# Acoustic Switch device for a Splitter Acoustic wave made by Waveguides and Resonator

Ilham El-Atmani¹, Ilyass El Kadmiri [1-2], Driss Bria¹, Aissam Khaled¹, Mounsif Ech Cherif El Kettani³ and Pierre Maréchal³

ilham.el-atmani@ump.ac.ma

¹ Laboratory of Materials, Waves, Energy and Environment, Team of Waves, Acoustic, Photonic and Materials, Faculty of Sciences, Mohamed First University, Oujda, Morocco

² Laboratory of Computer Science and Interdisciplinary Physics, ENSF, Sidi Mohamed Ben Abdellah University, Fez, Morocco

³ Laboratory of Waves and Complex Media, Le Havre Normandy University, France

**Abstract.** In this paper, we present the propagation of acoustic waves in switchable device within a specific frequency range. Our proposed system comprises an input and three output lines (three channels), initially acting as an amplitude divider which splits an incident acoustic wave to the three output channels. We demonstrate theoretically that the introduction of a resonator of length $d_2$ along the waveguide of length $d_c$, connecting the output channels to the input channel of our system, efficiently modulates the acoustic signal. Consequently, the input signal is entirely transmitted at the first output line $T_1 = 1$ while transmissions through the second and third channels, as well as reflection, are nullified $T_2 = T_3 = R = 0$. The theoretical analysis is based on the Transfer Matrix Method (TMM), enabling us to calculate the three transmission rates $T_1, T_2, T_3$ and the reflection rate $R$. The length and position of the resonator control the acoustic wave, achieving an acoustic switching phenomenon with important efficiency.

**Keywords:** Acoustic switch, Waveguide, Resonator, Transfer Matrix Method.

## 1 Introduction

Acoustic switches are crucial in modern signal processing and communication systems, attracting significant attention due to their potential applications in various fields such as telecommunications, underwater communication, and medical imaging [1-7]. These switches offer the ability to manipulate acoustic signals, redirecting them along desired channels or modulating their amplitudes with high precision and efficiency. In recent years, the intensity of research has increased to explore novel designs and mechanisms for acoustic switches, with the intention of improving their functionality, adaptability, and integration into useful systems. An acoustic switch, introduced by Pai Wang et al. [8], that is part of new type of switchable acoustic metamaterials, they scattered resonating units within an elastic matrix. They utilized an elastic beam to control the propagation of elastic waves effectively. Through numerical analysis, they demonstrated the evolution of locally resonant band gaps and how deformation in their system could
activate or deactivate these band gaps. This approach opens the way for the develop-
ment of adaptive switches. Among various studies, Sahab et al. [9] proposed a novel
category of architectural materials aimed to control sound transmission. Their system
comprises an array of elastomeric helices showing a considerable attenuation of waves
(band gaps) within specific frequency ranges. However, these band gaps are suppressed
when the helices confront axial enlargement, thereby enabling the development of a
new type of acoustic switching process. Furthermore, Alagoz et al. [10] investigated a
wave-controlled sonic crystal switch device, which demonstrates a reverse switching
effect based on destructive interferences between waves. This acoustic device, com-
prising two-dimensional square lattice sonic crystal, reduces the transmission of acous-
tic waves from input to output by utilizing control waves. Moreover, Vanhille et al.
[11], introduced a new device utilizing the linear effects of bubbly liquids, capable of
functioning as an acoustic switch. This device is engineered to manipulate acoustic
waves using other acoustic waves. They demonstrated of how acoustic waves can
switch other acoustic waves; The system operates based on the attenuation caused by
the bubbles, which is influenced by the resonance relationship between the bubbles and
the pulsation of the traveling acoustic waves. In addition, Liang et al. [12] designed an
acoustic transistor to manipulate sound in a manner similar to the manipulation of elec-
tric current. The acoustic transistor functioning as a three-terminal device possesses the
crucial ability of utilizing a minor monochromatic acoustic signal to control a much
larger output signal within a large frequency range.

The objective of our recent work is to study an acoustic switch with several output
channels. The proposed acoustic system consists of waveguides and resonator with one
input line and three output transmission lines. We organize our paper as follows: Firstly,
we use the transfer matrix method to calculate the transmission and reflection rates.
Then, in the Results and Discussions section, we investigate the impact of the wave-
guide length d_c, which has no effect on switching phenomenon. Therefore, we introduce
a resonator on the waveguide length d_c, and by adjusting its length d_2 as well as its
position, we demonstrate the achieving of the acoustic switching phenomenon. Finally,
we conclude by summarizing our results and discussing their implications.

