomenon known as the thermoacoustic effect. It is shown that this effect can be exploited for the design of systems with exotic acoustic scattering properties through two experimental demonstrations. The first example showcases a balanced asymmetric transmitter with transmission coefficients inverse of each other, yielding a nonreciprocity factor of 18 dB, without reflections. The second example shows a coherent perfect absorber, where maximum absorption is achieved for a wide range of temperature gradients by controlling the relative amplitudes and phasing of incoming waves.

I. INTRODUCTION

Nonreciprocal systems are a subject of intense re-
search, as they provide new ways to control wave prop-
agation and absorption. Breaking reciprocity is a key to
one-way propagation, a necessary feature for the design
of efficient acoustic isolators, one-way mirrors or nonre-
ciprocal topological insulators¹. In recent years, strate-
gies to break reciprocity made use of nonlinearities²,³
moving media⁴⁻⁶, spatiotemporal modulation⁷, nonlin-
ear bianisotropy⁸ or active control⁹,¹⁰. Recent reviews
covering the subject can be found in Refs. ¹¹,¹².

The thermoacoustic effect is another way to break reci-
procity. Due to the viscous and thermal coupling be-
tween an oscillating fluid and a porous medium submit-
ted to a temperature gradient, this effect may lead either
21 to the amplification or to the damping of an incident
22 wave, depending on the sign of the temperature gradi-
ent. For an ”in-depth” yet accessible explanation of the
24 thermoacoustic effect, the reader is referred to Ref. 13,
25 while Refs. 14 and 15 will provide recent reviews cover-
ing the various thermoacoustic machines developed in the
26 last decades. The thermoacoustic effect is therefore an
28 inherently nonreciprocal phenomenon. In recent years,
29 only a few studies have taken advantage of this effect to
30 obtain exotic systems such as a thermoacoustic diode¹⁶
31 exhibiting a 16 dB power-transmission ratio thanks to a
32 four-stages thermoacoustic unit, or a system achieving
33 spontaneous PT-symmetry breaking through the balance
34 of thermoacoustic amplification and damping¹⁷. Periodic
35 arrangements of thermoacoustic cores have also recently
36 been shown to be a form of Willis material exhibiting non
37 reciprocal and even couplings¹⁸,¹⁹. This description of-
fers a different way to describe one-dimensional periodic
39 thermoacoustic networks, and should ultimately provide
40 access to new means of controlling waves.

In this mostly experimental work, we propose to ex-
plore further some of the possibilities offered by a peri-
odic array of thermoacoustic cells to control wave propa-
gation. The theoretical description of the acoustic prop-
agation in such a system is provided in Sec. II, leading to
the expression of its theoretical transmission and reflec-
tion coefficients. Section III is dedicated to presenting
the experimental setup for the measurement of the scat-
tering matrix of two ThermoAcoustic Networks (TAN),
designed to demonstrate experimentally two exotic scat-
tering phenomena observed at specific frequencies.

In Section IV, a first configuration is presented, ex-
hibiting reflectionless asymmetric transmission. More
precisely, a specific array of thermoacoustic cells is de-
signed so that, at some specific frequency, the balance
of thermoacoustic amplification/damping is achieved in
such a way that the two-port is reflectionless but non-
reciprocal, giving rise to transmission coefficients being
inverse with each other. The experimental result show a
nonreciprocity factor (ratio of transmission coefficients)
of 18 dB with almost no reflection at a frequency of 57 Hz.

In Section V, a second configuration is presented in
the form of a Coherent Perfect Absorber which keeps the
nonreciprocal property inherent to the thermoacoustic
coupling. Such a system does not let any wave out, when
excited simultaneously from both ends with impinging
waves of appropriate relative amplitude and phase, due to
a combination of wave interference and absorption along
the system<sup>20,21</sup>. Experimental results demonstrate over
99.8 % absorption for impinging waves of relative ampli-
tudes from 1:1 to 25:1 at an almost constant frequency
of 70 Hz for increasing temperature gradients applied on
each cell.

II. THERMOACOUSTIC NETWORK DESCRIPTION

Figure 1(a) shows a representation of the elementary
unit of a periodic network of thermoacoustic cells [shown
in Fig. 1(c)]. It is composed of the usual elements con-

<footnote>a) Now at: Centre de Transfert de Technologies du Mans, CTTM, 2 rue Thalès de Milet, 72000 LE MANS.</footnote>
<figure><img src="image_4.png" /><figcaption>FIG. 1. (a) Unit cell of the ThermoAcoustic Network (TAN) showing the stack (ST) in which the thermoacoustic effect takes place, sandwiched between a hot heat exchanger and a cold heat exchanger (HHX and CHX resp. at temperatures $T_H$ and $T_C$) imposing the temperature difference $\Delta T = T_H - T_C$ between its ends, and followed by a Thermal Buffer Tube (TBT) along which the temperature cycles back to $T_C$. (b) Pore geometries of both stacks (square pores for the asymmetric transmission in Sec.IV, circular pores for the coherent perfect absorber in Sec.V), and the heat exchangers (honeycomb modeled as circular pores) (c) Experimental setup for the measurement of the scattering matrix of the thermoacoustic network, showing the position of the anechoic terminations, the two sources providing the excitation signals and the four microphones (m1 to m4) required.</figcaption></figure>

78

79 stituting a thermoacoustic core. The main element is

80 a porous material where the thermoacoustic effect takes

81 place, customarily called the stack²² (denoted by ST).

