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Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered Helmholtz resonators
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Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered Helmholtz resonators a
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Jun Lan , Yifeng Li , Huiyang Yu , Baoshun Li , Xiaozhou Liu
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College of Computer Science and Technology, Nanjing Tech University, Nanjing 211800, China b Key Laboratory of Modern Acoustics, Ministry of Education, Institute of Acoustics and School of Physics, Nanjing University, Nanjing 210093, China
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Article history: Received 1 November 2016 Received in revised form 16 January 2017 Accepted 26 January 2017 Available online xxxx Communicated by C.R. Doering Keywords: Acoustic metamaterial Helmholtz resonator Nonlinear effects Perturbation method
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We theoretically investigate the nonlinear effects of acoustic wave propagation and dispersion in a cylindrical pipe with periodically arranged Helmholtz resonators. By using the classical perturbation method in nonlinear acoustics and considering a nonlinear response up to the third-order at the fundamental frequency, the expressions of the nonlinear impedance Z NHR of the Helmholtz resonator and effective nonlinear bulk modulus B neff of the composite structure are derived. In order to confirm the nonlinear properties of the acoustic metamaterial, the transmission spectra have been studied by means of the acoustic transmission line method. Moreover, we calculate the effective acoustic impedance and dispersion relation of the system using the acoustic impedance theory and Bloch theory, respectively. It is found that with the increment of the incident acoustic pressure level, owing to the nonlinearity of the Helmholtz resonators, the resonant frequency ω0 shifts toward the lower frequency side and the forbidden bandgap of the transmission spectrum is shown to be broadened. The perturbation method employed in this paper extends the general analytical framework for a nonlinear acoustic metamaterial. © 2017 Elsevier B.V. All rights reserved.
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1. Introduction In recent years, a type of acoustic metamaterial based on a cylindrical pipe with ordered lattices has been widely studied [1]. In previous studies, the ordered lattices which are periodically connected to the cylindrical pipe are usually side holes [1–3], Helmholtz resonators (HRs) [4–8], or membranes [3,5,9–11]. This type of acoustic metamaterial can freely achieve negative- or zerobulk modulus B and/or mass density ρ . For instance, negative bulk modulus property could be achieved by a cylindrical pipe with periodically distributed side holes or HRs. Negative mass density property could be designed by a cylindrical pipe with periodically arranged membranes. All of these studies are mainly focused on the linear domain, the nonlinear research in this type of acoustic metamaterial has lagged far behind than that in the linear domain. Most researches on the nonlinear metamaterial are mainly about electromagnetic and optical waves, many intriguing nonlinear phenomena such as tenability [12,13], parametric shielding of electromagnetic fields [14] and parametric down-conversion (PDC) [15], as well as second- or third-harmonic generation [16–18] have already been investigated. In acoustics, the nonlinearities of acous-
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E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.physleta.2017.01.036 0375-9601/© 2017 Elsevier B.V. All rights reserved.
tic metamaterial cannot be ignored, particularly for the metamaterial based on the cylindrical pipe which has resonance phenomena in the tube or lattice and enhances the nonlinear effects in finite space [19,20]. Therefore, the nonlinearities in periodical structure along a cylindrical pipe must be paid more attention, and the nonlinear phenomena are considered to be the key to future research into the development of metamaterials [21,22]. The effects of an array of HRs on the propagation of nonlinear acoustic waves in a long tunnel were first investigated in 1992 by Sugimoto [23], it was found that appropriately designed resonators could effectively prevent the emergence of shock waves by introducing dispersion into acoustic waves. Theoretical and experimental studies of the nonlinear effects in propagation of acoustic Bloch waves in periodic waveguides were performed by Bradley [20], in which the nonlinearity of the system for a local second harmonic field has been investigated. Then, in 2007 Richoux et al. studied the nonlinear properties of the acoustic wave propagation based on the periodic lattices made of HRs [24]. The analytical study and experimental results showed that localized nonlinearities lead to the frequency bandgaps were amplitude dependent. In addition, the nonlinear effects in an acoustic metamaterial based on two types ordered lattices of side holes and membranes were studied by Fan et al. [25]. The nonlinearities of the membranes changed the characteristics of frequency bands and the nonlinear effects
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Fig. 1. (a) The model of the acoustic metamaterial based on a cylindrical pipe with periodically arranged Helmholtz resonators (HRs). (b) The picture and dimensions of a signal cell. (c) The equivalent mechanics prototype of a unit cell.
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of the side holes were very small and can be neglected. These researches show that the nonlinearities introduce an interaction between nonlinear effects and spatial periodicity. However, these works on the nonlinear acoustic metamaterial rarely discussed the influences of the nonlinear type (cubic, . . . ) on the propagation of acoustic waves, and the general expressions for the nonlinear bulk modulus and impedance of acoustic metamaterial have not been obtained. All of these questions need to be deeply investigated and resolved. In this paper, we design and analyze an acoustic metamaterial based on a cylindrical pipe with periodically arranged HRs. A straightforward homogenization procedure leads to generate an expression of the continuity equation for the composite acoustic metamaterial. Then based on the continuity equation and using the classical nonlinear perturbation method [17,26,27] which considering the nonlinear responses up to the third-order at the fundamental frequency, the expressions of the effective nonlinear bulk modulus B neff of the system and nonlinear impedance Z NHR of the HR that determine all the wave characteristics in the medium could be obtained. The nonlinear effects of the acoustic metamaterial are deeply investigated with the acoustic transmission line method (ATLM) [6], acoustic impedance theory and nonlinear Bloch theory. The results show that the localized nonlinearity due to the HRs can produce a shift in the position of the resonant frequency, and the forbidden bandgaps of the transmission spectra are amplitude dependent on the incident acoustic pressure level.
