Space-time caustics

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of linear differential equations modelling wave propagation in spatially inhomogeneous media at caustic (turning) points. Here the formalism is adapted to determine a class of asymptotic solutions at caustic points for those equations modelling wave propagation in media with both spatial and temporal inhomogeneities. The analogous Schrodinger equation is also considered.

Dispersive waves and caustics

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of linear, non-dispersive wave equations at turning points. The formalism is adapted to include those equations which model dispersive waves.

On the asymptotic series solution of some higher order linear differential equations at turning points

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of second-order “wave type” differential equations at turning points. The formalism also applies to higher order linear differential equations, as we make explicit here illustrating with some4th order equations of physical significance.

Langer's method for a second order linear ordinary differential equation with a double pole

If a second order ordinary differential equation has a simple or a double pole at a point zfl, then the standard Liouville-Green approximation could sometimes be valid near that point. In this paper we present an asymptotic series solution that is always valid near a double pole. A solution for that of a simple pole is also indicated. Asymptotic validity is proved.

On the effects of boundary-layer growth on flow stability

The stability of small travelling-wave disturbances in the flow over a flat plate is discussed. An iterative method is used to generate an asymptotic series solution in inverse powers of the Reynolds number Rx = Ux/v to the power one half. The neutral-stability boundaries given by the first two terms of this series are obtained and compared with experimental data. It is shown that the parallel flow approximation leads to a valid solution at very large Reynolds numbers.