82 Two heat exchangers on either side allow a temperature

83 gradient to be applied along the stack, from a cold tem-

84 perature $T_C$ to a hot temperature $T_H$. They are therefore

85 referred to as the cold heat exchanger (CHX) and the hot

86 heat exchanger (HHX), respectively. Lastly, the empty

87 waveguide along which the temperature returns to $T_C$,

88 is called the Thermal Buffer Tube (TBT). The TBT en-

89 sures the continuity of the temperature from cell to cell,

90 such that the temperature field is identical in all cells.

91 Assuming a $e^{iwt}$ temporal convention, the plane

92 wave propagation through the four elements compos-

93 ing the unit cell is described by Rott's theory of linear

94 thermoacoustics²³, such that the relation at position $x$

95 between the acoustic pressure $p$ and volume flow rate $U$

96 oscillating at angular frequency $\omega$ is governed by

$$ 
\begin{aligned}
\frac{dp}{dx} &= -\frac{i\omega\rho_0}{S\phi(1-f_\nu)}U, \\
\frac{dU}{dx} &= -\frac{i\omega S\phi}{\gamma P_0}[1+(\gamma-1)f_\kappa]p \\
&\quad + \frac{f_\kappa - f_\nu}{(1-f_\nu)(1-\sigma)}\frac{1}{T_0}\frac{dT_0}{dx}U,
\end{aligned}
\quad (1) 
$$

97 where $\rho_0$ and $P_0$ represent the mean values of the density

98 and the pressure of the fluid (air at atmospheric pressure

99 in this case) and $\sigma = \nu/\kappa$ its Prandtl number, i.e. the

100 ratio of its kinematic viscosity $\nu$ to its thermal diffusivity

101 $\kappa$. These quantities depend implicitly on the mean tem-

102 perature $T_0(x)$, and therefore are position-dependent. $\gamma$

103 denotes the fluid's specific heat ratio. $S\phi$ is the open

104 cross-section area available for the acoustic propagation,

105 with $\phi$ the porosity of the constituents of the cell. Con-

106 sidering $\phi = 1$ allows to use Eq.(1) for empty waveguides

107 (i.e. the TBT), whereas $\phi < 1$ in the heat exchangers

108 and the stack.

109 $f_{\nu,\kappa}$ are frequency- and geometry-dependent functions

110 that describe the viscous and thermal coupling between

111 the oscillating gas and the surrounding solid walls²²,²⁴.

112 They depend directly on the ratio of the characteristic

113 dimension $r$ of the channel to the viscous and thermal

114 acoustic boundary layer thicknesses $\delta_{\nu,\kappa}$ given as $\delta_{\nu} = \sqrt{2\nu/\omega}$ and $\delta_{\kappa} = \sqrt{2\kappa/\omega} = \delta_{\nu}/\sqrt{\sigma}$. The expression of

115 these functions varies with the geometry of the channel

116 considered²⁵, see Fig. 1(b). For cylindrical channels of

117 radius $r$, they are given as

118

$$ f_{\nu,\kappa} = \frac{2}{(i-1)r/\delta_{\nu,\kappa}} \frac{J_1[(i-1)r/\delta_{\nu,\kappa}]}{J_0[(i-1)r/\delta_{\nu,\kappa}]} \quad (2) $$

119 where $J_n$ is the $n$th order Bessel function of the first
120 kind. For rectangle channels of dimension $2a \times 2b$, their
121 expression is given as the series

$$
\begin{align}
f_{\nu,\kappa} &= 1 - \left(\frac{8}{\pi^2}\right)^2 \sum_m \sum_n F_{m,n}^{\nu,\kappa}, \nonumber \\
\frac{1}{F_{m,n}^{\nu,\kappa}} &= m^2 n^2 \left[ 1 - i \frac{\pi^2}{2} \left(\frac{\delta_{\nu,\kappa}}{r}\right)^2 \frac{b^2 m^2 + a^2 n^2}{(a+b)^2} \right] \tag{3}
\end{align}
$$

122 where $m, n = 1, 3, 5, \dots$ and with the equivalent radius
123 $r = 2ab/(a+b)$.

124 Note that an empty waveguide (i.e., the Thermal
125 Buffer Tube) is described as a single large pore, with
126 its characteristic dimension that of the waveguide.