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2. Theoretical description of the metamaterial system
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A typical acoustic metamaterial based on a cylindrical pipe with ordered HRs is shown in Fig. 1. The inner diameter of the cylindrical pipe is l = 30 mm and the unit cell length of the metamaterial is L = 70 mm. Each HR is composed of a neck and a cavity, the sectional area and length of the neck are s = 7.85 × 10−5 m2 and l1 = 13.5 mm, respectively. The size of the cavity is determined by the inner cross section sc = 2.5 × 10−3 m2 and length l2 = 50 mm. The dimension of the HR is much smaller than the wavelength of incident wave, hence we can assume the acoustic pressure and density of the air inside the HRs are spatially uniform and the air movement in the neck can be as a whole [24,28]. Since the acoustic wavelength (at 292.5 Hz, λ ≈ 1.18 m in air) is much longer than the distance between two adjacent HRs (L λ), this system can be regarded as a homogenized medium [7]. A straightforward homogenization procedure leads to the HRs modify the continuity equation of air in the pipe [2], which can be expressed as
∂(ρ0 v 1 ) ρ0 v 2 s ∂ ρ0 = (1) sw − sw , ∂x L ∂t where ρ0 is the density of air, sw is the cross section area of the −
pipe and v 1 , v 2 are the acoustic velocities in the waveguide and HRs, respectively. It is worth noting that in this acoustic metamaterial the other acoustic nonlinear effects due to the intrinsic air behavior and Eulerian description of the movement are considered to be neglected compared to the nonlinearity of the HRs. Consequently, only the nonlinearity of the HRs is taken into account in this metamaterial [24,29]. Therefore, in Eq. (1) only the acoustic velocity v 2 in the HRs needs to consider the nonlinear effects. According to the constitutive equation p /ρ = B 0 /ρ0 and harmonic expression of acoustic pressure p = pa e − j (ωt −kx) , where ρ represents the density variable, B 0 = ρ0 c 02 is the bulk modulus of air and c 0 is the sonic velocity in air, Eq. (1) in frequency domain can be simplified to
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∂ v1 sv 2 jω − − =− p. ∂x Lsw B0
(2)
According to the Newton’s law, the motion equation of air in the HR neck can be obtained
Mm
dv 2 dt
+ R m v 2 − f (ξ ) = sp ,
(3)
where M m = ρ0 l1 s and R m = R vis + R rad are the mechanical mass and resistance acting as acoustic mass and resistance due to the HR neck, respectively. Because of radiation, the short neck may become “longer”, the effective length of short neck must be corrected as l1 = l1 + 1.7a 1 [30], where a1 is the radius of the short neck. f (ξ ) = −1/C m v 2 dt is the nonlinear restoring force term, where C m = V 0 /ρ0 c 02 s2 is the mechanical capacitance of the cavity, with the volume notation of V 0 = scl2 . The mechanical resistance is con√ sisted of two parts: the viscous losses R vis = a1 ρ0l1 ηω/2s at the walls of HR and the radiation losses R rad = (1/4)ρ0 c 0 (ka1 )2 s at the open end of HR neck tube [28], where η is the coefficient of kinematic viscosity (approximately 1.5 × 10−5 m2 /s in air), ω and k are the angular frequency and wave vector, respectively. With a new notation of restoring force F = −ξ/C m = v 2 / j ω C m , where ξ is the air displacement in the neck, the motion Eq. (3) can be rewritten as 2
d F dt 2
+δ
dF dt
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+ ω02 f (ξ ) = −ω02 sp ,
(4)
√
where δ = R m / M m is the damping coefficient and ω0 = 1/ C m M m is the resonant angular frequency. A HR is a tube–cavity system
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oscillating in a simple mass–spring model, the air in the tube moves as a unit and acts as the mass element and the air in the cavity acts as the spring. A general description of the nonlinear behavior of a HR is derived by only taking into account the restoring force of the spring and the nonlinearity of damping term can be neglected [24,28,31]. In this paper, the nonlinear behavior of HR is discussed by considering the cubic nonlinear term of the restoring force. The relative change of the pressure in the HR cavity due to displacement ξ of the air in the neck induces a restoring force which can be written in a cubic nonlinear form f (ξ ) = −[ξ + aξ 2 + bξ 3 ]/C m . Using the Taylor expansion the quadratic and cubic nonlinear coefficients can be expressed as a = (γ + 1)s/2V 0 and b = (γ + 1)(γ + 2)s2 /6V 02 , respectively, where γ ≈ 1.4 is the specific heat ratio of the air. By substituting f (ξ ) into Eq. (4) and writing in terms of the restoring force F , a third-order nonlinear motion equation can be obtained
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d2 F
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dt 2
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+δ
dF dt
2 0F
+ω
2
3
2 0 sp ,
− α F + β F = −ω
where α = c ω = dω with c = aC m and d = An important feature can be noted from Eq. (5) that the spring is softer or harder depend on the value of displacement ξ . This makes it different from Duffing’s nonlinearity in which the spring hardening (or softening) no matter which sign the displacement is. For the general case, Eq. (5) usually cannot be solved exactly by using the classical method of successive approximations in nonlinear acoustics [24,29]. In such situation, it is often adequate to solve this equation through the use of classical perturbation theory [17,26, 27]. In order to solve Eq. (5) systematically in terms of a perturbation expansion, we replace the acoustic pressure p by λ p, where λ is a continuously varying parameter ranging from zero to unity that characterizes the strength of acoustic pressure and the value λ = 1 corresponds to the actual physical situation. We seek a solution to Eq. (5) in the form of a power series in λ 2 0, β
2 0,
2 bC m .
F (t ) = λ F (1) (t ) + λ2 F (2) (t ) + λ3 F (3) (t ).
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p = pa e
− j (ωt −kx)
= pa B (ω)e
− j ωt
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Λ
=
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¨ (1 )
F
˙ (1 )
(t ) + δ F
2 (1 ) (t ) 0F
(t ) + ω
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¨ (3 )
F
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˙ (3 )
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F (1) (t ) =
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αF
pa B (ωn )e
− j ωn t
, (8a)
n=−Λ
2
(1 )
(t ) F
= 0,
(2 )
(8b)
(t ) (8c)
Λ
F (1) (ωn )e − j ωn t ,
(9a)
n=−Λ
F (2) (t ) =
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(7)
Equations (8a)–(8c) are the first- (linear), second- and thirdorder responses of the restoring force F , respectively. Now we look for a steady-state solution to Eqs. (8) of the form
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2 (3 ) (t ) − 2 0F
(t ) + δ F (t ) + ω 3 + β F (1) (t ) = 0.
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2 0s
F¨ (2) (t ) + δ F˙ (2) (t ) + ω02 F (2) (t ) − α F (1) (t )
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Λ
= −ω
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.
Substituting Eq. (6) into (5) and equating terms of the same order in λ, the system of coupled equations can be expressed as
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pa B (ωn )e
− j ωn t
n=−Λ
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(6)
In this study, the acoustic pressure p can be expressed as a discrete sum of Λ components, each of which with an angular frequency ωn
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(5)
F (2) (ωr )e − j ωr t ,
(9b)
r
F
(3 )
(t ) =
s
F
(3 )
(ωs )e
− j ωs t
,
(9c)
3
where ωr = ωn + ωm , ωs = ωn + ωm + ω p and the summations are taken over both positive and negative frequencies, with n, m, p each taking the value between ±Λ.
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2.1. The linear metamaterial system
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As the first step toward the study of nonlinear effects of the acoustic metamaterial, the linear system is primarily investigated. The motion equation and linear response of the restoring force F at the fundamental frequency ωn are presented in Eqs. (3) and (8a), respectively. Inserting Eq. (9a) into (8a), the expression for each ωn can be written as
−ωn2 F (1) (ωn ) − j ωn δ F (1) (ωn ) + ω02 F (1) (ωn ) = −ω02 spa B (ωn ), (10) from which the restoring force F at each
ω2 spa B (ωn ) F (1) (ωn ) = 0 , D (ωn )
ωn can be derived as
F
(12)
2 0 sp a B (
ω)
D (ω)
e − j ωt =
2 0 sp
ω
D (ω)
ω, Eq. (12) can be explicitly
v 2 = jωC m F = jω
v2
(13)
V 0 ω02 p B 0 sD (ω)
.
(14)
j ω V 0 ω0
B0
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1−
V R ω02 D (ω)
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2
.
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(16)
where V R = V 0 / Lsw is the volume ratio of resonator cavity to waveguide section. According to Ref. [2], the effective bulk modulus of the system is defined as the ratio of the velocity gradient −∂ v 1 /∂ x to the change of acoustic pressure ∂ p /∂ t = − j ω p, hence
B leff
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(15)
. 2
V R ω0 ∂ v1 1 − = − jω 1− p, ∂x B0 D (ω)
1
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B 0 sD (ω)
=
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98
Combining Eqs. (2) and (14), the continuity equation of the acoustic metamaterial can be simplified to
1
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.
Then, in the linear situation, the effective impedance Z LHR of the HR is derived from Eq. (14),
=
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95
Considering the expression of the restoring force F , the acoustic velocity v 2 in the HR can be given by
Z LHR =
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For a continuous angular frequency written as
p
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90
n=−Λ
ω
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Λ ω02 spa B (ωn ) − jωn t (t ) = e . D (ωn )
F (t ) =
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84
(11)
where the denominator is defined as D (ωn ) = ωn2 − ω02 + j δ ωn . By substituting Eq. (11) into (9a), the linear response of restoring force F can be obtained (1 )
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(17)
The expression of the effective bulk modulus is identical to that in Ref. [7]. However, in Ref. [7] the bulk modulus is obtained by mapping the electromagnetic system into the acoustic analogues. Equation (17) shows that this acoustic √ metamaterial has two characteristic frequencies ω0 and ωn (≈ ω0 1 + V R ), and the real part of the effective linear bulk modulus of the system is negative in the frequency range of ω0 < ω < ωn . Fig. 2 shows the calculated results for the dispersion of the effective modulus of the system with respect to frequency from 200 to 700 Hz, in which the solid and broken curves stand for the
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where the indices n and m take any values between ±Λ. Equation (18) can be transformed into a set of independent equations and the second-order response F (2) (ωr ) of restoring force F at each frequency ωr can be expressed as
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F
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(2 )
(ωr ; ωn , ωm ) = −
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Fig. 2. Real and imaginary parts of the effective linear bulk modulus of the acoustic metamaterial.
