We analyze overlapping Schwarz waveform relaxation for linear advection reaction diffusion equations. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result also depends on the number of subdomains, because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. The superlinear convergence result however is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis.