Waves in wood: free vibrations of a wooden pole

摘要

Elastic waves in materials with cylindrical orthotropy are considered, this being a plausible model for a wooden pole. For time-harmonic motions, the problem is reduced to some coupled ordinary differential equations. Previously, these have been solved using the method of Frobenius (power-series expansions). Here, Neumann series (expansions in Bessel functions of various orders) are used, motivated by the known classical solutions for homogeneous isotropic solids. This is shown to give an effective and natural method for wave propagation in cylindrically Orthotropic materials. As an example, the frequencies of free vibration of a wooden pole are Computed. The problem itself arose from a study of ultrasonic devices as used in the detection Of rotten regions inside wooden telegraph (utility) poles and trees; some background to these Applications is given.