【作者】 郁高坤；

【导师】 王新龙；

【作者基本信息】 南京大学 ， 声学， 2011， 博士

【摘要】 局域共振在物理上可以用弹簧振子受外力驱动的模型来描述。在没有外界驱动但是存在初始扰动情形下,局域共振结构具有局域化的振动模式,这种振动模式在空间的分布是局域化的,甚至可以不存在空间分布(例如声亥姆霍兹共鸣器)。当存在外界驱动时,局域共振结构的响应具有共振或者反共振特性。由局域共振单元构成的复合结构,比如非线性链,可以用来描述孤子传输,能量局域化等等。而由非局域共振构成的复合结构,比如周期结构,只会产生布拉格共振。本文主要是研究局域共振所致的若干声与振动现象,着重探索共振在声传输以及孤子激发方面所起的作用。第一章论述了共振结构中的线性现象和非线性现象,由于局域共振与非局域共振存在紧密联系,二者密不可分,因此主要从共振所产生的现象来论述,而本文所着重研究的局域共振所致的声与振动现象在相应的论述部分都进行了说明。对线性现象而言,主要从禁带特性以及共振传输特性来论述,其中论文所讨论的声共振隧穿现象属于共振传输部分。对共振结构中的非线性现象的论述主要从非线性响应,孤子形成的机制以及孤子的激发方式来阐述共振的作用,其中论文第四章所研究的局域共振激发孤子属于孤子激发部分,而论文第五章所研究的非线性声孤子模型则属于孤子形成机制部分。第二章着重讨论局域共振的若干理论。以膜共振为例子讨论了局域化模式对传播模式(平面波)的影响,诸如传输特性,相位特性以及声延迟特性；从阻抗的角度讨论了模式耦合引起的并联共振,具体为：(1)不同亥姆霍兹共鸣器置于管中同一位置时的耦合共振,(2)相同亥姆霍兹共鸣器置于管中不同位置(有一个小的间距)的耦合共振。这种耦合共振可以引起声的共振隧穿现象,即通过时间域上的局域共振与空间域上的长度共振相互作用形成(论文第三章)；讨论了禁带中的局域振动,由于缺少辐射阻尼,振动的幅度可以达到很大,这与耦合共振是完全不同的。这种共振的优点可以用来共振激发孤子,具体为论文的第四章。第三章研究了局域共振导致的声共振隧穿现象。以声亥姆霍兹共鸣器为例子,着重讨论了周期亥姆霍兹共鸣器陈列中的声共振隧穿现象,这实际上并联共振引起共振传输现象的推广。由于这些传输峰的位置处于单个亥姆霍兹共鸣器的禁带中,因此这种共振隧穿主要是针对局域共振结构而言的。与完全由长度共振引起共振隧穿现象不同的是,这种类型的共振是由时间域的局域共振和空间域的长度共振共同作用的结果。经过分析发现,甚至完全由时间的域局域共振也可以产生共振隧穿现象。由于时间域局域共振没有势垒宽度的概念,因此群速度对势垒宽度的依赖关系在局域共振构成的复合结构中不再存在,取而代之的是群速度依赖于亥姆霍兹共鸣器的共振频率与布拉格共振频率的接近程度。在由两种不同共振频率共鸣器构成的周期陈列中,群速度甚至仅仅取决于两个不同的亥姆霍兹共振频率。进一步,我们从阻抗的角度对该问题进行分析,当声波频率大于或者小于亥姆霍兹共振频率时,共鸣器的呈现抗性或者容性,相应的会产生一个相位差,这个相位差不是由传播距离引起的,而完全由局域共振引起的。该相位差与空间传播距离引起的相位差互相补偿,最终导致共振传输。共鸣器之间的间距起到的是耦合作用,在两种不同共鸣器陈列构成的周期陈列中,我们观察到长度耦合不仅可以导致共振隧穿,在某些情况下可以产生宽禁带,即抑制共振隧穿。第四章研究了非线性局域共振及其孤子激发的问题,提出了局域共振激发孤子的概念,孤子的这种激发方式具有能量转换效率高,可控性强,相较于其它激发方式,具有明显的优势。具体是通过在半无限长的非线性链中引入缺陷,形成局域化的缺陷模,通过这种模式的共振从驱动端吸收能量,存储,然后局域化模式的振动进入非线性振动的状态,当吸收的能量达到某一阈值时,存储的能量将以孤子的形式向外辐射,更有趣的是,局域化模式存储的能量几乎完全转化为行波孤子的能量,所以说这种发射具有高效性,由于存在吸收过程,孤子的发射的平均间隔对驱动强度有明显的依赖关系,这为孤子的发射提供了一种控制方法。通过分析发现这种局域共振激发孤子的方式在离散系统中具有普适性。进一步,我们将这种共振激发孤子的概念推广到连续系统,通过对驱动边界条件作适当的简化,我们得到了一种物理可以实现的模型。在此基础上,我们提出了一种共振激发水波孤子的物理模型。第五章探讨了声孤子存在的可能性,提出了声孤子的共振激发以及非传播性声孤子的物理模型。在深入理解水槽孤子模型的基础,利用复合声材料构成了一个低频阻带系统,该声学系统存在一个截止频率,声波在截止频率附近非线性响应呈现“软弹簧”特性,因此该声学系统完全类似于水槽系统,更有趣的是,在截止频率附近,声学振动没有横向模式,并且速度扰动是二阶量,而密度扰动是一阶量,如果在边界处施加刚性边界条件,则可以忽略速度扰动量,该系统完全呈现密度扰动,因此在论文中我们称这种类型的孤子为密度型声孤子。为了补偿开口结构引起的辐射阻和摩擦阻,我们在模型中引入质量流来补偿阻尼衰减,最后我们得到了直接驱动薛定谔方程。通过对实际物理参数的估计,我们有理由认为这种密度型声孤子的实现是完全有可能的。本文工作的主要创新点在于：1.讨论了由局域共振(时间域)和空间尺度上的共振相互作用产生的声共振隧穿现象,并分析了相应的共振隧穿特性,同时指出其与空间尺度上共振隧穿的差别。2.利用局域共振来激发孤子,这种激发具有低幅度,高效率的特点,对孤子发射可以进行一些调控,当将该思路推广到连续系统时,我们发现这种方法提供了如何在连续系统中激发NLS孤子,并给出一种物理模型来共振激发水波孤子。3.提出了一种实现声孤子共振激发以及非传播型声孤子的模型,并从理论上给出了可行性的论证。更多还原

【Abstract】 Local resonance can be modeled as a spring oscillator excited by an external driving force, and it is always spatially localized, but for some special cases, such as Helmholtz resonance, there is no spatial distribution of local resonance. The composite structures consisting of local resonant elements, such as the nonlinear chains, have been applied to describe soliton transmission and energy localization, etc. For the composite structures with non-local resonance, such as periodic structures, the propagation of linear waves is forbidden in the band gap induced by Bragg resonance. Several acoustical and vibra-tory phenomena resulting from local resonance have been studied in this dissertation, especially for acoustic resonant transmission and soliton generation.In chapter 1 of the dissertation, a brief review is given with regard to linear and non-linear phenomena in resonant structures. Since local resonance is closely bound up with non-local resonance, we mainly focus on phenomena resulting from the two resonances and give some emphasis on several acoustical and vibratory phenomena resulting from local