SINGULAR PERTURBATION ANALYSIS OF ACOUSTIC-GRAVITY WAVES

A singular perturbation analysis of acoustic-gravity waves is presented in which not only the dependent but also the independent variables are expanded in power series of a small parameter ε. The series expansion of the independent variables enables us to correct, at each stage of the expansion, the forward characteristic lines of the governing equations and in turn, to determine the crossing of adjacent characteristics, if such crossings indeed occur for the chosen initial conditions. For waves propagating upwards in the vertical z direction and satisfying the condition that at z = 0, u = u 0 sin ωt, it is shown that a shock develops in the flow. The distance at which the shock takes place is given for different values of the index of refraction and the wave form obtained is compared to the usual linear solution.