On Webster’s horn equation and some generalizations

Abstract

Sound waves along a rigid axisymmetric tube with a variable cross-section are considered. The governing Helmholtz equation is solved using power-series expansions in a stretched radial coordinate, leading to a hierarchy of one-dimensional ordinary differential equations in the longitudinal direction. The lowest approximation for axisymmetric motion turns out to be Webster’s horn equation. Fourth-order differential equations are obtained at the next level of approximation. Comparisons with existing asymptotic theories for waves in slender tubes are made.