ACTA ACUSTICA UNITED WITH Vol. 104 (2018) 235 – 242
ACUSTICA DOI 10.3813/AAA.919165
Second-Harmonic Generation in Acoustic Waveguides Loaded with an Array of Side Holes Jiangyi Zhang1) , Vicente Romero-García1) , Georgios Theocharis1) , Olivier Richoux1) , Vassos Achilleos1) , Dimitrios J. Frantzeskakis2)
1)
2)
Laboratoire d’Acoustique de l’Université du Maine – CNRS UMR 6613, Le Mans, France. [email protected] Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
Summary We study analytically and numerically second-harmonic generation in a one-dimensional weakly lossy nonlinear acoustic metamaterial composed of an air-filled waveguide periodically loaded by side holes. Based on the transmission line approach, we derive a lossy nonlinear dispersive lattice model which, in the continuum limit, leads to a nonlinear, dispersive and dissipative wave equation. The latter is studied by means of a perturbation method, which leads to analytical expressions for the first and second harmonics, in very good agreement with numerical results. PACS no. 43.25.-x, 43.25.Jh
Author's complimentary copy
1. Introduction The study of metamaterials in different fields of wave physics has seen an explosion of interest during the last years, leading to significant developments both from basic research and applications point of view. Metamaterials are artificially engineered structures, exhibiting physical properties not found in nature [1, 2, 3]. In the case of acoustics, Liu et al. [4] proposed the first acoustic metamaterial, originally called locally resonant sonic material, based on an array of coated spheres presenting negative mass density. Later, Fang et al. [5] investigated an acoustic metamaterial composed by an array of Helmholtz resonators embedded in a waveguide, presenting negative bulk modulus. Although such systems have been previously analyzed in the context of acoustics by Sugimoto [6] and Bradley [7], the novelty in this case is to characterize the system by using the effective properties. The linear properties of such acoustic metamaterials including the effect of viscothermal losses [8] have been exploited over the last decade, showing several possibilities to control wave propagation in acoustics [9, 10, 11, 12, 13, 14, 15, 16]. In addition to the more standard case of linear metamaterials, the study of nonlinear ones has been receiving increased attention during the last years [17, 18, 19, 20]. However, in the case of acoustic metamaterials, the presence of nonlinearity is less studied, and only a few works have exploited the combined role nonlinear effects and
Received 25 August 2017, accepted 27 November 2017.
© S. Hirzel Verlag · EAA
other fundamental features of the system, such as dispersion or losses. The combination of nonlinearity and dispersion has revealed different effects, such as the formation of solitons [21, 22, 23], the self-demodulation effect [24, 25] and the generation of higher harmonics [26, 27, 28]. More recently, in a similar setup with negative mass density, the formation of envelope (bright and gap) solitons has been analytically and numerically studied in the presence of viscothermal losses [29]. In this work, we analytically and numerically study the second-harmonic generation in an one-dimensional (1D) weakly lossy nonlinear acoustic metamaterial with negative bulk modulus composed by an air-filled waveguide periodically loaded by side holes. The main motivation is to study the combined effects of dispersion, nonlinearity and dissipation in an acoustic metamaterial with negative bulk modulus (our setup). The model used in this work could pave the way to study the linear and nonlinear propagation in double negative metamaterials considering the different physical effects playing role as dispersion, nonlinearity and dissipation. The nonlinearity is activated here by using high-amplitude incident waves. Based on the electroacoustic analogy and the transmission line approach, we derive a weakly lossy lattice model describing the system, which, in the continuum limit, leads to a nonlinear, dispersive and dissipative wave equation. By using a perturbative scheme we derive analytical expressions for the first and second harmonics, in the presence of losses. The analytical results are found to be in very good agreement with direct numerical simulations in the framework of the lattice model.
235
ACTA ACUSTICA UNITED WITH Vol. 104 (2018)
ACUSTICA
Zhang et al.: Second-harmonic generation in waveguides
The article is structured as follows. In Section 2 we introduce the electro-acoustic analogue modeling based on the transmission line (TL) approach, which has been shown to be a powerful tool for studying electromagnetic and acoustic metamaterials [29, 30, 31, 32, 33, 34]. In Section 2.1, we describe the setup and introduce the 1D nonlinear dissipative lattice model. Then, in Section 2.2, we obtain a nonlinear dispersive and dissipative wave equation, stemming from the continuum limit of the lattice model. Section 2.3 describes the linear properties of the proposed metamaterial. In Section 3, we present the analytical and numerical results regarding the second harmonic generation. Finally, in Section 4, we summarize and present our conclusions.
