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作者(外文):Jyun-Ting Lu
論文名稱(外文):Research of the Dynamic Modulus of the Bubble-in-Water Acoustic Metamaterial
指導教授(外文):Pi-Gang Luan
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本論文探討水中氣泡聲子晶體 (bubble-in-water sonic crystal) 在水中的行為。由於水和氣泡阻抗差異極大,因此氣泡是很好的散射體,在低頻下具有強共振的現象。在我們的模型中,低頻下氣泡的行為相當於徑向振盪的諧振子 (harmonic oscillator) 與單極子 (monopole) 波源。一個受入射波聲波影響的氣泡不僅沿著徑向作膨脹與收縮運動,同時也受阻尼力的影響。利用水中氣泡近似模型分析低濃度下等效彈性模量 (bulk modulus) 以及散射函數 (scattering function),發現在很寬的頻率範圍內,氣泡水介質會產生負的等效彈性模量 (bulk modulus),而介質的等效質量密度仍為正,這對應到了氣泡聲子晶體帶隙 (gap) 範圍,使得波無法在該介質傳播。此外,我們也透過計算由氣泡所構成的層狀結構之穿透率,發現在共振和布拉格散射 (Brag scattering) 的機制耦合 (coupling) 下,氣泡聲子晶體的色散關係和離子晶體中的聲子-極化子 (phonon polariton) 形式相同。我們的理論除了可解釋氣泡聲子晶體的長波長行為,更進一步可用以研究具有隨機分佈氣泡的聲波屏的穿透率的頻率響應 (response)。數值模擬上採用傳遞矩陣法 (transfer matrix) 和多重散射法 (multiple scattering method)。由於波在無序性介質(disorder media) 中的傳播一直是波動學中重要的課題之一,而我們的單氣泡散射模型可以簡單解釋在低頻下的各向同性 (isotropic) 散射 (氣泡的存在相當於水中的雜質),因此更容易幫助我們理解聲波局域化現象 (acoustic localization)。我們的模型有助於研究海洋聲學,例如可藉此模型分析海中氣泡對聲波的散射,以發展出探測海洋的有效方法。再者,我們甚至能將其對應到其他類似氣泡在水中這種介質類型的聲波系統上。
In this thesis, we discuss the acoustic behaviors of the bubble-in-water sonic crystal in water. Due to the large impedance mismatch between water and bubbles, bubbles are very efficient acoustic scatterers and have strong resonances at low frequencies. In our model, a bubble under the influence of low frequency incident acoustic wave vibrates radially as a driven oscillator as well as a monopole acoustic source. In addition to the expansion and contraction motion in the radial direction, there is also damping force acting on the bubble. We use this simple model of bubbles in water to analyze the effective dynamic bulk modulus and scattering function at low concentration. It can be found that in a wide range of frequency, the bubble-in-water medium has an effective negative elastic modulus, while the effective density of the medium is still positive. This frequency range corresponds to the bandgap of the bubble-in-water sonic crystal, thus why the acoustic waves cannot propagate in this medium. In addition, we also calculate the transmittance of the acoustic waves through the layered structure composed of bubbles and found that under the coupling effect of resonance and Bragg scattering mechanism, the dispersion relationship of the bubble phonon crystal is the same as that of the phonon-polariton in the ion crystal. In addition to explaining the long-wavelength behavior of the bubble sonic crystal, our theory can also be used for studying the transmission response in frequency to a point source inside or outside the bubble screen (a slab or a shell) of randomly distributed bubbles. Our numerical simulations are implemented by using the transfer matrix method. Since wave propagation in disorder media has always been one of the most important topics in wave theory, and our bubble scattering model provides simple ways to analyze the low-frequency isotropic scattering (the bubbles play the roles of impurities in water), so it is very helpful for us to understand the mechanism of acoustic localization. Our model can help to study marine acoustics. For example, using our model to analyze the scattering of the sound waves by ocean bubbles, we may develop efficient method to detect the ocean. Moreover, we can even generalize our model and apply it to other types of sound wave systems.
摘要 I
Abstract II
致謝 III
目錄 VI
圖目錄 V
一、緒論 1
1-1光子晶體 1
1-2超材料 4
1-3水中的氣泡 8
1-4波的多重散射與侷域性 10
二、多重散射法Multiple Scattering Theory 13
三、氣泡(Bubble liquid)模型 18
3-1水中單個氣泡散射函數和本徵頻率 18
3-2等效彈性係數和等效質量 20
3-3等效體積模量及等效質量密度 21
3-4等效介質之色散關係 24
四、一維傳遞矩陣法 25
4-1一維等效介質傳遞矩陣推導 25
4-2一維等效介質球殼結構傳遞矩陣推導 27
4-3聲波系統中的能流 29
五、隨機分布下氣泡散射問題 31
5-1水中氣泡的多重散射 31
5-2 Leslie L. Foldy’s model 32
5-3 圖解法 (diagrammatic techniques) 35
5-3-1 單向散射路徑 (one-way multiple scattering) 35
5-3-2 雙向散射路徑 (two-way multiple scattering) 37
六、數值模擬與分析 39
6-1等效氣泡屏對頻率的響應 39
6-2等效模型及Foldy模型之色散關係比較 43
6-3不同等效模型下穿透率對頻率的響應 44
6-4有序及無序氣泡分佈下之穿透率比較 46
七、結論與未來展望 48
參考文獻 49

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