127 Equations (1) can be used to express the transfer ma-
128 trix **M** relating the pressure **p** and flow rate velocity **U**
129 between two positions **x**₁ and **x**₂²²,²⁶, such that:

$$
\begin{pmatrix} p(x_2) \\ U(x_2) \end{pmatrix} = \mathbf{M} \begin{pmatrix} p(x_1) \\ U(x_1) \end{pmatrix} = \begin{bmatrix} M_{11}M_{12} \\ M_{21}M_{22} \end{bmatrix} \begin{pmatrix} p(x_1) \\ U(x_1) \end{pmatrix}. \quad (4)
$$

130 It is worth noting that when the temperature is constant
131 $dT_0/dx = 0$ (as is the case in the heat exchangers HX),
132 Eqs. (1) fall back to the regular case of lossy propagation,
133 yielding the transfer matrix over the length $d = x_2 - x_1$:

$$
\mathbf{M}(\Delta T = 0) = \begin{bmatrix} \cos(kd) & -iZ_c \sin(kd) \\ -\frac{i}{Z_c} \sin(kd) & \cos(kd) \end{bmatrix}, \quad (5)
$$

134 with $Z_c = \rho_0 c_0 / S \phi [(1 - f_\nu)(1 + (\gamma - 1)f_\kappa)]^{-1/2}$ the
135 lossy characteristic impedance of the waveguide and
136 $k = \omega / c_0 [(1 + (\gamma - 1)f_\kappa) / (1 - f_\nu)]^{1/2}$ the wavenumber,
137 where $c_0$ is the adiabatic speed of sound.

138 The global transfer matrix **T** relating pressure and vol-
139 ume velocity at either end of a network composed of N<sub>cell</sub>
140 identical cells is obtained by multiplying sequentially the
141 transfer matrices of each constitutive element, such that

$$
\mathbf{T} = (\mathbf{M}_{\text{TBT}} \mathbf{M}_{\text{HHX}} \mathbf{M}_{\text{ST}} \mathbf{M}_{\text{CHX}})^{N_{\text{cell}}}. \quad (6)
$$

142 Incidentally, the acoustic pressure and volume velocity
143 in the system can also be expressed as functions of the
144 two counter-propagating pressure waves $p^{\pm}$, such that
145 $p^{+} = (Z_c p + U)/2$ and $p^{-} = (Z_c p - U)/2$. There-
146 fore, the scattering matrix **S**, describing the relation be-
147 tween the incoming ($p^{-}(x_2)$; $p^{+}(x_1)$) and the outcoming
148 ($p^{-}(x_1)$; $p^{+}(x_2)$) acoustic waves on either sides of an el-
149 ement, can be deduced from Eq.(4) and is given by

$$
\begin{pmatrix} p^{-}(x_1) \\ p^{+}(x_2) \end{pmatrix} = \mathbf{S} \begin{pmatrix} p^{-}(x_2) \\ p^{+}(x_1) \end{pmatrix} = \begin{bmatrix} t^{-} & r^{+} \\ r^{-} & t^{+} \end{bmatrix} \begin{pmatrix} p^{-}(x_2) \\ p^{+}(x_1) \end{pmatrix}. \quad (7)
$$

150 wherein the transmission coefficients $t^{\pm}$ and the reflection
151 coefficients $r^{\pm}$ are related to the transfer matrix coeffi-
152 cients $T_{ij}$, $(ij) \in [1, 2]$ through the relations

$$
\begin{align}
t^{-} &= 2/\Omega, \nonumber \\
r^{+} &= (-T_{11} - T_{12}/Z_c + Z_c T_{21} + T_{22})/\Omega, \tag{8} \\
r^{-} &= (T_{11} - T_{12}/Z_c + Z_c T_{21} - T_{22})/\Omega, \nonumber \\
t^{+} &= 2(T_{11}T_{22} - T_{12}T_{21})/\Omega, \nonumber
\end{align}
$$

153 with $\Omega = T_{11} - T_{12}/Z_c - Z_c T_{21} + T_{22}$.

154 With these theoretical tools now established for the
155 description of the scattering properties of a 1-dimensional
156 network of thermoacoustic cells, the following focuses on
157 the presentation of the setup used to characterize the
158 experimental scattering properties of networks designed
159 with these tools.

III. EXPERIMENTAL SETUP

161 The experimental setup for the measurement of the
162 scattering properties of a network of thermoacoustic cells
163 is presented in Fig. 1(c). The central section is the scat-
164 terer to characterize, that consists of a periodic network
165 of identical thermoacoustic cells. On either side of the
166 network, two identical measurement sections allow for
167 the measurement of its scattering coefficients.

A. Description of the TAN

169 The thermoacoustic network consists of a waveguide
170 of rectangular $2a \times 2b = 1 \text{ cm} \times 5 \text{ cm}$ cross-section con-
171 taining a regular layout of 15 identical thermoacoustic
172 cells. Each cell is 19.5 cm long, resulting in a nearly 3 m
173 long two-port. Each cell, as represented in Fig. 1(a),
174 contains a stack, sandwiched between a cold and a hot
175 heat exchanger controlling the temperature difference
176 $\Delta T = T_H - T_C$ applied along the stack.

