Interaction between an elastic wave and a single pinned dislocation

Acoustic, and more generally elastic, waves in solids are damped by several mechanisms, among which dislocation motion is believed to play an important role. This is because an elastic wave interacts with a dislocation causing it to oscillate in response, and the resulting transfer of energy from wave to dislocation damps the acoustic vibrations. Recently, improved experimental techniques as well as improved numerical methods have been able to probe in some detail this interaction, isolating the effect of a single dislocation, and at this stage the theory, in its analytic form, is not sufficiently developed to provide quantitative comparison with experimental data and computer simulations. There is thus a need for an improved theoretical study of this issue. In this paper, we consider the interaction of transverse $(T)$ and longitudinal $(L)$ polarized waves in a homogeneous and isotropic, three dimensional, continuum linear elastic medium interacting with a dislocation segment pinned at both ends. An elastic wave incident upon such a dislocation segment is scattered, and the resulting scattered wave is characterized by its scattering amplitudes, that account for possible $T\text{\ensuremath{-}}L$ mode conversions. Such scattering amplitudes are explicitly calculated. As a consequence, it is possible to calculate the resulting interference patterns of incident with scattered wave, such as have been observed in recent experiments [Shilo and Zolotoyabko, Phys. Rev. Lett. 91, 115506 (2003)]. The energy loss per cycle is also calculated using the optical theorem and results are shown to be in qualitative agreement with the results of numerical experiments [Greaney et al., Comput. Mater. Sci. 25, 387 (2002)].