Solve the two-way wave equation

Consider the two-way wave equation

$u_{tt} − u_{xx} = 0$

for $t ≥ 0$ and $x ≥ 0$ with initial data $u(x, 0) = u_0(x)$ and $u_t(x, 0) = 0$ for $x ≥ 0$ and boundary data $u_x(0, t) = 0$ for $t > 0$

Draw a diagram of the problem in the $x − t$ plane.
Solve the problem in terms of $u_0$. Describe the interaction of the left-moving wave with the $x = 0$ boundary (one sentence).


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