From bdf972e31e48788c0a75d9fed32ba3100c130ca1 Mon Sep 17 00:00:00 2001 From: flavis Date: Sun, 24 May 2020 11:18:38 +0200 Subject: [PATCH] update theo and num --- lecture.cls | 2 +- sose2020/ana/lectures | 2 +- sose2020/num/uebungen/num4.pdf | Bin 146539 -> 146547 bytes sose2020/num/uebungen/num4.tex | 2 +- sose2020/theo/uebungen/theo5.pdf | Bin 117429 -> 108982 bytes sose2020/theo/uebungen/theo5.tex | 39 +++++++++++++++++-------------- 6 files changed, 24 insertions(+), 21 deletions(-) diff --git a/lecture.cls b/lecture.cls index cec1f4c..12979f2 100644 --- a/lecture.cls +++ b/lecture.cls @@ -213,7 +213,7 @@ } % replace all relations with align characters (&) and add the needed padding \regex_replace_all:nnN - { (\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}{.*?}{.*?}|\c{stackrel}{.*?}{.*?}&|&\c{neq}|\c{neq}&) } + { 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Ansatz: $R(r) = \frac{\tilde{R}(r)}{r}$. + \item Sei $c > 0$ und $\omega^2 = c$. Ansatz: $R(r) = \frac{\tilde{R}(r)}{r}$. Aus der DGL für $R$ folgt \begin{align*} - &\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\tilde{R} - r \frac{\partial \tilde R}{\partial r^2}}{r^2} \right) + \frac{c}{v^2} \frac{\tilde R }{r} = 0 \\ - \implies &\frac{1}{r} \left( - \frac{\partial^2 \tilde R}{\partial r} - + \frac{c}{v^2} \tilde R \right) = 0 \\ - \implies & - \frac{\partial^2 \tilde R}{\partial r^2} + \frac{c}{v^2} \tilde R = 0 \\ - \implies & \tilde{R} = A\exp\left(\sqrt{\frac{c}{v^2}}r\right) + B\exp\left(-\sqrt{\frac{c}{v^2}}r\right) \\ - \implies & R = \frac{A\exp\left(\sqrt{\frac{c}{v^2}}r\right) + B\exp\left(-\sqrt{\frac{c}{v^2}}r\right)}{r} + &\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{r \frac{\partial \tilde R}{\partial r} - \tilde{R}}{r^2} \right) + \frac{\omega^2}{v^2} \frac{\tilde R }{r} = 0 \\ + \implies &\frac{1}{r} \left( \frac{\partial^2 \tilde R}{\partial r} + + \frac{\omega^2}{v^2} \tilde R \right) = 0 \\ + \implies & \frac{\partial^2 \tilde R}{\partial r^2} + \frac{\omega^2}{v^2} \tilde R = 0 \\ + \implies & \tilde{R} = A \sin\left( \frac{\omega}{v}r \right) + B \cos\left( \frac{\omega}{v}r \right) \\ + \implies & R = \frac{A \sin\left( \frac{\omega}{v}r \right) + B \cos\left( \frac{\omega}{v}r \right)}{r} + \intertext{Mit der Vorraussetzung $R$ bei $r=0$ stetig folgt $B=0$:} + &R = \frac{A}{r} \sin \left( \frac{\omega}{v}r \right) .\end{align*} Damit folgt \[ - q = T \cdot R = \frac{A_0}{r} \cos(\sqrt{c} t - \delta) - \left( A\exp\left(\sqrt{\frac{c}{v^2}}r\right) + B\exp\left(-\sqrt{\frac{c}{v^2}}r\right)\right) + q = T \cdot R = \frac{A_0}{r} \cos(\omega t - \delta) \sin\left(\frac{\omega}{v} r\right) .\] \end{enumerate} \end{aufgabe}