Surface elastic waves on semi-infinite (001) plane of null stress in cubic homogeneous crystal

Elastic surface waves exits on surface null stress of semi infinite homogeneous material. The energy of surface waves is confined up to few wavelength depth on surface . The displacement amplitude decays exponently with depth. To satisfy null stress , the wave is composed of three partial wave with three constants for linear combination of displacement . In symmetric direction such as [100] wave is purely in sagittal plane. The particle displacement in surface wave is elliptical with one of the displacement component imaginary or 90 degree out of phase with in plane component . In non symmetric direction there is component of displacement perpendicular to sagittal plane . At around 30 degree surface wave velocity becomes equal to quasi transverse bulk wave and exactly at 45 degree surface wave becomes transverse bulk wave propagating in [110] direction. In [110] direction , there is also a supersonic velocity . The wave of this velocity are of pseudo surface wave branch . The displacement in this supersonic branch in [110]direction are purely in sagital plane with elliptical polarization . The pseudo surface wave solution arise from surface wave solution such that when there is second higher velocity when real and imaginary component of stress traction at surface can be made null with a complex velocity . The real and imaginary part of surface traction at this angle are zero for a real velocity below quasi transverse threshold but above this velocity also surface traction can be made to zero by taking propagation vector complex such that it decay in direction of propagation .