  Efforts at modelling the propagation of seismic waves in half-spaces with
continuously varying properties have been mostly focused on shear-horizontal
waves. Here a sagittaly polarized (Rayleigh type) wave travels along a symmetry
axis (and is attenuated along another) of an orthotropic material with
stiffnesses and mass density varying in the same exponential manner with depth.
Contrary to what could be expected at first sight, the analysis is very similar
to that of the homogeneous half-space, with the main and capital difference
that the Rayleigh wave is now dispersive. The results are illustrated
numerically for (i) an orthotropic half-space typical of horizontally layered
and vertically fractured shales and (ii) for an isotropic half-space made of
silica. In both examples, the wave travels at a slower speed and penetrates
deeper than in the homogeneous case; in the second example, the inhomogeneity
can force the wave amplitude to oscillate as well as decay with depth, in
marked contrast with the homogeneous isotropic general case.
