In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order $O(k^{-5/6})$ where $k$ is the wave number in Helmholtz.

翻译：本文给出了折叠焦散处一阶高斯光束的误差估计，该光束近似了由亥姆霍兹方程控制的时谐波。对于所研究的焦散，精确解可通过艾里函数构造，并且高斯光束参数存在显式公式。通过精确比较，我们证明焦散上的逐点误差阶为$O(k^{-5/6})$，其中$k$为亥姆霍兹方程中的波数。