# Roller-Integrated Acoustic Wave Detection Technique for Rockfill Materials
## 2 Methodologies
### Theoretical Analysis  
#### 2.1.1 Lamb's Problem  
The elastic fluctuation problem caused by dynamic line load or point load on the surface or interior of a semi-space is a type of problem with great significance in the fluctuation field of elastic solids. The original research can trace back to the classics of [PERSON] (1930). In the early 1930s, [PERSON] developed a general method for solving Lamb problems, using this displacement field of the Fourier transform, with the inverse transformation obtained by using Laplace transform. After that, [PERSON] (1930) made further improvements. [PERSON] (1933) has analyzed the flexible circular plate problem under a uniformly distributed load on an elastic semi-space, and the problem can be solved by the Rahm solution integration of point load problems. In 1953, [PERSON] (1953) and [PERSON] (1956) published two papers to expand [PERSON]'s solution by considering the influence of the pressure distribution on the circular contact area on the surface of the semi-space. For the dynamic response of saturated soil, [PERSON] (1953) first established the phenomenological theory of two-phased medium fluctuation problems. The establishment of the theory made it possible to study the vibration characteristics of the foundation on the saturated soil foundation in the real sense. Since [PERSON] established the wave equation of saturated porous mediain 1956, [PERSON] [38] and [PERSON] et al. [39] carried out a thorough study of [PERSON]'s problem when there were horizontal and vertical forces on the surface of saturated soil. [PERSON] ignored the viscous fluid in saturated soil, utilized the Hankel transform and Cagniard method, adopted the Helmholtz decomposition proposed by [PERSON], and was the first to study [PERSON]'s problem in saturated semi-space. [PERSON] et al. [40], hypothesizing that viscosity was included in the dynamic permeability coefficient, adopted the integral transformation method to solve simultaneous coupled equations, avoided the introduction of potential functions, and gave the integral form solution of saturated elastic semi-space under low-frequency harmonic concentrated force. [PERSON] [41] studied the steady response of axisymmetric soil under vertical load, and developed the amplitude curve of surface displacement. On the basis of these fundamental solutions, the dynamic interaction between saturated soil and various types of foundations has also been extensively studied [42; 43; 44]. [PERSON]'s problem has been extensively studied internationally, and significant progress has been made. The analysis methods of such problems are constantly improving, and the solution to the problem is more refined. In this paper, the interaction between the soil and the vibratory wheel was considered as [PERSON]'s problem of saturated soil. In order to solve this problem, based on the vertical displacement solutions in the case of normal concentrated load on the surface of the space, an innovative differential pulse excitation method (DPEM) was proposed to solve the vertical displacement in the semi-space under simple harmonic loading. The isotropic and homogeneous semi-space is shown in Figure 1, and its surface is the X-Y plane. The coordinate axis \(z\) points to the interior of the semi-space. There is a normal concentrated load of \(QH(t)\) at the origin \(o\). Since the load is symmetrical to the \(z\) axis, the problem of wave motion in the infinite elastic semi-space is axial symmetry. Based on a cylindrical coordinate system \((r,\theta,z)\), \(u_{z}\) by the displacement potential \(\varphi\) and \(\chi(\psi=0)\) of the cylindrical coordinate system symmetric situation are expressed as follows:  
\[u_{z}=\frac{\partial\varphi}{\partial z}\ -\ \frac{1}{r}\frac{\partial}{ \partial r}\left(r\frac{\partial\chi}{\partial r}\right). \tag{1}\]  
Based on the study of [PERSON] [32] and other researchers [35; 38; 39; 40], the vertical displacement of the surface is as follows:  
\[u_{z}(r,0,t)=\left\{