# wave equation D'Alembert's solution

Consider the wave equation $$u_{tt} = c^2u_{xx}$$

with

$$u(x,0) = \begin{cases} \sin x, && 2\pi < x < 3\pi \\ 0, && x < 2\pi,\ x> 3\pi \end{cases}$$

What initial velocity function $$u_t(x,0)$$ will ensure that the solution is a wave travelling to the left, with no wave going to the right?

I'm struggling to understand the theory around $$u_t(x,0)$$ and how that would affect the wave. I drew the graph for $$u(x,0)$$ and substituted values for $$t$$ but can't seem to link it to $$u_t(x,t)$$.

• Hint: A left-traveling wave has the form $u(x,t) = f(x+ct)$ – Dylan Mar 16 at 14:49