  We present results for the dynamic structure factor of the S=1/2 bond
alternating Heisenberg chain over a large range of frequencies and
temperatures. Data are obtained from a numerical evaluation of thermal averages
based on the calculation of all eigenvalues and eigenfunctions for chains of up
to 20 spins. Interpretation is guided by the exact temperature dependence in
the noninteracting dimer limit which remains qualitatively valid up to an
interdimer exchange $\lambda \approx 0.5$. The temperature induced central peak
around zero frequency is clearly identified and aspects of the crossover to
spin diffusion in its variation from low to high temperatures are discussed.
The one-magnon peak acquires an asymmetric shape with increasing temperature.
The two-magnon peak is dominated by the S=1 bound state which remains well
defined up to temperatures of the order of J. The variation with temperature
and wavevector of the integrated intensity for one and two magnon scattering
and of the central peak are discussed.