2. Electro-Acoustic Analogue Modeling
Author's complimentary copy
2.1. Setup and model We consider low-frequency nonlinear wave propagation in the acoustic metamaterial shown in Figure 1a. The structure is composed by a waveguide of radius r periodically loaded with an array of side holes of radius rH and length lH . The distance between two consecutive side holes is d. The frequency range considered is well below the first cut-off frequency, therefore the problem is considered as one-dimensional. We adopt the electro-acoustic analogy, where the voltage corresponds to the acoustic pressure and the current to the volume velocity flowing through the waveguide’s cross-sectional area [35, 36]. Our aim is to derive a nonlinear discrete wave equation, describing wave propagation in an equivalent electrical transmission line (TL). To do this, we consider the unit-cell circuit of the equivalent TL model of our setting, shown in Figure 1b. The unit-cell circuit is composed by two parts. The first one, corresponding to the waveguide, is modeled by the acoustic masses Mω , the resistance Rω and shunt acoustic compliances Cω . In the linear regime, the acoustic masses and acoustic compliances are given by Mω0 = ρ0 d/S and Cω0 = Sd/(ρ0 c02 ), where ρ0 , c0 and S are respectively the density, the speed of sound and the cross-section area of the waveguide. The resistance Rω = Im(kZc )d describes the viscothermal losses, where the wavenumber and acoustic characteristic impedance of the waveguide are given by [8]: 0005 ω 1−j k= 1+ (1) (1 + (γ − 1)/ P r) , c0 s ρ0 c0 Zc = S
0005
1−j 1+ (1 − (γ − 1)/ P r) . s
(2)
Here, γ is the specific heat ratio, P r is the Prandtl number, and s = ωρ0 r2 /η, with η being the shear viscosity. The second part of the unit-cell circuit, corresponding to the side hole, is modeled by a shunt RM circuit composed by the series combination of an acoustic masses MH
236
2rH
lH
r
(a)
d un
(b)
M L !0001
pn-1
un +1 uH
R0001 C0001
L MHH
RH d
waveguide
pn
side holes
Figure 1. (Colour online) (a) Acoustic metamaterials composed by a waveguide loaded with an array of side holes. (b) Corresponding unit-cell circuit.
and a resistance RH . In the regime where the geometric characteristics of the side holes are much smaller than the wavelength λ of the sound wave, i.e., for krH 1, the corresponding acoustic masses and resistance are given by MH = ρ0 lH /SH and RH = ρ0 ω2 /(2πc0 ), where SH is the area of the side holes [37]. Here we should mention that we approximate the frequency-dependent viscothermal and radiation losses by a resistance with a constant value at the frequency of the wave excitation (cf. see last two paragraphs in Section 2.3 for more details). Here, we should mention that we consider the response of side holes to be linear while the propagation in the waveguide weakly nonlinear. At high acoustic level, generally the response of the side holes is nonlinear due to nonlinear losses as a consequence of a jet formation at the locations of the side holes and the formation of annular vortices dissipating part of the acoustic energy. However, these effects can be minimized considering holes with smoothed walls [38, 39]. Therefore the assumption of the linear behavior of the holes is a good approximation as far as the boundaries of the holes are smoothed, an aspect that has to be taken into account in the design of the experimental setup. On the other hand, it is well known that, due to the compressibility of air, the wave celerity cNL is considered amplitude dependent and thus the propagation in the waveguide weakly nonlinear. Therefore, we assume the acoustic compliances Ca to be nonlinear, depending on the pressure p [23], while the acoustic masses Ma linear. 0007 Approximating the celerity as 0004 cNL ≈ c0 1 + β0 p/ρ0 c02 , where β0 is the nonlinear parameter for the case of air, the pressure-dependent acoustic compliances Cω can be expressed as Cω = Cω0 − Cω pn , where 2β0 Cω = Cω0 . (3) ρ0 c02 Next, we use Kirchhoff’s voltage and current laws to derive an evolution equation for the pressure pn in the n-th cell of the lattice. In particular, Kirchhoff’s voltage law for two successive cells yields d pn−1 − pn = Mω un + Rω un , (4) dt
ACTA ACUSTICA UNITED WITH ACUSTICA
Zhang et al.: Second-harmonic generation in waveguides
pn − pn+1 = Mω
d un+1 + Rω un+1 . dt
(5)
Subtracting the two equations above, we obtain the differential-difference equation (DDE), δˆ2 pn =
0005 Mω
d + Rω dt
(un − un+1 ) ,
(6)
where δˆ2 pn ≡ pn+1 − 2pn + pn−1 . Then, Kirchhoff’s current law yields un − un+1 = Cω
d pn + uH , dt
(7)
where uH is the current through MH RH branch. The auxiliary Kirchoff’s voltage law in the output loop of the unitcell circuit reads uH = Qˆ −1 pn ,
d Qˆ = MH + RH . dt
(8)
Then, substituting Equation (7) and Equation (8) into Equation (6), and recalling that the acoustic compliances Cω depends on the pressure, we obtain the following evolution equation for the pressure, 0005 MH 0005
d + RH dt
2
d pn dpn δˆ2 pn − Mω Cω0 2 − Rω Cω0 dt dt
(9)
d2 p2n 1 dp2n 1 + Mω Cω 2 + Rω Cω 2 2 dt dt 0005 d − Mω + Rω pn = 0. dt
Our results are obtained for a temperature of 18◦ C and an air-filled waveguide. We use the following parameter values: d = 0.05 m, r = 0.025 m, rH = 0.0025 m and lH = 0.005 m, β0 = 1.2, c0 = 343.26 m/s, ρ0 = 1.29 kg/m3 , γ = 1.4, P r = 0.71, η = 1.84 10−5 kg/m/s.
