Problem about the disturbance of wave equation What is meant by "disturbance" of a wave equation? "Consider a vibrating infinite string whose displacement is given by the wave equation $$U_{tt}=U_{xx},$$ means (c=1). Suppose at $t=0$, the string is subjected to a disturbance exactly in the intervals $[1,2]$ and $[4,5]$. At $t=10$, describe the positions on the string which will be affected by the initial disturbance."

By "disturbance" one usually means a sudden energy input, here in terms of initial values that "disturb" the before non-vibrating string.

Since wave signals (distubrances, singularities) propagate with finite speed (see also this question and the references therein, it will need some time until the disturbance can influence the solution $U$ at a certain point.

Hint 1 to your (homework?) question: You are asked to find the domain of influence of the initial signal at a time $t=10$, which will be a bounded interval. Hint 2: The disturbances are propagating with the wave speed (in your case, $c=1$).