Contour integral solutions of the parabolic wave equation

TL;DR

This paper introduces a systematic method for constructing solutions to the 2D parabolic wave equation using complex contour integrals, capturing various high-frequency wave phenomena and their transitions.

Contribution

It provides a unified framework for describing classical wave phenomena through contour integral solutions, including new insights into wave localization near algebraic curves.

Findings

Solutions exhibit far-field localisation near algebraic curves.

Framework describes caustics, whispering gallery, and creeping waves.

Analysis of solutions near cubic parabolas and inflection points.

Abstract

We present a simple systematic construction and analysis of solutions of the two-dimensional parabolic wave equation that exhibit far-field localisation near a given algebraic plane curve. Our solutions are complex contour integral superpositions of elementary plane wave solutions with polynomial phase, the desired localisation being associated with the coalescence of saddle points. Our solutions provide a unified framework in which to describe some classical phenomena in two-dimensional high frequency wave propagation, including smooth and cusped caustics, whispering gallery and creeping waves, and tangent ray diffraction by a smooth boundary. We also study a subclass of solutions exhibiting localisation near a cubic parabola, and discuss their possible relevance to the study of the canonical inflection point problem governing the transition from whispering gallery waves to creeping waves.