On Waves in a Linear Elastic Half-Space with Free Boundary

The problem of linear elasticity for free harmonic (periodic) and solitary bell-shaped (nonperiodic) waves in an isotropic half-space with stress-free plane boundary is considered. The half-space is made of either conventional (classical structural) or nonconventional (nonclassical auxetic) material. Two cases of wave damping are studied: rapid (surface wave) and periodic (nonsurface wave). The following conclusions on a free harmonic wave are drawn: a surface wave exists in materials of both classes, but the ratio of the wave velocity to the velocity of a transverse plane wave in auxetic materials is somewhat lower than in conventional materials; a nonsurface wave cannot be described by the approach applied to conventional materials, but can theoretically exist in auxetic materials where there are two wave velocities. For a solitary (bell-shaped) wave, the assumption that the wave velocity depends on the wave phase is substantiated and some constraint is imposed on the time of travel of the wave and the way the wave velocity varies with time. The following conclusions are drawn: a rapidly damped bell-shaped wave cannot be described by the approach for both classes of materials, whereas a periodically damped bell-shaped wave can be described.