%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\schrift{11pt}
\input format

\def\baselinestretch{1.1}

\begin{document}

\pagestyle{empty}

\Blatt{4}{WS 2003/2004, 23.11.2004}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\item[11.]
{\bf Produktregeln}
\par\smallskip
Zeigen Sie für die Funktionen $f,g: \Real^3 \to \Real$ und die
Vektorfelder $\vek{a},\vek{b}: \Real^3 \to \Real^3$:

\begin{eqnarray}
  \nabla (fg) & = & f (\nabla g) + g (\nabla f) \\
  \nabla ( \vek{a} \cdot \vek{b} ) & = & \vek{a} \times (\nabla \times \vek{b})
  + \vek{b} \times ( \nabla \times \vek{a} ) + 
  (\vek{a} \cdot \nabla) \vek{b} + (\vek{b} \cdot \nabla) \vek{a} \\
  \nabla \cdot (f \vek{a}) & = & f (\nabla \cdot \vek{a}) + \vek{a} \cdot
  (\nabla f) \\
  \nabla \cdot (\vek{a} \times \vek{b}) & = & 
  \vek{b} \cdot (\nabla \times \vek{a}) - 
  \vek{a} \cdot (\nabla \times  \vek{b}) \\
  \nabla \times (f \vek{a}) & = & f (\nabla \times \vek{a}) + 
  (\nabla f) \times \vek{a} \\
  \nabla \times (\vek{a} \times \vek{b}) & = & 
  (\vek{b} \cdot \nabla) \vek{a} -
  (\vek{a} \cdot \nabla) \vek{b} + \vek{a} (\nabla \cdot \vek{b})
  - \vek{b} ( \nabla \cdot \vek{a} )
\end{eqnarray}

Hinweis: Verwenden Sie den $\varepsilon$-Tensor mit
\begin{displaymath}
  \varepsilon_{ijk} = \left\{ \begin{array}{cl}
    1  & \mbox{für gerade Permutationen $(i,j,k) \in \{1,2,3\}^3$} \\
    -1 & \mbox{für ungerade Permutationen} \\
    0  & \mbox{sonst}
    \end{array} \right.
\end{displaymath}
und zeigen Sie zunächst die Identität 
\begin{displaymath}
  \sum_k \varepsilon_{ijk} \varepsilon_{klm} = 
  \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} \,.
\end{displaymath}


\item[12.]
{\bf Zweite Ableitungen}
\begin{eqnarray}
  \nabla \cdot (\nabla \times \vek{a}) & = & 0 \\
  \nabla \times (\nabla f) & = & 0 \\
  \nabla \times (\nabla \times \vek{a}) & = & 
  \nabla (\nabla \cdot \vek{a}) - \nabla^2 \vek{a}
\end{eqnarray}


\vfill

\rule[.5mm]{\textwidth}{0.1mm}\\
{\footnotesize Bitte wenden.}

\newpage 


\item[13.]
{\bf Schallwellen}
\par\smallskip

\begin{enumerate}
\item Linearisieren Sie die Eulergleichungen
  \begin{eqnarray*}
    \partial_t \varrho + \div (\varrho\, \vek{v}) & = & 0 \\
    \partial_t v_i + \sum_{j=1}^3 ( \partial_j v_i ) v_j & = & - \frac{1}{\varrho} \partial_i p \,,
  \end{eqnarray*}
um $\varrho=\varrho_0$, $p=p_0$ und $\vek{v}=\vek{0}$, indem Sie
$\varrho=\varrho_0+\Delta\varrho$, und $p=p_0 + \Delta p$ setzen und
Terme zweiter Ordnung in $\Delta \varrho$, $\Delta p$ und $\vek{v}$
vernachlässigen. Beachten Sie, dass für Schallwellen in Gasen die
adiabatische Zustandsgleichung
\begin{displaymath}
  \frac{p}{p_0} = \left( \frac{\varrho}{\varrho_0} \right)^\gamma
\end{displaymath}
gilt und für Flüssigkeiten und Festkörper
\begin{displaymath}
  \varrho = \varrho_0 (1 + \varkappa \, \Delta p ) \,.
\end{displaymath}
\item Wie erhält man eine Wellengleichung für $\Delta \varrho$?
\item Berechnen Sie die Schallgeschwindigkeit in Luft und in Wasser
  (die Kompressibilität von Wasser beträgt $\varkappa = 50 \cdot 10^{-6}
  \mathrm{bar}^{-1}$).
\end{enumerate}



\par\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{deflist}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfill 

\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der "Ubungen am Dienstag, den 30.11.2004 um 9.30 Uhr im Raum HS3
(Garching).\\
"Ubungsleitung: Moritz Franosch, mail@Franosch.org, http://www.physik.tu-muenchen.de/lehrstuehle/T35/.

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
