Measure the speed of the pulse. The pulse which starts from rest travels at the same speed as the pulses which start as jerks. Measure the speed of the pulse as a function of its height. Then make a 25 cm high pulse. Then a 50 cm pulse.
Large amplitude pulses travel at the same speed as small amplitude pulses.
The speed of the pulse should be independent of its amplitude. Since the speed of pulses is independent of their size you can improve the accuracy of your timing by sending a pulse and allowing it to travel down the string to your partner and back 5 times. The speed of the pulse is the 10 times the length of the cord, d, divided by the time for 10 one way trips. To time ten one-way trips start the pulse as you count zero, then count one, two, three, four, five counting once each time the pulse returns. Note it is really important that when you start the pulse you count ZERO, then on the first return ONE, and so on.