Author's complimentary copy
2.2. Continuum limit For our analytical considerations, we focus on the continuum limit of Equation (9), corresponding to n → ∞ and d → 0 (but with nd being finite); in such a case, the pressure becomes pn (t) → p(x, t), where x = nd is a continuous variable, and ∂p d2 ∂2 p + ∂x 2 ∂x2 d3 ∂ 3 p d4 ∂ 4 p ± + + O(d5 ). 3! ∂x3 4! ∂x4
pn±1 = p ± d
(10)
Thus, the difference operator δˆ2 is approximated by δˆ2 pn ≈ d2 pxx +d4 pxxxx /12, where subscripts denote partial derivatives, and terms of the order O(d5 ) and higher are ne-
Vol. 104 (2018)
glected. This way, Equation (9) turns to the following partial differential equation (PDE), 0005 0005 ∂ ∂ 2 p d4 ∂ 4 p ∂2 p d2 2 + MH + RH − M C ω ω0 ∂t 12 ∂x4 ∂x ∂t2 ∂2 p2 1 ∂p2 ∂p 1 (11) − Rω Cω0 + Mω Cω 2 + Rω Cω ∂t 2 2 ∂t ∂t 0005 ∂ − Mω + Rω p = 0. ∂t It is convenient to express our model in a dimensionless form using the normalized variables τ, χ, and p, which are defined as follows: τ is time in units of ωB−1 , where ωB = πc0 /d is the Bragg frequency; χ is space in units of c0 /ωB 1 is a and p/P0 = P , where P0 = ρ0 c02 and 0 < formal small parameter. Then, Equation (11) is expressed in the following dimensionless form, 0001 (∂τ + γH ) Pττ − Pχχ − ζPχχχχ + γω Pτ (12) 0002 2 2 2 2 − β0 (P )ττ − β0 γω (P )τ + m Pτ + m γω P = 0, where SH d , π 2 lH S RH , γH = ωB MH m2 =
π2 , 12 Rω S γω = . πρ0 c0 ζ=
(13)
It is interesting to identify various limiting cases of Equation (12). First, in the linear limit (β0 = 0, or p2 1), in the absence of side holes (m2 → 0, γH → 0), without considering viscothermal losses (γω → 0) and higher order spatial derivatives, Equation (12) is reduced to the linear wave equation, Pττ − Pχχ = 0. In the linear limit, but in the presence of side holes, in the long wavelength approximation and without considering viscothermal losses (γω → 0), radiation losses (γH → 0), and higher order spatial derivatives (ζ → 0), Equation (12) takes the form of the linear Klein-Gordon equation [40, 41], Pττ −Pχχ +m2 P = 0. Finally, including only the nonlinearity and discarding losses, the side holes, and higher order spatial derivatives, Equation (12) is reduced 0004to the 0007 wellknown Westervelt equation, Pττ − Pχχ − β0 P 2 ττ = 0, which is a common nonlinear model describing 1D acoustic wave propagation [42]. 2.3. Linear limit We first study the linear limit of Equation (12) and the respective dispersion relation. Assuming the propagation of plane waves, of the form p ∝ exp[i(kχ − ωτ)], we obtain the following dispersion relation connecting the wavenumber k and frequency ω, D(ω, k) = −iω(−ω2 + k 2 − ζk 4 + m2 ) − iωγH γω + γH (−ω2 + k 2
(14)
− ζk 4 ) + γω (−ω2 + m2 ) = 0, where the terms including γH and γω accounts for the radiation and viscothermal losses. The term ζk 4 accounts for
237
ACUSTICA
The real and imaginary parts of the dispersion relation (including losses), Equation (15), are shown in Figure 2 (thin blue solid lines), which are almost the same as the lossless dispersion relation [Equation (15) with γω → 0 and γH → 0] shown as thick blue solid lines in Figure 2, since we consider a weakly lossy medium. In addition, in order to verify our theory, we compare this linear dispersion relation with the one obtained by using the Transfer Matrix Method (TMM) [7]. The thin red dashed lines in Figure 2 show the results for the lossy dispersion relation obtained by using the expression given by the TMM method,
Author's complimentary copy
0004 0007 0004 0007 0004 0007 Zc sin kd , cos kph d = cos kd + i 2ZH
(16)
where k and Zc are given by Equations (1) and (2) respectively, and ZH = iωph MH + RH is the input impedance of the side holes. For the lossless case, k, Zc and ZH are respectively reduced to k = ω/c0 , Zc = ρ0 c0 /S and ZH = iωph MH , and the corresponding results are shown as thick red dashed lines in Figure 2. The dispersion relation resulting from the continuum approximation (TLM) has a very good agreement with the one obtained by using the TMM, as shown in Figure 2, especially in the lowfrequency regime. We note here that while the TLM dispersion relation the losses are approximated by using constant parameters, in the TMM method the loses are frequency dependent. Therefore, the agreement between the two dispersion relations validates our assumption of frequency independent losses.
