Theoretical Study of Symmetric Wave Propagation in a Semi-Infinite Elastic Isotropic Cylinder

Abstract

The boundary conditions at the traction-free interface of a semi-infinite elastic cylinder can, with one exception, be satisfied exactly only if an infinite number of modes of propagation is considered. For the low-frequency range (diameter<wavelength), there exists only a single mode of propagation with a real propagation constant [the L(0,1) mode]; however, there is also an infinite number of modes of propagation with complex propagation constants. All of these modes are required to satisfy the boundary conditions. An approximate solution is found for the problem of the L(0,1) mode impinging on a traction-free interface by using only a finite number of modes of propagation. The reflection coefficient for the L(0,1) mode is given. The accompanying generation of higher-order modes with complex propagation constants is shown to cause high-amplitude end resonances at isolated frequencies. Results are given for approximations with 3, 5, 7, and 9 modes.