177 The stack is a porous material, characterized by its
178 porosity $\phi$ and the radius of its pores $r$. In the following,
179 two different materials are used to demonstrate two dif-
180 ferent kinds of scatterers, see Fig. 1(c). For the purpose
181 of demonstration of an asymmetric transmitter in Sec-
182 tion IV, the 15 mm long stack is carved out of cordierite
183 catalytic substrate, consisting of many parallel square
184 channels of equivalent radius $r = 585 \ \mu\text{m}$ (corresponding
185 to 400 cells per square inch), with a porosity of $\phi = 0.85$.
186 For the demonstration of a coherent perfect absorber in
187 Section V, the 14 mm long stack is made of a pile of 60
188 fine stainless wire meshes (mesh number 45 and wire di-
189 ameter 160 $\mu\text{m}$, ref.104738, Gantois Industries). Though
190 irregular in shape, the pores of such a material can sat-
191 isfactorily be modeled as of circular cross section. A fit
192 of experimental data gives an evaluation of the average
193 pore radius $r \approx 170 \ \mu\text{m}$ and a porosity $\phi \approx 0.80$.

194 The heat exchangers are used to ensure a uniform tem-
195 perature distribution over the rectangular 1 cm × 5 cm
196 cross-section of the waveguide on each side of the stack.
197 They are made of slices of 1/8” aluminium honeycomb.
198 Here too, the hexagonal pores are modeled as circular,
199 fitted from measurements to a pore radius $r \approx 770 \ \mu\text{m}$
200 and a porosity $\phi \approx 0.945$. The hot temperature is im-
201 posed to the hot heat exchangers through the 1 mm-
202 thick stainless steel walls of the waveguide by 10 W
203 polyimide heaters (KHLVA-0502/10-P, Omega Engineer-
204 ing) applied on each face of the waveguide, supplied by
<table><caption>TABLE I. Dimensions of the different elements of the thermoacoustic cell. Dimensions for the stack are given both for Reflectionless Asymmetric Transmission and Coherent Perfect Absorber configurations.</caption><thead><tr><th>Element</th><th>pore shape</th><th>pore size</th><th>porosity</th><th>length</th></tr></thead><tbody><tr><td>ST (§IV)</td><td>square</td><td>585 µm</td><td>0.85</td><td>15 mm</td></tr><tr><td>ST (§V)</td><td>mehsgrids</td><td>170 µm</td><td>0.80</td><td>14 mm</td></tr><tr><td>HX</td><td>honeycomb</td><td>770 µm</td><td>0.945</td><td>15 mm</td></tr><tr><td>TBT</td><td>rectangular</td><td>1 × 5 cm</td><td>1</td><td>150 mm</td></tr></tbody></table>

controlled DC power supplies (EA-PS 3065-10B, ElektroAutomatik). The cold temperature is imposed on the cold heat exchangers through the 1 mm-thick stainless steel walls of the waveguide, thanks to two 10.4 W Peltier cooler modules (ET-063-08-15, Adaptive, European Thermodynamics) on each side of the waveguide, supplied by a controlled DC power supply (CPX200DP, Aim-TTi Instruments). Water circulating in aluminum tubing in direct contact with their hot side evacuates the heat extracted by the Peltier modules.

A representation of the geometry of the pores of the different porous media (stacks and heat exchangers) used for this study is provided in Fig. 1(b).

Finally, the empty 15 cm TBT separates the hot heat exchanger from the cold heat exchanger of the following cell, limiting the amount of direct thermal leakage between cold and hot sources. The resulting temperature distribution is considered one-dimensional in the unit cell, homogeneous in the heat exchangers and varying linearly in the stack and the TBT, as is depicted in Fig. 1(a). Geometrical properties of the thermoacoustic unit cell are summarized in Table I.

## B. Measurement setup

For the need of the measurement of the scattering matrix by means of the two-sources, four-microphones method$^{27}$, this thermoacoustic network is itself connected to another section of waveguide at each side. As illustrated in Fig. 1(c), each of these measurement sections consists of a straight 1.2 m waveguide with the same cross-section, on which a pair of 1/4" microphones (Brüel & Kjær 4136 with Nexus amplifier 2690) are flush mounted. Each microphone pair is separated by a distance of 65 cm, allowing measurement in the 10 Hz to 220 Hz range provided that they have been precisely calibrated (both in amplitude and phase) to limit the errors on the measured and calculated quantities$^{28}$. A preliminary relative calibration has therefore been made using a small cavity coupler.

A moving coil loudspeaker (Visaton FRS 8) is side mounted near the end of the measurement section to provide the step-by-step harmonic excitation. The speaker is placed in a small enclosure and coupled to the waveguide via a tube of centimetric diameter. The measurements

are conducted with low enough excitation amplitudes that nonlinear effects of propagation can be discarded.

