###### Abstract  
The permeability of saturated soils has great influence on the velocities and attenuation characteristics of fast compressional wave P1, low compressional wave P2, and shear wave S in saturated soils, respectively. In three different cases, namely zero, finite, and infinite permeability, the wave equations and theoretical velocities of P1, P2, and S wave in saturated soils are given based on the _u-w-p_ equation, respectively. According to the solutions of the wave equations, the real velocities and attenuation coefficients of three waves are redefined, respectively. In different saturated soils, the influences of the permeability and the loading frequency on the wave velocities and attenuation are discussed, respectively. Moreover, the suitable application scope of the _u-p_ equation is discussed based on different permeabilities and loading frequencies. saturated porous media; wave velocity; frequency dispersion; attenuation; permeability 10  
water to simulate saturated porous media, which imposed a high frequency loading. He successfully verified the existence of the three waves in saturated porous media for the first time. [PERSON] [14] and [PERSON] et al. [15] separately conducted similar experiments on actual saturated sands and gravels. They observed the slow compressional wave, P2 wave, in saturated soils as well. [PERSON] [16] tested the velocities of the three waves in saturated porous media through experiments. The velocities were in a good agreement with the theory velocities of the Biot theory. In fact, the propagation of the three waves in saturated soils showed regularity. The P1 wave propagates in both of the solid and fluid phases with the same velocity because of the compressibility of the saturated porous medium [17]. The diffusion process of the fluid in the voids of the solid skeleton generates the P2 wave. The velocity of the P1 wave is faster than the velocity of the P2 wave. The two waves show amplitude attenuation during the propagation procedure. Moreover, the strong viscous coupling in saturated soils with a low permeability [12] or under a low frequency loading makes the velocity of the P2 wave attenuate to zero rapidly [3]. For the S wave, because a pore fluid cannot carry a shear force, the S wave only propagates in the solid skeleton and the propagation characteristics are only dependent on the shear strength of the solid skeleton. Present research on wave propagation behaviors in saturated soils mainly focus on the velocities, frequency dispersion, velocity attenuation characteristics, etc. In addition, the effect of the physical parameters, such as the loading frequency and permeability, on the wave propagation characteristics are analyzed [18]. [PERSON] [19] analyzed the wave attenuation characteristics in saturated porous media under the earthquakes. [PERSON] [20], and [PERSON] and [PERSON] [21] researched wave propagation characteristics in transverse isotropy saturated porous media, respectively. [PERSON] and [PERSON] [22] established the relations of wave propagation characteristics, attenuation characteristics, and porosity in frozen soils. In addition, the propagation characteristics of saturated soils based on the assumption of an uncompressed solid skeleton were studied [23; 24; 25; 26]. [PERSON] [27] acquired the velocities of two compressional waves in saturated porous media using the analytical method, and the velocities had good agreement with those measured from experiments [13]. [PERSON] and [PERSON] [28] obtained the velocities of two compressional waves under extreme permeability (zero and infinite). [PERSON] et al. [29; 30] analyzed the influence of the pore path, porosity, and the volume modulus of the solid skeleton on the propagation characteristics of the P1 and P2 wave. [PERSON] et al. [31] systematically introduced the relations of parameters, such as porosity, permeability, and fluid viscosity, and the wave propagation characteristics of saturated soils.