[(LSqLS-qperpendicular to)-L-perpendicular to], which is related to the eff
ect of damping of spin waves in a two-dimensional classical ferromagnetic X
Y model, is considered. The damping rate Gamma (q) is calculated using the
leading diagrams due to the quartic-order deviations from the harmonic spin
Hamiltonian. The resulting four-dimensional integrals are evaluated by ext
ending the techniques developed by Gilat and others for spectral density ty
pes of integrals, Gamma (q) is included into the memory function formalism
due to Reiter and Solander, and Menezes, to determine the dynamic structure
function S(q,omega). For the infinite sized system, the memory function ap
proach is found to give nondivergent spin-wave peaks, and a smooth nonzero
background intensity ("plateau" or distributed intensity) for the whole ran
ge of frequencies below the spin-wave peak. The background amplitude relati
ve to the spin-wave peak rises with temperature, and eventually becomes hig
her than the spin-wave peak, where it appears as a central peak. For finite
-sized systems, there are multiple sequences of weak peaks on both sides of
the spin-wave peaks whose number and positions depend on the system size a
nd wave vector in integer units of 2 pi /L. These dynamical finite-size eff
ects are explained in the memory function analysis as due to either spin-wa
ve difference processes below the spin-wave peak or sum processes above the
spin-wave peak. These features are also found in classical Monte Carlo-spi
n-dynamics simulations.