## 2 Acoustic switch design

In this paper, we present an acoustic device designed to operate as an acoustic switch.
The device comprises a resonant system with a length denoted as "d_c" which is con-
nected to four semi-infinite waveguides (Fig. 1). Constructed with waveguides of
lengths d₁ and d₃ and grafted by one resonator, the length d₂ needs to be determined.
Such a system with an appropriate resonator length d₂ shows a perfect switching mech-
anism; the input signal is completely redirected to the first output line. The character-
istic acoustic impedance is denoted as Z = ρc/S, where ρ = 1.2 kg/m³ is the density of
air and c = 340 m/s is the sound speed in air at T = 20°C, S is the cross-section of each
guide and resonator. The transmission and reflection rates are calculated using the
transfer matrix method (TMM).
<figure><img src="image_1.png" /><figcaption>Fig. 1. Schematic illustration of an acoustic switch.</figcaption></figure>

# 3 Theoretical formalism

## 3.1 Transfer Matrix Method (TMM)

We assume that the entrance of the device illustrated in Fig. 1 receives an incident plane acoustic wave $P_{inc}(x) = A_{inc}e^{ikx} + B_{inc}e^{-ikx}$ from the left side, where $A_{inc}$ and $B_{inc}$ are the amplitudes of the incident field, $k = \omega/c$ is the wavenumber in air. The constraints pressure $P(x, t)$ and vibration velocity $V(x, t)$ are related by the relation $V_i = -\nabla P_i/(i\omega\rho_i)$. The elements $M_{mn}$ of the matrix are produced by applying Hooke's laws and the continuity conditions to the formulations of pressure and velocity for an acoustic wave traveling through the acoustic switch. Therefore, for the proposed system, the TMM allows to relate $A_{0,inc}$ and $B_{0,inc}$, the amplitudes of the incoming waves and the amplitudes of the outgoing waves, respectively:

$$ \begin{pmatrix} A_{0,inc} \\ B_{0,inc} \end{pmatrix} = \begin{pmatrix} M_{11} & M_{12} & M_{13} & M_{14} & M_{15} & M_{16} \\ M_{21} & M_{22} & M_{23} & M_{24} & M_{25} & M_{26} \end{pmatrix} \begin{pmatrix} A_1 \\ B_1 \\ A_2 \\ B_2 \\ A_3 \\ B_3 \end{pmatrix} \quad (1) $$

where $A_i$ ($i = \{0, 1, 2, 3\}$) are the amplitudes of the progressive acoustic waves in the four channels, and $B_i$ ($i = \{0, 1, 2, 3\}$) are the amplitudes of the regressive acoustic waves in the four channels.
## 3.2 Transmission and Reflection rates

The matrix elements offer a means to calculate transmission and reflection rates. These coefficients offer numerical evaluations of wave behavior in distinct regions or components of the system [13-17]. They give useful information on the wave transmission or reflection process, revealing details about the general functioning and distinctive features of the system under study. The transmission and reflection coefficients can be calculated by the following relations:

$$t_1 = \frac{A_1}{A_0} \begin{matrix} B_1=0 \\ B_2=0 \\ B_3=0 \end{matrix}, \quad t_2 = \frac{A_2}{A_0} \begin{matrix} B_1=0 \\ B_2=0 \\ B_3=0 \end{matrix}, \quad t_3 = \frac{A_3}{A_0} \begin{matrix} B_1=0 \\ B_2=0 \\ B_3=0 \end{matrix}, \quad r = \frac{B_0}{A_0} \begin{matrix} B_1=0 \\ B_2=0 \\ B_3=0 \end{matrix} \quad (2)$$

The transmission and reflection rates associated with the output lines can be written as:

$$T_1 = |t_1|^2, \quad T_2 = |t_2|^2, \quad T_3 = |t_3|^2, \quad R = |r|^2, \quad (3)$$

In the studied case of an unattenuated device, the energy conservation gives:

$$R + T_1 + T_2 + T_3 = 1 \quad (4)$$

# 4 Results and discussion

## 4.1 Acoustic splitter waves without resonator

In this paragraph, we examine an acoustic wave splitter that consists of three output channels connected by a waveguide of length $d_c$. In Fig. 2, we show the transmission curve $T_1$ (red color) through the output 1, $T_2 = T_3$ through outputs 2 and 3 (green color) and reflection R (black color) as a function of the reduced frequency. It is noted that even when the length $d_c$ is varied, the transmission and reflection rates don't approach unity. This discovery emphasizes the absence of the acoustic switch phenomenon, in which modifications to the waveguide length $d_c$ don't effectively direct the acoustic wave towards the intended channel 1. The insertion of a resonator is taken into consideration as a solution to this limitation, as it is essential to modify the characteristics of the acoustic waves to produce the switching process.

