Modern Developments in Asymptotic Ray Theory

应用于亥姆霍兹方程的射线方法已经发展并应用于高频绕射和散射问题，取得了无与伦比的成功。这个理论是数学光学的基础，也是几何绕射理论的基础。它依赖于用振幅展开式和单相项来描述波，由此得到这些量的偏微分方程式。首席研究员已经确定了这种方法不适用的一类新的主题问题，计划通过在相位中包含广义项和来大幅修改现有理论，然后将新的广义理论应用于这些问题。

Ray methods applied to the Helmholtz equation have been developed for, and applied to, high-frequency diffraction and scattering problems with unparalleled success. This theory underpins mathematical optics and is fundamental to the geometrical theory of diffraction. It relies upon describing waves in terms of an amplitude expansion and a single phase term, from which partial differential equations for these quantities follow.The lead investigator has identified a novel class of topical problems for which this approach does not work, and the plan is to modify substantially the existing theory by the inclusion of a generalised sum of terms in the phase, and then to apply the newly generalised theory to these problems.