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− 2α 2
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Fig. 3. Transmission coefficient of the negative bulk modulus (B-NG) metamaterial with 50 units which has a forbidden band between two characteristic frequencies f 0 and f n .
F
2.2. The nonlinear metamaterial system In this section, we discuss the nonlinearity of the B-NG acoustic metamaterial. The linear analysis part above has discussed the first-order response of the system at the fundamental frequency. Using the similar approach, the second- and third-order responses of the restoring force F at the fundamental frequency or for the second- and third-harmonic generation accounting for the secondand third-order nonlinearities can be obtained. The third-order nonlinear motion equation of the HR is presented in Eq. (5), and Eqs. (8b) and (8c) are the expressions of the second- and third-order responses of the restoring force F , respectively. Using Eqs. (9b) and (12), Eq. (8b) can be written as 2 ¨ (2 ) ( r F
−ω ωr ) − j ωr δ F˙ (2) (ωr ) + ω02 F (2) (ωr ) ω4 s2 pa2 B (ωn ) B (ωm ) 0 , =α D (ωn ) D (ωm ) (nm)
+ c ω 3
×
1 D (ωn + ωm )
+
+
F (2) (2ωn ; ωn , ωn ) = −
c ω06 s2 pa2 B (ωn )2 D (ωn )2 D (2ωn )
8 3 3 0 s pa
(3ωn ; ωn , ωn , ωn ) = dω + 2c 2 ω010 s3 pa3
D (ωn
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1 D (ωm + ω p )
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105
.
(21)
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(22)
.
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B (ωn )
129
D (ωn )3 D (3ωn )
ωn ) D (2ωn )
124 125
3
B (ωn )3
)3 D (3
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For the third-order restoring force responsible for the thirdharmonic generation, Eq. (21) is rewritten for ωn = ωm = ω p (3 )
79
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D (ωn ) D (ωm ) D (ω p ) D (ωs ) D (ωn + ω p )
78
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B (ωn ) B (ωm ) B (ω p )
1
77
86
(20)
In combination with Eq. (9), Eqs. (11), (19) and (21) provide general expressions for the first-, second- and third-order responses of the restoring force F , respectively. Following we will provide the more explicit expressions for some specific combinatorial frequencies for the second- and third-order nonlinearities. i) F (3) (3ωn ; ωn , ωn , ωn ) is the third-order response of restoring force F for the third-harmonic generation and F (3) (ωn ) is the nonlinear response at the fundamental frequency ωn . ii) F (2) (2ωn ; ωn , ωn ) is the second-order response of restoring force F for the second-harmonic generation and F (2) (ωn ) is the nonlinear response at the fundamental frequency ωn . For example, for the second-order restoring force responsible for the secondharmonic generation, in case of ωm = ωn , Eq. (19) can be expressed as
F (18)
(nmp )
76
85
ω ωn ) B (ωm ) B (ω p ) , D (ωn ) D (ωm ) D (ω p ) D (ωn + ωm ) (nmp )
75
84
6 3 3 0 s pa B (
10 3 3 0 s pa
73
83
(ωs ; ωn , ωm , ω p ) B (ωn ) B (ωm ) B (ω p ) = dω08 s3 pa3 D (ωn ) D (ωm ) D (ω p ) D (ωs ) 2
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82
(3 )
2
70
81
(nmp )
real and imaginary parts, respectively. Using Eq. (15) and ATLM, the effective impedance Z leff of the linear negative bulk modulus (B-NG) acoustic metamaterial with 50 units can be calculated. Then, the sound pressure reflection coefficient r p can be obtained with the formula r p = ( Z leff − Z )/( Z leff + Z ), where Z = ρ0 c 0 /sw is the distributed impedance of the pipe. Consequently, the acoustic intensity transmission coefficient T can be calculated to be T = 1 − |r p |2 . The evolution of the transmission spectrum is shown in Fig. 3. From the curve we can see that the system exhibits one forbidden bandgap during the frequency range from f 0 (292.5 Hz) to f n (550.0 Hz) which corresponds to the regime of the negative bulk modulus.
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where the factor D (ωn + ωm ) in the second term on the right-hand side of Eq. (20) results from the second-order response F (2) (ωr ) at frequency ωr = ωn + ωm which produces in combination the frequency ωr . In order to account for the various contributions, the factor should be traversed over all possible combinations of indices n and m. Then from Eq. (20), the third-order response of restoring force F at each ωs accounting for the second- and third-order nonlinearities can be written as
24
35
(19)
.
(nmp )
21
34
(nm)
D (ωn ) D (ωm ) D (ωn + ωm )
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71
−ωs2 F¨ (3) (ωs ) − j ωs δ F˙ (3) (ωs ) + ω02 F (3) (ωs ) ω6 s3 pa3 B (ωn ) B (ωm ) B (ω p ) 0 = −β D (ωn ) D (ωm ) D (ω p )
20
33
c ω06 s2 pa2 B (ωn ) B (ωm )
The parentheses of (nm) indicate that the sum of ωn + ωm is constant, while the indices n and m are variable and Eqs. (18) and (19) sums over all possible values of ωr . The third-order response of the restoring force can be derived in the similar method, combining Eqs. (9c), (12), (19), and (9b), Eq. (8c) can be expressed as
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.
(23)
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Here, for the nonlinear response it is important to note that F (2) (ωr ; ωn , ωm ) has no nonlinear response at the fundamental frequency ωn . However, the nonlinear response F (3) (ωn ) at the fundamental frequency is result from both cases, the one is the interaction of a field with itself at frequency ωn which leads to a self-phase modulation and the other one is the interaction of two different fields at frequencies ωn and ωm which leads to a cross-phase modulation [17]. For the first case, we set ωm = ωn and ω p = −ωn in Eq. (21), resulting in
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10 11
76
F
(3 )
(ωn ; ωn , ωn , −ωn )
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=
8 3 3 s pa B ( n ) 4 3 0 3 D ( n ) D (− n ) 3
ω ω
ω ω
77
c
2
2 0
ω
D (0)
+
2
2 0
ω +d . 3 D (2ωn ) 2 c
78 79
(24)
80 81
For the other case (cross-phase modulation), considering
82
ω p = −ωm in Eq. (21), which gives
ω08 s3 pa3 B (ωn ) F (3) (ωn ; ωn , ωm , −ωm ) = 6 D (ωn )2 D (ωm ) D (−ωm ) 2 2 c ω0 c 2 ω02 2 2 2 c 2 ω02 × + + +d . 3 D (ωn + ωm ) 3 D (ωn − ωm ) 3 D (0)
83
Fig. 4. (Color online.) Real and imaginary parts of the effective linear and nonlinear bulk modulus for the metamaterial.
(25)
Note that the coefficients number 3 and 6 on the right-hand sides of Eqs. (24) and (25) are all resulting from the summation to account for the possible number of permutations. Now we assume that only one single incident frequency ωn exists in the total nonlinear responses. According to Eqs. (6), (9), (11), and (24), the expression of the restoring force F by considering the total nonlinear responses expanded up to the third-order at the fundamental frequency can be obtained Λ Λ ω02 spa B (ωn )e− jωn t ω08 s3 pa3 B (ωn )e− jωn t F (t ) = +3 D (ωn ) D (ωn )3 D (−ωn ) n=−Λ n=−Λ 2 2 4 c ω0 2 c 2 ω02 (26) × + +d , 3 D (0) 3 D (2ωn )
where on the right-hand side of Eq. (26), the first term is the linear response part, the second term is the nonlinear response part. Note that the nonlinear response F (2) (ωn ) at the fundamental frequency ωn does not exist and only the nonlinear response F (3) (ωn ) is considered in Eq. (26). However, the nonlinear response F (3) (ωn ) at the fundamental frequency ωn contains the secondand third-order nonlinearities. Thus the restoring force F which contains linear and nonlinear parts is solved. According to the relation v 2 = j ω C m F , and then one explicit expression of the velocity, as a function of a continuous angular frequency ω , can be given by
ω02 p j ω ω08 sV 0 pa2 p v 2 = jωC m F = jω +3 B 0 s D (ω) B 0 D (ω)3 D (−ω) 2 2 4 c ω0 2 c 2 ω02 + +d . × 3 D (0) 3 D (2ω) V0
(27)
From Eq. (27), the effective impedance of the HR which includes nonlinear response component can be calculated
V 0 ω02 Z NHR = 1 jω B 0 s D (ω) 2 2 j ω ω08 sV 0 pa2 4 c ω0 2 c 2 ω02 + 3 . d + + B 0 D (ω)3 D (−ω) 3 D (0) 3 D (2ω) (28) The acoustic velocity v 2 in the HR driven by the acoustic pressure p in the pipe has been obtained in Eq. (27), by substituting
it into the continuity Eq. (2), the continuity equation of this metamaterial system can be rewritten
∂ v1 1 1 V R ω02 1 3V R ω08 s2 pa2 − = − jω − − ∂x B0 B 0 D (ω) B 0 D (ω)3 D (−ω) 2 2 2 2 4 c ω0 2 c ω0 × + + d p. 3 D (0) 3 D (2ω)
B neff
=
1 B0
×
1−
V R ω02 D (ω)
4 c 2 ω02
+
−
2 c 2 ω02
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91
(29)
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)3 D (−
ω)
+d .