resonance that we study in the dissertation. A brief summary of linear phenomena is given, such as band gap and resonant transmission, where acoustic resonant tunneling in chapter 3 is related to resonant transmission. For nonlinear phenomena, the role of resonance is illustrated by the nonlinear vibration response, the physical mechanism for soliton existence and soliton generation, where soliton generation by local resonance in chapter 4 is related to the mechanism of soliton generation and a acoustic soliton model in chapter 5 is related to the mechanism of soliton existence.In chapter 2 of the dissertation, several types of local resonance have been dis-cussed. Firstly, the response of a membrane under plane wave incidence is illustrated by the amplitude and phase of the transmission coefficient and group delay. Secondly, parallel resonance induced by the coupling between local resonant units has been ana-lyzed from the view point of acoustic impedance with two examples, one is the coupling between two different Helmholtz resonators connected to a tube at the same site, the other one is the coupling between the two same Helmholtz resonators connected to a tube in a small space interval. As a result of parallel resonance, acoustic tunneling can happen via an interaction between local resonance in time domain and length resonance in the spatial domain (see chapter 3). Finally, we discuss local resonance in the band gap. The vibration of the localized mode could be amplified very much due to the lack of radiation damping. Different from parallel resonance induced by coupling, this kind of resonance can be used to generate solitons, which is discussed in chapter 4 of the dissertationIn chapter 3 of the dissertation, we discuss the acoustic resonant tunneling phe-nomenon in local resonant structures, such as the array of Helmholtz resonators. This phenomenon results from parallel resonance. Since the frequencies of resonant trans-mission peaks lie in the band gap of a single Helmholtz resonator, this kind of resonant tunneling phenomenon means that acoustic waves tunnels the band gap of a single Helmholtz resonator. Different from the resonant tunneling phenomenon caused by the half wavelength resonance, this resonant tunneling phenomenon results from the interaction between local resonance in time domain and Bragg resonance in the spatial domain, and could even be induced by two different local resonances. Since local res-onance is the resonance in time domain, the group velocity of the resonant tunneling phenomenon has no dependence on barrier thickness as in quantum mechanics, and it is determined by the resonant frequency difference of Helmholtz resonance and Bragg resonance, particularly, it could even be determined by the two local resonant frequen-cies in the array of two different Helmholtz resonators. Further more, we investigate this phenomenon from the view point of acoustic impedance. When the frequency of an incident wave is above or below the resonant frequency of a Helmholtz resonator, the impedance of a Helmholtz resonator exhibits the property of reactance or capacitance, as a result, a phase difference has been formed, which is not caused by the distance of wave propagation, but by local resonance. This phase difference compensates with the phase diffference caused by the distance of wave propagation and then the resonant tunneling phenomenon happens. It should be noted the coupling between Helmholtz resonators not only induces