  We study the spin excitations and the transverse susceptibility of a
two-dimensional antiferromagnet doped with a small concentration of holes in
the t-J model. The motion of holes generates a renormalization of the magnetic
properties. The Green's functions are calculated in the self-consistent Born
approximation. It is shown that the long-wavelength spin waves are
significantly softened and the shorter-wavelength spin waves become strongly
damped as the doping increases. The spin wave velocity is reduced by the
coherent motion of holes, and not increased as has been claimed elsewhere. The
transverse susceptibility is found to increase considerably with doping, also
as a result of coherent hole motion. Our results are in agreement with
experimental data for the doped copper oxide superconductors.
