1.College of Computer Science and Technology, Nanjing Tech University, Nanjing 211800, China 2.Key Laboratory of Modern Acoustics, Ministry of Education, Institute of Acoustics, Nanjing University, Nanjing 210093, China
Fund Project:Project supported by National Natural Science Foundation of China (Grant No. 61571222) and the Six Talent Peaks Project in Jiangsu Province, China
Received Date:21 December 2020
Accepted Date:08 March 2021
Available Online:07 June 2021
Published Online:05 August 2021
Abstract:Owing to the low energy density of sound energy in nature, it is difficult to realize the local enhancement effect of sound field in air. Therefore, it is of great significance to explore new physical mechanisms and methods to achieve sound field enhancement. In recent years, artificial Mie resonance structure as a kind of acoustic metamaterial has attracted considerable attention, which has a variety of resonant modes, such as monopolar, dipolar, quadrupolar and higher multipolar modes. Compared with local resonance, acoustic Mie resonance mode has strong acoustic interaction, which can effectively enhance the acoustic field by the coupling of the Mie resonance. In this paper, we design an acoustic metamaterial composed of multiple-cavity unit cells, which is capable of realizing sound field enhancement. The multiple-cavity unit is circular in external shape and it is composed of a circular central cavity and twelve resonators. The twelve resonators are evenly distributed around the circular central cavity, with three resonators combined into a group. This exotic function arises from the compound monopole Mie resonance introduced by mutual coupling between the system structure and the monopole Mie resonance of each unit cell. Symmetric and asymmetric metamaterials are constructed by arranging several multiple-cavity unit cells in different forms. These two kinds of metamaterials can be used to achieve sound field enhancement with different effects. The results show that due to the symmetry of metamaterial structure, the symmetric metamaterials with square, circle, rectangle and regular hexagon shapes can realize the sound field enhancement, which is independent of the direction of incident wave. However, for the asymmetric metamaterial with equilateral triangle shape, the sound intensity in the center of the system varies with incident direction, which indicates that the designed asymmetric metamaterial has a strong dependence on the direction of incident wave. These two kinds of metamaterials constructed in this research can possess a number of potential applications such as in sound insulation, acoustic sensor, noise location, acoustic communication and asymmetric acoustic device. These two kinds of metamaterials constructed in this research can possess a number of potential applications such as in sound insulation. Keywords:Mie resonance/ symmetric metamaterial/ asymmetric metamaterial/ sound field enhancement
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2.理论分析本文设计的多腔型基本单元的结构如图1(a)所示, 该基本单元的外部形状是1个圆形, 其由1个圆形中心腔和12个谐振腔共同组成, 可以分为4个相同的部分, 其中12个谐振腔以3个为一组围绕中心腔的四周均匀分布. 中心腔和谐振腔内填充的媒质均为空气, 腔壁选取的材料为环氧树脂. 腔壁的厚度为t, 中心腔的开口宽度均为a, 谐振腔的开口宽度均为w, 圆形中心腔的半径为r1, 基本单元的外部圆形的半径为R1. 所有的结构参数分别为t = 3 mm, a = 5 mm, w = 4 mm, r1 = 2.7 cm和R1 = 5 cm. 环氧树脂的材料参数分别为密度${\rho _1} = 1050\;{\rm{ kg/}}{{\rm{m}}^3}$、杨氏模量E = 5.08 GPa和泊松比v = 0.35; 背景媒质空气的密度和声速分别为ρ0 = 1.21 kg/m3和c0 = 343 m/s. 本文在r1和R1之间引入两个具有相同场的虚拟层, 基于声散射理论可以计算出多腔型基本单元的各阶Mie共振所对应的共振频率. 将入射声源视为平面波${p_{{\rm{in}}}} = {p_0}{{\rm e}^{{{\rm i}}(kr\cos \phi - \omega t)}}$, 其中${p_0}$为声压振幅, i表示虚部, k是空气中的波数, $\omega $是角频率, $\phi $是相位, r是半径. 声压分布可以用Bessel (Jm)函数和Hankel (Hm)函数表示: 图 1 (a)多腔型基本单元的结构示意图; (b)特征频率为672 Hz时, 基本单元在单极子Mie共振模式下的声强分布图; (c)当频率为672 Hz的声波从结构左侧垂直入射时, 多腔型基本单元的声强分布图 Figure1. (a) Structural diagram of multiple-cavity unit; (b) sound intensity distribution of the unit in monopole Mie resonance mode at the characteristic frequency of 672 Hz; (c) sound intensity distribution of the unit when the sound wave with frequency of 672 Hz is normally incident from the left side.
