本研究主要是從理論及實驗上探討低頻聲波通過任意體積分率之氣泡幕衰減特性。在理論的部分是利用賴（1999）與Tien（2001）將低體積分率之計算方式推導至任意體積分率之模式。在傳統上的氣泡液體（Bubble liquids）之動量方程式中，加入氣泡間交互作用的影響，重新整理出適用於任意體積分率之聲波傳遞方程式，再由液體與氣泡幕間速度及壓力連續之邊界條件，求出聲波通過氣泡幕之穿透係數及反射係數。實驗的部分則著重於低頻聲波通過氣泡幕之特性研究，探討頻率為（500Hz~90kHz ）。在低體積分率時，本文實驗與賴（1999）和Tien（2001）及傳統理論計算所得聲波通過氣泡幕之穿透及反射係數相當吻合；而在高體積分率時，本文實驗除了在較低頻部份（500Hz~1kHz ）有較大的差異（因實驗儀器限制），在較高頻部份（1kHz~90kHz ）皆有相當一致性。高體積分率之條件下，理論上在比共振頻率略低處有較大的穿透係數值。但與實驗結果並不一致。

The attenuation of sound propagating through a bubble screen was investigated both theoretically and experimentally in this study. The theoretical results of Lia (1999) and Tien (2001) were applied. By adding an additional virtual mass force into the momentum equation of the bubbly liquid, a modified Helmholtz equation valid for the sound propagation in the bubbly liquid at finite gas-volume fraction is reformulated. The low frequencies, 500Hz~90kHz , of the sound was focused for studying the sound propagating through a bubbly liquid. The experimental results agree with the theory deduced by Lia (1999) and Tien (2001) and the classical theory. There are excellent agreements of experimental results and theoretical results in the higher frequency range 1kHz~90kHz , but deviations in the lower frequency range 500Hz~1kHz .Theoretically, there are higher transmission coefficients in the frequencies lower than resonant frequency of the cases of the higher gas-volume fraction. But it is not consistent with the experimental results.

1.Biesheuvel, A. and van Wijngaarden, L. (1984)“Two-phase flow equations for a dilute dispersion of gas bubbles in liquid,”J. Fluid Mech., Vol.148, pp. 301-318.
2.Caflishch, R. E., Miksis, M. J., Papanicolaou, G. C., and Ting, L. (1985)“Effective equation for wave propagation in bubbly liquids,”J. Fluid Mech., Vol. 153. pp. 259-273.
3.Caflishch, R. E., Miksis, M. J., Papanicolaou, G. C., and Ting, L. (1985)“Wave propagation in bubbly liquids at finite volume fraction,” J. Fluid Mech., Vol. 166. pp. 1-14.
4.Carstensen, E. L., and Foldy, L. L. (1947)“Propagation of sound through a liquid containing bubbles,”J. Acoust. Soc. Am., Vol. 19, pp.481-501.
5.Commander, K. W. and Prosperetti, A. (1989)“Linear pressure waves in bubbly liquids：Comparison between theory and experiments,”J. Acoust. Soc. Am., Vol. 85, pp. 732-746.
6.Duraiswami, R., Prabhukumar, S., and Chahine, G.. L. (1987)“Bubble counting using an inverse acoustic scattering method,” J. Acoust. Soc. Am., 104(5), pp. 2699-2716.
7.Feuillade, C. (1996)“The attenuation and dispersion of sound in water containing multiply interacting air bubbles,”J. Acoust. Soc. Am., Vol. 99, pp. 3412-3430.
8.Foldy, L. L. (1945)“The multiple scattering of waves,”Phys. Rev., Vol. 67, pp. 107-119.
9.Fox, F. E., Curley, S. R., and Larson, G. S. (1955)“Phase velocity and absorption measurements in water containing air bubbles,” J. Acoust. Soc. Am., Vol. 27, pp. 534-539.
10.Keller, J. B. and Kolodner, I. I. (1956)“Damping of underwater explosion bubbles oscillations,”J. Applied Physics, Vol. 27, pp. 1152-1161.
