The model of soil mechanics for a fluid-saturated medium in which the liquid phase is assumed to be ideal, the solid phase is isotropic elastic, and the compression modulus of solid particles in point contact tends to infinity, can be expressed as [12, 44, 49, 50]. μ Δ u + λ + μ ∇ ∇ · u + 1 − n ∇ p f + B U ˙ − u ˙ = ρ 1 u ¨ n ∇ p f − B U ˙ − u ˙ = ρ 2 U ¨ 1 − n ∇ · u + n ∇ · U − n E w p f = 0 (1)

Where, Δ (= ∇ 2 ) is the Laplace operator in the Cartesian Coordinate, ∇ is the Hamiltonian operator. u , u ˙ , and u ¨ denote the solid phase’s displacement, velocity, and acceleration vector, respectively. U , U ˙ and U ¨ represent the absolute displacement, velocity, and acceleration vector of the liquid phase separately. λ and μ are the classical Lame constants, which are functions of the Poisson’s ratio υ and the elastic modulus of the solid phase E, λ = E υ / 1 + υ 1 − 2 υ and μ = E / 2 1 + υ . n is the porosity. ρ 1 and ρ 2 are defined to describe the solid and liquid density per unit volume, in which ρ 1 = 1 − n ρ s and ρ 2 = n ρ w , ρ s and ρ w are the solid and fluid mass densities separately. p f is the true pore pressure. E w refers to the bulk modulus of liquid. k (= K / ρ w g ) is the dynamic permeability coefficient of the solid skeleton, in which K (m/s) is the permeability coefficient that satisfies Darcy’s law, and g is the gravitation acceleration. B represents the dissipation coefficient, which is a third-order diagonal matrix. B = d i a g b x , b y , b z , and b x = b y = b z = n 2 / k in the isotropic medium.

Considering Helmholtz’s resolution, we introduce scalar potential functions ( ϕ s , ϕ w ) and vector potential functions ( ψ s , ψ w ) to describe the displacements of solid- and liquid-phase, which can be written as follows [11, 50, 55]. u = ∇ ϕ s + ∇ × ψ s U = ∇ ϕ w + ∇ × ψ w (2)

Insertion of Equation 2 in Equation 1 yields the wave equation expressed by potential function, as can be shown in the following form [50]. ρ 1 ϕ ¨ s − λ + 2 μ Δ ϕ s = p f − ρ 2 ϕ ¨ w ρ 1 ψ ¨ s − μ Δ ψ s = − ρ 2 ψ ¨ w n p f − B ϕ ˙ w − ϕ ˙ s − ρ 2 ϕ ¨ w = 0 ρ 2 ψ ¨ w + B ψ ˙ w − ψ ˙ s = 0 1 − n Δ ϕ s + n Δ ϕ w − n E w p f = 0 (3)

The in-plane wave problem in a fluid-saturated medium is a P-SV wave problem in the xoz plane. Assuming the displacements u and U are independent of the coordinate y. The scalar potential functions ϕ s = ϕ s x , z , t , and ϕ w = ϕ w x , z , t in the xoz plane. The vector potential functions ψ s =(0, ψ s x , z , t ,0), and ψ w =(0, ψ w x , z , t ,0). The components of solid-phase displacement ( u x , u z ) and the components of liquid-phase displacement ( U x , U z ) can be written in the form of potential functions, as shown in Equations 4a, 4b; [1, 20]. The potential function expressions of normal stress ( σ z z ) and shear stress ( σ x z ) are written in Equation 4c by the plane strain character. From the fifth of Equation 3, the potential function expression of pore fluid pressure ( p f ) can be given by the third of Equation 4c. u x = ∂ ϕ s ∂ x − ∂ ψ s ∂ z u z = ∂ ϕ s ∂ z + ∂ ψ s ∂ x (4a) U x = ∂ ϕ w ∂ x − ∂ ψ w ∂ z U z = ∂ ϕ w ∂ z + ∂ ψ w ∂ x (4b) σ z z = λ + 1 − n n E w ∇ 2 ϕ s + 2 μ ∂ 2 ϕ s ∂ z 2 + ∂ 2 ψ s ∂ x ∂ z + E w ∇ 2 ϕ w σ x z = 2 μ ∂ 2 ϕ s ∂ x ∂ z + μ ∂ 2 ψ s ∂ x 2 − ∂ 2 ψ s ∂ z 2 p f = 1 − n n E w ∇ 2 ϕ s + E w ∇ 2 ϕ w (4c)

Assuming the plane harmonic wave solutions of the potential functions in the following forms [11]. ϕ s = A s e i ω t − k P · r ϕ w = A w e i ω t − k P · r ψ s = B s e i ω t − k S · r ψ w = B w e i ω t − k S · r (5)

Where, A s and A w are the potential function amplitudes of solid- and liquid-phases for P-wave, respectively. B s and B w represent the potential function amplitudes of solid- and liquid-phases for S-wave separately. k P and k S denote the propagation directions of P- and S-waves (wave vector). k P and k S are the values of vectors ( k P and k S ), with k P and k S representing the wave numbers of P- and S-waves, respectively. r denotes the position vector. i = − 1 . ω is the circular frequency of a wave.

