Objective: This work focused on the effect of “linear-rise” amplitude modulated waveform on the time evolution of some typical electrical and plasma parameters in a pulsed radio-frequency capacitively coupled argon discharge. Methods: The pulse-on phase of a square wave pulse is divided into two distinct stages, namely, the “linear rise” phase (T1 phase) and “constant amplitude” phase (T2 phase). The study focused on exploring the impact of varying T1 durations on the temporal evolutions of these parameters. In the experiment, the electron density was measured time-resolved by a hairpin probe. Phase-resolved optical emission spectroscopy was employed to determine the spatial-temporal distribution of the electron impact excitation dynamics. Additionally, we determined the amplitudes and the relative phase of the discharge voltage and current, as well as the power deposition by analyzing the waveforms obtained from a voltage and a current probe. Results: It was found that with the increase of T1, the critical voltage value required for the plasma ignition becomes lower, and the RF power and the light intensity overshoot less significantly. Upon the overshoot, the plasma when the light intensity exhibits the maximum is dominated by the “overshoot” mode. After the pulse is turned off, the electron density under different T1 durations first decreases with the same rate, and then the decay rate is decreased with the increase of T1. This is because for a longer T1, the neutral gas temperature gets lower, leading to a higher density of neutral gas, so the electron diffusion loss is suppressed. Conclusion: Compared to the square pulse modulated radio frequency capacitively coupled plasmas, the “linear rise” amplitude modulation approach introduces external control knobs to modulate the plasma parameters, which benefits practical material processing. This paper is an experimental measurement implemented in an electropositive gas Ar discharge, and is expected to be extended to the plasmas operated in complex electronegative gases in the future.