11.Keller, J. B. and Miksis, M. (1980)“Bubble oscillations of large amplitude,”J. Acoust. Soc. Am., Vol. 68, pp. 628-633.
12.Kolaini, A. R., Dandenault, P. and Ruxton, A. D. (1995)“Passive and active acoustical measurements of laboratory breaking waves,”Sea Surface Sound 94’,Proc. of the Ⅲ International Meeting on Natural Physical Process Related to Sea Surface Sound, World Scientific.
13.Lamarre, E. and Melville W. K. (1992)“Instrumentation for the measurement of void fraction in breaking waves：Laboratory and field results,”IEEE J. of ocean. Eng., Vol. 17, pp. 204-215.
14.Lax, M. (1951)“Multiple scattering of waves,”Rev. Modern Phys., Vol. 23, pp.287-310.
15.Leighton, T. G., White P. R. and Schneider, M. F. (1998)“The detection and dimension of bubble entrainment and comminution” J. Acoust. Soc. Am., Vol. 103(4), pp. 1825-1835.
16.Macpherson, J. D. (1957)“The effect of gas bubbles on sound propagation in water,”Proc. Phys. Soc. London Sec. B, Vol. 70, pp. 85-92.
17.Maxwell, P. L.（1891）Treatise on Electricity and Magnetism 1, Chapter 9, Dover Pub.
18.Medwin, H. and Daniel, A. C. (1990)“Acoustical measurements of bubble production by spilling breakers,” J. acoust. Soc. Am., Vol. 88, pp. 408-412.
19.Silberman, E. (1957)“Sound velocity and attenuation in bubbly in water,”J. Fluid Mech., Vol. 1, pp. 249-275.
20.Streeter, V. L.（1948）Fluid dynamics, pp.78-81.
21.Su, M. Y. and Cartmill, J.（1994）“Breaking wave measurement by a void fraction technique,”Proceed. 2nd Int. Symp. Ocean Wave Measurement and Analysis, pp. 951-962.
22.Su, M. Y. and Wesson, J. C.（2001）“Bubble measurement techniques and bubble dynamics in coastal shallow water,”Advances in Coastal and Ocean Engineering, Vol. 7, pp. 77-124, World Scientific, Singapore.
23.Su, M. Y., Wesson, J. C., Burge, R. and Teague, W. J.（1999）“Temporal variation of bubble void fraction in the littoral zone,”Submitted to J. Geophys. Res., Aug., 1999.
24.Twersky, V. (1962)“On scattering of waves by random distribution. I. Free-surfeace scatterer formalism,”J. Math. Phys., Vol. 33, pp. 700-715.
25.Twersky, V. (1964)“Acoustic bulk parameters of random volume distributions of small scatterers,”J. Acous. Soc. Am., Vol. 36, pp. 1314-1329.
26.Tien, T. M.（2001）“Sound propagation through a bubble screen at finite gas-volume fraction,”Master Thesis, National Cheng Kung University, Tainan, Taiwan.
27.Van Wijngaarden, L.（1972）“One-dimensional flow of liquids containing small gas bubbles,”Ann. Rev. Fluid Mech., Vol. 4, pp. 27-44.
28.Waterman, P. C. and Truell, R. (1961)“Multiple scattering of waves,”J. Math. Phys., Vol. 2, pp. 331-339.
29.Wu, J.（1987）“Bubbles in the near-surface ocean：a general description,”J. Geophys. Res. Vol. 93, pp. 587-590.
30.Wu, J.（1994a）“Films drops produced by air bubbles bursting at the surface of seawater,”J. Geophys. Res., No. c8, pp. 16403-16407.
31.Wu, J.（1994b）“Bubbles in the near-surface ocean, J. Physical Oceangraphy, Vol. 24, pp. 1955-1965.
32.賴文傑（1999）“聲波通過氣泡幕衰減特性研究”，國立成功大學水利及海洋工程研究所碩士論文。