Low-frequency forbidden bandgap engineering via a cascade of multiple 1D

21 October 2018 Volume 124 Number 15 Low-frequency forbidden bandgap engineering via a cascade of multiple 1D superlattices DOI: 10.1063/1.5049514 Low-frequency forbidden bandgap engineering via a cascade of multiple 1D superlattices Sai Zhang, Yan Zhang, Wei Lu, Guanghua Hu, Bai-qiang Xu, and Wenwu Cao Citation: Journal of Applied Physics 124, 155102 (2018); doi: 10.1063/1.5049514 View online: /10.1063/1.5049514 View Table of Contents: /toc/jap/124/15 Published by the American Institute of Physics Low-frequency forbidden bandgap engineering via a cascade of multiple 1D superlattices Sai Zhang, 1,2 Yan Zhang, 1 Wei Lu, 1 Guanghua Hu, 1 Bai-qiang Xu, 1,a) and Wenwu Cao 2,a) 1 Department of Physics, Jiangsu University, Zhenjiang 212013, China 2 Department of Mathematics and Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, USA (Received 23 July 2018; accepted 13 September 2018; published online 15 October 2018) Low-frequency forbidden bandgap (LFB) in solid-uid superlattice (SFSL) provides a novel degree of freedom to control the propagation of low frequency acoustic waves. However, its narrow angle range seriously restricts its practical applications. To engineer the angle range of LFB, the transmis- sion coefcient of a cascade of multiple 1D superlattices was calculated using the transfer matrix method and also veried by nite element simulations. It was found that the LFB with ultra-wide angle range can be created using only 4 cells of a two-phased cascade structure and can be adjusted by changing the solid lling fraction of each sub-SFSL. By introducing two or more 1D superlatti- ces into the layered cascade structure, the LFB with multiple angle ranges and/or ultra-wide angle range can be engineered for low frequency acoustic waves whose wavelengths are much larger than the lattice constant. Such acoustic low-frequency forbidden bandgap materials are useful for making perfect acoustic low-frequency lters with broad bandwidth in selective angle ranges, which are useful in subwavelength acoustic devices. Published by AIP Publishing. /10.1063/1.5049514 I. INTRODUCTION The propagation of acoustic waves or elastic waves in composite structures has attracted much attention due to its fundamental importance in wave propagation and its promis- ing application potential in modern acoustic technologies. 112 In the last two decades, low-frequency forbidden bandgap (LFB) engineering has been successfully realized using locally resonant sonic crystals. 1315 The fundamental princi- ple is to use localized resonant structures with a lattice cons- tant much smaller than the relevant wavelength to create spectral bandgaps. Recently developed acoustic space-coiling metamaterials offer a novel approach for low-frequency or subwavelength airborne sound insulation. 16,17 For example, Cheng et al. fabricated an ultra-spare space-coiling metasur- face for high reection of low-frequency sound based on arti- cial Mie resonances. 16 Krushynska et al. proposed a spider web-structured labyrinthine and successfully realized subwa- velength bandgap with broad bandwidth. 17 In addition, one dimensional Bragg sonic crystals made of alternating elastic solid and uid layers, also called solid-uid superlattices (SFSLs), have also been demonstrated to have the ability to create LFB for waterborne sound. 1823 In a SFSL system, LFB exists with a broad bandwidth in the frequency range lower than the rst Bragg bandgap, which is part of non-Bragg bandgaps originated from the transmission zeros instead of Bragg resonance. 18,19 Obviously, such LFB provides a novel degree of freedom to engineer bandgaps for acoustic waves of long wavelength and have many potential applications in subwavelength underwater acoustic devices. For instance, a subwavelength asymmetric acoustic transmission device based on LFB has been realized in Refs. 20 and 21. It should be pointed out that two characteristics determine whether the LFB in SFSL can play a role in practical applications: the angle position and the angle range, which have been investigated in Refs. 22 and 23, respectively. Among them, the latter is par- ticularly important. In order to obtain an LFB with a rela- tively broad angle range in SFSL systems, an effective strategy was proposed. 