Spatial evolution of harmonic vibrations in linear elasticity

In the present paper we consider a prismatic cylinder occupied by an anisotropic homogeneous compressible linear elastic material that is subject to zero body force and zero displacement on the lateral boundary. The elasticity tensor is strongly elliptic and the motion is induced by a harmonic time-dependent displacement specified pointwise over the base. We establish some spatial estimates for appropriate cross-sectional measures associated with the harmonic vibrations that describe how the corresponding amplitude evolves with respect to the axial distance at the excited base. The results are established for finite as well as for semi-infinite cylinders (where alternatives results of Phragmén-Lindelöf type are obtained) and the exciting frequencies can take appropriate low and high values. In fact, for the low frequency range the established spatial estimates are of exponential type, while for the high frequency range the spatial estimates are of a certain algebraic type.