  We report a neutron scattering study of the long-wavelength dynamic spin
correlations in the model two-dimensional $S=1/2$ square lattice Heisenberg
antiferromagnets Sr$_2$CuO$_2$Cl$_2$ and Sr$_2$Cu$_3$O$_4$Cl$_2$. The
characteristic energy scale, $\omega_0 (T/J)$, is determined by measuring the
quasielastic peak width in the paramagnetic phase over a wide range of
temperature ($0.2 \alt T/J \alt 0.7$). The obtained values for $\omega_0 (T/J)$
agree {\it quantitatively} between the two compounds and also with values
deduced from quantum Monte Carlo simulations. The combined data show scaling
behavior, $\omega \sim \xi^{-z}$, over the entire temperature range with
$z=1.0(1)$, in agreement with dynamic scaling theory.
