  In this paper, we consider two types of damped wave equations: the weakly
damped equation and the strongly damped equation. We recover the initial
velocity from the trace of the solution on a space-time cylinder. This inverse
problem is related to Photoacoustic Tomography (PAT), a hybrid medical imaging
technique. PAT is based on generating acoustic waves inside of an object of
interest and one of the mathematical problem in PAT is reconstructing the
initial velocity from the solution of the wave equation measured on the outside
of object. Using the spherical harmonics and spectral theorem, we demonstrate a
way to recover the initial velocity.
