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).

Answer

Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer

1 Attachment

  • 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.

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

The answer is accepted.