臺灣博碩士論文加值系統

波的局域化研究在波的研究領域中一直是個很重要的研究課題。早在西元1958年，美國的科學家P. W. Anderson便提出電子在無序金屬當中傳導的理論。他指出，一但金屬摻入足夠多的雜質，電子就會由原來的傳導態轉變成侷限態。後來人們體認到這種由導體轉變成絕緣體的相變是因電子的波動特性所造成的，也就是說，電子波在無序的金屬中會遭受到雜質的多重散射，並且互相干涉。既然純粹是導因於電子的波動現象，人們就去猜測，古典波應該也會有類似的局域化現象。所以在八零年代初期開始，有關光波的局域化研究於焉展開。由於這整個研究的歷史發展背景，通常局域化就是意指由無序所導致的波的侷限現象。而在本論文中所要探討的是聲波在含氣泡水中的局域化研究。
由於局域化是由於波的多重散射後的干涉所引起的，而在散射的過程中有無窮多種可能的散射途徑。為了將這些種種的散射途徑全部考慮進去，嚴格的自恰法（self-consistent method）是一種很好的解決方式。此法早在西元1945年就由L. L. Foldy提出了理論表述，而在本論文中我們使用電腦將嚴格的數值解計算出來。結果表明，不管是氣泡無序分布或整齊分布的情況，都發現有聲波的局域化現象，並且局域化發生的頻率範圍相重疊。此外我們使用平面波展開的方法（先前已由M. S. Kushwaha對此工作做出貢獻）去計算氣泡分布整齊情況下的聲波能帶結構，驗證了此種情況所產生的局域化在本質上是一個能隙。其實早在我們之前就有科學家提出另外一種方法去實現古典波的局域化：在一個完美的晶體中，每個晶格點在原來的週期位置做「些許」的擾動，就會在原來的能隙中產生局域化的態。我們的數值結果支持這種論點，似乎對於古典波來說，能隙和局域化存在著某種唇齒相依的關係，很多文獻也提出了此種論點，不過並沒有人下很明確的的定論。 有一個令我們好奇的地方是，即使我們的氣泡是完全隨機分布的，能隙也沒有如預期般的消失，聲波的局域化程度還是很強。
到目前為止也有一些文獻提出波局域化與散射體的共振可能具有相當的關連性，而我們的結果也表明局域化發生的地方湊巧就在氣泡共振峰的旁邊。為了近一步了解局域化與共振散射的關連性，我們又計算了對於不同的「氣泡與水的體積比」的聲波能帶結構，發現在高頻的部份也產生了新的能隙。此結果似乎透露出局域化與共振散射並沒有絕對的關連性。
除了一般文獻所提出的局域化與能隙、共振散射可能具有關連性外，我們又另外提出了一個嶄新的觀點：局域化與散射體的相位協同性（phase coherence）可能具有關連性。氣泡在受到外場作用會做振動（pulsation），進而發射出散射波，所以除了真正的聲源外，每個氣泡都可以看做是第二個聲源。經由電腦的數值計算我們發現，一但局域化發生時，氣泡產生了協同性的運動（collective behavior）。如果我們將非局域化轉變到局域化的過程視為是一個種變過程的話，那麼這種氣泡的協同行為正好可以看成是系統對稱破缺後所產生的新的有序態。吾人認為，這是本論文在關於波局域化之研究上的一大新意。
在本論文的最後，我們提出了一個「迷之三角形－局域化、共振散射與散射體的相位協同性之相互關係」，意謂這三者之間所存在的曖昧關係，有待未來進一步再加以研究的目標和方向。

The idea of wave localization originates from people's understanding about the electron's conduction in disorder solids. In 1958, Philip Warren Anderson suggested that no electron diffusion can take place at all when there is an enough amount of impurities introduced into the solids. Electrons are localized in the random potential, and this metal-insulator phase transition induced by impurities is known as Anderson localization. This effect finds its nature in the interference of electron waves undergoing multiple scattering by impurities. Inspired by this discovery, people turned their attention to think about the question, "Can classical waves be localized, too?" Then the idea of light localization started to evolve in the beginning of the 1980's, and was followed by a large number of literature on the related topics. Due to the whole historical background of the progress for wave localization, the term 'localization' is defined as a disorder induced phenomenon.
In order to take account of all the orders in multiple scattering, the self-consistent method is used to derived the pressure field. The results denote that the localzaition phenomena can be observed for both cases, disorder and order: the frequency ranges of the localization window are overlapping. On the other hand, the plane-wave method is used to justify the band gap for the order case. We must emphasize that this band structure computation could independently prove the accuracy of the results from our self-consistent method. In fact, there had been some scientists suggesting an alternative
pathway to localization: electromagnetic wave(EM) localization may be more easily achieved in a weakly disordered system of almost periodically arranged dielectric scatterers in the frequency regime around a band gap. From this suggestion some scientists also agreed that plausibly a connection between the band gaps in a periodic system and the regions of localized states in a random system for classical waves exists, at least for weak disorder. But no one dared to give any definite conclusion. There is one unexpected situation which interests us. Our numerical results show that the band gap is not closed even though the bubble distribution is completely random. Inside the gap, the localization is still strong.
Not only the connection mentioned above, but also the scatterer resonance is suggested in many literature to be connected to localization. Moreover, by treating the bubble as a secondary source, the collective behavior of bubbles are observed when localization occurs. The collective behavior seems to be connected to localization, too. The possible relation between localization, scatterer resonance, and collective behavior of bubbles is a unresolved problem and is needed to be clarified in the future.

Chapter 1 Introduction..............................4
Chapter 2 Self-consistent method....................9
2.1 General formulation for wave propagation in
inhomogeneous media.......................9
2.2 Acoustic scattering by an air-bubble......10
2.3 Multiple scattering of acoustic waves.....15
Chapter 3 Acoustic energy transmission..............18
3.1 Disorder and order........................19
3.2 Frequency responce........................20
3.3 Acoustic energy distribution in space.....21
Chapter 4 Acoustic band structure...................24
4.1 Bloch theorem.............................24
4.2 Review of Kushwaha's work.................27
4.3 Numerical result..........................29
Chapter 5 Collective behavior of bubbles............32
5.1 Pulsating phase of bubbles................33
5.2 Order parameter...........................39
Chapter 6 Discussion................................42
Appendix A Scattering function of an air-bubble......48
Appendix B First Brillouin zone......................50
Bibliography..........................................51

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