A wave $f(x + ct)$ travels along a semi-infinite string $(0 < x < \infty)$ for $t < 0.$ Find the vibrations $u(x, t)$ of the string for $t > 0$ if the end $x = 0$ is fixed
Answer: $f (x + ct)$ for $x > ct$; $f (x + ct) − f (ct − x)$ for $x < ct$
My attempt: We have that $u$ satisfies $u_{tt}=c^2u_{xx}$ and, in this case, $u(0,t)=0$ because one end is fixed (Dirichlet condition). I think I am having difficults to understand what the question is asking. What does it means "A wave $f(x+ct)$..."? How can we fit the other conditions the wave must satisfy?