There are very few exact solutions available for wave propagation in plane‐layered materials that are pierced by a cylindrical hole. This is unfortunate since these problems occupy a central role in geophysics. Exact solutions for the cases of an empty hole and an infinitely rigid hole, where the geometry is separable so that the classical transform method is applicable, are presented. It is hoped that the exact solutions presented herein will provide a check on more general numerical methods.