---
tags: HW03
---
# ***Computational Hydraulics***
## **Group 01:** N88071053 洪豪男 N88091011 黃彥鈞
## **HW03:**derive the wave equation: from the second oder form to the equation system form.

**The wave equation**

$u_{tt} = c^2\,u_{xx}\,,\quad -\infty < x < \infty$

initial data $\left\{
\begin{array}{l}
\ u(x,0) = u_0(x)\\
\ u_t(x,0)=u_1(x)\
\end{array}\right .$

with$\quad v=u_x,\quad w=u_t$

$\quad\quad\quad\quad\quad\Downarrow$

$\dfrac{\partial}{\partial t}
\left(\begin{array}{c} v \\ w\end{array}\right) +
\dfrac{\partial}{\partial x}
\left(\begin{array}{c} -w \\ -c^2v\end{array}\right) = 0$

with $v(x,0) = u'_0(x)\,,\quad w(x,0)=u_1(x)$

and equation system,

$\dfrac{\partial}{\partial t}
\left(\begin{array}{c} w_1 \\ w_2\end{array}\right) +\left(\begin{array}{c} \lambda_1&0 \\ 0&\lambda_2\end{array}\right)
\left(\begin{array}{c}\dfrac{\partial w_1}{\partial x}\\\dfrac{\partial w_1}{\partial x}\end{array}\right)=\left(\begin{array}{c} 0 \\ 0\end{array}\right)$

$\dfrac{\partial}{\partial t}
\left(\begin{array}{c} v \\ w\end{array}\right) +
\left(\begin{array}{c} -w_x \\ -c^2v_x\end{array}\right) = 0$

$\Rightarrow\dfrac{\partial}{\partial t}
\left(\begin{array}{c} v \\ w\end{array}\right) +
\left(\begin{array}{cc} 0 & -1 \\ -c^2 & 0\end{array}\right)
\left(\begin{array}{c} v_x \\ w_x\end{array}\right) = 0$