Rayleigh's (1926, Appendix A) method for the approximate calculation of potential flow from the open end of a semi-infinite flanged cylinder is applied to obtain analytical representations of Green's function describing the generation of sound waves within a flanged cylinder by sources located in the neighbourhood of the open end. Detailed results are given for the circular cylinder considered by Rayleigh and extension made to a flanged cylinder of rectangular cross-section. The validity of various approximations is assessed by comparison with the exact solution available (by the method of conformal transformation) for potential flow from a two-dimensional, flanged duct. The results are used to compare the profiles and the pressure gradients of compression waves generated when a high-speed train enters tunnels of circular and rectangular cross-sections.