I have a problem which I slightly understand but need some clearing up. The question reads:
Suppose we have the PDE $$u_{tt}=u_{xx}$$ Which represents the displacement of an infinite vibrating string. Suppose there is a disturbance exactly in the intervals $[1,2]$ and $[4,5]$ at $t=0$. $\boldsymbol{a)}$ At $t=10$, describe the positions of the string which will be affected by the initial appearance. $\boldsymbol{b)}$ Repeat for the semi-infinite string with fixed boundary at $x=0$.
Now I know that I am looking for the Domain Of Influence. Thus, since the information propogates along the string at speed $1$ based off of the Wave Equation, should the Domain of Influence be $[-9, 15]$ with four waves at $[-9, -8]$, $[11, 12]$, $[-6,-5]$ and $[14,15]$, and the height of the initial disturbance remaining the same?
Similarly, for b) should the Domain of Influence be $[5, 15]$, with four waves at $[9,10]$, $[11, 12]$, $[5,6]$ and $[14,15]$, with the waves at $[9,10]$ and $[5,6]$ being upside down and the other two regular with the same initial height?
Can someone tell me if I've got this horrificly wrong or if I'm on the right track?