238
1000
1000
800
800
Frequency (Hz)
the influence of the periodicity of the lattice (originating from the term δˆ2 pn ) to the dispersion relation. Although this term appears to lead to instabilities for large values of k, both Equation (12) and Equation (14) are used in our analysis only in the long-wavelength limit, where k is sufficiently small. Without considering the losses and higher-order spatial derivatives, Equation (14) is reduced to −ω2 + k 2 + m2 = 0, the familiar dispersion relation of the linear Klein-Gordon model [40, 41]. For low frequencies, i.e., for 0 ≤ ω < m, there is a band gap, and for m < ω < ωB , there is a pass band, with the dispersion curve ω(k) having the form of hyperbola. Since all quantities in the above dispersion relation are dimensionless, it is also relevant to express them in physical units. In particular, taking into regard that the frequency ωph and wavenumber kph in physical units are connected with their dimensionless counterparts through ω = ωph /ωB and k = kph c/ωB , Equation (14) is written as 0005 ωph + γH −i ωB 0005 ω2 2 2 4 4 kph c0 kph c0 ωph ph − 2 + 2 −ζ 4 −i γω (15) ωB ωB ωB ωB ωph 2 m + m2 γω = 0. −i ωB
Frequency (Hz)
Zhang et al.: Second-harmonic generation in waveguides
600 400
400 Hz
200
0
(a)
0
10 5 Re{k} (1/m)
15
TLM TMM TLM(L) TMM(L)
600 400
300 Hz
200 0
(b)
100 Hz 0
5 Im{k} (1/m)
10
Figure 2. (Colour online) (a) and (b) respectively show the real and imaginary parts of the complex dispersion relation. Blue solid lines (Red dashed lines) show the results obtained by using Transmission line method, Equation (15) (Transfer matrix method, Equation (16)). The thin (thick) solid and dashed lines correspond to the lossy cases (lossless cases). The black points in Figure 2 show the frequencies used in the simulations, 400 Hz, 300 Hz and 100 Hz.
1
T,R, α
ACTA ACUSTICA UNITED WITH Vol. 104 (2018)
T R α
0.5
0
0
200
400 600 Frequency (Hz)
800
Figure 3. (Colour online) The frequency dependent transmission |T | (black dashed line), reflection |R| (blue dot-dashed line) and absorption α = 1 − |R|2 − |T |2 (red continuum line) coefficients for a lattice of 1.7 m (∼ 34 cells).
To further validate this assumption, especially for the case of high frequencies (having in mind the generation of the second harmonic), it is relevant to obtain the frequency dependence of the transmission and reflection coefficients, T and R, as well as the absorption coefficient, α, for finite lattice made of 34 cells (1.7 m), as shown in Figure 3, using the TMM method (see details in Appendix A). First, we observe that the transmission and reflection coefficients are in agreement with the dispersion relation shown in Figure 2. Moreover, we find that the absorption coefficient (red continuum line in Figure 3), can be approximated by a constant value in the range of frequencies considered in this work. Therefore, our constant loss assumption –for the frequencies of interest– is well affirmed.
3. Second Harmonic Generation in lossy dispersive acoustic media In this Section, we study analytically and numerically the second harmonic generation including the effect of the losses.
Zhang et al.: Second-harmonic generation in waveguides
3.1. Analytical Results Our analytical methodology relies on a perturbation scheme. First, we express P as an asymptotic series in , namely P = p1 + p2 +
2
p3 + . . . .
