  We investigated the spin-wave propagation in a micro-structured yttrium iron
garnet waveguide of $40$ nm thickness. Utilizing spatially-resolved Brillouin
light scattering microscopy, an exponential decay of the spin-wave amplitude of
$(10.06 \pm 0.83)$ $\mu$m was observed. This leads to an estimated Gilbert
damping constant of $\alpha=(8.79\pm 0.73)\times 10^{-4}$, which is larger than
damping values obtained through ferromagnetic resonance measurements in
unstructured films. The theoretically calculated spatial interference of
waveguide modes was compared to the spin-wave pattern observed experimentally
by means of Brillouin light scattering spectroscopy.
