# Fiber-Optic System for Monitoring Pit Collapse Prevention
## 2 Materials and Methods  
Let us assume that the geometric model of the main working element of an optical fiber (OF) is in the form of a cylindrical sample with a base radius \(r_{0}\) and length \(l\). The refractive index of material of the fiber-optic element is accepted in the general case \(n=\frac{k}{k_{0}}=\frac{c}{\theta}=\frac{\lambda_{0}}{\lambda}\), where \(k_{0}=\frac{\omega}{c}=\frac{2\pi}{\lambda_{0}}\); \(k=\frac{\omega}{v}=\frac{2\pi}{\lambda}\); \(k=\frac{\omega}{\theta}=\frac{2\pi}{\lambda}\) are the wave vectors of the falling and refracted waves, respectively. The physical model of the wave processes studied in the fiber-optic element will be constructed using the methods of classical electrodynamics and wave optics. The parameter of the d'Alembert wave equation for a input wave (falling from air at an angle \(\theta\) onto the cross-sectional surface of a fiber-optic element) is \(\overrightarrow{\bigtriangledown^{2}u}-\frac{1}{c^{2}}\frac{\partial^{2}u}{ \partial t^{2}}=0\), and wave, refracted at an angle \(\gamma\) in the fiber-optic element, is \(\overrightarrow{\bigtriangledown^{2}u}-\frac{1}{v^{2}}\frac{\partial^{2}u}{ \partial t^{2}}=0\). Its general solution is taken in the form  
\(\frac{1}{c}=\sqrt{\epsilon_{0}\mu_{0}}\), \(\overrightarrow{u}=\overrightarrow{u_{0}}\cdot\exp(\mathrm{i}ot-( \overrightarrow{k_{0}}\overrightarrow{r}))\) \(\frac{1}{c}=\sqrt{\epsilon_{0}\mu_{0}}\); \(\overrightarrow{u_{0}}=\overrightarrow{u_{0}}\cdot\exp(i\omega t-( \overrightarrow{k_{0}}\overrightarrow{r}))\). Vector classical wave function \(\overrightarrow{u}(\overrightarrow{r};t)\) is characterized by the amplitude \(\overrightarrow{u_{0}}\) and the spatial phase \(\phi=(\overrightarrow{k}\overrightarrow{r})=\frac{\omega}{\theta^{2}}( \overrightarrow{\theta}\overrightarrow{r})\). The vector \(\overrightarrow{u}(\overrightarrow{r};t)\) acts as both the electric field strength vector \(\overrightarrow{E}(\overrightarrow{r};t)\) and the wave's magnetic field strength vector \(\overrightarrow{H}(\overrightarrow{r};t)\). The spatial phase of the falling wave is taken as equal to \(\phi_{0}=(\overrightarrow{k_{0}}\overrightarrow{r})=\frac{\omega}{c}( \overrightarrow{c}\overrightarrow{r})\). The spatial phase of the refracted wave is taken as equal to \(\phi_{0}=(\overrightarrow{k}\overrightarrow{r})=\frac{\omega}{v^{2}}( \overrightarrow{r})\). The selection of the directions of single vectors \(\overrightarrow{u_{0}}\), \(\overrightarrow{t}\) is based on the directions of the rate vectors \(\overrightarrow{c}\), \(\overrightarrow{v}\) and the wave vectors \(\overrightarrow{k}_{0}=\frac{\omega}{c}\overrightarrow{v}_{0}\), \(\overrightarrow{k}=\frac{\omega}{v}\overrightarrow{r}\). The direction of the \(\overrightarrow{c}\) vector is based on the radius-vector of the point on the surface of \(\overrightarrow{r}\) light-sensitive element, accepting \((\overrightarrow{r_{0}}\overrightarrow{r})=r\cdot cos\alpha_{1},( \overrightarrow{r}\overrightarrow{r})=r\cdot cos\alpha_{2}\), \(\psi=\theta-\gamma\), \(\psi=\alpha_{2}-\alpha_{1}\), the Equation \(cos\psi=\frac{cos\theta(\sqrt{n^{2}-sin^{2}\theta}+sin^{2}\theta)}{n}\) is obtained. Then, the spatial difference of these waves phases is \(\varphi-\varphi_{0}=\frac{\omega}{c^{2}}(ncos\psi-1)(\overrightarrow{c} \overrightarrow{r})\), where \(n=\frac{sin\theta}{sin\gamma r}\), \(r\) is the distance from the point of intersection of the light beam (a wave incident on the surface of the cross-section of the optical fiber) with the surface of a given section, placed in this model at the origin of coordinates to a fixed point in space.