3.1.4 Shock speed I

**Figure 25:** Jump across shock.
$\textstyle \parbox{3.0cm}{\makebox[3.0cm]{}}$ $\includegraphics[width=8.0cm]{images/pde1d_jump.ps}$ $\textstyle \parbox{3.0cm}{\makebox[3.0cm]{}}$

A shock with speed $\bgroup\color{DEFcolor}$s$\egroup$ travels over an infinitesimal time $\bgroup\color{DEFcolor}$\Delta t$\egroup$ an infinitesimal distance

$\begin{displaymath} \Delta x=s \Delta t \end{displaymath}$

(164)

(see Fig. 25). Integration of the PDE

$\begin{displaymath} \frac{\partial q}{\partial t} + \frac{\partial f \! \left( q \right)}{\partial x} = 0 \enspace , \end{displaymath}$

(165)

over $\bgroup\color{DEFcolor}$\Delta t$\egroup$ and $\bgroup\color{DEFcolor}$\Delta x$\egroup$ results in

$\begin{displaymath} \int_{x}^{x + \Delta x} \left[ q \! \left( x , t + \Delta ... ...\! \left( q \! \left( x , t \right) \right) \right] dt = 0 \end{displaymath}$

(166)