23 However, for most SFSLs, the angle range of LFB is determined by the intrinsic properties of the constituent materials and thus is usually very limited. It is still a challenge to achieve low-frequency forbidden bandgap materials with multiple angle ranges and/or ultra-wide angle range, which are useful for making perfect acoustic low- frequency lters with broad bandwidth in selective angle ranges and are useful in subwavelength acoustic devices, such as acoustic diodes. The current work aims to utilize the idea of bandgap engineering to design material structures that have LFB with multiple incident angle ranges and/or ultra-wide angle range for underwater acoustic waves. First, a nite one-dimensional cascade structure containing multiple 1D superlattices is pro- posed, in which the acoustic wave propagation will be theo- retically studied based on the transfer matrix method and veried by the nite element simulations. Then, we show that the LFB with multiple incident angle ranges and/or ultra- wide angle range in the cascade structure can be achieved. Finally, the inuence of constituent materials and geometric parameters of the designed structure on the formation of LFB is investigated. a) Authors to whom correspondence should be addressed: and JOURNAL OF APPLIED PHYSICS 124, 155102 (2018) 0021-8979/2018/124(15)/155102/8/$30.00 124, 155102-1 Published by AIP Publishing. II. STRUCTURE DESIGN AND THEORETICAL METHOD As schematically shown in Fig. 1, the cascade structure we considered is made of two sub-SFSLs having the same lattice constant D with cell numbers m 1 and m 2 , respectively. The total number of layers is N= m 1 +m 2 . Each sub-SFSL can produce its own LFB with different angle ranges. If the LFBs of the two sub-SFSLs are not overlapped in certain incident angle range, acoustic waves cannot pass through the cascade system in these two separated angle ranges creating a LFB with multiple angle ranges; if they are partially over- lapped in certain angle range, acoustic waves within a wider angle range cannot pass through the entire structure, thus an ultra-wide LFB could be obtained. In the structure shown in Fig. 1, element type 1 and type 3 are two different solid materials while type 2 and type 4 are both water. The whole structure is immersed in water. The lling fractions of material 1 and material 3 are f c1 and f c2 , respectively. In homogeneous linear elastic material without body force, the equation of motion for velocity vector can be written as 2 t 2 rC2(rC2) ( 2)r(rC1), (1) where is the mass density of material, and the terms and are the two Lam constants, respectively. In uid, does not exist; thus, the motion equation in uid can be written as 2 t 2 C0 r(rC1) 0: (2) The velocity eld in solid can be separated into scalar and vector potentials according to Helmholtz decomposition, which satises the following relations: rrC2, (3a) rC1 0, (3b) where and describe the longitudinal and the transverse waves, respectively. Assume that a longitudinal acoustic plane wave was sent from the top with wave vector k and incident angle at z=0. At the time t, the wave function of the longitudinal wave nj can be written as nj A nj e i(2ftC0k L j xsin L j C0k L j zcos L j ) B nj e i(2ftC0k L j xsin L j k L j zcos L j ) , (j 1C0 4, n 1C0 N), (4) where the rst and the second terms on the right-hand side of Eq. (4) represent the forward and reected longitudinal waves, respectively, k L j 2f=c L j is the wave vector, L j arcsin(k=k L j sin) is the oblique incident angle of the longitudinal wave, and f and c L j are the frequency and phase- velocity of the longitudinal wave, respectively. The indices n and j are the cell number and material type, respectively. When =0 C14 , the transverse wave nj may be excited in the solid layer, which can be written as nj C nj e i(2ftC0k T j xsin T j C0k T j zcos T j ) D nj e i(2ftC0k T j xsin T j k T j zcos T j ) , (j 1or3,n 1 C0 N), (5) where the rst and the second terms on the right-hand side of Eq. (5) represent the forward and reected transverse waves, respectively, k T j 2f=c T j is the wave vector, T j arcsin(k=k T j sin) is the oblique incident angle, and c T j is the phase-velocity of the transverse wave. Under the boundary conditions of the normal velocity continuity and the stress balance at the interfaces of solids and uids 24 and using the transfer matrix method, 25 we can derive the transfer matrix T (1) n for the nth cell (1nm 1 )of FIG. 1. Schematic of the cascade structure studied, which consists of two sub-SFSLs with the same lattice constant D. Materials 1 and 3 are two differ- ent solids and materials 2 and 4 are both water. TABLE I. Material properties for the solid and fluid constituents of the cascade structure. 