  The matrix sign function is shown to provide a simple and direct method to
derive some fundamental results in the theory of surface waves in anisotropic
materials. It is used to establish a shortcut to the basic formulas of the
Barnett-Lothe integral formalism and to obtain an explicit solution of the
algebraic matrix Riccati equation for the surface impedance. The matrix sign
function allows the Barnett-Lothe formalism to be readily generalized for the
problem of finding the surface wave speed in a periodically inhomogeneous
half-space with material properties that are independent of depth. No partial
wave solutions need to be found; the surface wave dispersion equation is
formulated instead in terms of blocks of the matrix sign function of i times
the Stroh matrix.
