Waves in wood: elastic waves in cylindrically orthotropic materials

We consider elastic waves in materials with cylindrical orthotropy, this being a plausible model for wood. 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). We use Neumann series (expansions in Bessel functions of various orders), motivated by the known classical solutions for homogeneous isotropic solids. This gives an effective and natural method for wave propagation in cylindrically orthotropic materials. The problem itself arose from a study of ultrasonic devices as used in the detection of rotten regions inside wooden telegraph (utility) poles.