雖然在包含兩種不可混合可壓縮且具黏滯性流體的彈性孔隙介質中已被證明會有三種不同類型的膨脹波產生，即俗稱之P1、P2、P3波；但三個膨脹波的運動模式和相對飽和度之間對應的關係仍不明確。因此，本文利用Lo et al.（2005）所提出的孔彈性力學理論，求出三個膨脹波對應之正交座標系統（normal coordinates）。正交座標系統提供一個理論的基礎，將運動模式以六個相關係數（X1～X6）表示，而六個相關係數與慣性阻力、黏滯阻力、和介質之彈性係數三者相關。
利用數值方法計算在未壓密十一種土壤六個相關係數的值。其結果顯示，波速最快的膨脹波（P1波），所對應的運動模式為固體相和兩相流體作同向運動，且不受水飽和度變化影響，和傳統Biot理論所提出在飽和孔隙介質中的快速波（fast compressional wave）相符。速度第二快的波（P2波），在固體相和流體1相（非潤濕流體，本文中為空氣）傳遞作同向運動，而和流體2相（潤濕流體，本文中為水）作反向運動［模式（III）］；然而當水飽和度超過某一個值後，在固體相的傳波方向和兩相流體皆作反向運動［模式（IV）］。從模式（III）過渡到模式（IV）之間，對應到保水曲線中以毛細現象為主的區域和接近飽和時空氣進入區之間。模式（IV）對應到傳統Biot理論中的慢速波（slow compressional wave），而模式（III）只有在毛細壓力變動下的含有兩相流體的彈性孔隙系統中存在。傳波速度最小的波（P3波），在水飽和度大於某值後（例如在坋質壤土，當S>0.19後），其膨脹波運動模式由兩相作反向傳波運動的孔隙流體所主導，這樣的結果和此波是由毛細壓力變化所造成之結論一致。

Numerical simulations of dilatational waves in an elastic porous medium containing two immiscible viscous compressible fluids indicate that three types of wave occur（P1, P2 and P3）, but the modes of dilatory motion corresponding to the three waves remain uncharacterized as functions of relative saturation. In this paper, we address this problem by deriving normal coordinates for the three dilatational waves based on the general poroelasticity equations of Lo et al. 2005. The normal coordinates provide a theoretical foundation with which to characterize the motional modes in terms of six connecting coefficients（X1～X6）that depend in a well defined way on inertial drag, viscous drag, and elasticity properties.
Using numerical calculations of six connecting coefficients for eleven unconsolidated soil, we confirm that the dilatational wave whose speed is greatest corresponds to the motional mode in which the solid framework and the two pore fluids always move in phase, regardless of water saturation, in agreement with the classic Biot theory of the fast compressional wave in a water-saturated porous medium. For the wave which propagates second fastest that the solid framework moves in phase with water, but out of phase with air［Mode（III）］, if the water saturation is below some critical value, whereas the solid framework moves out of phase with both pore fluids［Mode（IV）］above this water saturation. The transition from Mode（III）to Mode（IV）corresponds to that between the capillarity-dominated region of the water retention curve and the region reflecting air-entry conditions near full water saturation. Mode（IV） corresponds exactly to the slow compressional wave in classic Biot theory, whereas the Mode（III） is possible only in a two-fluid system undergoing capillary pressure fluctuations. For the wave which has the smallest speed that water saturation more than some critical value, the dilatational mode is dominated by the motions of the two pore fluids, which are out of phase, a result that is consistent with the proposition that this wave is caused by capillary pressure fluctuations.

1.Bear, J., Dynamics of Fluids in Porous Media, Dover, Mineola, N. Y., 1988.
2.Beresnev, I. A., and P. A. Johnson, Elastic-wave stimulation of oil production: a review of methods and results, Geophysics, Vol. 59, no. 6, pp. 1000-1017, 1994.
3.Berryman, J. G., Confirmation of Biot’s theory, Applied Physics Letters, Vol. 37, no. 4, pp. 382-384, 1980.
4.Berryman, J. G., Effective conductivity by fluid analogy for a porous insulator filled with a conductor, Physical Review B, Vol.27, no. 12, pp. 7789-7792, 1983.
5.Berryman, J. G., Thigpen, L., and Chin, R. C. Y., Bulk elastic wave propagation in partially saturated porous solids, Journal of the Acoustical Society of America, Vol. 84, no. 1, pp. 360-373, 1988.
6.Biot, M. A., Theory of propagation of elastic waves in a fluid saturated porous solid, I. Low-frequency range, Journal of the Acoustical Society of America, Vol. 28, no. 2, pp. 168-178, 1956a.
7.Biot, M. A., Theory of propagation of elastic waves in a fluid saturated porous solid, II. Higher frequency range, Journal of the Acoustical Society of America, Vol. 28, no. 2, pp. 179-191, 1956b.
8.Biot, M. A., and Willis, D. G., The elastic coefficient of the theory of consolidation, Journal of Applied Mechanics, Vol. 24, pp. 594-601, 1957.