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3V R ω08 s2 pa2 D (ω
85
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From Eq. (29), the effective nonlinear bulk modulus of the system which contains nonlinear effects can be derived as
1
84
99
(30)
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It is note that the third term in the square bracket on the righthand side of Eq. (30) is the nonlinear part, and if without this term Eq. (30) arrives at the expression of the effective linear bulk modulus Eq. (17). We can draw from Eqs. (28) and (30) that the effective nonlinear impedance of HR and bulk modulus are all depended on the acoustic amplitude pa of incident acoustic pressure. Fig. 4 shows the calculated dispersion results of the effective linear and nonlinear bulk modulus of the B-NG acoustic metamaterial. For the nonlinear calculation, Eq. (30) has been used and the incident acoustic pressure level is 160 dB. By comparing the two curves for the real parts of the linear (black color) and nonlinear (red color) bulk modulus, we can obviously observe that there exists a left shift of the zero crossing point in the nonlinear case, which means that the resonant frequency f 0 is decrescent. The effective impedance Z NHR of the HR has been obtained considering the total nonlinear responses up to the third-order. The acoustic pressure amplitude pa introduced in Eq. (28) should be depended on the HR position, but we can apply only one global pressure amplitude value for all units as outlined in Ref. [24]. By applying the method of ATLM, the effective nonlinear impedance Z neff of the B-NG acoustic metamaterial with 50 units can be calculated. Thus, the theoretical nonlinear transmission spectra of the system with the different input acoustic pressure levels (130, 140, 150, and 160 dB) can be obtained and plotted in Fig. 5. From the curves we can see that, comparing with the linear transmission spectrum, the nonlinearity makes the resonant frequency f 0 shifted toward lower frequencies with increasing input acoustic pressure level, which broadens the forbidden bandgap’s width. It is well known that the acoustic impedance theory and Bloch theory are used to explain the unique characteristics of the acoustic metamaterial [10]. Following, we will carry out the theoretical
102
3 D (0)
3 D (2ω)
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(c) 160 dB. It is found that the acoustic resistance (solid curve) is smaller than the acoustic reactance (broken curve) in the frequency range of f 0 < f < f n , which means that the most of the incident energy in this frequency range is reflected back to the source and the forbidden band occurs in the acoustic metamaterial. It is also noted that when the acoustic pressure level is increasing, the resonant frequency f 0 corresponding to the lower limit of the bandgap moves toward the lower frequency while the upper limit frequency f n of the bandgap keeps constant. These results are exactly consistent with the transmission coefficient as shown in Fig. 5. Thus, the intrinsic parameters Z leff and Z neff can be used to characterize the properties of the acoustic metamaterial. Let us continue to discuss the nonlinear effects of the acoustic metamaterial using the Bloch theory, the nonlinear Bloch dispersion equation for the B-NG metamaterial can be written as [1,25]
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Fig. 5. (Color online.) Nonlinear transmission spectra for the negative bulk modulus (B-NG) metamaterial with 50 units, the curves show that the nonlinearity makes the lowest side of the forbidden bands shifted toward lower frequencies with increasing incident acoustic pressure level.
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analyses to understand the observed nonlinear behaviors by using these two theories. We know that the acoustic resistance (real part of Z leff or Z neff ) of the metamaterial system produces the transmission of the acoustic wave while the acoustic reactance (imaginary part of Z leff or Z neff ) leads to the reflection of the sound [6]. Fig. 6 shows the calculated dispersion curves of the acoustic resistance and acoustic reactance for the metamaterial system with three different incident acoustic pressure levels: (a) Linear, (b) 140 dB, and
cos(qL ) = cos(kL ) +
j ρ0 c 0 2sw Z NHR
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sin(kL ),
(31)
where q is the Bloch wave number. According to the dispersion theory, in the frequency range of | cos(qL )| > 1 the acoustic wave propagates in a forbidden band. Fig. 7 shows that with increasing incident acoustic pressure level, the resonant frequency f 0 which corresponds to | cos(qL )| = 1 moves toward the lower frequency and the characteristic frequency f n stays unchanged, as a consequence the forbidden bands become wider. In the frequency range of f 0 < f < f n , such a stop band is ascribed to the traditional physical mechanism of the absorption due to the local resonance
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Fig. 6. (Color online.) Calculated dispersion curves of acoustic resistance (solid curve) and reactance (broken curve) for the negative bulk modulus (B-NG) metamaterial with the different incident acoustic pressure levels: (a) Linear, (b) 140 dB, and (c) 160 dB.
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paper provides a potential possibility for applications to investigate other nonlinear phenomena of acoustic metamaterial.
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This research work was supported by the National Natural Science Foundation of China (grant Nos. 61571222, 11104142, and 11474160), by the Natural Science Foundation of Jiangsu Province, China (No. BK20161009), sponsored by Six Talent Peaks Project of Jiangsu Province, China, and Qing Lan Project of Jiangsu Province, China.
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Fig. 7. (Color online.) cos(qL ) of the negative bulk modulus (B-NG) metamaterial with increasing input acoustic pressure level.
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of the HRs. Furthermore, it can be seen from the inset zoom part of the curves in Fig. 7, that for the linear or small incident acoustic pressure level (130 dB), the resonant frequency f 0 ≈ 292.5 Hz, while for the acoustic pressure level 160 dB, f 0 ≈ 278.0 Hz, the corresponding increased percentage of bandwidth for the nonlinear metamaterial compared to the linear one is about 5.6%. Therefore, the nonlinear acoustic behavior of the dispersion relation of the acoustic metamaterial is only visible at the resonant frequency, which is due to the nonlinearity of the HRs. The analyzed results of Bloch theory are coincident to that obtained with the acoustic impedance theory and ATLM analysis described above.