(17)
Then, substituting Equation (17) into Equation (12), we obtain a hierarchy of equations at various orders in . The leading order equation, at O( 0 ), 0005 2 0005 ∂ p1 ∂2 p1 ∂p1 ∂4 p1 ∂ + γH − − ζ + γ (18) ω ∂τ ∂τ ∂τ 2 ∂χ 2 ∂χ 4 ∂p1 +m2 + m2 γω p1 = 0, ∂τ possesses plane wave solutions of the form, 0004 0007 p1 = A cos θ ,
(19)
where A is the wave amplitude, θ = ωτ − k(ω)χ, while parameters k and ω satisfy the dispersion relation D(ω, k) [cf. Equation (14)]. The equation at the next order, O( 1 ), is 0005 2 0005 ∂p2 ∂4 p2 ∂ p2 ∂2 p2 ∂ − − ζ + γ + γH ω ∂τ ∂τ ∂τ 2 ∂χ 2 ∂χ 4 ∂p2 + m2 + m2 γω p2 (20) ∂τ 0007 0004 0007 0004 0007 20004 2 = −β0 − 2iω + γH A 2ω +iγω ω cos 2ωτ −2k1 χ . The solution of Equation (20) is the sum of the particular p solution p2 of the inhomogeneous equation (forced wave, g steady state) and the general solution p2 of the homogeg p neous equation (free wave), namely p2 = p2 + p2 , with −β0 (−2iω + γH )A2 (2ω2 + iγω ω) D(2ω, 2k) 0004 0007 · cos 2ωτ − 2k1 χ , 0004 0007 g g p2 = p2 (χ = 0) cos 2ωτ − k2 χ , p
Author's complimentary copy
p2 =
(21) (22)
where k2 is the wavenumber at the second harmonic frequency taken from the dispersion relation. Assuming that the second harmonic vanishes at x = 0 and thus p2 (τ, χ = 0) = 0, we may set g
p2 (χ = 0) =
β0 (−2iω + γH )A2 (2ω2 + iγω ω) . (23) D(2ω, 2k)
Thus the full solution for p2 is written as g
p
p2 = p2 + p2 =
2β0 (−2iω + γH )A2 (2ω2 + iγω ω) D(2ω, 2k) 0003 Δk 0006 0004 0007 · sin χ sin 2ωτ − kef f χ , 2
(24)
where Δk is the detuning parameter that describes the asynchronous second harmonic generation, Δk = k(2ω) − 2k(ω) = k2 − 2k1 , and kef f is the effective wave number, kef f = (1/2)[k(2ω) + 2k(ω)].
ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 104 (2018)
The presence of β0 in the amplitude of p2 shows that, obviously, the second harmonic is generated due to nonlinearity. Since the forced and free waves have different phase velocities, i.e., 2k(ω) = k(2ω), the phase-mismatching introduces a beating in space [26, 27, 43] for the amplitude of the second harmonic –cf. the term sin(Δk/2χ) in Equation (24). The position of the maximum of the beating xc (n) can be related to the second harmonic phasemismatching frequency by xc (n) = π/Δkn = π/|k(nω) − nk(ω)|, indicating that as Δkn increases the beating spatial period, and also its maximum amplitude decrease. 3.2. Numerical simulations We now present results of direct numerical simulations in the framework of the nonlinear discrete model, Equation (9), with and without losses (i.e., RH = 0, Rω = 0). The system is excited using a sinusoidal time-depended boundary condition (a driver) at x = 0, with an amplitude of 1000 Pa. The length of the lattice is chosen to be long enough in order to avoid reflections from the right end during the evolution. Using a 4th-order Runge Kutta integrator, we study the propagation for three different driving frequencies. The first case corresponds to a fundamental and a second harmonic, both belonging to the propagating band (Figure 4a and 4b). For the second case, we choose a smaller fundamental frequency lying in the band gap, while the second harmonic lies in the pass band (Figure 4c and 4d). Third, we consider a case where both the fundamental and the second harmonic are in the band gap (Figure 4e and 4f). We start with the first case where the driver operates at f = 400 Hz, and thus both the fundamental component (p1 ) and the generated second-harmonic component (p2 ) are in the pass band. During the weakly nonlinear wave propagation, in the absence of losses (RH = 0, Rω = 0, γH = 0, γω = 0), the zeroth-order solution p1 travels with a constant amplitude and a wave vector k(ω), as shown in Figure 4a (see solid blue line). The second harmonic solution p2 is composed by a forced wave with k = 2k(ω) and a free wave propagating with k(2ω). The phase mismatch, 2k(ω) = k(2ω), introduces beatings in space for the second harmonic, which is shown with the thick blue line in Figure 4b. We also performed simulations including both the weak viscothermal and radiation losses. The results plotted with the thin (red) lines in Figure 4a and 4b show that the amplitude of both the fundamental and the second harmonics are weakly attenuated. Note that in all cases (with and without losses) our analytical results, shown by the blue stars and red circles, are found to be in a very good agreement with the numerical findings. When the driving frequency is in the band gap, but close to the cut-off frequency, the generated second harmonic will be located in the pass band. To study such a case, we choose a driver at f = 300 Hz. In the lossless case, the fundamental component p1 decreases exponentially (thick (blue) line in Figure 4c), because the corresponding wavenumber k(ω) is imaginary (black point in
239
ACTA ACUSTICA UNITED WITH Vol. 104 (2018)
(a)
(c)
0.5
x (m)
1
500
0
1.5
0
1 x (m)
0.5
x (m)
1
1.5
4
0
0
0.5
1 x (m)
1.5
0.5
1 x (m)
1.5
(f)
0.15
2
5
500
0
1.5
p (Pa)
10
0 0
0.5 (d)
p (Pa)
p (Pa)
15 (b)
(e)
1000 p (Pa)
numerics numerics (L) analytics analytics (L)
500
0 0
Zhang et al.: Second-harmonic generation in waveguides
1000 p (Pa)
p (Pa)
1000
ACUSTICA
0.1
0.05 0
0.5
1 x (m)
1.5
0
0
Author's complimentary copy
Figure 4. (Colour online) Harmonic generation in the presence of dispersion and (viscothermal and radiation) losses. For the lossy (lossless) case, thin red lines and red circles (thick blue lines and blue stars) stand respectively for the numerical and analytical results. (a) f = 400 Hz, p1 in the pass band; (b) f = 400 Hz, p2 in the pass band; (c) f = 300 Hz, p1 in the band gap; (d) f = 300 Hz, p2 in the pass band; (e) f = 100 Hz, p1 in the band gap; (f) f = 100 Hz, p2 in the band gap. Numerical results are in a good agreement with the analytical ones.
