Wave reflection from the free end of a cylinder with an arbitrary cross-sectio

Elastic wave scattering at the free end of a cylinder due to an incident monochromatic wave is investigated. The cross-section may have an arbitrary geometry with any number of distinct elastic rectilinear anisotropic materials comprising its planar pro®le. The governing equations are based on a semi-analytical ®nite element method in which the cross-sectional behavior is modeled by general two-dimensional ®nite elements with the axial dependence and time left unspeci®ed at the outset. First, all the modal data for the cylinder are established. Two eigenproblems are posed for this purpose, that are obtained by inserting a wavelike solution form into the governing equations. These eigenproblems allow all propagating waves and end modes for the cylinder to be determined. Propagating modes are traveling waves with energy transport capabilities, while the end modes are standing vibrations which, in contrast, do not transport any energy into the interior of the cylinder. These eigendata are the basis for representing the wave re¯ection phenomenon at the free end. The amplitudes of the traveling waves and end modes that satisfy traction-free end conditions may be determined by least-squares minimization or by a virtual work method. Four cylinders with di€erent cross-sectional geometries were considered to illustrate the analysis procedure and reveal some physical insight into the frequency dependent wave re¯ection phenomena in them