Wave Equation in the Formulation of the Cauchy Problem with Respect to Spatial Variables for the Prediction of Catastrophic Events

To determine a priori the unknown structure of the Earth and the forecast of catastrophic events such as earthquakes, it is suggested to apply the formulation of the Cauchy problem when both the Cauchy stress vector and the displacement vector as a function of the boundary and time coordinates t are given on the same surface. The advantages of this formulation are demonstrated by the example of the solution of the dynamic problem for a semi-infinite rod where there are no initial conditions along the entire length of the rod, and only Cauchy conditions at the end of the rod imitating the Earth's surface are given. To construct a finite-difference scheme with the second order of approximation accuracy, both the Cauchy conditions and the wave equation are used at the end of the rod. The calculated dependences of displacements, displacement velocities along the rod, comparison of numerical and analytical solutions are given.