Figure 2b), leading to a strong attenuation of p1 . Since the viscothermal and radiation losses are sufficiently small, we observe almost no difference between the lossless case (thick blue line) and the lossy one (thin red line) in Figure 4c. During propagation, the second harmonic is generated, with a frequency located in the pass band as shown in Figure 4d. Note that the beatings are absent since only the free wave with single wavenumber k(2ω) is propagating. In this case, we observe a small decrease of the amplitude of p2 due to the weak viscothermal and radiation losses (thin (red) line in Figure 4d), in comparison to the lossless case (thick (blue) line in Figure 4d). Finally, in Figure 4e and 4f) we show results corresponding to the third case, i.e., when both the fundamental component p1 and the second harmonic component p2 are in the band gap. Here we choose a frequency f = 100 Hz for the driver. In this case, the amplitude of p1 decreases exponentially and faster, as compared to Figure 4c, since the imaginary part of k(ω) is larger (black point in Figure 2b). The second harmonic is generated at the beginning of the waveguide, but its amplitude eventually decreases to zero, because its frequency is still in the band gap. Since both p1 and p2 are in the band gap with relatively large imaginary wavenumbers, the weak viscothermal and radiation losses do not have an important contribution in the evolution, and there is no visible different between the lossless [thick (blue) lines] and lossy [thin (red) lines] propagation, as shown in Figure 4e and 4f). Note that, in all the cases considered here, numerical results are found to be in a very good agreement with our analytical findings presented in Section 3.1 (blue stars and red circles in Figure 4). At this point, we should mention that during the nonlinear wave propagation, third har-
240
monic is also produced in cascade. However we only consider the second harmonic in this work, because the third harmonic is too small compared to the generated second harmonic. For example, in the lossless case, when our driver is a sinusoidal wave with an amplitude of 1000 Pa and a frequency of 400 Hz, the maximum amplitude of the generated second harmonic is about 14.03 Pa, while the amplitude of the third harmonic is about 0.0307 Pa.
4. Conclusions In conclusion, we have theoretically and numerically studied second harmonic generation in a 1D weakly lossy nonlinear acoustic metamaterial. The considered structure, composed by an air-filled waveguide loaded with a periodic array of side holes, exhibited viscothermal losses (due to viscous and thermal effects) and radiation losses (due to the side holes). Based on the electro-acoustic analogy, and using the transmission line approach, we derived a nonlinear discrete model describing the propagation of pressure waves. Then, we used a perturbation scheme to analyze the nonlinear dispersive and dissipative wave equation stemming from the long-wavelength limit of the discrete lattice. We have thus derived approximate analytical expressions for the first and second harmonic traveling in the metamaterial. Numerical results were also presented, using a driver, i.e., a sinusodial source, on one end of the waveguide with sufficiently high amplitude. The numerical results were found to be in a very good agreement with the analytical ones. For the lossless cases, we have shown that during the nonlinear propagation in the metamaterial, the generated higher harmonics could be controlled by tuning the dispersion relation –for instance,
Zhang et al.: Second-harmonic generation in waveguides
the beatings of second harmonic due to the phase mismatch introduced by the dispersion effect. We also studied the effects of viscothermal and radiation losses on the second harmonic generation in this acoustic metamaterial with negative bulk mudulus. Recently, the second harmonic generation in an one-dimensional, nonlinear acoustic metamaterial with negative mass density, composed of an air-filled waveguide periodically loaded by clamped elastic plates have been analytically and numerically reported [28]. These preliminary results pave the way to study t he nonlinear properties of effective double negative acoustic metamaterials, i.e., an air-filled waveguide periodically loaded with clamped elastic plates and side holes. It would also be interesting to study the nonlinear wave propagation in higher-dimensional acoustic metamaterials. Acknowledgements Dimitrios J. Frantzeskakis (D.J.F.) acknowledges warm hospitality at Laboratoire d’Acoustique de l’Université du Maine (LAUM), Le Mans, where most of his work was carried out.
