Arthur D. Gorman

Caustics and the parabolic wave equation

Abstract Caustics associated with the parabolic approximation to the reduced Helmholtz equation are considered using the modified Lagrange manifold formalism.

Time-evolution of a caustic

The Lagrange manifold formalism is adapted to study the time-evolution of caustics associated with high frequency wave propagation in media with both spatial and temporal inhomogenities.

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.

On caustics associated with the linearized vorticity equation

The linearized vorticity equation serves to model a number of wave phenomena in geophysical fluid dynamics. One technique that has been applied to this equation is the geometrical optics, or multi-dimensional WKB technique. Near caustics, this technique does not apply. A related technique that does apply near caustics is the Lagrange Manifold Formalism. Here we apply the Lagrange Manifold Formalism to determine an asymptotic solution of the linearized vorticity equation and to study associated wave phenomena on the caustic curve.

On caustics associated with horizontal rays and vertical modes

Abstract Wave propagation in a medium with slowly varying inhomogeneities can sometimes be modelled by a combination of normal mode theory in one direction and ray theoretical methods in the other directions. Caustics can occur in the ray theory ansatz. Here we consider those caustics using a Lagrange manifold approach.

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 some 4th order equations of physical significance.

On the time-dependent parabolic wave equation

One approach to the study of wave propagation in a restricted domain is to approximate the reduced Helmholtz equation by a parabolic wave equation. Here we consider wave propagation in a restricted domain modelled by a parabolic wave equation whose properties vary both in space and in time. We develop a Wentzel-Kramers-Brillouin (WKB) formalism to obtain the asymptotic solution in noncaustic regions and modify the Lagrange manifold formalism to obtain the asymptotic solution near caustics. Associated wave phenomena are also considered.

Addendum to a paper of Craig and Goodman

In [1], Craig and Goodman develop the geometrical optics solution of the linearized Korteweg-deVries equation away from caustic, or turning, points. Here we develop an analogous solution valid at caustic points.