Each measurement section ends with a low frequency anechoic termination. Such a termination consists of a larger cavity (1.5 cm × 7 cm) with an adjustable length closed by stainless wire mesh screens$^{29,30}$. The reflection coefficient of such terminations has a measured amplitude lower than 4 % over the frequency range of interest, which is a sufficiently low value for the needs of this study.

## IV. REFLECTIONLESS BALANCED ASYMMETRIC TRANSMISSION

A first setup of the thermoacoustic network is sought to demonstrate reflectionless balanced asymmetric transmission. That is, we search a geometrical configuration of the network and properties of the stack configuration for which, at a given frequency $f_1$ and a temperature difference $\Delta T_1$, its transmission coefficients are inverse of each other (and obviously non-unity) and the reflection coefficients are close to zero. The targeted properties of S can be summarized as

$$ (|S_{ij}(f_1, \Delta T_1)|) = \begin{bmatrix} |t^-| & \sim 0 \\ \sim 0 & |t^+| = 1/|t^-| \end{bmatrix}. \quad (9) $$

The chosen experimental configuration is built around 15 mm long cordierite substrate stacks with square pores of half width $r = 585$ µm and porosity $\phi = 0.85$, offering the best theoretical results within the experimental constraints.

## A. Theoretical results

The amplitude of the theoretical scattering coefficients of the system are shown with lines in Fig. 2(a-b) for increasing temperature differences $\Delta T = 0$ K, 50 K, 82 K and 106 K (from blue to yellow), evaluated for frequencies from 10 Hz to 250 Hz. The phase of the coefficients (not represented here) behaves as one would expect (i.e. the phase of the transmission coefficients is almost a linear function of frequency, that reflects the propagation of the wave through the two-port with a speed close to the adiabatic speed of sound.)

For $\Delta T = 0$ (blue curves), the system behaves as expected for a passive reciprocal and (almost) symmetric system, with transmission coefficients identical in both directions: both continuous and dashed blue lines are indistinguishable. Their amplitudes are less than unity $|t^{\pm}(\Delta T = 0)| < 1$ due to the propagation losses, mainly in the porous material constituting the stack and to a lesser extent in the heat exchangers. The reflection coefficients are also identical with magnitude $0 < |r^{\pm}| < 1$ (identical blue lines in Fig. 2). The variations of the reflection coefficients with the frequency are indicative of a Fabry-Perot-like interference behavior of the wave in
<figure><img src="image_5.png" /><figcaption>FIG. 2. Moduli of the (a) transmission coefficients and (b) reflection coefficients of the scattering matrix S of the 15 cells thermoacoustic network for various temperature differences $\Delta T = 0$ K, 50 K, 82 K and 106 K (from blue to yellow, lines represent the model and markers show the measurement data points). Reflectionless asymmetric transmission is expected to happen for $\Delta T_1 = 106$ K (yellow curves) at $f_1 = 57$ Hz (marked by the vertical dashed line) with $|t^-| = 1/|t^+|$ and $|r^\pm| \ll 1$. This last temperature difference was not achievable experimentally.</figcaption></figure>

297 the system, with $r^\pm$ dropping at a frequency related to
298 the total length of the network $f_L \approx 2c_0/L$ and its mul-
299 tiples. Note that, the system being composed of highly
300 porous materials, the transmission coefficients are larger
301 than the reflection coefficients in the absence of heating.

302 For increasing temperature differences $\Delta T > 0$, both
303 properties of reciprocity and symmetry of the network
304 are lost. The transmission coefficients of the system are
305 not identical anymore, highlighting a direction of propa-
306 gation where the wave is amplified (in the direction of the
307 positive temperature gradient in the stack) shown with
308 continuous lines. In the opposite direction, the transmit-
309 ted wave are attenuated, more strongly for higher tem-
310 perature differences on the considered frequency range,
311 as shown with dashed lines in Fig. 2(a). The behav-
312 ior of the reflection coefficients, though slightly different
313 in both directions, remains mainly driven by the inter-
314 ference of the waves with the periodic structure of the
315 system in Fabry-Perot-like resonances. Note that, with
316 the increase of the temperature gradient, the amplitude

317 of the reflection coefficients becomes larger, especially at
318 lower frequencies where they can become greater than
319 unity.

320 For a temperature difference of $\Delta T_1 = 106$ K (shown in
321 yellow), the system is expected to behave as a reflection-
322 less asymmetric transmitter at the frequency $f_1 = 57$ Hz.
323 Its reflection coefficients should drop considerably (see
324 Fig. 2(b)), while its transmission coefficients should be
325 inverse of each other $|t^-| = 0.26 = 1/|t^+|$ with $|t^+| = 3.9$,
326 as pointed in Fig. 2(a).

## B. Experimental results

328 Experimental data points corresponding to the mod-
329 eled conditions are presented with markers in Fig. 2, re-
330 spectively crosses for the coefficients $t^+$ and $r^+$, and cir-
331 cles for $t^-$ and $r^-$.

