本文應用孔彈性力學理論 (poroelasticity) 分析彈性波在飽和土壤及未飽和土壤中之傳波特性。飽和土壤之傳波頻率方程式是依據Biot (1956a, b) 模式所建立；未飽和土壤之傳波頻率方程式則是引用Lo et al. (2005) 模式。這兩個傳波頻率方程式可由數值模擬解出，進而得知十一種不同土壤 [砂土 (sand)、壤質砂土 (loamy sand)、砂質壤土 (sandy loam)、壤土 (loam)、坋質壤土 (silt loam)、砂質黏壤土 (sandy clay loam)、黏質壤土 (clay loam)、坋質黏壤土 (silty clay loam)、砂質黏土 (sandy clay)、坋質黏土 (silty clay) 和黏土 (clay)，依美國農業部 (USDA) 土壤分類] 在飽和及未飽和情況下且震盪頻率為50、100、150、200和250 Hz之傳波速度及傳波衰退係數。

在飽和土壤中 (孔隙完全由水或空氣所填滿)，P1 (Biot快速膨脹波) 波的傳波速度與傳波頻率無關，但與土壤和孔隙流體特性有關；P1波在各種飽和含水土壤中的傳波速度之差異不顯著，但在各種飽和含空氣土壤 (全乾土壤) 中的傳波速度之差異較明顯，而且在前者的速度遠高於在後者的速度；P1波的傳波衰退係數約與傳波頻率兩次方成正比。P2波 (Biot慢速膨脹波) 的傳波速度遠小於P1波的傳波速度，但P2波的傳波衰退係數遠大於P1波的傳波衰退係數，而且前述兩者均與土壤材質有密切之關係；P2波的傳波速度及衰退係數均約與傳波頻率的1/2次方成正比。

在未飽和土壤中 (孔隙中水和空氣並存)，除了前述P1和P2波之外，存在第三個膨脹波P3 (波速最慢之膨脹波)，此波是由於毛細壓力變動所產生。數值模擬結果顯示P1波波速與潤濕流體 (水) 的彈性係數有很大的相關，與非潤濕流體 (空氣) 的彈性係數影響較小。除了砂土與壤質砂土，其它九種土壤的P1波波速在特定飽和度有突然高起之現象，其背後控制之物理機制主要是受兩個與非潤濕流體相關儲水因子所影響，其比值直接控制此現象。本文結果亦顯示當頻率越大，P2波與P3波之波速則越大，但是P1波波速並不與頻率相關。當飽和度低於0.9時，P2波之波速會隨飽和度增加而減少，P3波則隨飽和度增加而增加。這三種波的衰退係數大小隨頻率而改變，P1波於波速突然高起之處衰退係數會變極小甚至為零，P2和P3波有相同的量化因子 (quality factor) 其值為常數。P1波的傳波衰退係數約與傳波頻率兩次方成正比，P2和P3波的傳波速度及衰退係數均約與傳波頻率的1/2次方成正比。

Elastic wave phenomena in porous media containing compressible viscous fluids are of considerable interest to diverse engineering applications. The behaviors of dilatational wave propagation and attenuation through different types of saturated soils whose pore spaces are completely occupied either by water or by air were first investigated in the present study based on the celebrated Biot model equations of poroelasticity. Then, applying the Lo et al. model [2005], we examined the effects of two immiscible pore fluids (water and air) on the behavior of elastic waves. The excitation frequency discussed both in the saturated and unsaturated soils was at the seismic range (50, 100, 150, 200 and 250 Hz). The existence of three different modes of dilatational wave in an unsaturated porous medium was demonstrated analytically and numerically.

In saturated soils, we show that the phase velocity of P1 wave (the Biot fast wave) is equal to the square root of the ratio of an effective bulk modulus to an effective density of the fluid-containing porous medium, regardless of excitation frequency and pore fluid. It is also found that the phase velocity of the P1 wave does not vary significantly in the water-saturated case whereas the variation becomes very apparent in the air-saturated case. In reference to the P2 wave (the Biot slow wave), its attenuation is demonstrated to be greater than that of the P1 wave, but its phase velocity is less than that of the P1 wave.

In unsaturated soils, our numerical results show that the phase velocity of the P1 wave whose magnitude is greatest is closely related to elastic properties of the wetting fluid. Although the phase velocity of the P1 wave is independent of the excitation frequency, the phase velocities of the P2 (second fastest) and P3 (slowest) waves are showed to be positively proportional to the excitation frequency. At certain soils, the phase velocity of the P1 wave has an abrupt increase, the physical mechanism behind which is controlled by the ratio of two storativity factors. When the saturation degree of water is below 0.9, the phase velocity of the P2 wave decreases as water saturation increases whereas that of the P3 wave increases. The attenuation of these three waves varies with the excitation frequency. The attenuation of the P1 wave is almost equal to zero at the water saturation in which the P1 wave has an abrupt increase in its phase velocity. The P2 and P3 waves are found to have the same constant value of the quality factor. Regardless of either saturated or unsaturated soils, the attenuation coefficient of the P1 wave is approximately proportional to the square of the excitation frequency. The magnitude of the phase velocity of attenuation coefficient of the P2 wave (saturated or unsaturated soils) and P3 wave (unsaturated soils) increases with the square root of the excitation frequency.

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