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
- The questioner was satisfied and accepted the answer, or
- The answer was disputed, but the judge evaluated it as 100% correct.
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.
- answered
- 135 views
- $10.00
Related Questions
- A linear ODE
- Beginner Differential Equations - Growth Rate Question
- How should I approach this question?
- Week solution of the equation $u_t + u^2u_x = f(x,t)$
- Differential equations
- Equipartition of energy in one dimensional wave equation $u_{tt}-u_{xx}=0 $
- Differential equations, question 4
- Pointwise estimate for solutions of the laplace equation on bounded domains