## 2 Theoretical Derivation of Analytical Solutions for Blasting Stress Fields in Adjacent Tunnels
### Blasting Stress Field of a Single-Hole Cylindrical Charge Based on [PERSON]'s Short-Cylinder Solution
\begin{array}{l}\overset{\rightarrow}{u}=\overset{\rightarrow}{u}_{P}+ \overset{\rightarrow}{u}_{SV}=\Big{(}\frac{F_{1}(\phi)}{R}\frac{d}{dt}\Big{\{} p(t-\frac{R}{c_{p}})\Big{\}}+\frac{G_{1}(\phi)}{R}q(t-\frac{R}{C_{p}})\Big{)}\sin\phi\\ +\Big{(}\frac{F_{2}(\phi)}{R}\frac{d}{dt}\Big{\{}p(t-\frac{R}{c_{s}})\Big{\}}+ \frac{G_{2}(\phi)}{R}q(t-\frac{R}{c_{s}})\Big{)}cos\phi\\ \overset{\rightarrow}{\omega}=\overset{\rightarrow}{\omega}_{P}+\overset{ \rightarrow}{u}_{SV}=-cos\phi\Big{(}\frac{F_{1}(\phi)}{R}\frac{d}{dt}\Big{\{} p(t-\frac{R}{c_{p}})\Big{\}}+\frac{G_{1}(\phi)}{R}q(t-\frac{R}{C_{p}})\Big{)}\\ +\Big{(}\frac{F_{2}(\phi)}{R}\frac{d}{dt}\Big{\{}p(t-\frac{R}{c_{s}})\Big{\}}+ \frac{G_{2}(\phi)}{R}q(t-\frac{R}{c_{s}})\Big{)}\sin\phi\\ \overset{\rightarrow}{v}=\overset{\rightarrow}{v}_{\text{SH}}=\frac{K(\phi) }{R}\frac{d}{dt}\Big{\{}s(t-\frac{R}{c_{s}})\Big{\}}\end{array}  \tag{10}\]  
By differentiating the displacement field with respect to time, the vibration velocity field of the short-cylinder charge is derived as follows:  
\[ \begin{array}{l}\overset{\rightarrow}{V}_{r}=\frac{\partial u}{dt}=\Big{(} \frac{F_{1}(\phi)}{R}\frac{d^{2}}{dt^{2}}\Big{\{}p(t-\frac{R}{c_{p}})\Big{\}}+ \frac{G_{1}(\phi)}{R}\frac{d}{dt}\Big{\{}q(t-\frac{R}{c_{p}})\Big{\}}\Big{)} \sin\phi\\ +\Big{(}\frac{F_{2}(\phi)}{R}\frac{d^{2}}{dt^{2}}\Big{\{}p(t-\frac{R}{c_{s}}) \Big{\}}+\frac{G_{2}(\phi)}{R}\frac{d}{dt}\Big{\{}q(t-\frac{R}{c_{s}})\Big{\}} \Big{\}}\cos\phi\\ \overset{\rightarrow}{V}_{\text{z}}=\frac{\partial\omega}{dt}=-cos\phi\Big{(} \frac{F_{1}(\phi)}{R}\frac{d^{2}}{dt^{2}}\Big{\{}p(t-\frac{R}{c_{p}})\Big{\}}+ \frac{G_{1}(\phi)}{R}\frac{d}{dt}\Big{\{}q(t-\frac{R}{c_{p}})\Big{\}}\Big{)}\\ +\Big{(}\frac{F_{2}(\phi)}{R}\frac{d^{2}}{dt^{2}}\Big{\{}p(t-\frac{R}{c_{s}}) \Big{\}}+\frac{G_{2}(\phi)}{R}\frac{d}{dt}\Big{\{}q(t-\frac{R}{c_{s}})\Big{\}} \Big{\}}\sin\phi\\ \overset{\rightarrow}{V}_{\theta}=\frac{\partial v}{\partial t}=\frac{K(\phi )}{R}\frac{d^{2}}{dt^{2}}\Big{\{}s(t-\frac{R}{c_{s}})\Big{\}}\end{array}  \tag{15}\]  
where \(\overset{\rightarrow}{V}_{r}\), \(\overset{\rightarrow}{V}_{z}\) and \(\overset{\rightarrow}{V}_{\theta}\) represent the radial, axial, and hoop vibration velocities, respectively, in units of m/s. During blasting operations, the relationships between the stresses (\(\sigma_{r}\), \(\tau_{z}\), \(\tau_{\theta}\)) generated by p-waves, SV-waves, and SH-waves and the corresponding particle vibration velocities (\(\overset{\rightarrow}{V}_{r}\), \(\overset{\rightarrow}{V}_{z}\), \(\overset{\rightarrow}{V}_{\theta}\)) can be expressed as follows [38]:  
\[\left\{ \begin{array}{l}\sigma_{r}=\overset{\rightarrow}{V}_{r}\rho C_{p} \\ \tau_{z}=\overset{\rightarrow}{V}_{\theta}\rho C_{s}\\ \tau_{\theta}=\overset{\rightarrow}{V}_{\theta}\rho C_{s}\end{array} \right. \tag{16}\]  
where \(\rho\) is the rock density, kg/m\({}^{3}\). By synthesizing Equations (13)-(16), the stress field induced by a short-cylinder charge blasting can be derived as follows:  
\[\begin{array}{l}\sigma_{r}=\Big{(}\frac{F_{1}(\phi)}{R}\frac{d^{2}}{dt^{2}} \Big{\{}p(t-\frac{R}{c_{p}})\Big{\}}+\frac{G_{1}(\phi)}{R}\frac{d}{dt}\Big{\{} q(t-\frac{R}{c_{p}})\Big{\}}\Big{)}\rho C_{p}\sin\phi\\ +\Big{(}\frac{F_{2}(\phi)}{R}\frac{d^{2}}{dt^{2}}\Big{\{}p(t-\frac{R}{c_{s}}) \Big{\}}+\frac{G_{2}(\phi)}{R}\frac{d}{dt}\  
\[\tau_{\theta}=\frac{\rho C_{s}K(\phi)}{R}\frac{d^{2}}{\mathrm{d}t^{2}}\bigg{\{}s(t -\frac{R}{C_{s}})\bigg{\}} \tag{19}\]