The initial launch can sometimes be identified (if you're quick and have good eyesight or are low level) by a type of corkscrew smoke trail created by the primary charge being fired and the rocket motor kicking in while the beast stabilises and all the fins and other junk align themselves.
Rocket motor only burns for a few seconds but accelerates the missile to supersonic speeds after which it runs ballistically to a distance of around 14000 - 15000' (SAM7).
Hope this helps.
BTW, the tone heard by the operator is actually a low frequency vibration felt through a buzz bar (not bus bar) against the cheek.