Alternate solution for the cylindrical Helmholtz vector equation applied to helical elastic guided waves in pipes

Abstract

Elastic helical guided wave propagation in pipes that has recently gained importance in applications related to tomography and structural health monitoring is analyzed using an alternate formalism. Closed form exponential function based solutions for the Helmholtz vector equation in cylindrical polar coordinates are derived. Relationship of these alternate solutions for the Helmholtz vector equation with the traditional integer order Bessel function based formulation – that has been established for the corresponding solutions of Helmholtz scalar equation in prior literature – is presented. The solutions are single valued at every point in the physical space, and therefore, unlike traditional non-integer order Bessel function based methods, the formulation presented herein preserves the physical uniqueness of the field quantities involved in the wave propagation. The alternate solutions, when applied to the boundary value problem of an isotropic elastic pipe with stress free boundaries, yield a formulation for helical guided wave propagation. A class of helical guided wave modes that have a constant helix angle across the wall thickness of the pipe is predicted. Dispersion characteristics for guided wave propagation such as phase velocity curves; displacement profiles for some points of interest on the phase velocity curves, for select helical angles are presented. The results are compared against traditional notions about helical guided wave propagation.