Appendix A1. Transfer Matrix Method For a finite lattice of N cells (here, N = 34), the total transmission matrix can be expressed as 0005 0005 0004 0007N p2 p1 = mω mH mω u2 u1 0005 0005 p2 T11 T12 (A1) , = u2 T21 T22 with
0005
Author's complimentary copy
cos(kd/2) jZc sin(kd/2) mω = j cos(kd/2) Z sin(kd/2) 0005 c 1 0 mH = , (iωMH + RH )−1 1
,
(A2) (A3)
where p1 (u1 ) and p2 (u2 ) are the pressure (volume velocity) at the input and output of the settings. Then the transmission coefficient (T ) and reflection one (R) can be calculated respectively as 2 , T = T11 + T12 /Zc + T21 Zc + T22 T11 + T12 /Zc − T21 Zc − T22 . R = T11 + T12 /Zc + T21 Zc + T22
ACTA ACUSTICA UNITED WITH ACUSTICA
Vol. 104 (2018)
[2] R. A. Shelby, D. R. Smith, S. Schultz: Experimental verification of a negative index of refraction. Science 292 (2001) 5514. [3] A. A. Houck, J. B. Brock, I. L. Chuang: Experimental observations of a left-handed material that obeys Snell’s Law. Phys. Rev. Lett. 90 (2003) 137–401. [4] Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, P. Sheng: Locally resonant sonic materials. Science 289 (2000) 1734. [5] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang: Ultrasonic metamaterials with negative modulus. Nat. Mater. 5 (2006) 452. [6] N. Sugimoto, T. Horioka: Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators. J. Acoust. Soc. Am. 97 (1995) 1446. [7] C. E. Bradly: Time harmonic acoustic Bloch wave propagation in periodic waveguides. Part I. Theory. J. Acoust. Soc. Am. 96 (1994) 1844. [8] C. Zwikker, C.W.Kosten: Sound absorbing materials. Elsevier, New York, 1949. [9] C. E. Bradly: Time harmonic acoustic Bloch wave propagation in periodic waveguides. Part II. Experiment. J. Acoust. Soc. Am. 96(3) (1994) 1854–1862. [10] L. Solymar, E. Shamonina: Waves in Metamaterials. Oxford University Press, New York, 2009. [11] G. Theocharis, O. Richoux, V. R. GarcÃŋa, A. Merkel, V. Tournat: Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures. New J. of Phys 16(9) (2014) 093017. [12] G. Ma, M. Yang, S. Xiao, Z. Yang, P. Sheng: Acoustic metasurface with hybrid resonances. Nature Mater. 13 (2014) 873–878. [13] V. Romero-García, G. Theocharis, O. Richoux, A. Merkel, V. Tournat, V. Pagneux: Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators. Sci. Rep. 6 (2016) 19519. [14] A. Merkel, G. Theocharis, O. Richoux, V. Romero-García, V. Pagneux: Control of acoustic absorption in onedimensional scattering by resonant scatterers. Appl. Phys. Lett. 107, (2015) 244102. [15] N. Jiménez, W. Huang, V. Romero-García, V. Pagneux, J.-P. Groby: Ultra-thin metamaterial for perfect and quasiomnidirectional sound absorption Appl. Phys. Lett. 109, (2016) 121902. [16] Noé Jiménez, Vicent Romero-García, Vincent Pagneux, Jean-Philippe Groby: Quasiperfect absorption by subwavelength acoustic panels in transmission using accumulation of resonances due to slow sound. Phys. Rev. B 95, (2017) 014205.
(A4)
[17] Mikhail Lapine, Ilya V. Shadrivov, Yuri S. Kivshar: Colloquium: Nonlinear metamaterials, Rev. Mod. Phys. 86, (2014) 1093.
(A5)
[18] M. I. Shalaev, S. A. Myslivets, V. V. Slabko, A. K. Popov: Negative group velocity and three-wave mixing in dielectric crystals. Optics letters 36(19) (2011) 3861–3863.
The absorption coefficient α is α = 1 − |T |2 − |R|2 .
[19] Jacob B. Khurgin: Optical parametric oscillator: Mirrorless magic. Nat Photon. 1 (2007) 446–447.
References
[20] A. K. Popov, V. V. Slabko, V. M. Shalaev: Second harmonic generation in left-handed metamaterials. Laser Physics Letters. 3(6) (2006) 293.
[1] V. G. Veselago: The electrodynamics of substances with simultaneously negative values of and µ. Soviet physics uspekhi 10(4) (1968) 509.
