  Our earlier papers explore the nature of large wave vector spin waves in
ultrathin ferromagnets, and also the properties and damping of spin waves of
zero wave vector, at the center of the two dimensional Brillouin zone, with
application to FMR studies. The present paper explores the behavior of spin
waves in such films at intermediate wave vectors, which connect the two
regimes. For the case of Fe films on Au(100), we study the wave vector
dependence of the linewidth of the lowest frequency mode, to find that it
contains a term which varies as the fourth power of the wave vector. It is
argued that this behavior is expected quite generally. We also explore the
nature of the eigenvectors of the two lowest lying modes of the film, as a
function of wave vector. Interestingly, as wave vector increases, the lowest
mode localizes onto the interface between the film and the substrate, while the
second mode evolves into a surface spin wave, localized on the outer layer. We
infer similar behavior for a Co film on Cu(100), though this evolution occurs
at rather larger wave vectors where, as we have shown previously, the modes are
heavily damped with the consequence that identification of distinct eigenmodes
is problematical.
