# D'Lembert method

#### Markov

##### Member
Consider

\begin{aligned} & {{u}_{tt}}=9{{u}_{xx}},\text{ }x\in \mathbb{R},\text{ }t>0, \\ & u(x,0)=\left\{ \begin{align} & 1,\text{ }x\in [1,2] \\ & 0,\text{ }x\notin [1,2] \\ \end{align} \right. \\ & {{u}_{t}}(x,0)=0, \end{aligned}

then determine the points of the semiplane $t>0$ where $u(x,t)=0.$

Okay I know the D'Lembert's formula, but I don't know how to apply it since having $u(x,0)$ defined by two conditions.
Thanks!

#### Jester

##### Well-known member
MHB Math Helper
Consider

\begin{aligned} & {{u}_{tt}}=9{{u}_{xx}},\text{ }x\in \mathbb{R},\text{ }t>0, \\ & u(x,0)=\left\{ \begin{align} & 1,\text{ }x\in [1,2] \\ & 0,\text{ }x\notin [1,2] \\ \end{align} \right. \\ & {{u}_{t}}(x,0)=0, \end{aligned}

then determine the points of the semiplane $t>0$ where $u(x,t)=0.$

Okay I know the D'Lembert's formula, but I don't know how to apply it since having $u(x,0)$ defined by two conditions.
Thanks!
$u(x,0) = H(x-1)-H(x-2)$
Thank you Jester! I'm sorry but I'm a bit lost on how applying D'lembert's formula, do I need to apply it for $H(x-1)$ and $H(x-2)$ ? How to do so?