A wide class of analytical solutions of the Webster equation

Abstract This paper aims at presenting closed-form general analytical solutions of the Webster equation describing plane elastic or acoustic waves. The considered radius functions of non-uniform cross-sectioned rods or ducts are based on the triconfluent Heun functions and contain some optional parameters enabling us to set various profiles of the radius functions in a relatively wide range, while it is possible to employ the presented exact general analytical solution of the Webster equation for all selected profiles. If the radius functions are predetermined, then the derived general analytical solution can also be employed for their triconfluent Heun approximations, including certain polynomial ones. The applicability and correctness of the derived analytical solutions are demonstrated by calculations of natural frequencies and mode shapes for representative radius functions while the results based on approximate analytical solutions are verified numerically.