9.Biot, M. A., Mechanics of deformation and acoustic propagation in porous media, Journal of Applied Physics, Vol. 33, no. 4, pp. 1428-1498, 1962.
10.Brooks, R. H., and A. T. Corey, Hydraulic properties of porous media, Hydrology Paper 3, Civ. Eng. Dep., Colo. State Univ., Fort Collins, 1964.
11.Brutsaert, W., The propagation of elastic waves in unconsolidated unsaturatedgranular mediums, Journal of Geophysical Research, Vol. 69, no. 2, pp. 243-257, 1964.
12.Drew, D., L. Cheng, and R. T. Lahey, The analysis of virtual mass effects in two-phase flow, International Journal of Multiphase Flow, Vol. 5, no. 4, pp. 233-242, 1979.
13.Geller J. T. and Myer L. R., Ultrasonic imaging of organic liquid contaminants in unconsolidated porous media. Journal of Contaminant Hydrology, Vol. 19, no. 2, pp. 85-104, 1995.
14.Grag, S. K., and A.H. Nayfeh, Compressional wave propagation in liquid and/or gas saturated elastic porous media, Journal of Applied Physics, Vol. 60, no. 9, pp. 3045-3055, 1986.
15.Johnson, D. L., Recent developments in the acoustic properties of porous media, in Proceedings of the international school of physics “Enrico Fermi” Course XCIII, Frontiers in physical acoustics, edited by Sette, pp. 255-290, 1986.
16.Lo, W.-C., G. Sposito, and E. Majer, Immiscible two-phase flow in deformable porous media, Advances in Water Resources, Vol. 25(8-12), pp. 1105-17, 2002.
17.Lo, W.-C., G. Sposito, and E. Majer, Wave propagation through elastic porous media containing two immiscible fluids, Water Resources Research, Vol. 41, no. 2, pp. W02025, 2005.
18.Lo, W.-C., G. Sposito, and E. Majer, Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids, Transport in Porous Media, Vol. 68, pp. 91-105, 2007.
19.Lo, W.-C., Yeh, C.-L. and Tsai, C.-T., Effect of soil texture on the propagation and attenuation of acoustic wave at unsaturated conditions, Journal of Hydrology, Vol. 338, pp. 273-284, 2007.
20.Lo, W. C., G. Sposito, E. Majer and Yeh, C.-L., Motional modes of dilatational waves in elastic porous media containing two immiscible fluids, Advances in Water Resources, Vol. 33, no. 3, pp. 304-311, 2010.
21.Marion, J. B., Thornton, S. T., Classical dynamics of particles and systems. Fort Worth: Saunders College Publishers, 1995.
22.Mualem, Y., A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resources Research, Vol. 12, no. 3, pp. 513-522, 1976.
23.Rawls, W. J., J. R. Ahuja, and D. L. Brakensiek, Estimating soil hydraulic properties from soils data, Proceedings of Workshop on Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils, Riverside, CA, pp. 329-341, 1992.
24.Santos, J. E., J. M. Corbero, and J. Douglas, Static and dynamic behavior of a porous solid saturated by a two-phase fluid, Journal of the Acoustical Society of America, Vol.87, no. 4, pp. 1428-1438, 1990a.
25.Santos, J. E., J. Douglas, M. Corbero, and O. M. Lovera, A model for wave propagation in a porous media saturated by a two-phase fluid, Journal of the Acoustical Society of America, Vol.87, no. 4, pp. 1439-1448, 1990b.
26.Stoll, R. D., Acoustic waves in saturated sediments, Physics of sound in marine sediments, edited by L. Hampton, Springer, New York, pp. 19-39, 1974.
27.Tuncay K. and M. Y. Corapcioglu, Body waves in poroelastic media saturated by two immiscible fluids, Journal of Geophysical Research, Vol. 101, pp. 25149-25159, 1996.
28.Tuncay K. and M. Y. Corapcioglu, Wave propagation in poroelastic media saturated by two fluids, Journal of Applied Mechanics, Vol. 64, no. 2, pp. 313-320, 1997.
29.van Genuchten, M. T., A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America Journal, Vol. 44, no. 5, pp. 892-898, 1980.
30.Vogler E. T. and Chrysikopoulos C. V., Experimental investigation of acoustically enhanced solute transport in porous media, Geophysical Research Letters, Vol. 29, no. 15, 10.1029/2002GL015304, 2002.
31.Wang, H. F., Theory of linear poroelasticity with applications to geomechanics and hydrogeology, Princeton University Press, Princeton, 2000.
32.徐仁君，「未飽和孔隙介質受簡諧振盪下其彈性波傳遞及衰退行為之數值研究」，國立成功大學水利及海洋工程學系碩士論文，2009。
33.葉昭龍，「彈性波在飽和及未飽和土壤中傳波特性之影響評估」，國立成功大學水利及海洋工程學系碩士論文，2005。
34.楊佶欽，「解析去耦合孔彈性偏微分方程式特性之理論與數值研究」，國立成功大學水利及海洋工程學系碩士論文，2009。