Numerically inverting a class of singular Fourier transforms: theory and application to mountain waves

Many partial differential equations of physics are solved using Fourier transforms. Even when solutions to the transformed equations can be found analytically it is rare for their inverse transforms to be known in terms of simple functions. Instead, asymptotic and/or numerical approaches are commonly used to approximate the inverse transforms. A numerical technique is here developed to invert a one–dimensional Fourier transform with singularities on the real axis of the complex plane that are at worst simple poles, subject to the condition that the inverse transform vanishes as its transform variable becomes infinite in one direction. By way of demonstration, it is used to compute the respective solutions of Long and Wurtele for trapped and evanescent lee waves driven by flow over gently sloping hills.