# Roller-Integrated Acoustic Wave Detection Technique for Rockfill Materials
## 2 Methodologies
### Theoretical Analysis
#### 2.1.3 Analytical Solution  
In acoustics, the sound pressure and particle velocity often need to be computed. For the ease of calculation, plural forms are often used to express these physical quantities. The speed of the piston in the \(z\) direction can be written as \(u=u_{A}cos\omega t\), and the displacement of the piston in the \(z\) direction is as follows:  
\[\overline{u}_{z}^{0}=\int_{0}^{t}udt=u_{A}sin\omega t. \tag{7}\]  
When \(\tau=1.088\), Rayleigh will arrive, and \(\tau=c_{T}t_{0}/r=1.088\), therefore \(t_{0}=1.088r/c_{T}\). Since \(c_{T}=\sqrt{G/\rho}\), at time \(t_{0}\), the boundary condition between the soil and the air medium can be described by Equation (8):  
\[\overline{u}_{z}^{0}\big{|}_{t=t_{0}}=u_{z}(r,0,t_{0}). \tag{8}\]  
Substituting Equation (4) and Equation (7) into Equation (8), since \(k=\omega/c\), where \(k\) is the wave number, \(\omega\) is the angular frequency, and \(c\) is wave velocity, Equation (9) is obtained:  
\[k=\frac{1}{c}\arcsin\bigg{(}\frac{Q_{0}}{\pi Gru_{A}}cos\bigg{(}1.088\omega_{ 0}r\sqrt{\frac{\rho}{G}}\bigg{)}\bigg{)}. \tag{9}\]  
Substituting Equation (9) into Equation (6), the sound pressure expression is obtained:  
\[p(r_{P},t)=i\frac{a^{2}\rho_{0}u_{A}}{4r_{P}}\arcsin\bigg{(}\frac{Q_{0}}{\pi Gru _{A}}cos\bigg{(}1.088\omega_{0}r\sqrt{\frac{\rho}{G}}\bigg{)}\bigg{)}e^{i( \omega t-kr_{P})}, \tag{10}\]  
where \(u_{A}=p_{0}/\rho_{0}c_{0}\), \(p_{0}\) is the standard atmospheric pressure at ambient temperature, \(c_{0}\) is the propagation velocity of the acoustic wave in the air medium at ambient temperature, \(\rho\) is the mass density of the soil, and \(\mu\) is the shear modulus of the soil. After Equation (10) is carried on by FFT, the root mean square (RMS) amplitude can be obtained. In Equation (11), \(b_{0}\) is a constant:  
\[\text{A}=\frac{a^{2}\rho_{0}u_{A}}{4r_{P}}b_{0}\arcsin\bigg{(}\frac{Q_{0}}{ \pi Gru_{A}}cos\bigg{(}1.088\omega_{0}r\sqrt{\frac{\rho}{G}}\bigg{)}\bigg{)}. \tag{11}\]  
[PERSON] and [PERSON] [46] developed a nonlinear isotropic model for the elastic behavior of granular materials. In their research, the [PERSON]'s modulus is expressed as in Equation (12):  
\[E=Mp_{a}\Bigg{[}\bigg{(}\frac{I_{1}}{p_{a}}\bigg{)}^{2}+R\frac{J_{2}^{\tau}}{ p_{a}^{2}}\bigg{]}^{\lambda}. \tag{12}\]  
Based on [PERSON] and [PERSON], [PERSON] et al. [47] carried out a constitutive modeling research of dense gravelly soils subjected to cyclic loading. In their paper, modulus \(G\) is further expressed as Equation (13):  
\[G=G_{0}p_{a}\frac{(2.97-e)^{2}}{1+e}[(\frac{3p}{p_{a}}^{2})^{2}+9\frac{K_{0}} {G_{0}}\frac{J_{2}}{p_{a}^{2}}]^{m/2}, \tag{13}\]  
where \(G\) is the shear modulus, \(G_{0}\) and \(m\) are material constants, \(e\) is the current void ratio, and \(p_{a}\) is the atmospheric pressure. Substituting Equation (13) and \(\text{e}=1-\rho_{0}/\rho\) into (11), Equation (14) is obtained:  
\[\text{A}=\frac{a^{2}\rho_{0}u_{A}b_{0}}{4r_{P}}\arcsin\bigg{(}\frac{Q_{0}M_{0} }{\pi ru_{A}}\frac{2\rho^{2}-\rho\rho_{0}}{\left(1.07\rho+\rho_{0}\right)^{2} }cos\bigg{(}1.088\omega_{0}r\frac{\rho}{1.97\rho+\rho_{0}}\sqrt{M_{0}(2\rho- \rho_{0})}\bigg{)}\bigg{)}, \tag{14}\]  
where \(M_{0}=G_{0}p_{a}[(3p/p_{a})^{2}+9(K_{0}/G_{0})\left(J_{2}/p_{a}^{2}\right)]^{m /2}\), \(\rho\) is the mass density, and \(\rho_{0}\) is the dry density. A diagrammatic sketch of the compacting area in pulse load duration time (\(\Delta\tau\)) is shown in Figure 7. The velocity of the roller is \(v\). As mentioned above, the harmonic load is discretized into N parts (Figure 4). The duration of each pulse load is \(\Delta\tau\). The compaction area during the time interval \(\Delta\tau\) is illustrated as rectangle ABCD in Figure 7. At this point, the area ABCD can be considered to consist of \(N_{c}\) rows and \(M_{c}\) columns of small piston radiators. Then the radiation field generated by the interaction between the vibration wheel and the soil is composed of \(N_{c}\) rows and \(M_{c}\) columns of small piston radiators on the infinite baffle.