24 Material Mass density (kgm 3 ) Longitudinal velocity (ms 1 ) Transverse velocity (ms 1 ) Lead 11400 2160 780 Rubber 1200 2300 940 Epoxy 1100 2400 1100 PMMA 1180 2700 1300 Copper 8900 4710 2260 Steel 7800 6100 3300 Aluminum 2700 6260 3080 Water 1000 1500 155102-2 Zhang et al. J. Appl. Phys. 124, 155102 (2018) the upper sub-SFSL and T (2) n for the nth cell (m 1 +1nN) of the lower sub-SFSL, which are given in the Appendix. Thus, we obtain the total transfer matrix T 1,N Q m 1 n1 T (1) n C1 Q N nm 1 1 T (2) n connecting the upper and lower sides of the whole cascade system. The total transmission and reection rates can be obtained from the transfer matrix T 1,N , 26 T 1,N 1=t r r C3 e =t C3 r r e =t r 1=t C3 r C18C19 , (6) where t r and r e are the transmission and reection coef- cients, respectively, and T jt r j 2 and R jr e j 2 dene the transmission and reection rates. Based on the above method, we can calculate the transmission spectrum of acous- tic waves at arbitrary incidence for the given design in Fig. 1. Besides, in order to demonstrate the accuracy of the trans- fer matrix method, the theoretical results are also validated by the nite element simulations using the acoustic module in the COMSOL Multiphysics software. As the proposed design in Fig. 1 is innite along the x axis, we only simulated a portion of the structure and applied periodic Floquet boundary condi- tions to extend the domain to innity. 27 The incident eld is modeled by applying a background pressure eld dened as p inc e C0i(kxsinkzcos) in the water. In this theoretical study and numerical simulations, the dissipation properties of all materials are not considered for simplicity. The material parameters used in the calculations are given in Table I. III. RESULTS AND DISCUSSION The band structures of seven innite SFSLs are shown in Fig. 2, where the yellow area and the blue area represent the passband and stop band, respectively. In the calculations, the lattice constant is chosen as D=3mm, and the solid lling fraction of each SFSL is xed at 0.5. From Figs. 2(a)2(g), it is obvious that a broad LFB (see the blue area) exists in the low-frequency range in each sub-gure. For the incident waves with a frequency of 80kHz, its wave- length is about 6.25 times of the lattice constant D; thus, the FIG. 2. Band structures of innite SFSLs. (a) aluminum-water SFSL, (b) steel-water SFSL, (c)copper-water SFSL, (d) PMMA-water SFSL, (e) epoxy-water, (f) rubber-water SFSL, and (g) lead-water SFSL. Yellow and blue represent the passband and stop band, respectively. FIG. 3. Transmission spectrum for (a) PMMA-water SFSL and (b) epoxy- water SFSL. (c) Transmission spectrum for the cascade structure shown in Fig. 1. The solid materials 1 and 3 are epoxy and PMMA, respectively. (d) Transmission coefcient as a function of the incident angle calculated by the transfer matrix method and nite element method at the frequency of 80 kHz for the cascade structure designed in Fig. 3(c). 155102-3 Zhang et al. J. Appl. Phys. 124, 155102 (2018) LFBs are subwavelength bandgaps. Besides, we can nd that the angle position and angle range of LFB differ greatly with different solid constitutions of SFSLs. Usually, a larger angle position is accompanied by a wider angle range. It should be pointed out that, unlike the complete bandgaps in two or three dimensional sonic crystals, 13,17 there is clearly a limit for the angle range of LFB in 2-element SFSLs, which is not desired for practical applications. In practice, it is more meaningful to study the transmis- sion properties of nite SFSLs. The low frequency transmis- sion characteristics of two special 16-layer SFSLs, e.g., the PMMA-water SFSL and the epoxy-water SFSL, are shown in Figs. 3(a) and 3(b), respectively, where the transmission is calculated using the transfer matrix method as a function of frequency and incident angle. From Fig. 3(a), we can nd a conspicuous LFB (it should be called quasi-LFB) with the transmission coefcients close to zero within a certain angle range near 40, while another LFB occurs within a certain angle near 48 in Fig. 3(b). Obviously, the angle position and angle range of the LFBs in Figs. 3(a) and 3(b) corre- spond well with the bandgaps of innite SFSLs shown in Figs. 2(d) and 2(e), respectively. Besides, Gibbs-type oscilla- tions that take place with N1 (here N=16) peaks can be clearly seen in the rst Bragg passband. 