<figure><img src="image_2.png" /><figcaption>Fig. 2. Variation of the transmission and reflection rates as function of reduced frequency for different value of $d_c$.</figcaption></figure>
## 4.2 Acoustic splitter waves with resonator: effect of the resonator position

In the second part, a resonator is grafted at a distance $d_1$ along the length $d_c$ of the waveguide, where $d_c = d_1 + d_3$, with $d_1$ and $d_3$ representing the lengths of the acoustic waveguide on each side. As illustrated in Fig.3, the length of the resonator is fixed at $d_2 = 1$D, while the transmission curve $T_1$ (red color) through output 1, and the transmission $T_2 = T_3$ through outputs 2 and 3 (green color), are investigated by varying the length of the waveguide $d_1 = \{0.25D; 0.5D; 0.75D\}$. The addition of the resonator enhances the three transmission rates but it stills insufficient to attain the best acoustic switching ratio. In order to achieve this goal ($T_1 = 1$; $T_2 = T_3 = R = 0$), the length of the resonator $d_2$ must be precisely adjusted. Consequently, the study focuses towards exploring the effects of adjusting the resonator length $d_2$ to achieve a complete acoustic switching process.

<figure><img src="image_3.png" /><figcaption>Fig. 3. Variation of the transmission and reflection rates as a function of reduced frequency with $d_2 = 1$D, for different values of guide length $d_1 = \{0.25D; 0.5D; 0.75D\}$.</figcaption></figure>

## 4.3 Acoustic switch waves: effect of the resonator length $d_2$

In this part, we show in Figs 4.a and 4.c the transmission rates as a function of reduced frequency, while Figs 4.b and 4.d illustrate the maximum transmission as a function of the variation of the resonator length $d_2$. We observe a significant improvement in achieving an efficient acoustic switch for the reduced frequency interval $\Omega = [2.2; 4.0]$. The lengths of the acoustic waveguides $d_1$ and $d_3$ are set to 0.5D, resulting in a complete transmission of the acoustic wave towards the first output line (red color). Different factors contribute to modulate this acoustic wave switching process. Firstly, the resonator length $d_2$ is optimized to resonate at the system's specific frequency range, thereby facilitating selective energy transfer towards the desired channel (the first channel). Additionally, with $d_1$ and $d_3$ set to 0.5D, the acoustic waves are in a favorable phase, promoting constructive interferences that reinforce the wave's direction towards the first output line. By precisely adjusting these parameters, the correlation between the waveguide lengths $d_1$ and $d_3$ and the resonator length $d_2$ enables maximal transmission $T_1 = 1$ towards channel 1 (red color), resulting in a perfect acoustic switch (Figure 4.d). This observation shows the critical importance of parameters precision in optimizing system performance and opens the way for practical applications in the field of acoustic switching devices.
<figure><img src="image_4.png" /><figcaption>Fig. 4. Variation of the reduced frequency as a function of the transmission rates T₁ (red color) T₂, T₃ (blue color) through the output channels for: (a) d₂ = 1D and (c) d₂ = 0.5D. (b) and (d) Variation of the reduced frequency as a function of the resonator length d₂.</figcaption></figure>

4.4 Acoustic Switching wave rate in first Output channel: Effect of Resonator position d₁

In Fig.5, we present the maximum transmission T₁ that provides information about the
system's switch at various frequencies. To get the best switch, the system parameters
are adjusted using the analysis of these curves. The desired transmission thresholds
(T₁ > 0.9, T₁ > 0.99 and T₁ > 0.999) are used to define the switch rate in the cases that
are specified. Case 1 presents a transmission T₁ exceeding 0.9 on first output 1 (Fig.
5.a), while case 2 shows a transmission surpassing 0.99 on the same channel (Fig. 5.b).
Furthermore, case 3 illustrates a transmission exceeding 0.999 on first channel (Fig.
5.c). These curves illustrate the maximum transmissions for these rates concerning re-
duced frequency as function of the variation of the length guide d₁ (resonator position),
with the resonator length fixed at d₂ = 0.5D. We demonstrate that the switch rate serves
as a performance indicator, reflecting the system's ability to obtain specified transmis-
sion thresholds across channels.
<figure><img src="image_5.png" /><figcaption>Fig. 5. Schematic Illustration of acoustic switching efficiency.</figcaption></figure>

**5 Conclusion**

The aim of our paper was to introduce a model of an acoustic switch, a device designed
to manipulate acoustic waves. We presented the initial demonstration of our system to
modify and redirect acoustic waves effectively. Our results demonstrated the perfect
behavior of the acoustic switch which allows us to recover 100% of the input signal at
the first output channel. The resonator's introduction played a crucial role in influencing
acoustic wave propagation, which improved system efficiency and revealed the phe-
nomena known as the acoustic switch. Consequently, we established that the device
proposed functions effectively as an acoustic switch. Furthermore, its adaptability to
treat a wider range of frequencies by adjusting resonator characteristics and guide
lengths enhances its utility. This acoustic switch is interesting and may be useful for
applications which may require the control of acoustic waves by switching their prop-
agation, such as medicine and noise treatment.

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