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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 $u_t(x,t)$.

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

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