基于该多腔型基本单元, 设计了一系列由多个基本单元组成的复合结构, 用以实现复合结构的单极子共振. 如图2(a)所示, 将8个相同的多腔型基本单元组成了一个外部形状为正方形的对称型超构材料, 基本单元的结构参数与图1(a)的相同, 相邻两个基本单元中心点之间的距离d为15 cm. 图2(b)给出了该对称型超构材料在特征频率为669 Hz时所对应的复合单极子Mie共振模式下的声强分布. 在该频率点处, 设计的正方形对称型超构材料处于单极子共振模式, 可以将声能集中在系统的中心区域, 实现声场增强效应. 图 2 (a)由8个多腔型基本单元组成的正方形系统示意图; (b)特征频率为669 Hz时, 正方形系统在复合单极子Mie共振模式下的声强分布图 Figure2. (a) Schematic diagram of a square system composed of eight multiple-cavity units; (b) sound intensity distribution of the square system in the compound monopole Mie resonance mode at the characteristic frequency of 669 Hz.
上述的研究通过使用特征频率分析求得该正方形系统的单极子Mie共振模式所对应的共振频率为669 Hz. 图3(a)和图3(b)分别是频率为669 Hz的平面波从该正方形系统的左侧和右侧垂直入射时系统的声强分布图, 可以发现, 该正方形系统的复合单极子共振模式在669 Hz频率处被激发, 声场增强效应在此频率处实现. 同时, 声波从左侧和右侧垂直入射时声强的分布均相同, 说明该系统具有良好的对称性. 为了与多腔型基本单元作对比, 与其样品同尺寸同分布的8个硬边界圆形结构被放置在相同位置, 相应的声强分布如图3(c)所示. 通过对比图3(a)、图3(b)与图3(c), 可以发现在相同入射声波的情况下, 正方形对称型超构材料样品内部中心点处的声强明显比硬边界圆形结构的要大, 这意味着超构材料样品具有更好的声能聚焦效果, 有效地体现了该对称型超构材料在复合单极子共振模式下实现的声场增强效应. 图 3 声波从(a)左侧和(b)右侧垂直入射到8个多腔型基本单元组成的正方形系统时的声强分布图; (c)声波从左侧垂直入射到由8个硬边界圆形结构组成的系统时的声强分布 Figure3. Sound intensity distributions of the square system composed of eight multiple-cavity units when the sound wave is normally incident from (a) the left side and (b) the right side, respectively; (c) sound intensity distribution of the system composed of eight hard boundary circular structures when the sound wave is normally incident from the left side.
