Example 3 A person walks from place A to place B, and there are regular buses running between A and B, with equal intervals for departures from both places. He notices that a bus going to A passes by every 6 minutes, and a bus going to B passes by every 12 minutes. How often do the buses depart from their respective starting stations? (Assume that each bus travels at the same speed)
Analysis: Let the distance between two consecutive buses in the same direction be $s$, and the time be $t$ minutes, then the speed of the bus is $\frac{s}{t}$.
(1) A person's speed relative to the bus going to location A is $\frac{s}{6}$, so the person's speed is $\frac{s}{6}-\frac{s}{t}$.
(2) A person's speed relative to the bus going to location B is $\frac{s}{12}$, so the person's speed is $\frac{s}{t}-\frac{s}{12}$.
$$
\therefore \frac{s}{6}-\frac{s}{t}=\frac{s}{t}-\frac{s}{12} \text {. }
$$

Solving for $t$ gives $t=8$ (minutes).