Complex ray theory applied to instability of a two-dimensional wake

Abstract

Complex ray theory successfully describes temporal and spatial development of a two-dimensional wave packet into steady and semi-infinite distribution of very large disturbances in the wake accompanied by a region of reverse flows. Final formation of the steady state is caused by the existence, on integral path of propagation equations, of a logarithmic singularity, at which the group velocity and wave-number variation both vanish and the temporal growth rate is positive. Nearly singular frequencies around the complex frequency defined at the singularity pass through a close vicinity of the singular point while spending a very large time and then compose the distributed disturbances at the limit of large time. Propagation of the nearly singular frequencies can be pursued by an appropriate application of the complex-time method of integral to the region away from the singularity and the complex-coordinate method of integral to its neighborhood.