Substituting Equation 5 into Equation 3, we can obtain the dispersion equations of P- and S-waves. λ + 2 μ n E w ρ 1 ρ 2 k P ω 4 − λ + 2 μ ρ 1 + n E w ρ 2 + 1 − n 2 E w n ρ 1 − i b ω ρ 1 ρ 2 λ + 2 μ + E w n k P ω 2 + 1 − i b ω 1 ρ 1 + 1 ρ 2 = 0 (6a) i b μ ω ρ 1 ρ 2 − μ ρ 1 k S ω 2 + 1 − i b ω 1 ρ 1 + 1 ρ 2 = 0 (6b)

It can be seen from Equations 6a, 6b that the velocities and attenuation coefficients for two kinds of compressional waves (P₁- and P₂- waves) and one shear wave (S- wave) in an unbounded saturated medium are calculated. All three body waves are dispersed and attenuated, which are related to the properties of medium and wave frequency.

In the two-phase medium (i.e., the half-space z > 0), the incident P₁-wave gives rise to reflected waves of all three types, i.e., P₁-, P₂-, and SV- waves. The expressions for solid- and liquid-phases potential functions of P-wave ( ϕ s , ϕ w ) and SV-wave ( ψ s , ψ w ) are shown in Equation 8; [27]. The plane harmonic solutions of potential functions for different waves are shown in Equations 9a-9d; [27]. ϕ s = ϕ s 1 I + ϕ s 1 R + ϕ s 2 R ϕ w = ϕ w 1 I + ϕ w 1 R + ϕ w 2 R ψ s = ψ s R ψ w = ψ w R (8) ϕ s 1 I = A s 1 I ⁡ exp i ω t − k 1 x I x + k 1 z I z ϕ w 1 I = A w 1 I ⁡ exp i ω t − k 1 x I x + k 1 z I z (9a) ϕ s 1 R = A s 1 R ⁡ exp i ω t − k 1 x R x − k 1 z R z ϕ w 1 R = A w 1 R ⁡ exp i ω t − k 1 x R x − k 1 z R z (9b) ϕ s 2 R = A s 2 R ⁡ exp i ω t − k 2 x R x − k 2 z R z ϕ w 2 R = A w 2 R ⁡ exp i ω t − k 2 x R x − k 2 z R z (9c) ψ s R = B s R ⁡ exp i ω t − k s x R x − k s z R z ψ w R = B w R ⁡ exp i ω t − k s x R x − k s x R z (9d)

Where, ϕ s 1 I ( ϕ w 1 I ) is a potential function in the solid (liquid) of incident P₁-wave. A s 1 I and A w 1 I are the amplitudes of the corresponding potential functions. Similarly, ϕ s 1 R , ϕ s 2 R , and ψ s R are the solid-phase potential functions of the reflected P₁-, P₂-, and SV-waves, respectively. A s 1 R , A s 2 R and B s R correspond to the solid-phase potential amplitudes. ϕ w 1 R , ϕ w 2 R and ψ w R denote the liquid-phase potential functions of the reflected P₁-, P₂-, and SV-waves, separately. A w 1 R , A w 2 R and B w R are the liquid-phase potential amplitudes. k 1 x I and k 1 z I represent the components of the incident P₁-wave vector in the x and z directions. k 1 x R and k 1 z R are the components of the reflected P₁-wave vector in the x and z directions. Similarly, k 2 x R , k 2 z R , k s x R and k s z R are the components of the reflected P₂- and SV- wave vectors of the corresponding directions.

Following the geometric relationship of wave vectors, it can be seen that the wave vectors and their components of all waves satisfy the equalities Equation 10. Moreover, by Snell’s law, the x-components of the wave numbers for the incident and reflected waves are the same, as shown in Equation 11; [55, 56]. k 1 x I 2 + k 1 z I 2 = k 1 I 2 k 1 x R 2 + k 1 z R 2 = k 1 R 2 k 2 x R 2 + k 2 z R 2 = k 2 R 2 k s x R 2 + k s z R 2 = k s R 2 (10) k 1 x I = k 1 x R = k 2 x R = k s x R (11)

From Equations 6a, 6b, the relations between the various amplitudes in Equations 9 can be obtained as follows. δ 1 = A w 1 β A s 1 β = λ + 2 μ + 1 − n n E w k 1 β 2 − ρ 1 ω 2 ρ 2 ω 2 − E w k 1 β 2 , β = I , R (12a) δ 2 = A w 2 R A s 2 R = λ + 2 μ + 1 − n n E w k 2 R 2 − ρ 1 ω 2 ρ 2 ω 2 − E w k 2 R 2 (12b) δ s = B w R B s R = μ k s R 2 − ρ 1 ω 2 ρ 2 ω 2 (12c)

Where, δ ₁, δ ₂, and δ _s are the amplitude ratios of potentials related to liquid and solid phases for P₁-, P₂-, and SV-waves, respectively.