332 In the absence of a temperature gradient, measure-
333 ments confirm the reciprocity of the network, with the
334 superposition of the blue crosses and circles. With in-
335 creasing temperature differences, the scattering coeffi-
336 cients follow the predicted behavior. The transmission
337 coefficients are in very good agreement with the model.
338 The reflection coefficients, though slightly carrying the
339 perturbation observed at $\Delta T = 0$, still follow closely the
340 modeled evolution, with $|r^+|$ being larger than $|r^-|$.

341 The predicted temperature difference of $\Delta T_1 = 106$ K
342 proved to be not reachable with the experimental setup
343 due to heat input limitations. However, the data set
344 corresponding to the highest experimental temperature
345 difference $\Delta T = 82$ K (pink) is already worth of inter-
346 est. At the frequency of interest, namely ~ 57 Hz, the
347 experimental coefficients of reflection experience a min-
348 imum, with respectively $|r^+| \approx 0.04$ and $|r^-| \approx 0.02$,
349 while the transmission coefficients are not far to balance
350 with $|t^+| \approx 2.5$ and $|t^-| \approx 1/3.2$. These results show a
351 nonreciprocity factor (defined as the ratio of both trans-
352 mission coefficients) of 18 dB.

353 To further illustrate the operation of the reflection-
354 less balanced asymmetric transmitter, the propagation
355 of monochromatic waves is studied, in conditions cor-
356 responding to $\Delta T = 82$ K (pink markers) and at two
357 frequencies, with different scattering properties. Fig-
358 ure 3 shows temporal measurements of waves propagat-
359 ing through the network from either direction and at the
360 two frequencies indicated by the vertical lines in Fig. 2.
361 The first frequency $f = 57$ Hz corresponds to the drop of
362 the reflection coefficients. It is representative of the tar-
363 get result, though exact balance was not totally achieved
364 between the opposite directions of propagation. The sec-
365 ond frequency $f = 25$ Hz depicts the typical behavior of
366 the scatterer at other frequencies, where the asymmetric
367 transmission is perturbed by the strong reflection coeffi-
368 cients on both ends of the system.

369 In the case of $f = 57$ Hz (indicated by the dashed
370 line in Fig. 2), the experimental coefficients of reflec-
371 tion are small, respectively $|r^+| \approx 0.04$ and $|r^-| \approx 0.02$.
<figure><img src="image_6.png" /><figcaption>FIG. 3. Measured propagation of a continuous signals of unitary peak amplitude at 57 Hz and 25 Hz with $\Delta T = 82$ K by the four microphones in the setup, excited either from the left or from the right. Blue lines show upstream waves, and red lines downstream waves. The top sequences (a-b) are for 57 Hz that corresponds to the drop of the reflection coefficients, with the transmissions such that $|t^+t^-| \sim 1$, marked by the vertical dashed line in Fig. 2. Both upstream and downstream signal pairs are phased with a delay corresponding to the propagation of a traveling wave (TW) along the distance separating the microphones in either direction. The bottom sequences (c-d) at 25 Hz correspond to the dotted line in Fig. 2, with a strong attenuation in the $(-x)$-direction coupled with near unity reflections in both directions, inducing upstream quasi-standing waves (QSW) as demonstrated by the blue signals in phase.</figcaption></figure>

As a consequence, the measured upstream waves are quasi-traveling, as is demonstrated by the blue curves in Fig. 3(a-b), both having the same amplitude but being delayed by the time of flight between the microphones. Thanks to the anechoic terminations, the transmitted waves (in red, continuous then dashed lines) are purely traveling, and therefore measured by the two downstream microphones with a constant amplitude and a phase difference corresponding to the time of flight separating the measurement points. In this case, the amplitudes of the downstream waves are direct translation of the transmission coefficients: the wave propagating in the $(+x)$-direction is amplified by a factor of 2.5, whereas the wave propagating in the opposite direction is attenuated with a factor of 3.2 : 1.

In the case of $f = 25$ Hz indicated by the vertical dotted line in Fig. 2, the individual coefficients of $\mathbf{S}$ do not fulfill the target requirements of Eq. (9): both reflections

coefficients are indeed close to unity. As a consequence on corresponding temporal measurements (Fig. 3.c-d), the upstream waves (shown in blue) appear as quasi-standing waves, the signals measured at different distances of the network being almost in phase. Without knowledge of the amplitude of the impinging waves, further interpretation of the temporal signals in this case is rendered more difficult.

## V. COHERENT PERFECT ABSORPTION

A second setup of the thermoacoustic network is sought to demonstrate coherent perfect absorption. That is, we search a geometrical configuration of the network and properties of the stack configuration for which, at a given frequency $f_2$ and a temperature difference $\Delta T_2$, the output of the network is null. This translates in one of the eigenvalues $\lambda_{min}$ of its transfer matrix $\mathbf{S}$ going through a minimum, with the lowest value possible. If this condition is met, by exciting the network simultaneously from both ends with continuous signals at the frequency $f_2$ with relative amplitudes and phase difference given by the corresponding eigenvector $q_{min}$, the output waves should be of correspondingly small amplitudes. The target properties for the system may be summarized$^{31}$ as

$$ \mathbf{S}(f_2) = \mathbf{Q} \begin{bmatrix} \lambda_{max} & 0 \\ 0 & \lambda_{min} \sim 0 \end{bmatrix} \mathbf{Q}^{-1}. \quad (10) $$

where $\mathbf{Q}$ is the square matrix whose nth column is the eigenvector $q_n$ of $\mathbf{S}$.

