Solve the two-way wave equation in terms of $u_0$

Consider the two-way wave equation for $u(x, t), t ≥ 0$ and all $x$ with piecewise constant speed:

$u_{tt} − u_{xx} = 0$ for $x ≥ 0$
$u_{tt} − 9u_{xx} = 0$ for $x < 0$

At $x = 0$, $u$ and $u_x$ are continuous. Data $u_0(x)$ is given for $x ≥ 0$. $u(x, 0) = u_0(x)$ for $x ≥ 0$ and $u(x, 0) = 0$ for $x < 0$. $u_t(x, 0) = 0$ for all $x$.

Solve the problem in terms of $u_0$.
Describe the interaction of the left-moving wave for $x > 0$ with the $x = 0$ boundary (one sentence).

Partial Differential Equations Differential Equations Ordinary Differential Equations
Chdogordon Chdogordon
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Martin Martin
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  • Martin Martin
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    Solving and writing up the solution to this question took about three hours. Please consider setting the price at a more appropriate level depending on the question's difficulty.

  • Chdogordon Chdogordon
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    I'm sorry about that! Will do it next time :)

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