Modern Developments in Asymptotic Ray Theory

用于亥姆霍兹方程的射线方法已经被开发出来，并应用于高频衍射和散射问题，取得了无与伦比的成功。这个理论是数学光学的基础，也是衍射几何理论的基础。它依赖于用振幅扩展和单相项来描述波，由此产生这些量的偏微分方程。首席研究员已经确定了一类新的局部问题，这种方法不适用于这些问题，并且计划通过在阶段中包含一个广义和的术语来实质性地修改现有的理论，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.