The formation of shock waves in Krylov-Bogoliubov solutions of hyperbolic partial differential equations

Abstract

The longitudinal vibrations of an elastic column are considered for the case when the stress-strain relation of the material involves a small non-linear term, quadratic in the strain. When the boundary conditions are of fixed-end type, the system possesses an infinite number of internal resonances. The Krylov-Bogoliubov-Mitropolskii (KBM) method is extended to such cases, and is used to find the approximate solution for the elastic column with arbitrary initial conditions. It is shown that the solution always develops a singularity (shock wave) at a position and time which are calculated explicitly in terms of the initial data. When viscous damping is present in the system, an upper bound is found for the damping, above which the shock wave does not form.