Generation and propagation of acoustic solitons in a periodic waveguide of air-water metamaterials

In this study, we propose an equivalent circuit of a metamaterial 1D waveguide. The latter is made of a diphasic medium to induce both non-linearity and dispersion. The balance between these two effects makes it possible to obtain soliton waves not studied in the fluid-fluid metamaterial so far. The purpose of the present paper is to confront the numerical Runge Kutta-based solution to the Transmission Line based circuits. The latter is compared to the numerical solution obtained by a Finite element method (FEM) algorithm to validate the numerical solution. The obtained solution is proved to be in good agreement with FEM solution.


Introduction
Acoustic propagation is affected by several phenomena, such as nonlinearity and dispersion of the medium.The former transcribes the dependancy between the wave's magnitude and the resulted alteration to the intrinsic sound speed of the hosting medium, which induces an additional distorsion to a to the waveform profile.Consequently, the initial spectrum is assigned further frequency ranges and peaks within the frequency spectrum.Regarding the dispersion phenomenon, a bunch of frequencies traveling as a wave packet undergo velocity braking such the ones observed around Bragg band gaps ignited by phononic crystals [1].It is worth noting that since non-linearity provoke a sort of local acceleration to the wavefront, a suitable choice of the propagation medium enables one to counterbalance the excessive amount throughout dispersion.The shape of the launched wave is thus preserved in a certain frequency range and the corresponding wave is called Solitary-Wave or Soliton [2,3].
Several works have extensively studied both theoretically and experimentally the existence, the propagation as well as the stability of this type of vibration in different fields of physics, including nonlinear optics [4], condensed matter physics [5], plasma physics [6], and surface waves in water [7].The solitary wave equation in mechanical vibrations has also been studied in crystalline solids [8] and thin magnetic films [9].In mathematics, solitons are considered to be self-reinforcing solitary waves (a wave packet or pulse) that maintain their shape while propagating at a constant speed and amplitude [3].Some models were proposed to describe the propagation of soliton waves, namely the Sine-Klein-Gordon equation and the NLS (Nonlinear Schrödinger) equation [10], but the major analytical model is considered to be the one derived by Korteweg-de Vries (KdV) in 1895, described by a simple partial differential equation for shallow water waves [2].This model have not been extensively used before 1965, then gained further interest when Martin Krustal and Norman Zabusky investigated in detail the KdV equation to discover the natural existance of solitary waves as a solution.
Though many studies have been reported since then in various branches of physics, the propagation of solitons in fluid media remains poorly considered due to the weak intrinsic dispersion phenomenonand the natural nonlinearity exhibited by these media.Some studies aimed to overcome these previous drawbacks by implementing complex structures such as 1D acoustic metamaterials in the form of subwavelength resonators.In these type of propagating materials, the spatial periodicity introduces dispersion while the host material exhibits nonlinearity [11].Here, one can cite the pioneering work of Sugimoto et al [12,13], in which they presented an experimental and theoretical models to study the generation and propagation of low-frequency solitons in a 1D-metamaterial in the form of an air-filled waveguide composed of a periodic arrangement of Helmholtz-type resonators.
Besides, Zhang et al have suggested some configurations of 1D metamaterials, incorporating elastic membranes [14] or side holes [15], to study gap and dark solitons analytically using the transmission line (TL) approach; widely used in acoustics [16][17][18].Recently, Ioannou et al studied the propagation of a soliton-shaped pulse in an air-filled acoustic waveguide of periodically varying cross-section using the TL-approach [18].
On the other hand, adding air bubbles to a liquid is an effective way to not only significantly increase its ultrasonic nonlinearity but also change the dispersion relation into the propagating medium because the dimensions of these bubbles are subwavelength [19].This suggests that the air-water mixture is a promising candidate for exploring acoustic soliton waves [20].
Throughout the last five years, several papers summoned the solitons theory and applications.Deng et al [21] presented combined experimental, numerical and analytical studies to designe nonlinear metamaterials presenting gaps in amplitude for elastic solitons.Besides, a theoretical framework for programming static periodic topological solitons into programmable metamaterials has been also investigated [22].Peronne et al [23] reported relevent results on temporal distribution of the solitons is also analysed with the help of the inverse scattering method.Zhang et al [24] presented a mechanical micro-soliton excitation by optical fields in an optomechanical micro-resonator.El-Labany et al studied the oblique interaction of two ion acoustic solitons in a magnetized relativistic degenerate plasma [25].More recently, many works examined the existance of Ion Acoustic Solitons [26] and in two various temperature plasma [27].Jahangir et al [28] explored the interaction between two acoustic solitons with adiabatically trapped ions and Maxwellian electrons in dusty plasma.Jiao et al [29] have analysed observations of phase transitions in 2D multistable mechanical metamaterials initiated by collisions of soliton-like pulses within the metamaterial.
In this paper, we report on the propagation of soliton waves in a 1D fluid-fluid metamaterial based waveguide constituted by an array of air cylinders immersed in a water matrix.The presence of subwavelength cylinders makes it possible to add dispersive effects to the nonlinear propagating medium.We propose an equivalent model based on TL-approach to observe the existance and propagation of solitons.To validate our model, two further different methods are examined, namely, the integration of the KdV model along with a numerical Finite element method (FEM) based algorithm.