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3. Conclusion
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In conclusion, the nonlinear properties of the negative B-NG acoustic metamaterial based on the cylindrical pipe with periodically arranged HRs have been investigated theoretically. The second- and third-order nonlinear responses at the fundamental frequency have been analyzed in detail by using the perturbation solution. The theoretical analyses show that, with increasing incident acoustic pressure level, the nonlinearity of the system leads to resonator frequency shift toward the lower frequency side and broadens the width of the forbidden band. Compared with the other previous studies about the B-NG acoustic metamaterial, we made some significant improvements, such as getting the explicit expressions of the nonlinear acoustic impedance of the HR and effective nonlinear bulk modulus of the metamaterial system. The nonlinear model obtained with the perturbation solution provides a very close description of the interaction of the acoustic wave with the HRs. The analytical perturbation method employed in this
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Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered Helmholtz resonators a
a
a
Jun Lan , Yifeng Li , Huiyang Yu , Baoshun Li , Xiaozhou Liu
b
a
College of Computer Science and Technology, Nanjing Tech University, Nanjing 211800, China b Key Laboratory of Modern Acoustics, Ministry of Education, Institute of Acoustics and School of Physics, Nanjing University, Nanjing 210093, China
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Article history: Received 1 November 2016 Received in revised form 16 January 2017 Accepted 26 January 2017 Available online xxxx Communicated by C.R. Doering Keywords: Acoustic metamaterial Helmholtz resonator Nonlinear effects Perturbation method
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We theoretically investigate the nonlinear effects of acoustic wave propagation and dispersion in a cylindrical pipe with periodically arranged Helmholtz resonators. By using the classical perturbation method in nonlinear acoustics and considering a nonlinear response up to the third-order at the fundamental frequency, the expressions of the nonlinear impedance Z NHR of the Helmholtz resonator and effective nonlinear bulk modulus B neff of the composite structure are derived. In order to confirm the nonlinear properties of the acoustic metamaterial, the transmission spectra have been studied by means of the acoustic transmission line method. Moreover, we calculate the effective acoustic impedance and dispersion relation of the system using the acoustic impedance theory and Bloch theory, respectively. It is found that with the increment of the incident acoustic pressure level, owing to the nonlinearity of the Helmholtz resonators, the resonant frequency ω0 shifts toward the lower frequency side and the forbidden bandgap of the transmission spectrum is shown to be broadened. The perturbation method employed in this paper extends the general analytical framework for a nonlinear acoustic metamaterial. © 2017 Elsevier B.V. All rights reserved.
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1. Introduction In recent years, a type of acoustic metamaterial based on a cylindrical pipe with ordered lattices has been widely studied [1]. In previous studies, the ordered lattices which are periodically connected to the cylindrical pipe are usually side holes [1–3], Helmholtz resonators (HRs) [4–8], or membranes [3,5,9–11]. This type of acoustic metamaterial can freely achieve negative- or zerobulk modulus B and/or mass density ρ . For instance, negative bulk modulus property could be achieved by a cylindrical pipe with periodically distributed side holes or HRs. Negative mass density property could be designed by a cylindrical pipe with periodically arranged membranes. All of these studies are mainly focused on the linear domain, the nonlinear research in this type of acoustic metamaterial has lagged far behind than that in the linear domain. Most researches on the nonlinear metamaterial are mainly about electromagnetic and optical waves, many intriguing nonlinear phenomena such as tenability [12,13], parametric shielding of electromagnetic fields [14] and parametric down-conversion (PDC) [15], as well as second- or third-harmonic generation [16–18] have already been investigated. In acoustics, the nonlinearities of acous-
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E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.physleta.2017.01.036 0375-9601/© 2017 Elsevier B.V. All rights reserved.
tic metamaterial cannot be ignored, particularly for the metamaterial based on the cylindrical pipe which has resonance phenomena in the tube or lattice and enhances the nonlinear effects in finite space [19,20]. Therefore, the nonlinearities in periodical structure along a cylindrical pipe must be paid more attention, and the nonlinear phenomena are considered to be the key to future research into the development of metamaterials [21,22]. The effects of an array of HRs on the propagation of nonlinear acoustic waves in a long tunnel were first investigated in 1992 by Sugimoto [23], it was found that appropriately designed resonators could effectively prevent the emergence of shock waves by introducing dispersion into acoustic waves. Theoretical and experimental studies of the nonlinear effects in propagation of acoustic Bloch waves in periodic waveguides were performed by Bradley [20], in which the nonlinearity of the system for a local second harmonic field has been investigated. Then, in 2007 Richoux et al. studied the nonlinear properties of the acoustic wave propagation based on the periodic lattices made of HRs [24]. The analytical study and experimental results showed that localized nonlinearities lead to the frequency bandgaps were amplitude dependent. In addition, the nonlinear effects in an acoustic metamaterial based on two types ordered lattices of side holes and membranes were studied by Fan et al. [25]. The nonlinearities of the membranes changed the characteristics of frequency bands and the nonlinear effects
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Fig. 1. (a) The model of the acoustic metamaterial based on a cylindrical pipe with periodically arranged Helmholtz resonators (HRs). (b) The picture and dimensions of a signal cell. (c) The equivalent mechanics prototype of a unit cell.
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of the side holes were very small and can be neglected. These researches show that the nonlinearities introduce an interaction between nonlinear effects and spatial periodicity. However, these works on the nonlinear acoustic metamaterial rarely discussed the influences of the nonlinear type (cubic, . . . ) on the propagation of acoustic waves, and the general expressions for the nonlinear bulk modulus and impedance of acoustic metamaterial have not been obtained. All of these questions need to be deeply investigated and resolved. In this paper, we design and analyze an acoustic metamaterial based on a cylindrical pipe with periodically arranged HRs. A straightforward homogenization procedure leads to generate an expression of the continuity equation for the composite acoustic metamaterial. Then based on the continuity equation and using the classical nonlinear perturbation method [17,26,27] which considering the nonlinear responses up to the third-order at the fundamental frequency, the expressions of the effective nonlinear bulk modulus B neff of the system and nonlinear impedance Z NHR of the HR that determine all the wave characteristics in the medium could be obtained. The nonlinear effects of the acoustic metamaterial are deeply investigated with the acoustic transmission line method (ATLM) [6], acoustic impedance theory and nonlinear Bloch theory. The results show that the localized nonlinearity due to the HRs can produce a shift in the position of the resonant frequency, and the forbidden bandgaps of the transmission spectra are amplitude dependent on the incident acoustic pressure level.
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2. Theoretical description of the metamaterial system
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A typical acoustic metamaterial based on a cylindrical pipe with ordered HRs is shown in Fig. 1. The inner diameter of the cylindrical pipe is l = 30 mm and the unit cell length of the metamaterial is L = 70 mm. Each HR is composed of a neck and a cavity, the sectional area and length of the neck are s = 7.85 × 10−5 m2 and l1 = 13.5 mm, respectively. The size of the cavity is determined by the inner cross section sc = 2.5 × 10−3 m2 and length l2 = 50 mm. The dimension of the HR is much smaller than the wavelength of incident wave, hence we can assume the acoustic pressure and density of the air inside the HRs are spatially uniform and the air movement in the neck can be as a whole [24,28]. Since the acoustic wavelength (at 292.5 Hz, λ ≈ 1.18 m in air) is much longer than the distance between two adjacent HRs (L λ), this system can be regarded as a homogenized medium [7]. A straightforward homogenization procedure leads to the HRs modify the continuity equation of air in the pipe [2], which can be expressed as
∂(ρ0 v 1 ) ρ0 v 2 s ∂ ρ0 = (1) sw − sw , ∂x L ∂t where ρ0 is the density of air, sw is the cross section area of the −
