Acoustic waves in slender axisymmetric tubes

Acoustic waves in a rigid axisymmetric tube with a variable cross-section are considered. The governing Helmholtz equation is solved using Neumann series (expansions in Bessel functions of various orders) with a stretched radial coordinate, leading to a hierarchy of one-dimensional ordinary differential equations in the longitudinal direction. The lowest approximation for axisymmetric motion includes Webster's horn equation as a special case. Fourth-order differential equations are obtained at the next level of approximation. Good agreement with existing asymptotic theories for waves in slender tubes is found.