Wave equation and multiple boundary conditions

[Markov2]

I need to apply D'Lembert's method but in this case I don't know how. How to proceed?

Determine the solution of the wave equation on a semi-infinite interval $u_{tt}=c^2u_{xx},$ $0<x<\infty,$ $t>0,$ where $u(0,t)=0$ and the initial conditions:

$\begin{aligned} & u(x,0)=\left\{ \begin{align}
& 0,\text{ }0<x<2 \\
& 1,\text{ }2<x<3 \\
& 0,\text{ }x>3 \\
\end{align} \right. \\
& {{u}_{t}}(x,0)=0. \\
\end{aligned}$

 

[Sudharaka]

I need to apply D'Lembert's method but in this case I don't know how. How to proceed?

Determine the solution of the wave equation on a semi-infinite interval $u_{tt}=c^2u_{xx},$ $0<x<\infty,$ $t>0,$ where $u(0,t)=0$ and the initial conditions:

$\begin{aligned} & u(x,0)=\left\{ \begin{align}
& 0,\text{ }0<x<2 \\
& 1,\text{ }2<x<3 \\
& 0,\text{ }x>3 \\
\end{align} \right. \\
& {{u}_{t}}(x,0)=0. \\
\end{aligned}$

Hi Markov, :)

The d'Alembert's solution for the wave equation, \(u_{tt}=c^2u_{xx}\) can be written as,

\[u(x,t)=\frac{1}{2}u(x-ct,\,0)+\frac{1}{2}u(x+ct,\,0)-\frac{1}{2c}\int_{x-ct}^{x+ct}u_{t}(s,\,0)\,ds\]

Since, \(u_{t}(x,0)=0\) we get,

\[u(x,t)=\frac{1}{2}u(x-ct,\,0)+\frac{1}{2}u(x+ct,\,0)\]

Case I: When \(0<x-ct<2\mbox{ or }x-ct>3\)

\[u(x,t)=0\]

Case II: When \(2<x-ct<3\)

\[u(x,t)=1\]

Kind Regards,
Sudharaka.