We report on the response characteristics of plasmonic terahertz field-effect transistors (TeraFETs) fed with femtosecond and picosecond pulses. Varying the pulse width (tpw) from 10-15 s to 10-10 s under a constant input power condition revealed two distinctive pulse detection modes. In the short pulse mode (tpw << L/s, where L is the gated channel length, s is the plasma velocity), the source-to-drain voltage response is a sharp pulse oscillatory decay preceded by a delay time on the order of L/s. The plasma wave travels along the channel like the shallow water wave with a relatively narrow wave package. In the long pulse mode (tpw > L/s), the response profile has two oscillatory decay processes and the propagation of plasma wave is analogues to oscillating rod with one side fixed. The ultimate response time at the long pulse mode is significantly higher than that under the short pulse conditions. The detection conditions under the long pulse mode are close to the step response condition, and the response time conforms well to the analytical theory for the step function response. The simulated waveform agrees well with the measured pulse response. Our results show that the measurements of the pulse response enable the material parameter extraction from the pulse response data (including the effective mass, kinematic viscosity and momentum relaxation time).