There are only a few analytical 2+1D models for tsunami propagation, most of which treat tsunami generation from a static deformation field isolated from the kinematics of the rupture. This work examines the behavior of tsunami propagation in a simple setup including a source time function which accounts for a time description of the rupture process on the tsunami source. An analytical solution is derived in the wavenumber domain, which is quickly inverted to space with the fast Fourier transform. The solution is obtained in closed form in the 1+1D case. The inclusion of temporal parameters of the source such as rise time and rupture velocity reveals a specific domain of very slow earthquakes that enhance tsunami amplitudes and produce non-negligible shifts in arrival times. The results confirm that amplification occurs when the rupture velocity matches the long-wave tsunami speed, and the static approximation corresponds to a limit case for (relatively) fast ruptures.