pipe and v 1 , v 2 are the acoustic velocities in the waveguide and HRs, respectively. It is worth noting that in this acoustic metamaterial the other acoustic nonlinear effects due to the intrinsic air behavior and Eulerian description of the movement are considered to be neglected compared to the nonlinearity of the HRs. Consequently, only the nonlinearity of the HRs is taken into account in this metamaterial [24,29]. Therefore, in Eq. (1) only the acoustic velocity v 2 in the HRs needs to consider the nonlinear effects. According to the constitutive equation p /ρ = B 0 /ρ0 and harmonic expression of acoustic pressure p = pa e − j (ωt −kx) , where ρ represents the density variable, B 0 = ρ0 c 02 is the bulk modulus of air and c 0 is the sonic velocity in air, Eq. (1) in frequency domain can be simplified to
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∂ v1 sv 2 jω − − =− p. ∂x Lsw B0
(2)
According to the Newton’s law, the motion equation of air in the HR neck can be obtained
Mm
dv 2 dt
+ R m v 2 − f (ξ ) = sp ,
(3)
where M m = ρ0 l1 s and R m = R vis + R rad are the mechanical mass and resistance acting as acoustic mass and resistance due to the HR neck, respectively. Because of radiation, the short neck may become “longer”, the effective length of short neck must be corrected as l1 = l1 + 1.7a 1 [30], where a1 is the radius of the short neck. f (ξ ) = −1/C m v 2 dt is the nonlinear restoring force term, where C m = V 0 /ρ0 c 02 s2 is the mechanical capacitance of the cavity, with the volume notation of V 0 = scl2 . The mechanical resistance is con√ sisted of two parts: the viscous losses R vis = a1 ρ0l1 ηω/2s at the walls of HR and the radiation losses R rad = (1/4)ρ0 c 0 (ka1 )2 s at the open end of HR neck tube [28], where η is the coefficient of kinematic viscosity (approximately 1.5 × 10−5 m2 /s in air), ω and k are the angular frequency and wave vector, respectively. With a new notation of restoring force F = −ξ/C m = v 2 / j ω C m , where ξ is the air displacement in the neck, the motion Eq. (3) can be rewritten as 2
d F dt 2
+δ
dF dt
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+ ω02 f (ξ ) = −ω02 sp ,
(4)
√
where δ = R m / M m is the damping coefficient and ω0 = 1/ C m M m is the resonant angular frequency. A HR is a tube–cavity system
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oscillating in a simple mass–spring model, the air in the tube moves as a unit and acts as the mass element and the air in the cavity acts as the spring. A general description of the nonlinear behavior of a HR is derived by only taking into account the restoring force of the spring and the nonlinearity of damping term can be neglected [24,28,31]. In this paper, the nonlinear behavior of HR is discussed by considering the cubic nonlinear term of the restoring force. The relative change of the pressure in the HR cavity due to displacement ξ of the air in the neck induces a restoring force which can be written in a cubic nonlinear form f (ξ ) = −[ξ + aξ 2 + bξ 3 ]/C m . Using the Taylor expansion the quadratic and cubic nonlinear coefficients can be expressed as a = (γ + 1)s/2V 0 and b = (γ + 1)(γ + 2)s2 /6V 02 , respectively, where γ ≈ 1.4 is the specific heat ratio of the air. By substituting f (ξ ) into Eq. (4) and writing in terms of the restoring force F , a third-order nonlinear motion equation can be obtained
18
d2 F
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dt 2
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+δ
dF dt
2 0F
+ω
2
3
2 0 sp ,
− α F + β F = −ω
where α = c ω = dω with c = aC m and d = An important feature can be noted from Eq. (5) that the spring is softer or harder depend on the value of displacement ξ . This makes it different from Duffing’s nonlinearity in which the spring hardening (or softening) no matter which sign the displacement is. For the general case, Eq. (5) usually cannot be solved exactly by using the classical method of successive approximations in nonlinear acoustics [24,29]. In such situation, it is often adequate to solve this equation through the use of classical perturbation theory [17,26, 27]. In order to solve Eq. (5) systematically in terms of a perturbation expansion, we replace the acoustic pressure p by λ p, where λ is a continuously varying parameter ranging from zero to unity that characterizes the strength of acoustic pressure and the value λ = 1 corresponds to the actual physical situation. We seek a solution to Eq. (5) in the form of a power series in λ 2 0, β
2 0,
2 bC m .
F (t ) = λ F (1) (t ) + λ2 F (2) (t ) + λ3 F (3) (t ).
43
p = pa e
− j (ωt −kx)
= pa B (ω)e
− j ωt
46
Λ
=
49
¨ (1 )
F
˙ (1 )
(t ) + δ F
2 (1 ) (t ) 0F
(t ) + ω
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¨ (3 )
F
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˙ (3 )
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F (1) (t ) =
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αF
pa B (ωn )e
− j ωn t
, (8a)
n=−Λ
2
(1 )
(t ) F
= 0,
(2 )
(8b)
(t ) (8c)
Λ
F (1) (ωn )e − j ωn t ,
(9a)
n=−Λ
F (2) (t ) =
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(7)
Equations (8a)–(8c) are the first- (linear), second- and thirdorder responses of the restoring force F , respectively. Now we look for a steady-state solution to Eqs. (8) of the form
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2 (3 ) (t ) − 2 0F
(t ) + δ F (t ) + ω 3 + β F (1) (t ) = 0.
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2 0s
F¨ (2) (t ) + δ F˙ (2) (t ) + ω02 F (2) (t ) − α F (1) (t )
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Λ
= −ω
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.
Substituting Eq. (6) into (5) and equating terms of the same order in λ, the system of coupled equations can be expressed as
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pa B (ωn )e
− j ωn t
n=−Λ
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(6)
In this study, the acoustic pressure p can be expressed as a discrete sum of Λ components, each of which with an angular frequency ωn
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(5)
F (2) (ωr )e − j ωr t ,
(9b)
r
F
(3 )
(t ) =
s
F
(3 )
(ωs )e
− j ωs t
,
(9c)
3
where ωr = ωn + ωm , ωs = ωn + ωm + ω p and the summations are taken over both positive and negative frequencies, with n, m, p each taking the value between ±Λ.
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2.1. The linear metamaterial system
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As the first step toward the study of nonlinear effects of the acoustic metamaterial, the linear system is primarily investigated. The motion equation and linear response of the restoring force F at the fundamental frequency ωn are presented in Eqs. (3) and (8a), respectively. Inserting Eq. (9a) into (8a), the expression for each ωn can be written as
−ωn2 F (1) (ωn ) − j ωn δ F (1) (ωn ) + ω02 F (1) (ωn ) = −ω02 spa B (ωn ), (10) from which the restoring force F at each
ω2 spa B (ωn ) F (1) (ωn ) = 0 , D (ωn )
ωn can be derived as
F
(12)
2 0 sp a B (
ω)
D (ω)
e − j ωt =
2 0 sp
ω
D (ω)
ω, Eq. (12) can be explicitly
v 2 = jωC m F = jω
v2
(13)
V 0 ω02 p B 0 sD (ω)
.
(14)
j ω V 0 ω0
B0
82 83 85 87 88 89 91 92
1−
V R ω02 D (ω)
94
97 99 100 101 102 103 105 106 108 109
2
.
110 111 112
(16)
where V R = V 0 / Lsw is the volume ratio of resonator cavity to waveguide section. According to Ref. [2], the effective bulk modulus of the system is defined as the ratio of the velocity gradient −∂ v 1 /∂ x to the change of acoustic pressure ∂ p /∂ t = − j ω p, hence
B leff
81
107
(15)
. 2
V R ω0 ∂ v1 1 − = − jω 1− p, ∂x B0 D (ω)
1
80
104
B 0 sD (ω)
=
79
98
Combining Eqs. (2) and (14), the continuity equation of the acoustic metamaterial can be simplified to
1
78
96
.
Then, in the linear situation, the effective impedance Z LHR of the HR is derived from Eq. (14),
=
77
95
Considering the expression of the restoring force F , the acoustic velocity v 2 in the HR can be given by
Z LHR =
76
93
For a continuous angular frequency written as
p
75
90
n=−Λ
ω
74
86
Λ ω02 spa B (ωn ) − jωn t (t ) = e . D (ωn )
F (t ) =
73
84
(11)
where the denominator is defined as D (ωn ) = ωn2 − ω02 + j δ ωn . By substituting Eq. (11) into (9a), the linear response of restoring force F can be obtained (1 )
72
113 114 115 116 117 118 119 120
(17)
The expression of the effective bulk modulus is identical to that in Ref. [7]. However, in Ref. [7] the bulk modulus is obtained by mapping the electromagnetic system into the acoustic analogues. Equation (17) shows that this acoustic √ metamaterial has two characteristic frequencies ω0 and ωn (≈ ω0 1 + V R ), and the real part of the effective linear bulk modulus of the system is negative in the frequency range of ω0 < ω < ωn . Fig. 2 shows the calculated results for the dispersion of the effective modulus of the system with respect to frequency from 200 to 700 Hz, in which the solid and broken curves stand for the
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where the indices n and m take any values between ±Λ. Equation (18) can be transformed into a set of independent equations and the second-order response F (2) (ωr ) of restoring force F at each frequency ωr can be expressed as
1 2 3 4 5 6
F
7
(2 )
(ωr ; ωn , ωm ) = −
8 10 11 12 13 14 15 17
Fig. 2. Real and imaginary parts of the effective linear bulk modulus of the acoustic metamaterial.
18 19
− 2α 2
22 23 25 26 27 28 29 30 31 32
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
Fig. 3. Transmission coefficient of the negative bulk modulus (B-NG) metamaterial with 50 units which has a forbidden band between two characteristic frequencies f 0 and f n .
F
2.2. The nonlinear metamaterial system In this section, we discuss the nonlinearity of the B-NG acoustic metamaterial. The linear analysis part above has discussed the first-order response of the system at the fundamental frequency. Using the similar approach, the second- and third-order responses of the restoring force F at the fundamental frequency or for the second- and third-harmonic generation accounting for the secondand third-order nonlinearities can be obtained. The third-order nonlinear motion equation of the HR is presented in Eq. (5), and Eqs. (8b) and (8c) are the expressions of the second- and third-order responses of the restoring force F , respectively. Using Eqs. (9b) and (12), Eq. (8b) can be written as 2 ¨ (2 ) ( r F
−ω ωr ) − j ωr δ F˙ (2) (ωr ) + ω02 F (2) (ωr ) ω4 s2 pa2 B (ωn ) B (ωm ) 0 , =α D (ωn ) D (ωm ) (nm)
+ c ω 3
×
1 D (ωn + ωm )
+
+
F (2) (2ωn ; ωn , ωn ) = −
c ω06 s2 pa2 B (ωn )2 D (ωn )2 D (2ωn )
8 3 3 0 s pa
(3ωn ; ωn , ωn , ωn ) = dω + 2c 2 ω010 s3 pa3
D (ωn
87 89 90 91 92 93 94 95 96 97
100 101
1 D (ωm + ω p )
102 103 104
105
.
(21)
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123
(22)
.
126 127 128
B (ωn )
129
D (ωn )3 D (3ωn )
ωn ) D (2ωn )
124 125
3
B (ωn )3
)3 D (3
80
99
For the third-order restoring force responsible for the thirdharmonic generation, Eq. (21) is rewritten for ωn = ωm = ω p (3 )
79
98
D (ωn ) D (ωm ) D (ω p ) D (ωs ) D (ωn + ω p )
78
88
B (ωn ) B (ωm ) B (ω p )
1
77
86
(20)
In combination with Eq. (9), Eqs. (11), (19) and (21) provide general expressions for the first-, second- and third-order responses of the restoring force F , respectively. Following we will provide the more explicit expressions for some specific combinatorial frequencies for the second- and third-order nonlinearities. i) F (3) (3ωn ; ωn , ωn , ωn ) is the third-order response of restoring force F for the third-harmonic generation and F (3) (ωn ) is the nonlinear response at the fundamental frequency ωn . ii) F (2) (2ωn ; ωn , ωn ) is the second-order response of restoring force F for the second-harmonic generation and F (2) (ωn ) is the nonlinear response at the fundamental frequency ωn . For example, for the second-order restoring force responsible for the secondharmonic generation, in case of ωm = ωn , Eq. (19) can be expressed as
F (18)
(nmp )
76
85
ω ωn ) B (ωm ) B (ω p ) , D (ωn ) D (ωm ) D (ω p ) D (ωn + ωm ) (nmp )
75
84
6 3 3 0 s pa B (
10 3 3 0 s pa
73
83
(ωs ; ωn , ωm , ω p ) B (ωn ) B (ωm ) B (ω p ) = dω08 s3 pa3 D (ωn ) D (ωm ) D (ω p ) D (ωs ) 2
72
82
(3 )
2
70
81
(nmp )
real and imaginary parts, respectively. Using Eq. (15) and ATLM, the effective impedance Z leff of the linear negative bulk modulus (B-NG) acoustic metamaterial with 50 units can be calculated. Then, the sound pressure reflection coefficient r p can be obtained with the formula r p = ( Z leff − Z )/( Z leff + Z ), where Z = ρ0 c 0 /sw is the distributed impedance of the pipe. Consequently, the acoustic intensity transmission coefficient T can be calculated to be T = 1 − |r p |2 . The evolution of the transmission spectrum is shown in Fig. 3. From the curve we can see that the system exhibits one forbidden bandgap during the frequency range from f 0 (292.5 Hz) to f n (550.0 Hz) which corresponds to the regime of the negative bulk modulus.
69
74
where the factor D (ωn + ωm ) in the second term on the right-hand side of Eq. (20) results from the second-order response F (2) (ωr ) at frequency ωr = ωn + ωm which produces in combination the frequency ωr . In order to account for the various contributions, the factor should be traversed over all possible combinations of indices n and m. Then from Eq. (20), the third-order response of restoring force F at each ωs accounting for the second- and third-order nonlinearities can be written as
24
35
(19)
.
(nmp )
21
34
(nm)
D (ωn ) D (ωm ) D (ωn + ωm )
68
71
−ωs2 F¨ (3) (ωs ) − j ωs δ F˙ (3) (ωs ) + ω02 F (3) (ωs ) ω6 s3 pa3 B (ωn ) B (ωm ) B (ω p ) 0 = −β D (ωn ) D (ωm ) D (ω p )
20
33
c ω06 s2 pa2 B (ωn ) B (ωm )
The parentheses of (nm) indicate that the sum of ωn + ωm is constant, while the indices n and m are variable and Eqs. (18) and (19) sums over all possible values of ωr . The third-order response of the restoring force can be derived in the similar method, combining Eqs. (9c), (12), (19), and (9b), Eq. (8c) can be expressed as
9
16
67
130 131
.
(23)
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5
Here, for the nonlinear response it is important to note that F (2) (ωr ; ωn , ωm ) has no nonlinear response at the fundamental frequency ωn . However, the nonlinear response F (3) (ωn ) at the fundamental frequency is result from both cases, the one is the interaction of a field with itself at frequency ωn which leads to a self-phase modulation and the other one is the interaction of two different fields at frequencies ωn and ωm which leads to a cross-phase modulation [17]. For the first case, we set ωm = ωn and ω p = −ωn in Eq. (21), resulting in
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10 11
76
F
(3 )
(ωn ; ωn , ωn , −ωn )
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
=
8 3 3 s pa B ( n ) 4 3 0 3 D ( n ) D (− n ) 3
ω ω
ω ω
77
c
2
2 0
ω
D (0)
+
2
2 0
ω +d . 3 D (2ωn ) 2 c
78 79
(24)
80 81
For the other case (cross-phase modulation), considering
82
ω p = −ωm in Eq. (21), which gives
ω08 s3 pa3 B (ωn ) F (3) (ωn ; ωn , ωm , −ωm ) = 6 D (ωn )2 D (ωm ) D (−ωm ) 2 2 c ω0 c 2 ω02 2 2 2 c 2 ω02 × + + +d . 3 D (ωn + ωm ) 3 D (ωn − ωm ) 3 D (0)
83
Fig. 4. (Color online.) Real and imaginary parts of the effective linear and nonlinear bulk modulus for the metamaterial.
(25)
Note that the coefficients number 3 and 6 on the right-hand sides of Eqs. (24) and (25) are all resulting from the summation to account for the possible number of permutations. Now we assume that only one single incident frequency ωn exists in the total nonlinear responses. According to Eqs. (6), (9), (11), and (24), the expression of the restoring force F by considering the total nonlinear responses expanded up to the third-order at the fundamental frequency can be obtained Λ Λ ω02 spa B (ωn )e− jωn t ω08 s3 pa3 B (ωn )e− jωn t F (t ) = +3 D (ωn ) D (ωn )3 D (−ωn ) n=−Λ n=−Λ 2 2 4 c ω0 2 c 2 ω02 (26) × + +d , 3 D (0) 3 D (2ωn )
where on the right-hand side of Eq. (26), the first term is the linear response part, the second term is the nonlinear response part. Note that the nonlinear response F (2) (ωn ) at the fundamental frequency ωn does not exist and only the nonlinear response F (3) (ωn ) is considered in Eq. (26). However, the nonlinear response F (3) (ωn ) at the fundamental frequency ωn contains the secondand third-order nonlinearities. Thus the restoring force F which contains linear and nonlinear parts is solved. According to the relation v 2 = j ω C m F , and then one explicit expression of the velocity, as a function of a continuous angular frequency ω , can be given by
ω02 p j ω ω08 sV 0 pa2 p v 2 = jωC m F = jω +3 B 0 s D (ω) B 0 D (ω)3 D (−ω) 2 2 4 c ω0 2 c 2 ω02 + +d . × 3 D (0) 3 D (2ω) V0
(27)
From Eq. (27), the effective impedance of the HR which includes nonlinear response component can be calculated
V 0 ω02 Z NHR = 1 jω B 0 s D (ω) 2 2 j ω ω08 sV 0 pa2 4 c ω0 2 c 2 ω02 + 3 . d + + B 0 D (ω)3 D (−ω) 3 D (0) 3 D (2ω) (28) The acoustic velocity v 2 in the HR driven by the acoustic pressure p in the pipe has been obtained in Eq. (27), by substituting
it into the continuity Eq. (2), the continuity equation of this metamaterial system can be rewritten
∂ v1 1 1 V R ω02 1 3V R ω08 s2 pa2 − = − jω − − ∂x B0 B 0 D (ω) B 0 D (ω)3 D (−ω) 2 2 2 2 4 c ω0 2 c ω0 × + + d p. 3 D (0) 3 D (2ω)
B neff
=
1 B0
×
1−
V R ω02 D (ω)
4 c 2 ω02
+
−
2 c 2 ω02
87 88
91
(29)
92 93 94 95 96 97 98
)3 D (−
ω)
+d .