Structure of the waveguide
Let us consider a periodic arrangement of air cylinders in a water-like medium (figure 1).The wavelengths λ of interest are considered to be larger than the periodicity a, so the whole structure can be considered as a metamaterial.
We assume that the propagating medium is nonlinear and dispersive but not dissipative.The governing equation used is the Burgers model (equation ( 1)) [30].Since the filling factor of the cylinders is too weak , we can assume that the nonlinearity of the structure is only due to the water-like medium.Hence, the effective nonlinear coefficient is sensibly equal to the propagating medium.Furthermore, water is a weakly nonlinear (β = 3.5) but almost a non-absorbant medium.Hence, to observe the soliton wave without using a large number of cylinders and thus to ensure the put forward proof of concept, we consider larger nonlinearity coefficient (β = 10).

Theoretical equation
Soliton waves are propagated as a coupling of non-linear and dispersive phenomena within a medium.Let us first recall equation (1) that presents the nonlinear propagation of a wave in a non-dissipative and non-dispersive medium [31].
where p is being the pressure, c 0 the small signal sound speed and β is the nonlinear coefficient.Since our structure is periodic, all these characteristics are assumed to be x-position dependent.To take into account the dispersion, we must write the effective celerity as a function of frequency.The dispersion relation is obtained by the Rytov formula (equation ( 2)) using the transfer matrix method [32]: where, k 0 = ω c0 is the wavenumber, Z is the acoustic impedance of the waveguide and Z c is the acoustic impedance of the air cylinder.The Taylor series development of dispersion relation ω (k) around k = 0, and using equation (2) yields [33]: where α = a 2 24 c 0 .Equation ( 3) is defined as the low k-wavenumber dispersion relation.Substituting equation (3) into the Burgers equation, we obtain the KdV model (equation ( 4)) as follows [33]: In equation ( 4), the third term accounts for the nonlinear effects of the medium (due to the p ∂p ∂x expression).Moreover, the nonlinear coefficient β describes how nonlinear is the propagating structure.Besides, the last term discribes the dispersion phenomenon.Equation ( 4) provides exact solitary solutions.However, the nonlinear character makes analytical solutions not obvious to obtain.It has been proved that the solution of equation ( 4) can be approximated by [33]: where, γ = √ βP 12α , P and W are the soliton's width, amplitude, and velocity, respectively:

Electroacoustic analogy
In this section, we present a TL based equivalent circuit to simulate the propagation of the soliton wave.This approach is widely used in linear acoustics [34,35] and nonlinear ultrasounds [36,37].
Let us consider the unit cell of our structure.According to the electroacoustic analogy, the propagation along a waveguide is modeled by an electric circuit in which the electric voltage U and current I are, respectively, equivalent to the acoustic pressure p and the acoustic flux i.e. the volume velocity u across the waveguide's cross-section [17,38].Figure (1) represents the unit cell and its equivalent circuit [39,40].
Indeed, according to [38], the equivalent circuit of a single unit cell has two sub-circuits (figure 1).The left part represents a serial LC circuit, modeling the water part of the cell, consisting of a capacitor C ω and an inductance L ω , their expressions can be written as follows: where s is the cross-sectional area of the waveguide.The thermal and viscous effects are assumed to be negligible, which justifies the internal resistance of this part of the circuit being zero.The right part of the circuit is equivalent to an air cylinder, and takes the form of a parallel LC circuit [35,39] consisting of a capacitor C c and an inductance L c .Their expressions can be written as follows: where V c is the cylinder volume and ρ is the air density.Obtaining a derivative equation starts with the classical Kirchhoff law, which checks the voltage and electric current through two successive cells.If we start with the acoustic pressure p n (acoustic voltage) for the n th cell of the network, Kirchhoff 's law would be written as follows: where p n−1 and p n+1 are the input pressure of the nth cell and the output pressure of the (n + 1)th cell.
Combining the equations ( 7) and ( 8) leads to the following differential equation: On the other hand, the Kirchhoff laws applied to the electric current give: Replacing in (9) one can obtain: ( Equation ( 11) describes the propagation into a linear but dispersive medium.In order to take account of nonlinearity, the sound speed propagation should be written as [40,41]: Taking into account expression (12) in the definition of C ω and L ω leads to the following nonlinear wave equation: With δ = Vc V is the filling factor and ω 0 = √ L c C c the characteristic pulsation of one single air cylinder.Considering the same initial conditions of pressure and frequency, and resolving equation ( 13) using 4 order Runge-Kutta algorithm, we obtain (figure 2) a good agreement of the temporal profile of the soliton with the FEM method that will be explained and used in the next section.Nevertheless, some oscillations are noted after the Gaussian shape and are due to the spatial discretization used in the Runge-Kutta algorithm.
In the next section we present a numerical solution based on the Finite Elements Model.

