| Summary: | 碩士 === 國立中央大學 === 物理研究所 === 89 === Wave are everywhere around us. We need light and sound to sense our environment. We use radio waves and microwaves for long distance communications. Water waves are responsible for the ocean’s dynamic image. In the microscopic scale, quantum waves associated with electrons and atoms are important in maintaining the structure and stability of solids.
When propagating through media containing many scatterers, waves will be scattered by each scatterer. The scattered waves will be scattered again by other scatterers. This process is repeated to establish an infinite recursive pattern of wave scattering, called multiple scattering [1, 2]. Multiple scattering effects occur in our everyday life. Modulation of ocean ambient sound [3], acoustic scintillation from fish schools [4], light passing through the atmosphere, electron transporting in impured solids [5], sound propagating in the forest, and laser beam in colloidal liquid, and so on are only some examples of multiple scattering.
When waves propagates through media contains periodic arrays of scatterers, multiple scattering leads to band structures. In certain situations, it appears the complete forbidden band gaps within which no normal mode exists and wave cannot propagate in any direction. Though finite in numbers, as long as the number of scatterers is sufficiently large, the gap effects will remain and we can use this property to make new materials which can modulate waves. Like the silicon chips manipulating electronic waves, photonic crystals can be used to manufacture “photonic chips” to manipulate electromagnetic waves, and “sonic crystals chips” can manipulate acoustic waves.
On the other hand, when multiple scattering happens in random media, the phenomenon named wave localization may occur. In this situation, wave will not propagate anymore and it will be localized in the media.
Historically, the wave dispersion bands are first studied for electronic waves in solids, providing the basis for understanding the properties of conductors, semiconductors, and insulators [6]. In late 1980s, it became known that such a wave band phenomenon is also possible for electro-magnetic waves in media with periodically modulated refractive-indices [7]. Since then, optical wave bands have been extensively studied, yielding a rich body of literature. The theoretical calculations have proven to match well with the experimental observations [8]. In contrast, research on acoustic wave band structures has just started. Although theoretical computations of band structures have been documented for periodic acoustic structures [9], the experimental work was only recent, and to date only a limited number of measurements has been reported. One of the first observations was made on acoustic attenuation by a sculpture [10]. The authors obtained a sound attenuation spectrum, which was later verified by the band structure computation [11]. Recently, acoustic band structures have been further measured for acoustic transmission through two-dimensional (2D) periodic arrays of rigid cylinders placed in the air [12]. The authors demonstrated the properties of sound attenuation along two high-symmetry directions of the Brillouin zone of the arrays. They also observed a peculiar effect of deaf bands; within the bands, in spite of non-zero band states, wave propagation is prohibited due to particular symmetry of the states [12].
On another aspect, the concept of wave localization are also first proposed in electronic systems. In 1958, Philip Warren Anderson conceived that no electron diffusion can take place when there is an enough amount of impurities introduced into the solids [13], i.e. beyond a critical amount of impurity scattering the motion of the electrons will come to a halt. By analogy, the localization of electronic wave have extended to the localization of electromagnetic waves in the late 1980s [14]. Recently, the localization of light are studied comprehensively, such as in the subjects of white paint and random laser [15]. In contrast, the research on localization of acoustic waves are insufficient in the literature. The experimental work was accomplished only in one-dimensional(1D) systems [16] and absent in twodimensional(2D) and three-dimensional(3D) systems. Besides, only little theoretical analysis can be found [17].
Although the phenomenons of band gap(periodic media) and localization(random media) are discoverd in electronic waves, it can be extended deservedly to other types of wave. In other words, the band structure and localization are the nature of wave phenomenons.
In this thesis, we use a self-consistent treatment of multiple scattering to investigate waves propagation in the media with either ordered or random arrays of a finite number of scatterers in two dimensions (2D). The band structures of the corresponding ordered case is also calculated. In chapter 2, we present an multiplescattering algorithm, based on partial-wave concept and self-consistent method. In addition, the band structure calculaiton based on plane-wave method is also discussed. In chapter 3, we compare our simulation results with existing experiments. From the comparison, we can gain confidence of our numerical codes. In chapter 4, we focus on localization and band structure effects of acoustic wave in the system of air-filled cylinders in water. Finally, we point out some possible appalications.
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