## A. Theoretical results

Such a behavior is predicted to be achievable with the same thermoacoustic network by replacing the ceramic stack with a 14 mm stack with pores of radius $r_{th} \sim 160$ µm, and porosity $\phi = 0.8$. Such dimensions are achieved by stacking 60 fine stainless wire meshes (mesh number 45 and wire diameter 160 µm, ref. 104738, Gantois Industries). The properties of the stack fitted to emulate its experimental characterization are set to a porosity $\phi = 0.8$ and a pore radius $r_{exp} = 170$ µm.

Fig. 4(a) shows the expected moduli of the eigenvalues of the scattering matrix of the 15-cells system for increasing temperature difference on each cell varying from $\Delta T = 0$ K to 75 K with increments of 15 K (blue to yellow). One of the eigenvalues, represented with continuous lines, shows a minimum around 70 Hz for temperature differences up to 75 K. A zoom on these minima (marked by the triangles on Fig. 4(b)) shows a nearly constant small amplitude of the eigenvalue around $3 \cdot 10^{-2}$. This eigenvalue is hereafter designated as $\lambda_{min}$. The other eigenvalue $\lambda_{max}$ is shown with dashed lines. It is worth noting that the acoustic lasing effect expected to take place at the same frequency in a symmetrical system and
<figure><img src="image_7.png" /><figcaption>FIG. 4. (a) Theoretical amplitudes of the eigenvalues $|λ|$ of the scattering matrix $S$ of the 15-cells thermoacoustic network as functions of the frequency and for increasing temperature difference $ΔT = 0$ K to 75 K in 15 K increments (blue to yellow). The eigenvalue showing a drop for a frequency of $f_2 \sim 70$ Hz will be called $λ_{min}$ and is shown by a continuous line. (b) Relative magnitude of the input pressure waves $|p_{in}^-|/|p_{in}^+|$ versus their phase difference $φ = \angle(p_{in}^-/p_{in}^+)$ at the points shown in the zoomed insert. The trajectory followed versus $ΔT$ is traced in gray to ease reading: from $ΔT = 0$ K to 75 K, the amplitude ratio change from 1 : 1 to a 100 : 1 while the phase shifts from in phase to $-60°$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 5. (a) Amplitude ratio and phase of the input pressure waves as functions of the temperature difference $ΔT$ measured on the outer wall of the system, in the range 0 K to 134 K (blue to red). (b) Magnitude of the outcoming pressure waves (normalized such that $|p_{in}^+|^2 + |p_{in}^-|^2 = 1$ Pa$^2$). (c) Resulting measured absorption coefficient $α$ of the thermoacoustic network.</figcaption></figure>

438 translating in $λ_{max}$ peaking is not observed here (oppos-

439 site to what is shown in$^{32,33}$). This absence is due to the

440 asymmetry in construction of the unit cell in addition to

441 being nonreciprocal.

454 ability.

442 For each of the eigenvalues $λ_{min}$, the corresponding

443 eigenvector is evaluated and shown in the polar plot

444 (Fig. 4(c)). It is represented as the relative amplitudes

445 of the incident waves $|p_{in}^-|/|p_{in}^+|$ versus their phase differ-

446 ence $φ = \angle(p_{in}^-/p_{in}^+)$ yielding the minimal output from

447 the network, for each temperature difference. The acous-

448 tic input condition required to reach the coherent perfect

449 absorption shows an important shift of amplitude ratio,

450 which changes from 1 : 1 to a 100 : 1 while the phase

451 shifts from in phase to $-60°$ over the studied temper-

452 ature difference range. The trajectory followed by the

453 eigenvector is shown with a gray line to help with read-

## B. Experimental results

456 The scattering matrix $S$ of the two-port is measured

457 for several conditions of heating. From the measurement

458 of $S$, the eigenvalues and associated eigenvectors are com-

459 puted. Focusing on the frequency of interest (in the range

460 65 Hz < $f_{2,exp}$ < 70 Hz changing slowly with $ΔT$), the

461 relevant eigenvalue reaching a minimum, is selected with

462 its associated eigenvectors.

463 In Figure 5(a), the relative amplitudes and phase dif-

464 ferences of the eigenvectors producing the minima of
$\lambda_{min}$ are reported for increasing temperature differences up to $\Delta T = 134$ K. Note that the attainable temperature difference is higher in this configuration due to the different thermo-physical properties of the stack material (notably, we evaluated the thermal conductivity of the ceramics stack to be an order of magnitude greater than that of the stacked wiremesh). These results show a behavior very similar to the expected one [see Fig. 4(c)], with the amplitude ratio of incident waves varying gradually from 1 : 1 to a 16.2 : 1 while the relative phase shifts from $18^{\circ}$ to $-52^{\circ}$. (The gray line is a visual guide to improve readability.)

