Solution of the End Problem of a Liquid-Filled Cylindrical Acoustic Waveguide Using a Biorthogonality Principle

Abstract

This paper treats the forced motion of an isothermal, Newtonian liquid in a semi-infinite cylindrical waveguide. Its bounding wall is assumed rigid, allowing neither normal nor tangential fluid velocities at its inner surface. Small amplitude acoustic waves are considered to be driven by a steady periodic motion due to the rotationally symmetric deflection of an end plate or membrane. A series expansion of the waveguide eigenmodes is used to construct the solution for the motion anywhere within the guide. Based on a biorthogonality property of the eigenmodes, each coefficient of the series is shown to be directly calculable in terms of axial velocity and radial shear stress at the driver face. Also, results of Galerkin solutions, based on driver axial velocity and zero radial velocity, are given for comparison.