the resonant tunneling phenomenon but also forms wide band gap between two different local resonant frequencies.In the chapter 4 of the dissertation, we propose a resonant mechanism to generate discrete solitons. This method is manageable and high efficient in generating solitons. By introducing a mass impurity in a semi-infinite nonlinear chains, a localized mode has been formed around the impurity. Due to the resonance with this localized mode, the localized mode absorbs energy from the driving end and stores in it. With the increasing the amplitude of the localized mode, the vibration of the localized mode enters into the nonlinear state. When the absorbed energy arrives a threshold, the stored energy is radiated in form of solitons. It is interesting to observed that the energy stored in the localized mode is almost transferred to the propagating solitons, so local resonance in generating solitons is more efficient. Due to the process of absorption, the averaged emission interval of solitons has strong dependence on the driving amplitude, so local resonance is more manageable in generating solitons. We also discuss local resonance in other discrete systems. Further more, by extending the concept of local resonance to continuous systems and making a simplification of the driving boundary condition, we obtain a model which is realizable in physics. At the end, we proposed a model to generated water solitons by this concept.In the chapter 5 of the dissertation, a acoustic soliton model has been proposed. With well understanding the mechanism of generating the non-propagating soliton in the water trough and the progress in acoustic composite structures, we build an acoustic composite structure, which has a cut-off frequency at low frequency and the nonlinear response at this cut-off frequency exhibits "soft spring" property. So this acoustic com-posite structure is similar to the water trough. It is interesting that there is no transverse mode at the cut-off frequency, the order of velocity disturbances is second order, and the order of density disturbances is first order. If the rigid boundary condition is im-posed, we could neglect the effect of velocity disturbances, so this acoustic composite structure only exhibits density disturbances and we call acoustic soliton as density type in our dissertation. In order to compensate the dissipation in the structure caused by the radiation loss and viscosity loss in holes, density flow is introduced in our model, and then we obtain the damped-driven nonlinear Schrodinger equation. By estimating the physical parameters, we thought it is possible to generate acoustic solitons in this acoustic composite structure. The principal contributions of the present study are summarized as below:1. The acoustic resonant tunneling phenomenon produced by local resonance in time domain and Bragg resonance in the spatial domain are given in chapter 3. We discuss the corresponding property of the resonant tunneling phenomenon and illustrate differences with the resonant tunneling phenomenon caused by Fabry-Perot resonance.2. We propose a resonant mechanism to generate discrete solitons. This method is manageable and high efficient in generating solitons. we also extend this mechanism to continuous systems and a physical model based on this mechanism has been given to generate water solitons.3. We propose a soliton model in an acoustic composite structure and give some evidences for the existence of this soliton.更多还原