Let the liquid density ρ w = 0, and the bulk modulus of liquid E w = 0. Then, the solution in this paper can degenerate into the case of a P-wave incident on the free interface of a single-phase medium. Now, the potential amplitude ratios of the liquid-solid phase in the two-phase medium δ 1 = 0, δ 2 = 0, and δ s = 0. And the wave vector of the reflected P₁-wave is the same as that of the P₂-wave, i.e., k 1 R = k 2 R , k 1 z R = k 2 z R . When the two-phase medium is reduced to a single-phase medium, the velocity of P-wave V P = λ + 2 μ / ρ s , the velocity of SV-wave V S = μ / ρ s , which can be derived from the dispersion Equations 6a, 6b. The potential amplitude of the reflected P-wave A s R = A s 1 R + A s 2 R . Accordingly, Equation 14a, 14b can be simplified as μ V P 2 V S 2 k 1 I 2 − 2 k 1 x I 2 A s I + A s R + 2 μ k s x R k s z R B s R = 0 2 μ k 1 x I k 1 z I A s R − A s I + μ k s x R 2 − k s z R 2 B s R = 0 (17)

Equation 17 is further simplified to obtain a new expression, which is the same as the Equations of a single-phase medium in Stein and Wysession [56]. It can be seen that the reflection of the P-wave on the free surface of a single-phase medium is a special case in this paper.

To further verify the correctness of the formulas for reflection coefficient and surface displacement, Equation 17 is compared with the curve of P-wave incident on the free surface of a single-phase medium in Pujol [55]. The parameters for single-phase media are taken from Pujol [55], namely, V P / V S = 1.732 and υ = 0.25. The variations of amplitude ratios and surface displacements with the incident angle are shown in Figure 2 when the P-wave is incident on the interface. It is noted that the displacement components u _x and u _z in Figure 2b are normalized by a factor k 1 I , which represents the displacement intensity of the incident P-wave. If not specified, the surface displacements in the following figures are all normalized.

It can be seen from Figure 2 that the calculated results of Equation 17 are consistent with those of Pujol [55]. This is sufficient to demonstrate the correctness of the formula derived in this paper.

When P₁-wave propagates in a saturated half-space, specific solutions can be obtained using appropriate boundary conditions. The single control variable method is introduced to analyze the influence of boundary drainage on the surface response of half space. The values of the physical parameters of the saturated poroelastic half-space are selected from Section 5, and the other parameters are as follows: n = 0.1, υ = 0.2, and E _w/μ = 0.1. The frequency of incident wave f = 100 Hz. The curves in Figure 3 represent the displacement reflection coefficients and surface displacements with distinct boundaries.

It can be seen from Figure 3a that the displacement reflection coefficient of P₁-wave decreases with an increase in the incident angle before reaching its minimum value near 65ºunder different conditions. Moreover, when the incident angle θ _IP is greater than 36°, the displacement reflection coefficient under the impermeable interface is more than that of the permeable interface. Figure 3b shows that the displacement reflection coefficient of the P₂-wave is much less than those of other reflected P₁- and SV-waves, and the coefficient under a permeable interface is greater than that under an impermeable boundary. From Figure 3c, for the reflected SV-wave, the displacement reflection coefficient increases with a rise in the incident angle before attaining its maximum value near 45°. Also, the displacement reflection coefficient at an impermeable interface is more than that at a permeable interface if the θ _IP is within the range of 16º-90°. Given Figure 3d, u _x reaches its peak value at approximately 60°, while u _z reaches its peak value at 0°, and the peak value of u _z is larger than that of u _x . If θ _IP < 30°, the vertical and horizontal displacements (e.g., u _x and u _z ) under two boundary conditions are the same. However, if θ _IP > 30°, both displacements u _x and u _z (absolute values) increase slightly under the impervious interface. Accordingly, the boundary conditions have a certain effect on the surface response of half-space, and this effect manifests a considerable dependence on the incident angle.

As analyzed in Refs. [2, 50], all three body waves are dispersive and attenuated, and the velocities and attenuation are frequency-dependent. To illustrate the effects of wave frequency on the reflection, four different values of wave frequency are considered in this paper, i.e., f = 1, 10, 100, and 1000 Hz. The four typical frequencies are within the common frequency range used in engineering and experimental testing [58]. The soil parameters remain invariable, as described in Section 5.1. The boundary is completely permeable. Figure 4 shows the variations of displacement coefficients and surface displacements with the incident angle for different frequencies.

