Elastic surface waves guided by the edge of a slit

Surface waves propagating along the free surface of a homogeneous, isotropic, linearly elastic half-space, are shown to have the property that the normal displacement component at the free surface is governed by a reduced wave equation. This suggests a "membrane analogy", and a corresponding family of surface waves. Of particular interest is a three-dimensional surface wave, whose displacement components in the sagittal plane vary linearly with the co-ordinate normal to that plane, while the displacement component in the direction normal to the sagittal plane is uniform in that direction. This new wave arises when surface waves propagate along the free surfaces of a semi-infinite slit, parallel to the edge of the slit, with the classical Rayleigh wave velocity. It is also shown that a semi-infinite slit cannot support surface waves which decay with the distance from the edge of the slit.