Initiation of turbulence and chaos in non-equilibrium inhomogeneous media: Wave beams

We show that wave excitation and propagation in an inhomogeneous medium leads to the simultaneous appearance of a huge number of waves with different phase velocities. This phenomenon arises in any medium with inhomogeneous parameters, e.g. in fluid jets where the inhomogeneity appears as a result of the boundary layer. Because of fluctuations (however small) the waves become randomised, i.e. turbulence develops. We demonstrate that the eigenvalues depend essentially on the frequency of the perturbation and on the distance from the initial section of a jet or wave beam. We show how to find the continuous set of eigenvalues — complex wave numbers — and corresponding eigenfunctions for any given frequency. The implication of these results is that the transition to turbulence occurs, not through the excitation of a gradually increasing number of waves, as commonly supposed, but by the simultaneous excitation of a continuous wave spectrum. PACS numbers: 47.27.wg, 47.27.Cn, 47.20.Ft, 47.27.ed