It is clear from Figure 4 that the surface response is not sensitive to wave frequency. However, the displacement reflection coefficient of P₂-wave decreases as the frequency is reduced. This result matches the case of Rjoub [21]. So, the frequency is assumed to be 100 Hz when analyzing the effect of soil parameters on surface response next.

Since porosity mainly affects the loose degree of soil, it is instructive to investigate the effect of porosity on the displacement reflection coefficients and surface displacements. Except for the porosity, the soil parameters remain invariable, as described in Section 5.1. The frequency of the incident plane P₁-wave is also taken to be 100 Hz. The boundary is completely permeable. The variations with the incident angle of displacement coefficients and surface displacements are shown in Figure 5 in the case that the porosity n = 0.1, 0.2, 0.3, and 0.4, respectively.

It is shown in Figure 5a that the variations of displacement reflection coefficient for reflected P₁-wave with porosity are very complex. When the porosity n = 0.3 and 0.4, a special wave mode conversion occurs, namely, only P₂-and SV- waves are reflected, and the reflected P₁-wave is not generated. Under the case that n = 0.3, the displacement reflection coefficient of P₁-wave exhibits zero values at incident angles of 60° and 77°. The angles for incidence corresponding to wave mode conversion are 57ºand 79° with the instance that n = 0.4. If the porosity n = 0.1 and 0.2, this phenomenon disappears. Moreover, the displacement reflection coefficient for P₁-wave decreases with the increase of porosity when the incident angles θ _IP < 57ºor θ _IP > 79°. From Figures 5b, c, the displacement reflection coefficient for SV-wave (P₂-wave) increases (decreases) with the increase in porosity, and that for P₂-wave is the smallest of all three reflected waves as described in Section 5.1. It is noticed from Figure 5d that the horizontal displacement u _x increases with a rise in porosity. However, the porosity considered in this study has little impact on vertical displacement u _z . The effect of porosity on the surface response depends on the incident angle to a large extent.

The Poisson’s ratio mainly affects Lame constants (λ and μ), which reflect the consolidation status of the soil. To investigate the effects of Poisson’s ratio on the displacement reflection coefficients and surface displacements, the soil parameters remain constants as described in Section 5.1, except for Poisson’s ratio. The frequency of the incident plane P₁-wave f = 100 Hz. The boundary is completely permeable. Figure 6 shows the effects of Poisson’s ratio on the displacement reflection coefficients and surface displacements. In calculations, the Poisson’s ratio (υ) is taken to be 0.1, 0.2, 0.3, and 0.4.

It can be found from Figures 6a–c that the displacement reflection coefficient of P₁-wave increases with the increasing Poisson’s ratio at the same incident angle, while those of P₂- and SV-waves diminish with a rise of Poisson’s ratio. For all three reflected waves, the amplitude of variation is related to the incident angle. As observed in Figure 6d, the horizontal displacement u _x (the vertical displacement u _z ) decreases (increases) with the rise of Poisson’s ratio. When the Poisson’s ratio increases, the variation range of horizontal displacement is larger than that of vertical displacement, and the variation range depends on the incident angle.

The modulus ratio mainly affects the stiffness of the soil layer in the saturated half-space. The larger the modulus ratio is, the softer the soil layer is. For this reason, there is a need to study the effects of the modulus ratio on the displacement reflection coefficients and surface displacements. Except for the modulus ratio, the soil parameters are taken according to Section 5.1. The frequency f of the incident plane P₁-wave is taken as 100 Hz. The boundary is completely permeable. The modulus ratio E _w/μ = 0.1, 1.0, 10, and 100. Figure 7 depicts the displacement reflection coefficients and surface displacements as a function of incident angle for the above four values of modulus ratio.

It can be revealed from Figure 7 that the displacement reflection coefficients and surface displacements vary with the modulus ratio. As can be seen from Figures 7a–c, the displacement reflection coefficient of P₁-wave (P₂- or SV-wave) increases (decreases) with the increasing modulus ratio at the same incident angle. The variation amplitude is related to the incident angle. Moreover, when the modulus ratio E _w/μ = 100, the displacement reflection coefficient of P₁-wave increases towards −1.0, and that of SV-wave reduces to nearly 0, indicating that soft soil mainly transmits compression waves. All in all, the effect of incident angle on the reflection coefficients of P₂ and SV waves diminishes with the increase of the modulus ratio. Figure 7d shows us that the horizontal displacement u _x decreases with a rise in modulus ratio, while the vertical displacement u _z is less affected. For E _w/μ = 100, the peak displacement u _x decreases to 0.063. The extent of influence is decided by the incident angle.