I don't think there is a pulse as you describe. The thermostat achieves a steady state mostly open position at the intended operating temp of the engine (set point of the thermostat). If the engine becomes too hot then the thermostat will open even more to try to cool it down. In my experience bad thermostats fail open so the engine does not warm up enough or fast enough. If the engine is overheating it usually a clogged radiator, but could be other things such as bad head gasket or bad fan.

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1998 C230 330,000 miles (currently dead of second failed EIS, yours will fail too, turning you into the dealer's personal human cash machine)
1988 F150 144,000 miles (leaks all the colors of the rainbow)
Previous stars: 1981 Brava 210,000 miles, 1978 128 150,000 miles, 1977 B200 Van 175,000 miles, 1972 Vega (great, if rusty, car), 1972 Celica, 1986.5 Supra