Numerical model
In this section we show the results of the numerical integration of the KdV equation (equation ( 4)).We performed a simulation in time domain of a Gaussian-shaped plan propagation through the metamaterial-based waveguide depicted in figure 1.To avoid spurious reflections on the boundaries of the structure, we use perfectly matched layers To validate our FEM model, we compare the obtained numerical results to the exact solution presented by equation (5) immediately after the soliton exits the structure (figure 3).A good correspondance is to be noted between the two methods, except theweak oscillations observed forthe numerical solution.These artifacts are due to reflexions from cylinders, not considered for inexact solution case, since we have considered a homogeneous medium with effective values of characteristics.
Furthermore, we present in figure 4, the spatial propagation of the numerical solution at different moments.As we noticed, the wave shape remains almost unchanged during propagation, which allows us to conclude the good balance obtained between nonlinearity and dispersion.
In order to quantify the amount of energy exchanged between nonlinear and dispersion effects, we performed three simulations using our FEM algorithm.Figure 5 depicts the three curves using linear and nondispersive (LN), nonlinear and nondispersive (NN) and nonlinear but dispersive (ND) media with respect to time and position.It seems clear that NN curves are being shifted to the time's origin (t = 0) and   far from space's origin (x = 0) when the magnitude is growing.This is explained by the fact that speed increases with signal magnitude (equation ( 1)).However, it is worthy to notice that this shifting is 'corrected' by the presence of dispersion (ND curve) and then the curve regains a quasi-linear shape (LN curve).
Let us now experiment the collision of solitons.It is well known [42] that solitons' collision occures without profiling change.In fact, if two solitons propagate in opposite directions, each propagates freely and independently of the other, i.e. they do not interfere.
Figure 6 presents a soliton collision scenario.The upper curve shows two solitary waves propagating in opposite directions.The middle curve depicts the instant of their collision, one can see the increase in the amplitude, which proves that they are fully coupled.Yet, they separate without interfering, and each remains at its initial shape (lower curve).
In this section, we have validated our FEM algorithm as we use it to highlight different characteristics of obtained solitons in a fluid-fluid metamaterial.

Conclusion
In this paper, we have proposed a TL based equivalent circuit for studying the obtention and propagation of soliton waves in a fluid-fluid acoustic metamaterial.First, we proposed the equivalent circuit and then we derived the governing equation.The Runge-Kutta algorithm was used to obtain the wave temporal evolution.We presented the KdV equation and its approximate solution.The latter was compared to the numerical solution obtained by a FEM algorithm to validate the numerical solution.The obtained solution is proved to be in good agreement with FEM solution.

Figure 1 .
Figure 1.The theoretical model represents a 1D metamaterial based waveguide.It consists of a row of air cylinders of periodicity a and radius r in a water matrix (up), and the equivalent circuit of the unit cell of the structure (down).

Figure 2 .
Figure 2. Time evolution of a soliton pulse in the acoustic 1D waveguide obtained using the numerical method (FEM) (red curve) and by Transmission Line (TL) analogy in the acoustic waveguide, at the output of the structure (blue curve).

Figure 3 .
Figure 3.The FEM and exact solutions obtained after the last cylinder, in black the numerical solution, in red the KdV based exact solution.

Figure
FigureThe spatial propagation of the soliton wave computed using finite element method at different instants.

Figure 5 .
Figure 5. Numerical solution respectively to time (up) and position (down) of the solitary wave in three combinations of medium effects namely nonlinear and dispersive (black curves), nonlinear and nondispersive (red curve) and linear and nondispersive (blue).

Figure 6 .
Figure 6.Two solitons propagates in opposite direction, after their collision, no interference is noted and each one continue its propagation with no change.