28 It should be men- tioned that the LFB exists with a broad bandwidth below 240kHz and its corresponding wavelength is much larger than the lattice constant D, indicating its promising application potential in broadband low frequency lter and subwavelength devices. However, for the 1D superlattice in Figs. 3(a) and 3(b), their LFBs are restricted by the solids and have only a single narrow angle range. In Fig. 3(c), we display the transmission spectrum for the cascade structure designed in Fig. 1, where the upper and the lower sub-SFSLs are made of PMMA and epoxy with water, respectively. The lattice constant and solid lling frac- tion of each sub-SFSL are consistent with Figs. 3(a) and 3(b). The cell numbers m 1 and m 2 are both 8, and thus the total cell number of the cascade structure is still 16. From Fig. 3(c), it is intriguing to nd that two separated LFBs are indeed obtained near 40 and 48, respectively; thus, a lter of narrow passband is obtain near 43. Obviously, the two LFBs are the intrinsic bandgaps of each sub-SFSL, which are shown in Figs. 3(a) and 3(b). Therefore, we can engineer LFB with multiple angle ranges by simply assembling two SFSLs as expected. In addition, the transmission coefcients of acoustic waves in the cascade structure as a function of the incident angle at 80kHz is shown in Fig. 3(d). We can nd that the agreement between the transfer matrix calculations and nite element simulation results is excellent, which vali- dated the accuracy of the transfer matrix theory. Figure 4 shows the transmission coefcients for the cascade structure with different sub-SFSLs from the transfer matrix method. The shadow area with angle range from 45 to 51 in Figs. 4(a)4(f) represents LFB induced by the epoxy-water sub-SFSL. By introducing the lower sub-SFSL made of solid material 3 and water, another LFB with chang- ing angle position and angle range is produced. It is found that when the solid material 3 is relatively “harder” to water, the LFB is extremely narrow, as shown in Figs. 4(a)4(c) indicated by the arrows marked 1, 2, and 3. However, when the solid material 3 is relatively “softer” to water, wide LFBs FIG. 4. Transmission coefcient as a function of the incident angle for the cascade structure shown in Fig. 1 with different sub-SFSLs calculated by the transfer matrix method. The cell numbers m 1 and m 2 are both 8, and the inci- dent frequency is 80kHz. FIG. 5. Transmission coefcient as a function of the incident angle for the designed cascade structure in Fig. 4(e) with different cell numbers at 80kHz. 155102-4 Zhang et al. J. Appl. Phys. 124, 155102 (2018) can be obtained as shown in Figs. 4(d)4(f), indicated by the arrows marked 4, 5, and 6. Thus, material 3 with a lower lon- gitudinal velocity or transverse velocity will lead to another LFB with a larger angle position and a wider angle range. It is worth noting that Fig. 4(e) shows the transmission spectrum for a cascade design, for which the two sub-SFSLs have over- lapped LFBs. As a result, acoustic waves cannot pass through from 44.9 to 62.4, leading to an ultra-wide LFB of about 17.5. As a matter of fact, such ultra-wide LFB can be pre- dicted from the band structures shown in Figs. 2(e) and 2(f), where the two LFBs of sub-SFSLs have an overlapping area. In practice, in order to engineer an ultra-wide LFB using the proposed structure, an effective strategy is to cascade another sub-SFSL, of which the solid material 3 is relatively “soft” and its acoustic impedance is close to that of solid material 1, such as the case shown in Fig. 4(e). We further investigated the inuence of the cell number of each sub-SFSL on the formation of LFB for the designed cascade structure in Fig. 4(e); the results are shown in Fig. 5. It is found that even in the case of a single epoxy plate with a single rubber plate immersed in water see Fig. 5(a), the transmittance become exactly zero near 49.8 and 60.2 at 80kHz. When synchronously increasing the cell number of each sub-SFSL, the angle ranges shown in Figs. 5(b)5(d) increase on the basis of Fig. 5(a). It is also found that LFB with ultra-wide angle range can be created using only 4 cells of a t