“Oscillating” Riemann invariants of the hyperbolic systems of partial differential equations

Abstract

The phenomenon of wavenumber oscillations in a wave packet representing an RF pulse propagating in a dispersive medium was studied by numerical methods. An explanation is suggested in the context of nonlinear geometrical optics (NGO), where the wavenumber is a Riemann invariant that remains constant on characteristics of the corresponding equations. The wavenumber oscillations arise in the region of intersection of the characteristics of the equations. It is demonstrated that the intersection of characteristics is due to the approximation neglecting the interaction between narrow-band wave packets constituting the pulse. With the packet interaction ignored, the characteristics are straight lines. If the interaction is allowed for, the characteristics do not intersect and may significantly differ in shape from straight lines. Consequently, an external observer moving at a constant velocity crosses the same characteristic many times, thus perceiving the wave-number as an oscillating function.