1. Introduction

2. Model Simplification and Partition

3. Construction of Wave Functions

For steady state, the governing equation of SH waves in isotropic homogeneous and continuous media can be expressed as [9,10]

(1)

where W is the displacement function, k=ω/C_s, ω is frequency, is the shear wave velocity.

In polar coordinates, the corresponding stress component is

(2)

3.1. Wave Function in Region Ι

The zero-stress condition on the boundary C is

(3)

The standing wave function satisfying the governing Equation (1) and boundary condition (3) can be expressed as [11]

(4)

where A_m is the constant to be solved, W₀ is the displacement amplitude. p=π/(2θ₀). J_mp(·) is a Bessel function with mp ranks.

The wave function needs to satisfy the continuity condition on S₁ boundary [12,13]. According to the moving coordinate method, Equation (4) in the coordinate system can be expressed as

(5)

The corresponding stress can be expressed as

(6)

where

$$P_t^j\left(s\right)=J_{t-1}\left(k_1\left|s\right|\right){\left(\frac{s}{\left|s\right|}\right)}^{t-1}{e}^{i{\theta }_j}-J_{t+1}\left(k_1\left|s\right|\right){\left(-1\right)}^t{\left(\frac{s}{\left|s\right|}\right)}^{-\left(t+1\right)}{e}^{i{\theta }_j}+J_{t-1}\left(k_1\left|s\right|\right){\left(-1\right)}^t{\left(\frac{s}{\left|s\right|}\right)}^{1-t}{e}^{-i{\theta }_j}-J_{t+1}\left(k_1\left|s\right|\right){\left(\frac{s}{\left|s\right|}\right)}^{t+1}{e}^{-i{\theta }_j}$$

(7)

3.2. Wave Function in Region Ш

The boundary condition of the upper surface of sediment can be expressed as

(8)

The standing wave function satisfying condition (8) and Equation (1) can be expressed as

(9)

where B_m is the constant to be solved.

According to the moving coordinate method, Equation (9) in the coordinate system can be expressed as

(10)

The corresponding stress can be expressed as

(11)

3.3. Scattering Waves in RegionⅡ

In region Ⅱ, there are two scattering waves on each boundary because the scattering waves generated by the two depressions interfere with each other. The functions that satisfy the governing equation and the boundary condition can be expressed in the form of multipolar coordinates as [14,15,16]

(12)

(13)

(14)

(15)

The corresponding stress can be expressed as

(16)

(17)

where C_m, D_m are the constant to be solved. $H_m^{\left(1\right)}\left(\bullet \right)$ is Hankel function of the first kind, and

$$U_t^j\left(s\right)=H_{t-1}^{\left(1\right)}\left(k_1\left|s\right|\right){\left(\frac{s}{\left|s\right|}\right)}^{t-1}{e}^{i{\theta }_j}-H_{t+1}^{\left(1\right)}\left(k_1\left|s\right|\right){\left(-1\right)}^t{\left(\frac{s}{\left|s\right|}\right)}^{-\left(t+1\right)}{e}^{i{\theta }_j}+H_{t-1}^{\left(1\right)}\left(k_1\left|s\right|\right){\left(-1\right)}^t{\left(\frac{s}{\left|s\right|}\right)}^{1-t}{e}^{-i{\theta }_j}-H_{t+1}^{\left(1\right)}\left(k_1\left|s\right|\right){\left(\frac{s}{\left|s\right|}\right)}^{t+1}{e}^{-i{\theta }_j}$$

(18)

3.4. Incident and Reflected Waves

In the complex plane , the incident and reflected waves can be expressed as

(19)

The corresponding stress can be expressed as

(20)

where , and

$$P_t^j\left(s\right)=J_{t-1}\left(k_1\left|s\right|\right){\left(\frac{s}{\left|s\right|}\right)}^{t-1}{e}^{i{\theta }_j}-J_{t+1}\left(k_1\left|s\right|\right){\left(-1\right)}^t{\left(\frac{s}{\left|s\right|}\right)}^{-\left(t+1\right)}{e}^{i{\theta }_j}+J_{t-1}\left(k_1\left|s\right|\right){\left(-1\right)}^t{\left(\frac{s}{\left|s\right|}\right)}^{1-t}{e}^{-i{\theta }_j}-J_{t+1}\left(k_1\left|s\right|\right){\left(\frac{s}{\left|s\right|}\right)}^{t+1}{e}^{-i{\theta }_j}$$

(21)

4. Establishment of Equations

(22)

(23)

(24)

(25)

(26)

(27)

5. Results Analysis

(28)

(29)

6. Conclusions

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