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逼近亥姆霍兹方程控制的时谐波的误差估计。对于所研究的焦散，精确解可利用艾里函数构造，且高斯光束参数具有显式表达式。通过精确比较，我们证明了焦散上的逐点误差为 $O(k^{-5/6})$ 阶，其中 $k$ 为亥姆霍兹方程中的波数。