除此之外, 还可以利用该多腔型基本单元组成了其他不同形状的对称型超构材料, 验证不同的对称结构是否存在声场增强效应. 结构设计分别如图4(a)—(c)所示, 利用不同数量的多腔型基本单元设计了外部形状分别为圆形、矩形、正六边形的对称型超构材料. 其中圆形对称型超构材料的半径为d1 = 18 cm, 矩形和正六边形对称型超构材料中相邻两个基本单元中心点之间的距离分别为d2 = 16 cm和d3 = 20 cm, 并且这些多腔型基本单元的结构参数与图1(a)中的完全一致. 对于这3个系统, 圆形、矩形和正六边形超构材料对应的复合单极子Mie共振模式所对应的共振频率分别为648, 629和621 Hz. 在共振频率处, 当声波以1 Pa的强度从左侧垂直入射到这3个超构材料系统时的声强分布分别如图4(d)—(f)所示, 声场增强效应均存在于这3个对称型超构材料的中心区域, 其内部中心点处的声场强度分别达到了15, 20和20 Pa2. 由此可以推断, 由该多腔型基本单元构成的不同形状的对称型超构材料均可以实现声场增强效应, 这种灵活可调的声场增强系统可应用于许多特定场景, 例如声通信、声源或噪声定位等. 图 4 由多腔型基本单元组成的(a)圆形、(b)矩形和(c)正六边形对称型超构材料的结构示意图; 在复合单极子Mie共振模式下, (d)圆形超构材料、(e)矩形超构材料和(f)正六边形超构材料的声强分布图, 对应的工作频率分别为648, 629, 621 Hz Figure4. Schematic diagram of the (a) circular, (b) rectangular and (c) regular hexagonal symmetric metamaterials composed of multiple-cavity units. Sound intensity distributions of the (d) circular metamaterial, (e) rectangular metamaterial and (f) regular hexagonal metamaterial under the compound monopole Mie resonance mode with the working frequencies of 648, 629 and 621 Hz, respectively.
23.2.基于多腔型基本单元的非对称型超构材料 -->
3.2.基于多腔型基本单元的非对称型超构材料
除了利用该多腔型基本单元设计对称型超构材料, 本文还利用该基本单元构造了左右非对称超构材料. 如图5(a)所示, 设计了一个形状为等边三角形的左右非对称型的超构材料, 在此结构中相邻两个基本单元中心点之间的距离均为d4 = 28 cm. 如图5(b)所示, 频率为637 Hz的平面波以1 Pa的强度从左侧垂直入射, 该非对称型超构材料实现了声场的局部增强效应, 声能集中在系统内部的中心区域, 中心点处声强可高达100 Pa2, 实现了较强的声场增强效应. 为了探究该非对称型超构的声场增强效应是否受入射声波角度的影响, 绘制了结构内部点A处的声强随入射声波角度的变化情况. 如图5(c)所示, 点A处的声强分布关于0°对称, 当入射声波的角度趋于–90°和90°时, 点A处的声强趋于0 Pa2. 而当声波从左侧垂直入射时($\theta {\rm{ = }} {\rm{0}}^\circ $), 点A处的声强达到最大值, 随着角度的逐渐增大, 点A处的声强也随之减小, 因此, 该非对称型超构材料内部的声场增强效应与入射声波的角度有密切关系. 从图5(a)的几何图形可以发现, 当平面波沿着该非对称超构材料左侧以60°入射时, 相当于平面波从右侧垂直入射. 结合图5(b)的声强分布可知, 平面波从左侧垂直入射的情况下, 结构内部中心点处的声强约为100 Pa2, 而由图5(d)可知, 当平面波从右侧垂直入射的情况下, 结构内部中心点处的声强仅为10 Pa2. 因此, 设计的非对称超构材料对侧向垂直入射声波实现了高效的非对称声场增强效应, 且通过调节该等边三角形的倾角可对声场增强效应进行调控. 图 5 (a) 6个多腔型基本单元组成的等边三角形非对称型超构材料的结构示意图; (b) 当频率为637 Hz的声波从结构左侧垂直入射时, 非对称超构材料的声强分布图; (c) 非对称超构材料内部点A处的声强随入射声波角度的变化情况; (d) 当频率为637 Hz的声波从结构右侧垂直入射时, 非对称超构材料的声强分布图 Figure5. (a) Structural diagram of the equilateral triangular asymmetric metamaterial composed of six multiple-cavity units; (b) sound intensity distribution of the asymmetric metamaterial when the sound wave with frequency of 637 Hz is normally incident from the left side; (c) acoustic intensity curve at the center point A of the asymmetric metamaterial with different angles of incident wave; (d) sound intensity distribution of the asymmetric metamaterial when the sound wave is normally incident from the right side with frequency of 637 Hz.