[21] N. Sugimoto, M. Masuda, J. Ohno, D. Motoi: Experimental demonstration of generation and propagation of acous-
241
ACTA ACUSTICA UNITED WITH Vol. 104 (2018)
ACUSTICA
tic solitary waves in an air-filled tube. Phys. Rev. Lett. 83 (1999) 4053. [22] O. Richoux, B. Lombard, J. F. Mercier: Generation of acoustic solitary waves in a lattice of Helmholtz resonators. Wave Motion 56 (2015) 85–99. [23] V. Achilleos, O. Richoux, G. Theocharis, D. J. Frantzeskakis: Acoustic solitons in waveguides with Helmholtz resonators: Transmission line approach. Phys. Rev. E 91 (2015) 023204. [24] M. A. Averkiou, Y. S. Lee, M. F. Hamilton: Self-demodulation of amplitude and frequency modulated pulses in a thermoviscous fluid. J. Acoust. Soc. Am. 94(5) (1993) 2876–2883. [25] H. J. Vos, D. E. Goertz, N. de Jong: Self-demodulation of high-frequency ultrasound. J. Acoust. Soc. Am. 127(3) (2010) 1208–1217. [26] V. J. Sánchez-Morcillo, I. Pérez-Arjona, V. Romero-García, V. Tournat, V. E. Gusev: Second-harmonic generation for dispersive elastic waves in a discrete granular chain. Phys. Rev. E 88 (2013) 043203. [27] N. Jiménez, A. Mehrem, R. Picó, L. M. García-Raffi, V. J. Sánchez-Morcillo: Nonlinear propagation and control of acoustic waves in phononic superlattices. C. R. Phys. 17 (2016) 543–554. [28] J. Zhang, V. Romero-García, G. Theocharis, O. Richoux, V. Achilleos, D. J. Frantzeskakis: Second-Harmonic Generation in Membrane-Type Nonlinear Acoustic Metamaterials. Crystals 6(8) (2016) 86. [29] J. Zhang, V. Romero-García, G. Theocharis, O. Richoux, V. Achilleos, D. J. Frantzeskakis: Bright and Gap Solitons in Membrane-Type Acoustic Metamaterials. Phys. Rev. E 96 (2017) 022214. [30] F. Bongard, H. Lissek, J. R. Mosig: Acoustic transmission line metamaterial with negative/zero/positive refractive index. Phys. Rev. B 82 (2010) 094306.
Author's complimentary copy
[31] C. M. Park, J. J. Park, S. H. Lee, Y. M. Seo, C. K. Kim, S. H. Lee: Amplification of acoustic evanescent waves using metamaterial slabs. Phys. Rev. Lett. 107 (2011) 194301.
242
Zhang et al.: Second-harmonic generation in waveguides
[32] K. J. B. Lee, M. K. Jung, S. H. Lee: Highly tunable acoustic metamaterials based on a resonant tubular array. Phys. Rev. B 86 (2012) 184302. [33] R. Fleury, A. Alú: Extraordinary sound transmission through density-near-zero ultranarrow channels. Phys. Rev. Lett. 111 (2013) 055501. [34] T. D. Rossing, N. H. Fletcher: Principles of Vibration and Sound. Springer-Verlag, New York, United States of America, 1995. [35] L. E. Kinsler, A. U. Frey, A. B. Coppens, J. V. Sanders: Fundamentals of Acoustics. Whiley, New York, 1982. [36] M. Rossi: Acoustics and Electroacoustics. Artech House, Norwood, MA, 1988. [37] P. A. Kalozoumis, O. Richoux, F. K. Diakonos, G. Theocharis, P. Schmelcher: Invariant currents in lossy acoustic waveguides with complete local symmetry. Phys. Rev. B 92 (2015) 014303. [38] M. Atig, J. P. Dalmont, J. Gilbert: Termination impedance of open-ended cylindrical tubes at high sound pressure level. Comptes Rendus Mécanique 332(4) (2004) 299– 304. [39] J. M. Buick, M. Atig, D. J. Skulina, D. M. Campbell, J. P.Dalmont, J. Gilbert: Investigation of non-linear acoustic losses at the open end of a tube. J. Acoust. Soc. Am. 129(3) (2011) 1261–72. [40] M. Remoissenet: Waves called solitons. Springer, Berlin, Germany, 1999. [41] M. J. Ablowitz: Nonlinear Dispersive waves, asymptotic analysis and solitons. Cambridge texts in applied mathematics, Cambridge University Press, New York, 2011. [42] M. F. Hamilton, D. T. Blackstock: Nonlinear Acoustics. Academic Press, AIP, San Diego, California, United States of America, 1998. [43] A. Mehrem, N. Jiménez, L. J. Salmerón-Contreras, X. García-Andrés, L. M. García-Raffi, R. PicÃş, V. J. Sánchez-Morcillo: Nonlinear dispersive waves in repulsive lattices. Phys. Rev. E 96 (2017) 012208.