Maximum displacement of finite string

Question

If the transverse displacement of a string is given by $u(x,t)$, how would I go about calculating the maximum displacement of the midpoint of that string? Is this just when $u(x,t)$ is maximised? Where does the 'midpoint' come into it?

Answer 1

If the string length is $L$ and one end is $x=0$ the midpoint is $x=\frac L2$. You are assuming that the maximum displacement occurs at the midpoint. You maximize $u(\frac L2,t)$ over $t$.