Each data point in Fig. 5(a) is then used as an excitation condition for the two-port, which should then act as a coherent perfect absorber, with minimal outgoing waves. Fig. 5(b) reports the amplitudes of the wave exciting the system, normalized such that the amplitude of excitation is unitary $\sum |p_{in}|^2 = 1$.

From the measurement of the amplitudes of incident $p_{in}^{\pm}$ and outgoing $p_{out}^{\pm}$ waves, the absorption coefficient $\alpha$ of the system is expressed:

$$ \alpha = 1 - \frac{|p_{out}^{+}|^2 + |p_{out}^{-}|^2}{|p_{in}^{+}|^2 + |p_{in}^{-}|^2}. \quad (11) $$

The experimental absorption coefficient of the coherent perfect absorber is presented in Fig. 5(c) with square markers, and it is larger than 99.9% for most of the temperature range and close to the expected value of $\alpha_{th} = 1 - \lambda_{min}^2$, shown with round markers.

Fig. 6 presents observations of the behavior of the system in the time domain. observations of the behavior of the system in the time domain. Though no temporal measurements were made, reconstructed data are presented based on the measurements of the scattering matrix of the system and the deduction of its eigenvalues/vectors. On the basis of the highest temperature difference measured previously $\Delta T = 136$, the top figures 6(a-b) present the normalized incident waves exciting the system from either side with relative amplitudes and phase given by the eigenvectors reported in Fig. 5(a). The lower figures 6(c-d) present, with the same respective, the scattered components, i.e. the waves reflected and transmitted by the TAN. The amplitude and phase of this components are obtained from the measured scattering coefficients of the TAN. It appears clear how the specifically designed scatterer has reflection and transmission coefficients allowing, with the proper excitation defined by the eigenvectors of the system, outgoing components on each side to have the same amplitudes and opposite phases. The result of simultaneous excitation in these conditions, by application of the superposition principle shown by the green lines, is the cancellation of the scattered components resulting in quasi-null outgoing wave amplitudes. As is to be expected, the reconstructed amplitudes $p_{out}^{+} \approx 1 \times 10^{-3}$, $p_{out}^{+} \approx 1.5 \times 10^{-2}$ are close to the experimental ones shown on Fig. 5(b) (resp. the yellow diamond at $4 \times 10^{-3}$ and the yellow disk at $2 \times 10^{-2}$).

<figure><img src="image_9.png" /><figcaption>FIG. 6. Reconstructed time domain signals on either sides of the TAN in the case of the highest temperature in 5: $\Delta T = 134$, $f = 65.2$. (a)&(b) Incident waves respectively in the $(x)$ and $(-x)$-directions, with relative amplitudes $|p_{in}^{-}|/|p_{in}^{+}| = 16.2$ and a phase difference $\angle(p_{in}^{-}/p_{in}^{+}) = 50.29$ (yellow triangle in Fig. 5(a)). (c)&(d) Scattered components of the wave, obtained from the scattering matrix of the TAN. Blue lines show the components deriving from the wave impinging in the $(x)$-direction, red lines show the components deriving from the wave impinging in the $(-x)$-direction. The green lines are the outcoming waves, sums of both scattered components on each side resulting from the simultaneous excitation.</figcaption></figure>

## VI. CONCLUSION

Experimental demonstration has been provided of the capability for thermoacoustic networks to create nonreciprocal acoustic systems with a high potential of tunability for various scattering properties. In particular, a system has been characterized which demonstrates asymmetric transmission of 18 dB, without reflections, and with balanced amplification/damping in opposite directions. A second demonstration has highlighted the possibility to make a coherent perfect absorber, that keeps the nonreciprocal properties inherent to thermoacoustic systems. The system has been shown to keep its absorption properties over a large span of temperature differences, allowing its effectiveness for largely disparate conditions of excitation. All of these behaviors can be modeled and predicted by the well-established linear theory of thermoacoustics, requiring only a good geometrical characterization of the unit cell.

Beyond the design choices made in this study, thermoacoustics offers several possibilities to generate systems exhibiting single frequency, as well as broadband
542 behaviors where exotic nonreciprocal scattering can be
543 achieved. We believe this experimental study, as a proof
544 of concept, may open the route to new interests in using
545 thermoacoustics as a mean to generate tunable nonrecip-
546 rocal systems.

547 ACKNOWLEDGMENTS

548 The authors wish to thank Y. Auregan and V. Pagneux
549 for their remarks and suggestions about this work.

550 This work is supported by the French National Agency
551 for Research (SelfiXs Project No. ANR-18-CE92-0001).

552 AUTHOR DECLARATION

553 The authors have no conflicts to disclose.

554 DATA AVAILABILITY STATEMENT

555 The data that support the findings of this study are
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