86
90
3V R ω08 s2 pa2 D (ω
85
89
From Eq. (29), the effective nonlinear bulk modulus of the system which contains nonlinear effects can be derived as
1
84
99
(30)
100
It is note that the third term in the square bracket on the righthand side of Eq. (30) is the nonlinear part, and if without this term Eq. (30) arrives at the expression of the effective linear bulk modulus Eq. (17). We can draw from Eqs. (28) and (30) that the effective nonlinear impedance of HR and bulk modulus are all depended on the acoustic amplitude pa of incident acoustic pressure. Fig. 4 shows the calculated dispersion results of the effective linear and nonlinear bulk modulus of the B-NG acoustic metamaterial. For the nonlinear calculation, Eq. (30) has been used and the incident acoustic pressure level is 160 dB. By comparing the two curves for the real parts of the linear (black color) and nonlinear (red color) bulk modulus, we can obviously observe that there exists a left shift of the zero crossing point in the nonlinear case, which means that the resonant frequency f 0 is decrescent. The effective impedance Z NHR of the HR has been obtained considering the total nonlinear responses up to the third-order. The acoustic pressure amplitude pa introduced in Eq. (28) should be depended on the HR position, but we can apply only one global pressure amplitude value for all units as outlined in Ref. [24]. By applying the method of ATLM, the effective nonlinear impedance Z neff of the B-NG acoustic metamaterial with 50 units can be calculated. Thus, the theoretical nonlinear transmission spectra of the system with the different input acoustic pressure levels (130, 140, 150, and 160 dB) can be obtained and plotted in Fig. 5. From the curves we can see that, comparing with the linear transmission spectrum, the nonlinearity makes the resonant frequency f 0 shifted toward lower frequencies with increasing input acoustic pressure level, which broadens the forbidden bandgap’s width. It is well known that the acoustic impedance theory and Bloch theory are used to explain the unique characteristics of the acoustic metamaterial [10]. Following, we will carry out the theoretical
102
3 D (0)
3 D (2ω)
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(c) 160 dB. It is found that the acoustic resistance (solid curve) is smaller than the acoustic reactance (broken curve) in the frequency range of f 0 < f < f n , which means that the most of the incident energy in this frequency range is reflected back to the source and the forbidden band occurs in the acoustic metamaterial. It is also noted that when the acoustic pressure level is increasing, the resonant frequency f 0 corresponding to the lower limit of the bandgap moves toward the lower frequency while the upper limit frequency f n of the bandgap keeps constant. These results are exactly consistent with the transmission coefficient as shown in Fig. 5. Thus, the intrinsic parameters Z leff and Z neff can be used to characterize the properties of the acoustic metamaterial. Let us continue to discuss the nonlinear effects of the acoustic metamaterial using the Bloch theory, the nonlinear Bloch dispersion equation for the B-NG metamaterial can be written as [1,25]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Fig. 5. (Color online.) Nonlinear transmission spectra for the negative bulk modulus (B-NG) metamaterial with 50 units, the curves show that the nonlinearity makes the lowest side of the forbidden bands shifted toward lower frequencies with increasing incident acoustic pressure level.
19 20 21 22 23 24 25 26 27 28
analyses to understand the observed nonlinear behaviors by using these two theories. We know that the acoustic resistance (real part of Z leff or Z neff ) of the metamaterial system produces the transmission of the acoustic wave while the acoustic reactance (imaginary part of Z leff or Z neff ) leads to the reflection of the sound [6]. Fig. 6 shows the calculated dispersion curves of the acoustic resistance and acoustic reactance for the metamaterial system with three different incident acoustic pressure levels: (a) Linear, (b) 140 dB, and
cos(qL ) = cos(kL ) +
j ρ0 c 0 2sw Z NHR
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82
sin(kL ),
(31)
where q is the Bloch wave number. According to the dispersion theory, in the frequency range of | cos(qL )| > 1 the acoustic wave propagates in a forbidden band. Fig. 7 shows that with increasing incident acoustic pressure level, the resonant frequency f 0 which corresponds to | cos(qL )| = 1 moves toward the lower frequency and the characteristic frequency f n stays unchanged, as a consequence the forbidden bands become wider. In the frequency range of f 0 < f < f n , such a stop band is ascribed to the traditional physical mechanism of the absorption due to the local resonance
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95
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96
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99
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101
36
102
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103
38
104
39
105
40
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41
107
42
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Fig. 6. (Color online.) Calculated dispersion curves of acoustic resistance (solid curve) and reactance (broken curve) for the negative bulk modulus (B-NG) metamaterial with the different incident acoustic pressure levels: (a) Linear, (b) 140 dB, and (c) 160 dB.
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paper provides a potential possibility for applications to investigate other nonlinear phenomena of acoustic metamaterial.
1 2 3 5
This research work was supported by the National Natural Science Foundation of China (grant Nos. 61571222, 11104142, and 11474160), by the Natural Science Foundation of Jiangsu Province, China (No. BK20161009), sponsored by Six Talent Peaks Project of Jiangsu Province, China, and Qing Lan Project of Jiangsu Province, China.
7 8 9 10 11 12
References
14 16
Fig. 7. (Color online.) cos(qL ) of the negative bulk modulus (B-NG) metamaterial with increasing input acoustic pressure level.
19
22 23 24 25 26 27 28 29 30
of the HRs. Furthermore, it can be seen from the inset zoom part of the curves in Fig. 7, that for the linear or small incident acoustic pressure level (130 dB), the resonant frequency f 0 ≈ 292.5 Hz, while for the acoustic pressure level 160 dB, f 0 ≈ 278.0 Hz, the corresponding increased percentage of bandwidth for the nonlinear metamaterial compared to the linear one is about 5.6%. Therefore, the nonlinear acoustic behavior of the dispersion relation of the acoustic metamaterial is only visible at the resonant frequency, which is due to the nonlinearity of the HRs. The analyzed results of Bloch theory are coincident to that obtained with the acoustic impedance theory and ATLM analysis described above.
31 32
3. Conclusion
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Acknowledgements
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In conclusion, the nonlinear properties of the negative B-NG acoustic metamaterial based on the cylindrical pipe with periodically arranged HRs have been investigated theoretically. The second- and third-order nonlinear responses at the fundamental frequency have been analyzed in detail by using the perturbation solution. The theoretical analyses show that, with increasing incident acoustic pressure level, the nonlinearity of the system leads to resonator frequency shift toward the lower frequency side and broadens the width of the forbidden band. Compared with the other previous studies about the B-NG acoustic metamaterial, we made some significant improvements, such as getting the explicit expressions of the nonlinear acoustic impedance of the HR and effective nonlinear bulk modulus of the metamaterial system. The nonlinear model obtained with the perturbation solution provides a very close description of the interaction of the acoustic wave with the HRs. The analytical perturbation method employed in this
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